DATA-ADAPTIVE SPECTRAL ESTIMATION ALGORITHMS AND THEIR SENSING
APPLICATIONS
By
ODE OJOWU JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2013
c 2013 Ode Ojowu Jr.
⃝
2
I dedicate this to God, family and friends.
3
ACKNOWLEDGMENTS
This dissertation would not have been possible without the support of several
people. I would first of all like to thank my parents, my siblings and friends for the love
and moral support they have given me throughout the years.
I would also like to thank my advisor, Prof. Jian Li for taking me in as a student,
and taking the time and patience to guide me throughout this important phase of my
academic career; I will forever be grateful.
This dissertation also would not have been possible without the help of some of
my close colleagues, lab mates and friends at the Spectral Analysis Lab, which include:
William Rowe, Dr. Johan Karlsson, Dr. Duc Vu, Chris Gianelli, Kexin Zhao, Dr. Luzhou
Xu, Dr. Hao He, Dr. Jun Ling, Lim Deoksu, Qilin Zhang and Dr. Ming Xue. The daily
discussions and advice helped with my work tremendously.
I would finally like to thank my committee members, Prof. Henry Zmuda, Prof.
Jenshan Lin and Prof. Hugh Fan for their guidance and support, and also for taking the
time to be on my committee. I appreciate the sacrifice sincerely.
4
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER
1
REVIEW OF SPECTRAL ESTIMATION AND TECHNIQUES . . . . . . . . . . 13
1.1 Introduction: Spectral Estimation Problem . . . . . . . . . . . . .
1.1.1 Energy Spectral Density . . . . . . . . . . . . . . . . . . .
1.1.2 Power Spectral Density . . . . . . . . . . . . . . . . . . .
1.1.3 Power Spectral Density Estimation . . . . . . . . . . . . .
1.2 Periodogram: Non-parametric Method . . . . . . . . . . . . . . .
1.2.1 Resolution: Periodogram . . . . . . . . . . . . . . . . . . .
1.2.2 Filter-bank Interpretation: Periodogram . . . . . . . . . . .
1.3 Data-adaptive Approaches . . . . . . . . . . . . . . . . . . . . .
1.3.1 CAPON: Non-parametric . . . . . . . . . . . . . . . . . . .
1.3.2 Amplitude and Phase Estimation (APES): Non-parametric
1.3.3 Iterative Adaptive Approach (IAA): Non-parametric . . . .
1.3.4 SLIM and SPICE Algorithms: Non-parametric . . . . . . .
1.3.5 RELAX: Parametric . . . . . . . . . . . . . . . . . . . . . .
1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
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13
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DATA-ADAPTIVE TECHNIQUES FOR NETWORK FREQUENCY EXTRACTION
FROM DIGITAL RECORDINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1
2.2
2.3
2.4
Chapter Summary . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . .
Network Frequency Characteristics and Database
Extraction Algorithms . . . . . . . . . . . . . . . .
2.4.1 Frequency Domain Analysis (STFT) . . . .
2.4.2 IAA and TRIAA . . . . . . . . . . . . . . . .
2.4.3 Frequency Tracking . . . . . . . . . . . . . .
2.4.4 Matching the Extracted ENF to Database .
2.5 Experimental Results . . . . . . . . . . . . . . . .
2.5.1 Data1 Analysis . . . . . . . . . . . . . . . .
2.5.2 Data2 Analysis . . . . . . . . . . . . . . . .
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . .
5
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31
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51
3
DATA-ADAPTIVE RELAX FOR RFI SUPPRESSION FOR THE SYNCHRONOUS
IMPULSE RECONSTRUCTION (SIRE) RADAR . . . . . . . . . . . . . . . . . 53
3.1
3.2
3.3
3.4
3.5
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
SIRE Equivalent Sampling Scheme . . . . . . . . . . . . .
Existing RFI Suppression Methods . . . . . . . . . . . . . .
Proposed RFI Suppression Method: RELAX and Averaging
3.5.1 Modelling of RFI . . . . . . . . . . . . . . . . . . . .
3.5.2 RELAX Algorithm . . . . . . . . . . . . . . . . . . . .
3.5.3 Multi-snapshot RELAX Algorithm . . . . . . . . . . .
3.6 Autoregressive (AR) Modelling . . . . . . . . . . . . . . . .
3.7 Experimental Results . . . . . . . . . . . . . . . . . . . . .
3.7.1 Simulations . . . . . . . . . . . . . . . . . . . . . . .
3.7.2 Sniff Experimental Data . . . . . . . . . . . . . . . .
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
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53
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DATA-ADAPTIVE, SPARSE SUPER-RESOLUTION IMAGING FOR THE SIRE
FLGPR RADAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1
4.2
4.3
4.4
Chapter Summary . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Data Model: SIRE Impulse Based FLGPR . . . . . . .
Back-projection/Delay-and-sum (DAS) Based Methods
4.4.1 Back-projection/DAS . . . . . . . . . . . . . . .
4.4.2 Sparse: CLEAN Method . . . . . . . . . . . . .
4.5 Super-resolution Methods . . . . . . . . . . . . . . . .
4.5.1 Orthogonal Projection and Time Gating . . . .
4.5.2 SLIM . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 SPICE . . . . . . . . . . . . . . . . . . . . . . .
4.6 Numerical and Experimental Results . . . . . . . . . .
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
5
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82
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106
CONCLUDING REMARKS AND FUTURE WORK . . . . . . . . . . . . . . . . 107
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6
LIST OF TABLES
Table
page
1-1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2-1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-2 Parameters for the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2-3 Correlation coefficients of Algorithms (Data1) . . . . . . . . . . . . . . . . . . . 47
2-4 Standard Deviation of error for Algorithms (Data1) . . . . . . . . . . . . . . . . 47
2-5 Correlation coefficients of Algorithms (Data2) . . . . . . . . . . . . . . . . . . . 48
2-6 Standard Deviation of error for Algorithms (Data2) . . . . . . . . . . . . . . . . 48
3-1 ARL Parameters for Synchronous Reconstruction Radar. . . . . . . . . . . . . 57
3-2 Suppression Algorithm: RELAX + Averaging . . . . . . . . . . . . . . . . . . . 68
3-3 Suppression Algorithm: M-RELAX + Averaging . . . . . . . . . . . . . . . . . . 71
3-4 RFI Suppression (dB): File 1 (P~
. . . . . . . . . . . . . . . . . . . . . . . . 77
3-5
. . . . . . . . . . . . . . . . . . . . . . . . 77
= 1)
RFI Suppression (dB): File2 (P~ = 1)
4-1 SLIM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4-2 CG SPICE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4-3 Subspace approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7
LIST OF FIGURES
Figure
page
1-1 Synthetic aperture radar (SAR) imaging . . . . . . . . . . . . . . . . . . . . . . 14
1-2 Spectrogram: Estimating the Electric Network Frequency (ENF) in audio an
recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1-3 Bartlett window spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1-4 Spectra of two sinusoids with large frequency spacing . . . . . . . . . . . . . . 21
1-5 Spectra of two sinusoids with small frequency spacing . . . . . . . . . . . . . . 21
1-6 Spectrum: Comparison of adaptive methods to the periodogram . . . . . . . . 27
2-1 FDR Distribution in North America . . . . . . . . . . . . . . . . . . . . . . . . . 35
2-2 Segmentation of data for STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2-3 Matching extracted ENF to database (Data1) . . . . . . . . . . . . . . . . . . . 46
2-4 Power Spectrum of one Frame (Data2): poor resolution of FFT . . . . . . . . . 49
2-5 Power Spectrum of one Frame (Data2): strong interference signal . . . . . . . 49
2-6 Extracted ENF via Frequency Tracking . . . . . . . . . . . . . . . . . . . . . . . 50
2-7 Matching extracted ENF to database (Data1) . . . . . . . . . . . . . . . . . . . 51
2-8 Absolute error of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3-1 Synchronous Impulse Reconstruction (SIRE) equivalent time sampling . . . . . 58
3-2 Spectrum of SIRE sampling after interleaving compared to the spectrum of
regular sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3-3 Spectrum SIRE sampling pattern: One fast time pulse . . . . . . . . . . . . . . 59
3-4 Spectrum SIRE sampling pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3-5 RFI Suppression (dB): Averaging method (M realizations) for simulated SIRE
sampled RFI signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3-6 RFI suppression (SIRE sampling) - using RELAX with P (real-valued) sinusoids
estimated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3-7 RFI suppression - RELAX algorithms with P (real-valued) sinusoids estimated
and suppressed from sniff data . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3-8 Echo retrieval (File1) - RELAX with P (real-valued) sinusoids compared to
ideal echo signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8
3-9 Echo retrieval (File1) - RELAX with P (real) sinusoids combined with M-RELAX
with P~ = 1 real sinusoid, compared to ideal echo signal . . . . . . . . . . . . . 79
3-10 RFI suppression - AR modelling with order q compared to averaging for sniff
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3-11 Echo retrieval (File1) - AR modelling with order q, compared to ideal echo signal 80
4-1 Forward looking ground penetrating radar . . . . . . . . . . . . . . . . . . . . . 84
4-2 SIRE FLGPR: 2D aperture for SAR imaging . . . . . . . . . . . . . . . . . . . . 86
4-3 Time gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4-4 Subspace dimension (s) for high resolution imaging . . . . . . . . . . . . . . . 100
4-5 FLGPR SAR Imaging - detection of weak target . . . . . . . . . . . . . . . . . . 101
4-6 FLGPR Imaging - resolution improvement . . . . . . . . . . . . . . . . . . . . . 102
4-7 Orthogonal projection comparison . . . . . . . . . . . . . . . . . . . . . . . . . 103
4-8 Real data - SIRE FLGPR SAR Imaging . . . . . . . . . . . . . . . . . . . . . . 104
4-9 ROC comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DATA-ADAPTIVE SPECTRAL ESTIMATION ALGORITHMS AND THEIR SENSING
APPLICATIONS
By
Ode Ojowu Jr.
December 2013
Chair: Jian Li
Major: Electrical and Computer Engineering
Spectral analysis of signals, or the problem of spectral estimation revolves around
estimating the distribution of power over frequency of a random signal. It has useful
applications in various fields of study (including Speech analysis, Medicine, RADAR and
SONAR) due to the fact that the frequency content of an observed signal can provide
very useful information in these fields.
A well known method for estimating the spectral content of a signal is the Periodogram (developed by Arthur Schuster), which is a data-independent method of
estimation. This method is based on computing the Fourier transform of the signal which
can be computed efficiently using the Fast Fourier Transform (FFT) algorithm. This
method however, is limited by relatively poor resolution and high sidelobe problems,
which can lead to degradation in retrieval of the desired information present within the
signal.
Data-dependent (adaptive) techniques both non-parametric and parametric can
offer superior performance over data-independent methods like the periodogram
at a cost of increased computational complexity. These data-adaptive approaches
however, can lead to improved spectral resolution and lower sidelobes, which can
reveal more information about the signal under study. These advantages have led to
increased interest in data-adaptive approaches to the problem of spectral estimation.
This dissertation revolves around analyzing and applying robust adaptive techniques
10
to real-world problems in a unique, effective and efficient way to achieve superior
performance over their data-independent counterparts.
The introduction chapter briefly reviews the problem of spectral estimation as well
as some of the methods for spectral estimation. We start this dissertation in Chapter
2 with the basic problem of frequency estimation (harmonic retrieval). In this chapter,
adaptive techniques are used in the problem of harmonic retrieval in the presence of
strong interference. The focus is on the problem of digital audio forensics, where the
goal is to extract the embedded network frequency from a digital recording and compare
it to a known database for digital audio verification. In the presence of significant
interference, extracting the network frequency using the standard method (Periodogram)
is ineffective and proves to be challenging due to poor resolution and high sidelobe
problems. We therefore use a robust adaptive algorithm (Iterative Adaptive Approach
- IAA) to improve the spectral resolution and suppress sidelobes hence effectively
separating the network frequency from interference. A frequency tracking method based
on dynamic programming is used in addition to this data-adaptive method to extract
the Network frequency accurately and hence provide more reliability for the verification
process compared to the current standard, which is based on the data-independent
Fourier transform.
Chapters 3 and 4 are the focus of this dissertation. In these chapters, the remote
sensing tool known as the Synchronous Impulse Reconstruction (SIRE) Ultra-wideband
radar (currently being built by the Army Research Lab (ARL) for landmine detection)
is analyzed and studied. In Chapter 3, we once again apply an adaptive technique
for harmonic retrieval. The goal here is to effectively suppress Radio Frequency
Interference (RFI) picked up by this UWB radar which samples its returned signals
using an equivalent sampling scheme. This equivalent sampling scheme makes RFI
suppression difficult due to its irregular and under-sampled data (aliasing). The current
method for RFI suppression for this UWB radar is simply averaging multiple realizations
11
of the measured data. In this chapter, we model the aliased RFI signals as a sum of
sinusoids and estimate the aliased frequencies and amplitudes accurately using a
robust algorithm - RELAX. A direct implementation of this algorithm is computationally
intensive, therefore, an efficient method for implementation is presented in this chapter,
which takes advantage of this equivalent sampling and improves computation. As RFI
suppression is the goal, the estimates are used to reconstruct the aliased RFI samples
accurately and are then suppressed from the data without altering the desired radar
signals.
In Chapter 4, we focus on radar imaging for landmine detection for this SIRE
UWB radar. The standard method currently used for this radar is the data-independent
backprojection or delay-and-sum (DAS) approach. This method suffers from high
sidelobe problems and poor resolution. A recursive sidelobe minimization (RSM)
algorithm was recently proposed by the army research laboratory for effective sidelobe
reduction. This data-independent approach however, has the same resolution limitation
as the backprojection algorithm. As imaging resolution is important for separating
desired targets (mines) from clutter, this chapter, focuses on sparse super-resolution
imaging techniques for imaging. A new technique for imaging based on applying
data-adaptive approaches post significant data reduction as well as interference
reduction via an orthogonal projection is proposed in this chapter. This approach is
able to achieve an improvement in imaging resolution by a factor of approximately 2,
based on simulated experiments. Chapter 5 provides the concluding remarks and
possible future work.
The contents of Chapter 2 are published in IEEE transactions on information
forensics and security Volume 7, no. 4. The contents of Chapter 3 are published in the
International Journal of Remote Sensing and Applications (IJRSA) vol 3 Issue 1. The
contents of Chapter 5 are to be submitted for publication.
12
CHAPTER 1
REVIEW OF SPECTRAL ESTIMATION AND TECHNIQUES
1.1 Introduction: Spectral Estimation Problem
Most phenomena or signals that occur in nature or in practice are typically random
in nature, and are best modelled as random signals. Examples of such random signals
include but are not limited to speech/audio signals and thermal noise generated by
electronic devices. Due to the random fluctuation of these signals, they are best
characterized in terms of statistical averages. The autocorrelation function of a random
process is a statistical average used for characterizing these random signals in the time
domain. The power spectral density (spectrum) provides the frequency content of such
signals.
Spectral analysis of signals or the spectral estimation problem, involves estimating
the frequency content of a random signal. This is done by estimating the power
distribution over frequency from a stationary sequence of finite time samples, which
is known as the power spectrum of the signal [1–5].
Schuster in the late 19th century pioneered the most well-known spectral estimation
techniques called the Periodogram. This harmonic analysis approach allows for
detecting and measuring ”hidden periodicities” [6] in the observed data. Spectral
estimation can also be performed on non-stationary data, by dividing the data into
segments in time (each assumed to be stationary) [7],[8],[9]. A time-varying power
spectrum (image) can be displayed to provide information about the signal (also known
as the spectrogram [10]).
Power spectral estimation has applications in many fields [1–3, 11, 12]. Speech
signals which are periodic in nature are analyzed using the spectrogram. This frequency
domain analysis provides useful information that can lead to speech recognition and
generation. In the sensing fields of RADAR and SONAR, the spectral content of
received signals may provide information about the targets of interest [11, 13] in a
13
given scene of interest (see Fig. 1-1). Also the power spectrum of signals may provide
information about radio frequency interference in such a signal and hence lead to
effective suppression of the interference. In the field of MEDICINE, the power spectrum
of electroencephalogram (EEG) signals can be used to evaluate the different sleep
cycles in humans [14, 15]. These can are used to investigate and study narcoleptic
(disease characterized by the inability to properly regulate sleep-wake cycles) patients
[15]. More recently in audio analysis, the spectrogram of the audio signal can indicate
the presence of the electric network frequency (see Fig. 1-2), which can be used for
digital audio authentication [16].
SAR Image from Phase History data
0
−5
−10
−15
−20
−25
−30
−35
A
B
Figure 1-1. Synthetic aperture radar (SAR) imaging: (A) Photograph of object at 45o (B)
SAR image formed using Spectral estimation (FFT)
There are two broad approaches to spectral estimation. The first approach is
called the non-parametric method and the other is called the parametric method. The
non-parametric methods assumes no prior information about the data, where as the
parametric methods assumes a specific model of the data, which then results in a
problem of parameter estimation. The parametric methods are more accurate than the
classical non-parametric techniques, when the assumed model is accurate. However,
they perform poorly when there are inaccuracies in the data-model.
14
Spectrogram of pre−filtered audio signal (ENF Harmonic =180 Hz)
179.5
0
179.6
−5
179.7
−10
179.8
−15
179.9
−20
180
−25
180.1
−30
180.2
−35
180.3
−40
180.4
−45
180.5
0
500
1000
1500
2000
time (secs)
2500
3000
3500
−50
Figure 1-2. Spectrogram: Estimating the Electric Network Frequency (ENF) in audio an
recording for forensic analysis (see chapter 1 for more details)
In this chapter, the problem of power spectral density estimation of signals is briefly
described. Commonly used techniques for spectral estimation within these two broad
methods (non-parametric and parametric methods) for estimating the spectrum of a
signal, will be briefly discussed. The limitations of these methods in practice will also be
briefly discussed.
Some data-dependent (adaptive) algorithms (Capon, APES, IAA, SLIM and
SPICE which are non-parametric and RELAX which is paramteric) will be mentioned
along with their benefits [1, 2, 17] over the classical (data-independent) approaches in
practical scenarios. The core of this dissertation is effective and efficient application of
data-adaptive techniques to solving real-world problems.
Before delving into the problem of spectral estimation of random signals, let us
consider the case of spectral estimation of finite length deterministic signals. This
analysis is fairly straightforward as deterministic signals are predictable over time [18].
The results will then be extended to the case of random signals.
15
1.1.1 Energy Spectral Density
Consider a signal x[n] (discrete) with finite energy, that is,
E
=
∞
∑
n=−∞
|x[n]|2 < ∞
(1–1)
then its discrete time fourier transform (DTFT) exists and is given by:
X (ω ) =
∞
∑
n=−∞
x[n]e−jωn
(1–2)
where ω is the angular frequency variable measured in radians per sample. From
Parseval’s theorem equation (1–1) can be written as:
E
=
∞
∑
n=−∞
|x[n]|2 =
1 ∫ π |X (ω)|2
2π −π
(1–3)
From the equation above the energy spectral density of x[n] which is the distribution of
the energy of the signal of frequency is therefore defined as:
Sxx (ω ) = |X (ω )|2
(1–4)
Note that the energy spectral density Sxx (ω ) can be written as the Fourier transform
of the autocorrelation sequence rxx (k ) of the signal x[n]:
Sxx (ω ) =
∞
∑
n=−∞
rxx (k )e−jωk dω
(1–5)
x∗ [n]x[n − k ]
(1–6)
where
rxx (k ) =
∞
∑
n=−∞
The analysis above, is specifically for signals with finite energy (deterministic
signals). However, signals typically encountered in applications are characterized as
stochastic processes and do not have finite energy and hence do not posses a Fourier
transform. These random signals however, posses an average power and can be
described by their power spectral density.
16
1.1.2 Power Spectral Density
Consider a stationary stochastic process y [n], where E{y [n]}
= 0 for all n. The
autocovariance function (same as autocorrelation function for stationary stochastic
process with mean zero) of y [n] is given by
ryy (k ) = E{y ∗ [n]y [n − k ]}
(1–7)
where E{·} is the statistical average over all realizations. The power spectral density
(PSD) of y [n] is defined as (Wiener-Khintchine theorem [1] ):
ϕyy (ω ) =
∞
∑
n=−∞
ryy (k )e−jωk
(1–8)
This simply the fourier transform of the autocorrelation function. Note that the inverse
transform of this PSD gives ryy (k ) as shown below
[
]
∫ π
∞
∞
∑
1
1 ∫ π ϕ (ω)ejωk dω = ∑
jω (k−s)
ryy (s)
dω =
ryy (s)δks = ryy (k )
2π −π yy
2π −π e
s=−∞
s=−∞
were δ denotes the Kronecker delta function. Note that, the average power of the
stochastic process y [n] is given by the zero lag of the autocorrelation function ryy (0):
E{|y [n]|2 } = ryy (0) =
1 ∫ π ϕ (ω)dω
2π −π yy
(1–9)
This equation (1–9) leads to the motivation for defining the power spectral density in
(1–8). The PSD can also be defined as:
 2 
N
 1 ∑

ϕyy (ω ) = lim E
y [n]e−jωn N →∞
N 
n=1
(1–10)
which is equivalent to the definition in (1–8) under the assumption that the autocovaraince
sequence (ACS) ryy (k ) decays quickly.
1.1.3 Power Spectral Density Estimation
Obtaining the true power spectral density (PSD) ϕyy (ω ) of a random process is
impossible from a finite set of measurements. This is due to the fact that one will need
17
to compute an an infinite number of values from a finite set of data, which is an ill-posed
problem [1, 2].
The problem of spectral estimation, then becomes getting an estimate ϕ^yy (ω ) of
the true PSD ϕyy (ω ) of a random process from a finite sequence of observations of
the signal. If the signal is statistically stationary, the longer the observed sequence the
more accurate the estimate. However if the signal is statistically non-stationary, then one
cannot select and arbitrarily long data length for estimation. This is a major limitation on
the quality of the PSD estimate.
Recall that the PSD describes how the power of a signal is distributed in frequency.
This can then be interpreted physically as filtering the random signal through a
narrowband filter around a specific frequency of interest (ωo ). This process is then
repeated for all the frequencies of interest (−π ≤ ωo ≤ π). Fourier based methods
(computed efficiently using the Fast Fourier Transform (FFT)) of spectral estimation are
based on this technique [1] and are discussed next.
1.2 Periodogram: Non-parametric Method
As mentioned in the section above, the non-parametric methods of spectral
estimation provide an estimate of the power spectral assuming no prior information
of the data model. The periodogram which was introduced by Schuster in 1898 to detect
”hidden periodicities” in a signal, is a classical non-parametric method which is widely
used for spectral estimation. This fourier based method, along with its modified versions
are based directly on the definition in (1–10). The periodogram of a set of N samples of
random process {y [n]}N
n=1 is given as (the subscript yy in ϕyy (ω ) has been dropped for
notational simplicity):
2
N
∑
^ϕp (ω) = 1 y[n]e−jωn N
n=1
(1–11)
Note that (1–11) is essentially thesame as the (1–10) with the expectation and limit
operation removed. This ommission is due to the fact that only N samples of the signal
18
are available. The periodogram can be computed using the discrete fourier transform
of the available samples (which can be efficiently computed using the fast fourier
transform (FFT). This yields samples of the PSD estimate at frequencies ωk
k
= 2πk/N for
= 0, 1, . . . , N − 1).
Note that equation (1–11) can be written as:
ϕ^p (ω ) =
N
−1
∑
r^[k ]e−jωk
(1–12)
y [ n] y ∗ [ n − k ]
(1–13)
k=−N +1
where
1
r^[k ] =
N
N
−1
∑
k=−N +1
corresponds to the biased estimates of the ACS sequence. This is referred to as the
correlogram. The unbiased estimate of the ACS can also be used to compute the
correlogram.
One major limitation of the periodogram is limited spectral resolution, which is
discussed next.
1.2.1 Resolution: Periodogram
One key concept in spectral estimation is spectral resolution, which is the ability
to resolve or seperate closely spaced frequency components within a signal. The
resolution of the periodogram is one major drawback of this data-independent method of
spectral estimation. Note that the expected value of the periodogram can be written as:
E{ϕ^p (ω )} =
N
−1
∑
k=−N +1
E{r^[k ]}e
−jωk
=
N
−1
∑
k=−N +1
w[k ]r[k ]e−jωk
(1–14)
where (based on (1–13))



1 − |k|
N
w [k ] =

 0
for n = ±1, ±2, . . . , ±N
otherwise
19
(1–15)
is the Bartlett window and r[k ] is the true PSD. Equation (1–14) is the Fourier transform
of the product of two time sequences, which correspond to the convolution of their
individual Fourier transforms as given in (1–16).
∫
1
E{ϕ^p (ω )} =
2π ϕ(β )W (ω − β )
π
(1–16)
−π
where W (ω ) is the Fourier transform of the Bartlett window.
[
1 sin(ωN/2)
W (ω ) =
N sin(ω/2)
Figure 3 below shows W (ω ) for N
= 10 and N = 20.
]2
(1–17)
The 3dB (half-power) main lobe
0
N = 10
N = 30
−5
−10
dB
−15
−20
−25
−30
−35
−40
−3
−2
−1
0
1
ω (radians/sample)
2
3
Figure 1-3. Bartlett window spectrum: resolution limitation periodogram (window length
= N)
width is approximately equal to 4π/2N
= 2π/N radians per sample (1/N cycles per
sample). The spectral estimate of periodogram ϕ^(ω ) will not be able to resolve peaks in
the true PSD ϕ(ω ) that have less than 1/N cycles per sample separation. Increasing the
number of observed samples will improve the spectral resolution (not be confused with
zero-padding).
The estimated spectrum can be computed using the DFT (and efficiently using
the FFT as mentioned earlier). Increasing the number of available samples by
zero-padding (adding zeros to the end of the signal) can provide more detail in the
20
spectrum computed using the FFT. This results in the interpolation of spectrum, however
it does not change the spectral resolution as shown in Figure 1-4 and 1-5. Figures 1-4
= 20 samples) with frequency spacing
and 1-5 show the spectrum of sinusoids (N
ω = 2π × (0.06) and ω = 2π × (0.02) respectively. Each figure shows different
zero-padding factors.
The periodogram suffers from relatively poor resolution and high
1
Zeropad (32 samples)
0.5
0
0
5
10
15
20
25
30
1
Zeropad (128 samples)
0.5
0
0
20
40
60
80
100
120
Figure 1-4. Spectra of two sinusoids with frequency spacing ω
= 2π × (0.06)
1
Zeropad (32 samples)
0.5
0
0
5
10
15
20
25
30
1
Zeropad (128 samples)
0.5
0
0
20
40
60
80
100
120
Figure 1-5. Spectra of two sinusoids with frequency spacing ω
= 2π × (0.02)
sidelobe problems as seen in Figure 1-3. These reasons have led to investigation into
data-adaptive methods of spectral estimation that can provide improved resolution and
sidelobe suppression capabilities. In the next subsection some of these data-adaptive
21
algorithms are discussed. Prior to this discussion we will review the periodogram in
different light which leads to one of the well known data-adaptive algorithms known as
the CAPON algorithm [19].
1.2.2 Filter-bank Interpretation: Periodogram
Recall that the PSD is the power distribution over frequency of the signal, which
as mentioned earlier can be interpreted as passing the signal through a bank of
narrowband filters (at different frequencies) and computing the output power (which
is then divided by the bandwidth of the filter). In this light, the periodogram estimator
ϕ^p (ω ) at a given frequency ω can be written as:
2
2
−1
N
N
1 ∑
∑ ∗
2
jω (N −n) ϕ^p (ω ) =
y
[
n
]
e
=
N
h
[
k
]
y
[
N
−
k
]
= N |z (N )|
ω
N
n=1
n=0
(1–18)
where
z (N ) =
and


 ejωk
∗
hω [ k ] =


0
∞
∑
n=0
hω [ k ] y [ N − k ]
for k
= 0, 1 , . . . , N − 1
=
(1–20)
otherwise
Note that the periodogram can be interpreted as filtering the signal y
through a narrowband pass filter hω
(1–19)
=
{y [k ]}N
k=0
{hω [k ]}N
k=0 and selecting just a single output
z (N ) of the filtering process hH
ω y for power calculation at the specified frequency (
{·}∗ and {·}H correspond to the conjugate (scalar) and conjugate transpose (vector)
operation). This fact leaves the periodogram with a large variance irregardless of the
data length (N ). The output power divided by the bandwidth (PSD) is then calulcated as
E|z [n]|2 / = |z [N ]|2 /, where = 1/N cycles per sample is the filter’s bandwidth.
Modified versions of the periodogram such as the Bartlett and Welch which
segment (non-overlapping and overlapping respectively) the stationary sequence
22
in question and average the periodograms of the segments can be used to reduce
the variance [2]. In terms of the filter-interpretation, these methods can be seen as
computing the power with more than one sample (number of segments). However, the
periodogram is computed using a reduced length of the data, hence there is a trade-off
between statistical variance and resolution.
Some data-adaptive non-parametric methods have addressed the limitations
of the periodogram by designing a data-adaptive filter, to provide more accurate
PSD estimates with better resolution. In the next section methods like the CAPON,
APES (Amplitude and Phase Estimation), IAA (Iterative Adaptive Approach) which
are data data-adaptive non-parametric approaches are discussed. The data-adaptive
parametric approach known as RELAX (strictly for sinusoidal parameter estimation) is
also discussed.
1.3
Data-adaptive Approaches
In this section, we discuss some well known non-parametric data-adaptive
approaches (CAPON, APES) as well as recent non-parametric spectral estimators
(IAA, SLIM SPICE). These algorithms improve upon the periodogram spectral estimator
in terms of resolution and sidelobe reduction. A parametric approach specifically for
estimating parameters of line spectra (sinusoids) known as RELAX is also mentioned
and discussed in detail later on in Chapter 3.
1.3.1 CAPON: Non-parametric
From the last sub-section, the periodogram output at a specific frequency ω can be
interpreted as using a data-independent filter (bandpass filter) with an impulse response
−1
{hω [k ] = e−jωk }N
k=0 corresponding simply to the Fourier Transform vector.
Unlike the data-independent filter used in the periodogram, the CAPON method
[19–21] (also known as the minimum variance method) designs a data-dependent
(adaptive) bandpass filter hω
1
= {hω [k]}l−
k=0 to achieve some specific desired properties
23
(The CAPON method uses overlapping segments of length (l × 1) of the data to improve
statistical variance). These properties include:
Design a bank of filters hω that pass the frequency component (or sinusoid with
frequency) ω undistorted.
2.
Filter should also effectively suppress (or minimize) all out-of-bound (any other
frequencies) power within the signal.
This process can be expressed as follows. Let the output of the filter at any instant
1.
n = [0, 1, . . . , N − 1] be given by:
z [ n] =
l−1
∑
k=0
h∗ω [k ]y [n − k ] = hH
ω yn
(1–21)
= [y[n], y[n − 1], . . . , y[n − l + 1]]T . The total output power of the filter is
H
then given as E{|z [n]|2 = hH
ω Rhω . Where R = E{yn yn } is the covariance matrix of the
where yn
data vector. The CAPON filter is designed to meet the properties in the aforementioned
steps by minimizing the total output power of the filter subject to the constraint that the
frequency ω is filtered without distortion given by the optimization equation (1–22).
min hH
ω Rhω
hω
where a(ω )
=
subject to hH
ω a(ω ) = 1
(1–22)
{e−jω }ln=0 is the sinusoid component with frequency ω to be passed
undistorted. The resulting filter is given by:
hω =
R−1 a(ω)
aH (ω)R−1 a(ω)
(CAPON filter)
(1–23)
The PSD estimate can then be calculated as filter output power E{|z [n]|2 divided by the
bandwidth ≈ 1/(l).
ϕ^CAP ON (ω ) =
E{|z [n]|2
= aH (ω)Rl −1 a(ω)
24
(CAPON spectral estimate)
(1–24)
^ based on the M
The sample covariance matrix R
= N − l + 1 overlapping segments
(each of length l) of the data is used to estimate the covariance matrix and is given by:
M
−1
∑
^R = 1
yn ynH
M
n=0
(1–25)
A very similar algorithm to the CAPON algorithm known as the Amplitude and
Phase Estimation algorithm (APES) is described next.
1.3.2 Amplitude and Phase Estimation (APES): Non-parametric
Note that in the description of the CAPON algorithm, the filter design was based
on passing a single frequency, while suppressing all other out-of-bound frequencies.
CAPON achieves the suppression by minimizing the total output power. APES algorithm
[22],[23],[24] uses the same idea but suppressing out-of-bound frequencies is achieved
by designing a filter such that the filtered sequence is as close as possible to the
a sinusoidal signal at the given frequency ω in the least squares (LS) sense. The
optimization equation is given by :
min
α(ω ),hω
M
−1
∑
n=0
jωn 2
|hH
|
ω yn − α(ω )e
subject to hH
ω a(ω ) = 1
(1–26)
The cost function in (1–25) can be re-written as:
1
M
−1
∑
M n=0
jωn 2
∗
H
^
|hH
| = |hH
yω − α(ω)~yωH hω + α(ω)|2
ω yn − α(ω )e
ω Rhω − α (ω )hω ~
(1–27)
= |α(ω) − hHω ~yω |2 +hHω R^ hω − |hHω ~yω |2
Note that the second and third terms in (1–27) do not depend on α(ω ) and therefore the
minimization of this cost function with respect to α(ω ) is given by α
^(ω) = hHω ~yω where
∑
~yω = (1/M ) Mn=0−1 yne−jωn . The optimization problem for designing the filter hω is given
as:
^
min hH
ω Qω hω
hω
subject to hH
ω a(ω ) = 1
25
(1–28)
^ω
where Q
= R^ − ~yω ~yωH The APES filter is given by:
hω =
Q^ −ω 1 a(ω)
aH (ω)Q^ −ω 1 a(ω)
(APES filter)
(1–29)
The amplitude spectrum of APES algorithm is given by:
α
^(ω) =
a(ω)H Q^ −ω 1~y(ω)
aH (ω)Q^ −ω 1 a(ω)
(APES amplitude spectrum)
(1–30)
The APES and CAPON algorithms have been shown to provide higher resolution
compared to the classical non-parametric methods. The CAPON algorithm minimizes
the total output power subject to a constraint which tends to provide spectral estimates
that are biased downward due to the noise gain of the filter [23]. The APES algorithm
minimizes a least square function requiring the filter output to be as close as possible to
the a sinusoid. This provides more accurate spectral estimates.
However, in the cases where the data is not stationary for a long period of time (only
few snapshots are available), the APES and CAPON methods yield undesirable results.
The Iterative Adaptive Approach (IAA) algorithm improves on these algorithms by being
able to give good spectral estimates for a few snapshots (even a single snapshot), while
providing high spectral resolution, making it very suitable for practical applications. This
algorithm is discussed briefly in the next subsection and also in Chapter 1 where it is
used.
1.3.3 Iterative Adaptive Approach (IAA): Non-parametric
The IAA algorithm [25],[26],[27],[28] for spectral estimation is derived by minimizing
a weighted least squares cost function (described in Chapter 1). The spectral estimate
for the IAA algorithm for a single snapshot y is given below:
α
^(ω) =
a(ω)H Q^ −ω 1 y
aH (ω)Q^ −ω 1 a(ω)
(IAA amplitude spectrum)
(1–31)
This estimate looks similar to the APES estimate, with the main differences being
that the IAA algorithm is iterative and also the computation of the covariance matrix of
26
^ ωl
the noise is given as Q
= R^ −
∑K
H
i=0,i̸=l pi a(ωi )a(ωi ) , where R
= APAH and P is a
2 K
diagonal matrix with elements corresponding to {pi }K
i=0 = {|α(ωi )| }i=0 (powers at each
individual frequencies).
Note that unlike the CAPON and APES algorithms where the covariance matrices
are based on the data samples and computed once. The covariance matrix of the
IAA algorithm is dependent on the spectral estimate and hence refined iteratively, with
the initial estimates of the spectral powers computed using the periodogram. This
refinement allows the IAA algorithm to produce accurate spectral estimates with a single
snapshot and hence makes it useful for practical applications (where the available data
for estimation is usually limited to a single snapshot). Fig. 1-6 shows the spectrum of
spectrum
Periodogram (single snapshot)
0.2
0
spectrum
Periodogram
Sinusoid 1
Sinusoid 2
Sinusoid 3
0.4
−3
−2
−1
0
1
ω (rad/sample)
Adaptive Algorithm
2
3
IAA
0.4
0.2
0
−3
−2
−1
0
ω (rad/sample)
1
2
3
Figure 1-6. Spectrum: Comparison of adaptive methods to the periodogram
three sinusoids in white noise (SNR = 30dB) with frequencies ω1
ω2
= 0.63 rad/samp,
= 1.26 rad/samp and ω3 = 1.33 rad/samp. The CAPON and IAA estimates are poor,
due to ill-conditioning of the matrices. However with a single snapshot the IAA spectra is
capable of picking out the sinusoids.
A comparison of the periodogram to the IAA algorithm in this figure shows how this
adaptive technique improves over the periodgoram in terms of spectral resolution and
sidelobe suppression.
27
1.3.4 SLIM and SPICE Algorithms: Non-parametric
The Sparse Learning via Iterative Minimization (SLIM) [29] and the Sparse Iterative
Covariance-based Estimation (SPICE)[30] algorithms are two super-resolution
algorithms capable of providing high resolution estimates even in the presence of a
single snapshot and coherent sources similar to the IAA algorithm. They both estimate
the covariance matrix R iteratively similar to the IAA algorithm and are hence also useful
for practical applications. The SLIM approach is a maximum a posteriori approach
(MAP) based on the hierrachial model. The goal is to use a sparse prior to promote
sparsity in the estimates which is useful for certain applications. The SPICE algorithm
on the other hand minimizes a covariance cost function that yields sparse estimates.
These two algorithms empirically yield less accurate that the IAA approach. However
they provide sparse and higher resolution estimates compared to the IAA algorithm and
can be useful in certain applications. They are described in more detail in Chapter 4.
The algorithms mentioned above are all non-parametric methods that do not
assume a specific model for the data. Next we briefly mention parametric methods,
which assume a specific data model for PSD estimation. A robust algorithm (which is
later discussed in more detail in Chapter 3) RELAX [31]; which is specific for estimating
parameters of line-spectra (sinusoids) is discussed next.
1.3.5 RELAX: Parametric
Parametric methods unlike the non-parametric methods assume a specific model
for the observed data. These methods essentially estimate the PSD, by assuming
the data takes on a specific model and then estimates the parameters of the model.
Auto-regressive (AR) methods such as Yule, Prony, Forward-Backward Prony methods
are used for estimating parameters for continuous spectra and Eigen-analysis methods
(MUSIC, ESPRIT) are used for estimating parameters of line spectra (sinusoids). The
AR methods model the data as the output of a linear system driven by white noise
and proceed to estimate the parameters of that system. One major limitation of these
28
parametric methods is that they a subject to errors due to poor model specifications.
The Eigen-analysis methods (for line spectra) estimate frequency components of
sinusoids buried in noise by an eigen-decompostion of the autocorrelation matrix. These
methods tend to perform poorly in practical applications due to data model inaccuracies.
The RELAX algorithm is an algorithm that is robust, and that estimates parameters
of sinusoids in an iterative manner. It estimates the parameters of the sinusoid
accurately even with modelling errors and colored noise [31]. This algorithm is described
in more detail in Chapter 3 where Radio Frequency Interference (RFI) is modeled as
sum of sinusoids. The RELAX algorithm is used there for identifying and suppressing
the RFI signals.
1.4
Conclusions
In this section, a brief discussion on the problem of spectral estimation is presented.
The periodogram which is a data-independent algorithm for spectral estimation and
also widely applied in practical applications is briefly discussed. This algorithm is then
re-interpreted as a filtering process with a data-independent filter. This re-interpretation
has led to some data-dependent (adaptive) filters which provide improved spectral
estimates.
We discussed some well-known data-adaptive (non-parametric) algorithms
(CAPON,APES) and the advantages provided by these data-adaptive approaches
over the classical non-parametric methods. However these algorithms perform poorly
in the case when only one snapshot of data is available. More recent data-adaptive
(non-parametric) algorithms, which are robust and perform well even in the single
snapshot case were briefly mentioned and will be discussed in more detail in later
chapters. The improved robustness of these algorithms allows for useful applications
in a practical setting, while providing better spectral properties (high resolution, lower
sidelobes) over the commonly used periodogram.
29
A robust parametric algorithm known as the RELAX for sinusoidal parameter
estimation is also mentioned briefly (discussed in more detail in Chapter 3). This
algorithm is capable of accurate sinusoidal parameter estimation even in the presence of
colored noise making it suitable for practical applications.
In this dissertation we focus on solving specific real world problems by analyzing
these data-adaptive techniques and coming up with effective and efficient ways to apply
them to give superior performance to the standard data-independent approaches.
1.5
Notations
Notation: Throughout this dissertation, Boldface upper-case and lower-case letters
are used to denote matrices and vectors, respectively. See Table 1-1 for more details on
notation.
Table 1-1. Notations
x
X
a vector
a matrix
diag(x) a diagonal matrix with elements of x on the diagonal
(·)H
(·)T
(·)(n)
conjugate transpose of a matrix or vector
transpose of a matrix or vector
n th iteration of a scalar, vector or matrix in algorithm
||·||2
x
^
ℓ2 norm
estimate of scalar x
,
definition
30
CHAPTER 2
DATA-ADAPTIVE TECHNIQUES FOR NETWORK FREQUENCY EXTRACTION FROM
DIGITAL RECORDINGS
2.1 Chapter Summary
A novel forensic tool used for assessing the authenticity of digital audio recordings
is known as the Electric Network Frequency (ENF) criterion. It involves extracting
the embedded power line (utility) frequency from said recordings, and matching it to
a known database to verify the time the recording was made, and its authenticity. In
this chapter, a non-parametric, adaptive, and high resolution technique known as the
Time-Recursive Iterative Adaptive Approach (TRIAA), is presented as a tool for the
extraction of the ENF from digital audio recordings. A comparison is made between this
data dependent (adaptive) filter, and the conventional Short-time Fourier Transform
(STFT). Results show that the adaptive algorithm improves the ENF estimation
accuracy in the presence of interference from other signals. To further enhance the
ENF estimation accuracy, a frequency tracking method based on dynamic programming
will be proposed. The algorithm uses the knowledge that the ENF is varying slowly with
time to estimate with high accuracy the frequency present in the recording.
2.2 Introduction
The use of digital recorders has become more prevalent in the world today due to
the advancement in digital technology and the significant progress made in the field of
digital signal processing (DSP). Prior to the increased use of digital recorders, forensic
audio analysis relied on different techniques of audio authentication. For instance, the
magnetic signatures that are left by the erase, record or play heads on the magnetic
tape of analog recorders, can be used to verify the authenticity of such recordings.
When it comes to digital recordings, alterations can be made very easily without
leaving behind such imprints, because digital recorders produce a recording by
converting sound variations to a series of numbers, making authentication of these
recordings a lot more difficult [32]. The importance of being able to verify the authenticity
31
of a recording can be seen in litigation cases [16] [33], where digital recordings are
brought forward as evidence in a trial. Therefore, more reliable methods of verifying the
authenticity of digital recordings need to be researched.
The Electric Network Frequency (ENF) Criterion was proposed by Grigoras [16], [34]
to address the issue of digital audio authentication. The ENF criterion is based on
extracting the utility frequency or ENF from a digital audio recording, and matching
the extracted frequency estimate to a reference database in order to determine the
authenticity, and also time of the digital recording. This process is possible because,
in some cases, digital recorders (even some battery powered recorders [35]), can
pick up the audible sound that is generated by the oscillation of a power grid’s
alternating current at this frequency. The frequency of oscillation is approximately
60 Hz in the United States, whereas in Europe it oscillates at approximately 50 Hz. The
corresponding harmonics of this frequency might also be present in the digital recording.
The ENF criterion is based on two assumptions. Firstly, the ENF for interconnected
networks is the same at all points within the network. Secondly, the frequency varies
randomly within a given interconnection, and hence, are not repeatable over a long
period of time. [33]
There are three known methods of extracting the ENF over time from a digital
recording [16], [34]. They are termed:
•
time/frequency domain analysis - This method is based on computing the
spectrogram of the signal and visually comparing it to the database.
•
frequency domain analysis - This method is based on selecting the frequency
location corresponding to the maximum amplitude of the power spectrum of
segments (frames) of the data after applying a band-pass filter.
•
time domain analysis - This method is based on measuring the zero crossings
of the signal in the time domain after a bandpass filter has been applied to the
recording.
Recently in [36], a quadratic interpolation scheme was applied to the frequency
domain analysis method to estimate the spectral peak locations (frequencies) more
32
accurately. This reduces the estimation error resulting from the use of a fixed grid size in
the spectral estimation process.
Besides the time-domain analysis, the above methods estimate the ENF based on
computing the Fast Fourier Transform (FFT) of overlapping segments (frames) of the
data known as the Short-Time Fourier Transform (STFT) which is limited by the trade-off
between time resolution and frequency resolution [2]. Parametric methods such as the
Frequency Selective ESPRIT, which give superior resolution compared to the FFT, can
also be used successfully to extract the ENF from one frame to another. However, in the
presence of significant interference within a given frame, the parametric methods yield
poor frequency estimates because of their sensitivity to an assumed data model.
This chapter focuses on two methods of extraction. The first, builds upon the
frequency domain analysis with quadratic interpolation. However, in place of the FFT,
the spectrum is estimated for each segment of the data using a non-parametric and
high resolution adaptive algorithm known as the Iterative Adaptive Approach (IAA) [25].
In the presence of interfering signals with frequencies within the range of values the
ENF can take on, IAA yields more accurate estimates of the ENF compared to the FFT
as a result of the improved spectral resolution and interference suppression capability.
The second method involves applying a frequency tracking algorithm based on discrete
dynamic programming [37], which takes into account the slowly varying nature of the
ENF over time. This tracking algorithm is necessary because, in some frames of the
data, the maximum spectral peak might correspond to an interference signal rather
than the network frequency signal even within the acceptable ENF limits. The ENF is
then estimated inaccurately, which can result in a false diagnosis that the recording in
question has been edited.
It is worthwhile to point out that, in order for the proposed methods to work, the ENF
must be embedded in the recording, which is not always the case especially in some
battery operated recorders [35]. This is certainly a drawback of using the ENF criterion
33
for digital authentication. However, if the ENF is embedded in a digital recording, more
reliable methods of extraction need to be sought.
Table 2-1. Abbreviations
APES
Amplitude and Phase Estimation
ENF
Electric Network Frequency
ESPRIT Estimation of Signal Parameters by Rotational Invariance
FDR
Frequency Disturbance Recorder
FIAA
Fast Iterative Adaptive Approach
IAA
Iterative Adaptive Approach
QN-IAA
Quasi-Newton Iterative Adaptive Approach
STFT
Short-time Fourier Transform
TRIAA
Time-Recursive Iterative Adaptive Approach
Extraction can also be carried out using the harmonics of the ENF signal for the
frequency estimation process. In some cases, the harmonics may give better estimates
because of a higher signal-to-interference-and-noise ratio compared to the fundamental
frequency.
The remaining sections of this chapter are organized as follows. In Section 2.3, the
network characteristics and the network frequency database are described. In Section
2.4, the IAA and TRIAA algorithms are described along with the frequency tracking
algorithm for ENF extraction. In Section 2.5, the experimental results based on a set
of digital audio recordings are presented. Finally, Section 2.6 contains the conclusions
drawn from the results.
Abbreviations: The abbreviations are presented for easy reference in Table 2-1.
2.3
Network Frequency Characteristics and Database
The frequency at which alternating current is distributed to various customers from
power stations, corresponds to the utility frequency or ENF. For European and most
Asian countries the value of this frequency is 50 Hz, while the value is 60 Hz in North
America, and several countries in South America. Japan uses both frequencies (50 and
34
Figure 2-1. FDR Distribution in North America
60 Hz) for electricity distribution. This frequency is determined by the speed of rotation
of the turbines used to drive the generators at the various power plants [38]. Naturally,
the rotation speed is not constant and varies within a certain limit (approximately
±0.05 Hz) depending on the amount of load connected to the network, and amount
of power generated at a given time. Experiments carried out in some European
countries [16], [39], have shown that this frequency variation is random and unique
within specific geographic locations. This uniqueness in frequency variation within a
region, coupled with the fact that network frequency is not repeatable over a long period
of time is what makes the aforementioned ENF criterion possible.
A database of the network frequency is needed in order to match the extracted ENF
from a recording for verification. In [16], such a database is created by connecting the
sound card of a computer to a transformer which is then connected directly to an AC
power outlet. The database currently being built in North America involves deploying
several sensors termed frequency disturbance recorders (FDRs), which perform
accurate ENF measurements, up to about ±0.0005 Hz. The measured data collected
by the FDRs is transmitted over the internet to servers, where it can be analyzed
and stored in a system termed the Information Management System (IMS) [40]. This
collection forms the Frequency Monitoring Network (FNET).
35
There are two major interconnections in North America and three minor interconnections.
These regions have unsynchronized networks (frequency and phase) and are therefore
connected via High Voltage Direct Current Lines (HVDC) [41]. The Eastern and Western
Interconnections form the major interconnections, while the Quebec, Texas and Alaska
Interconnections form the minor. The Alaska Interconnection is isolated, in the sense
that it is not connected to any of the other interconnections. It is therefore generally
not considered to be part of the North American grid. Fig. 2-1 shows the distribution
of the FDRs in Western, Eastern, Quebec and Texas Interconnections. Frequency
measurements collected by the FDRs in these interconnections show that the frequency
pattern is different at a given time from one interconnection to another. However, the
frequency pattern is unique at different locations within each interconnection [42]. The
FNET system, therefore, provides a viable ENF database.
2.4
Extraction Algorithms
2.4.1 Frequency Domain Analysis (STFT)
Due to the fact that the ENF varies with time, the extraction process involves
analysing a non-stationary data sequence. STFT is a common method for time-frequency
analysis of signals. This analysis assumes the signal of interest is stationary within short
time windows (frames); the FFT of the signal is then computed for each frame. The
frequency domain analysis [16] method of extraction is based on this idea.
The process involves re-sampling the audio signal to a lower sampling rate, to
reduce the computational complexity of the analysis. A bandpass filter with a narrow
bandwidth is applied to the signal with center frequency 50/60 Hz as a preprocessing
step. The rest of the analysis is described as follows. Let,
z = [z0 , z1 . . . zN −1 ]T
(2–1)
denote the re-sampled and filtered discrete-time signal. This signal is then split into R
overlapping frames as shown in Fig. 2-2, with each frame having length M and a shift
36
from frame to frame of length T . Using the frequency domain analysis method, the ENF
of the rth frame is estimated by finding the frequency that maximizes the spectrum of
each frame which is computed using the FFT based periodogram.
In order to get a more accurate estimate of the frequency, quadratic interpolation is
used [36], [43]. This interpolation scheme, involves fitting a quadratic model of the form
log ϕ^(ω) = m(ω − ωk
max
− )2 + c
(2–2)
around the frequency point that maximizes the power spectrum:
ωkmax
= argmax ϕr (ωk )
(2–3)
ωk
= 2πk/K, k = 0, 1, . . . , K − 1 corresponds to the frequency grid point of a
frequency grid with size K, and ϕr (ωk ) is power spectrum of the rth frame.
where ωk
The value of ω that maximizes the model (2–2) is taken as the estimated peak
of the spectrum. This value is determined by fitting the model to the highest sample
of the power spectrum and the two adjacent points with corresponding frequencies
(ωk
max −
1 , ωkmax , ωkmax +1 ). This value of ω that maximizes the model is:
ω
= ωk + max
(2–4)
where
= 12 β
β− 1 − β1
(ωkmax+1 − ωkmax )
−1 − 2β0 + β1
βℓ , log ϕr (ωkmax +ℓ ),
ℓ = −1 , 0 , 1 .
(2–5)
(2–6)
The corresponding frequency estimate of the rth frame in Hz is given by:
f^(r) = 2π (ωkmax
+ ) Fs
where Fs is the sampling frequency (in Hz) of the signal.
37
(2–7)
Figure 2-2. Segmentation of data for STFT
The use of STFT will result in a trade-off between frequency resolution and time
resolution. For a given frame length, this trade-off can be optimized by applying a
rectangular window to each frame, which will provide the best spectral resolution at a
cost of higher side lobes compared to other spectral windows.
In order to get improved spectral resolution over FFT, one has to resort to using
parametric methods or data-dependent (adaptive) non-parametric methods for spectral
estimation. Parametric methods, on the one hand, are not robust against data model
errors. On the other hand, non-parametric adaptive methods are more robust, since they
do not assume a specific parametric data model. Well-known adaptive methods include
the Capon algorithm and the Amplitude and Phase Estimation (APES) algorithm. These
algorithms also provide higher resolution and lower sidelobes than the periodogram.
However, these methods are inadequate because they require multiple realizations
(snapshots) of the random signal, which is not the case with the current data, as only
one snapshot is available for frequency estimation. Spatial smoothing (segmenting and
spectral averaging of the data) can be used to improve the spectral estimates of the
Capon and APES algorithms in the one-snapshot case; but the cost of doing this will be
a degradation in the spectral resolution, which is not desirable. The wavelet transform
is also a common tool for time-frequency analysis. Contrary to the STFT, which uses a
38
fixed window size, the wavelet transform uses short windows at high frequencies and
longer windows at low frequencies. The wavelet transform is therefore not suitable for
our problem because we are interested only in a small range of frequencies.
IAA is a non-parametric data-dependent algorithm based on Weighted Least
Squares (WLS), originally presented in [25] for Direction of Arrival (DOA) estimation
in array processing. The IAA algorithm is capable of yielding high resolution and low
sidelobes even in the case of a single snapshot [25], hence making it suitable for
estimating the ENF in the presence of interferences.
2.4.2 IAA and TRIAA
The ENF can be extracted with high accuracy in the presence of interference using
the IAA algorithm for a given frame. The proposed ENF extraction process follows
(2–2)-(2–7), with the FFT spectral estimate ϕr replaced by the IAA spectral estimate.
The IAA and TRIAA [44] used for spectral estimation of non-stationary data will be
discussed in this section.
= [y0 , y1 . . . yM −1 ]T
denote a uniformly sampled stationary data sequence and A = [a(ω0 ), a(ω1 ) . . . a(ωK−1 )],
where a(ωk ) = [1, ejω , . . . , e(M −1)jω ]T corresponds to a steering (frequency) vector, and
ωk = 2πk/K, k = 0, 1, . . . , K − 1, corresponds to a frequency grid point of a frequency
grid with size K. Also let α = [α(ω0 ), α(ω1 ), . . . , α(ωK−1 )]T , with α(ωk ) denoting the
complex spectral estimates of y at ωk . The following data model can be formulated:
The spectral estimation problem can be set-up as follows. Let y
k
k
y = Aα
where the noise contributions of y are taken into account implicitly [25].
39
(2–8)
The IAA algorithm solves for the spectral estimates α by minimizing the following
quadratic cost function in (2–9) using weighted least squares (WLS):
||y − a(ωk )α(ωk )||2Q−1 (ωk )
(2–9)
where ||x||2Q−1 (ωk ) , xH Q−1 (ωk )x,
Q(ωk ) = R − pk a(ωk )aH (ωk )
(2–10)
R = APAH
(2–11)
and P , diag[p0 , p1 , . . . pK−1 ], with pk for k
= 0, . . . , K − 1, denoting the power estimate at
each frequency grid point, given by |α(ωk )|2 . R1 is the covariance matrix of the data and
Q(ωk ) is the covariance matrix of the interference and noise, where interference refers
to all the signals at frequency grid points other than the current grid point of interest ωk .
Minimizing the cost function in (2–9) with respect to the α(ωk ) for k
= 0, . . . , K − 1 gives
the following solution:
α
^(ωk ) =
aH (ωk )Q−1 (ωk )y
,
aH (ωk )Q−1 (ωk )a(ωk )
k
= 0, 1 , . . . , K − 1
(2–12)
The solution in (2–12) can be re-written as
α
^(ωk ) =
aH (ωk )R−1 y
,
aH (ωk )R−1 a(ωk )
k
= 0, 1 , . . . , K − 1
(2–13)
using the Woodbury matrix identity2 and (2–10). This prevents the computation of
the interference covariance matrix Q−1 (ωk ) for each frequency grid point. Note that
the computation of R−1 requires the knowledge of α(ωk ) and vice versa. Hence this
algorithm is solved in an iterative manner, with the estimate of α initialized using the
1
R = APAH + σ2 I for ill-conditioned matrices [45]
2
matrix inversion lemma
40
FFT. This iterative algorithm takes about 10 to 15 iterations to converge based on
experimental and numerical results.
Note also that without accounting for the interference from other frequency grid
points (without weighting), minimizing the cost function in (2–9) for K
= M gives the
Discrete Fourier Transform (DFT) of the signal:
α
^(ωk ) =
aH (ωk )y
M
,
k
= 0 , 1, . . . , M − 1.
(2–14)
The IAA algorithm described above is used for spectral estimation of stationary data.
Analogous to the STFT, the spectral content of a non-stationary data sequence, such as
(1), can be estimated using the TRIAA [44]. The signal is split into overlapping frames
similar to Fig. 2-2 and the IAA spectral estimate is computed for each frame. However,
to reduce the computational complexity, each subsequent frame after the first frame
is initialized with the spectral estimate of the previous frame instead of the FFT based
periodogram as described in the IAA algorithm. The resulting algorithm yields better
spectral resolution and lower side lobes than the STFT.
There is still a significant increase in the computational complexity when using the
TRIAA algorithm compared to using STFT for spectral estimation. This computational
complexity is reduced slightly by reducing the number of iterations in subsequent frames
for the TRIAA. This is because convergence of the estimated spectrum will occur in
fewer iterations given the current frame is initialized by the spectral estimate of the
previous frame. When the dataset is significantly large, the use of this algorithm is
still impractical. The bottle-neck of the TRIAA algorithm is in the computation of the
denominator in (2–13) for each frame.
In [46], [47] the Toeplitz structure of the covariance matrix R is exploited and the
computation of R−1 is performed using the Gohberg-Semencul (GS) factorization of
this matrix [2]. Moreover, the denominator is obtained via evaluating a polynomial.
This reduces the computational complexity of the denominator in (2–13) (which is the
41
bottleneck of the IAA algorithm) from O(M 2 K ) to O(M 2 ) floating point operations
(flops) [46] for a given frame, without a loss in performance. The algorithm is termed the
Fast IAA (FIAA), which is a significant improvement but still computationally expensive
for large datasets. The computational complexity of IAA and FIAA are O(M 2 K ) and
O(M 2
+ KlogK ), respectively, where M is the data length and K is the grid size, with
K >> M .
An approximate algorithm to the IAA algorithm with significantly faster computational
time is described in [48] and referred to as the Quasi-Newton IAA (QN-IAA). The
QN-IAA algorithm estimates the covariance matrix as if it were from a low-order (L)
autoregressive (AR) process, where L << M with M being the data (frame) length.
The inversion of the lower-order covariance matrix Q ∈ CL×L is carried out in place of
R ∈ CM ×M , yielding an approximate solution to the IAA spectral estimate (2–13) with
significant reduction in the computational complexity and just a slight degradation in the
resolution. The computational complexity of this algorithm is O(L2 + KlogK ).
The FIAA or QN-IAA can be used in a time-recursive manner for non-stationary
data as is the case with the ENF signal. This algorithm reduces the trade-off between
frequency resolution and time-resolution for a given frame length compared to the FFT
based periodogram during the ENF extraction process. The extraction process is the
same as the frequency domain analysis (2–2)-(2–7) with ϕr replaced by either of the
aforementioned algorithms.
However, even if a good algorithm is used for frequency estimation based on (2–7),
specific frames might be corrupted by interference signals with frequency components
within the ENF limits. This could lead to errors in frequency estimation, if the frequency
location corresponding to the maximum value of the estimated spectra belongs to an
interference signal. A robust method of tracking the ENF that exploits the slowly varying
nature of this frequency is needed. The next section describes the proposed frequency
tracking algorithm.
42
2.4.3 Frequency Tracking
A method of estimating the ENF by tracking it from one frame to another is
formulated here from a mathematical point of view. The proposed method uses discrete
dynamic programming [37] to find a minimum cost path. A cost function as shown in
this section is selected which takes into account the slowly varying nature of the actual
network frequency. This cost function penalizes significant jumps in frequency from
frame to frame and the corresponding path is used to estimate the ENF.
This algorithm involves finding the peak locations from the spectrum of each frame
and assigning costs based on the difference between a peak location in one frame and
a peak location in another frame. The magnitude of the assigned cost is related to the
difference in the frequency from one frame to another. The minimum cost path from the
first frame to the last frame is computed to estimate the ENF.
To estimate the number of relevant peaks (sinusoids) in a given frame, a model
order selection tool known as the Bayesian Information Criterion (BIC) is used. The BIC
for complex sinusoids in noise is given by (refer to [2] [49] for a full derivation):
)
(
2nr
∑
a(ωk )^α(ωk )||2 + 5(2nr )lnM.
BIC(nr ) = M ln ||y −
k=1
(2–15)
The number of peaks (real sinusoids) nr , is estimated as the minimizing argument of the
above BIC criterion. The first term in (2–15) is a Least-Squares data fitting term, which
decreases as the number of estimated peaks nr increases, where as, the second term
is a penalty term that prevents ’over-fitting’ of the data model. Once the nr largest peaks
and corresponding locations are determined in each frame, the frequency tracking
problem is formulated and solved as follows.
Assume that for a given frame r, a set of estimated peak locations (frequencies)
is denoted by r
fr ∈
= {Pr1 , Pr2 , . . .
Prnr }. We would like to find a path {fr }R
r=1 , such that
r and where the difference fr − fr−1 is as small as possible for r = 1, 2, ..., R.
This set corresponds to the estimated ENF over all frames and can be obtained as the
43
minimizing argument in the following optimization problem:
=
J
min
R
∑
fr ∈ r
r=1,...,R r=2
(fr − fr−1 )2 .
(2–16)
Calculating this cost using an exhaustive search is impractical. However, using dynamic
programming [37] the path that minimizes this cost can be computed recursively and
efficiently by minimizing the cost from a given frame j < R, to the last frame, denoted by
J (j, fj ).
J (j, fj ) =
R
∑
min
fr ∈r
r=j +1,...,R r=j +1
(fr − fr−1 )2 , fj ∈ j .
(2–17)
This optimal cost satisfies the recursive equation,
J (j, fj ) = min {(fj +1 − fj )2 + J (j + 1, fj +1 )}, fj ∈ j
fj +1
∈j +1
which can be calculated for j
(2–18)
= R − 1, R − 2, . . . , 1, with the initialization, J (R, fR ) =
0, fN ∈ N . Note that
J
= fmin
J (1, f1 ), fN ∈ N
∈
1
1
(2–19)
is the cost from the first frame to the last frame R and the set {fr }R
r=1 that minimizes this
cost function corresponds to the extracted ENF signal as mentioned above. Dynamic
programming has a computational complexity of O(R2max ), where R corresponds to the
total number of frames and max is the number of spectral peaks in the frame with the
maximum number of peaks.
2.4.4 Matching the Extracted ENF to Database
Once the ENF signal has been extracted, a method of matching the estimated
signal to the database signal is required. The goal is to find the location/time within
the database that is similar in pattern to the extracted ENF. In [36] a method based on
minimizing the squared error between the ENF and database is used for automated
44
matching. A method of correlation matching proposed in [50] for short digital recordings
(10-15 minutes) is used in place of this MSE method. The process of correlation
matching is described as follows. Assume that f
signal and d
= [f1 , f2 , . . . , fR ] is the extracted ENF
= [d1 , d2 , . . . , dL] corresponds to the database signal with L
> R. The
matching process requires finding lmax such that:
lmax
= argmax c(l),
l
l
= 1, 2, . . . , L − R
(2–20)
where c(l) is the correlation coefficient between f and the vector [dl , dl+1 , . . . , dl+R−1 ].
An important point to make is that, the maximum correlation coefficient c(lmax ) is
used here only for matching the estimated ENF to the database and comparing the
accuracy (reliability) of the different algorithms presented. Once a match has been
made, determining locations of edits to a recording should be based on the differences
between the ENF estimate and the database.
Table 2-2. Parameters for the Experiment
PARAMETERS
Data1 Data2
T (Time Shift)
1s
1s
M (Length of Frame)
20s
33s
R (Number of Frames)
1800
1800
2.5
Experimental Results
The algorithms presented in the previous section are applied to two different
digital audio datasets referred to as Data1 and Data2. The two datasets are recorded
simultaneously and therefore, should contain the same ENF pattern over time. The
first data set (Data1) is acquired by connecting an electric outlet via a voltage divider
directly to the internal sound card of a desktop computer, resulting in an ENF signal
with a rather high signal-to-interference-and-noise ratio. On the other hand, the second
dataset (Data2) is an actual speech recording played from a speaker and picked up by
the internal microphone of a laptop computer.
45
Each of these recordings are originally sampled at 44.1 kHz at a bit rate of 16 bits
per sample. Each dataset is re-sampled to 441 Hz, hence keeping only the fundamental
frequency (1st harmonic) and the two higher harmonics of the ENF. A bandpass
filter with a narrow bandwidth around the network frequency is applied to the data to
eliminate as much interference as possible without distorting the ENF signal. Based on
Fig. 2-2 each data set is split using the values shown in Table 2-2. This set-up results in
an ENF estimate every second for a total of 30 minutes for each dataset.
An increase in the frame length improves the signal-to-noise ratio of the signal [36]
and the spectral resolution at the cost of lower time resolution. Therefore, a larger frame
length is used for Data2 which has a weak ENF signal compared to Data1 which has a
strong ENF signal.
Figure 2-3. Matching extracted ENF to database (Data1 - scaled to 60 Hz)
Fig. 2-3 shows the extracted ENF signal (shifted by 0.05 Hz for illustration
purposes) from Data1, matched with the truth obtained from the FDRs, when the
data set has not been altered in any form (using STFT and (2–7)). Fig. 2-7 shows the
extracted ENF using the STFT based method and our proposed method (also shifted
for comparison purposes). Tables 2-3 and 2-5 give the maximum correlation coefficient
c(lmax ) of the various methods for Data1 and Data2, respectively, also when the signals
have not been altered. The maximum correlation coefficient values are used to compare
46
the accuracy of the algorithms and hence determine which is more reliable for ENF
estimation. We have also included similar MSE (actually standard deviation) analysis
in Tables 2-4 and 2-6 for the datasets, where the MSE is computed by averaging the
squared difference between the True ENF and the estimated ENF. It is important to point
out that the estimated ENF can sometimes have a constant offset [39], [50]. Therefore,
the correlation is the preferred method for accuracy measure. The datasets used for
this experiment do not have such an offset. They have also been made available at
http://www.sal.ufl.edu/download.html.
Table 2-3. Correlation coefficients of Algorithms (Data1)
Algorithm
Harmonic
STFT STFT(Track) TRIAA TRIAA(Track) F-ESPRIT
60 Hz
0.9912
0.9917
0.9895
0.9900
0.9800
120 Hz
0.9911
0.9949
0.9902
0.9946
0.9470
180 Hz
0.9968
0.9968
0.9961
0.9961
0.9962
Table 2-4. Standard Deviation of error for Algorithms (Data1)
Algorithm
Harmonic
60 Hz
120 Hz
180 Hz
STFT STFT(Track) TRIAA TRIAA(Track) F-ESPRIT
2.772e−3 2.650e−3 3.032e−3 2.919e−3
5.364e−3
2.774e−3
1.900e−3
2.145e−3
1.851e−3
2.822e−3
1.999e−3
2.198e−3
1.999e−3
6.570e−3
2.830e−3
2.5.1 Data1 Analysis
Fig. 2-3 shows the extracted harmonic (180 Hz) of the ENF signal scaled to 60
Hz and matched (using the location corresponding to the maximum correlation (2–20))
to the actual database frequency obtained from the FDRs. For each of the algorithms
used, the third harmonic gave the most accurate results for this dataset as shown in
Table 2-3. This is because for a fixed grid size, the estimation error when using the
third harmonic is reduced by a factor of three compared to the fundamental frequency.
Harmonics with frequencies higher than 180 Hz can be used for the estimation process
47
at a cost of increased computational complexity due to the increased sampling rate. Also
from Table 2-3, It can be seen that each of the STFT and TRIAA algorithms, produce
accurate estimates of the ENF using (2–7) because of the rather strong ENF signal. The
signal at the second harmonic is weak relative to the first and third harmonics, and in
a few frames the estimate was inaccurate. However, the frequency tracking algorithm
mitigated these inaccuracies successfully by tracking the correct spectral peaks.
The parametric method, frequency selective (F-ESPRIT) [2],[51] also yields
accurate estimates of the ENF for Data1 when the signal model assumes there is
only one sinusoid per frame. However, this method and other parametric methods are
not appropriate for ENF estimation in the presence of interference, because they are
sensitive to model assumptions.
For this dataset, the STFT yields slightly better results, compared to the adaptive
method (TRIAA). This can be explained by the fact that the periodogram is optimal for
estimating spectral lines (sinusoids) in the presence of white noise when they are well
resolved [2]. However, when there are interfering signals present, the poor resolution
of the periodogram will yield inaccurate estimates as is the case with Data2, a typical
digital recording.
Table 2-5. Correlation coefficients of Algorithms (Data2)
Algorithm
Harmonic
STFT STFT(Track) TRIAA TRIAA(Track) F-ESPRIT
120 Hz
0.9125
0.9857
0.9305
0.9907
0.8446
Table 2-6. Standard Deviation of error for Algorithms (Data2)
Algorithm
Harmonic
120 Hz
STFT STFT(Track) TRIAA TRIAA(Track) F-ESPRIT
7.948e−3 3.369e−3 7.225e−3 2.914e−3
1.086e−2
48
−7
2.5
x 10
True frequency (119.963 Hz)
FFT
IAA
Freq. estimate IAA (119.965 Hz)
Freq. estimate FFT (119.970 Hz)
squared magnitude
2
1.5
1
0.5
0
119.8
119.85
119.9
119.95
120
120.05
frequency (Hz)
120.1
120.15
120.2
Figure 2-4. Power Spectrum of one Frame (Data2): poor resolution of FFT
2.5.2 Data2 Analysis
For Data2, the second harmonic (120 Hz) is used to estimate the ENF, because
the first and third harmonics are too weak to be used for estimation. Table 2-5 shows
the maximum correlation coefficient values for the STFT and TRIAA using (2–7), the
frequency tracking algorithm using the spectral peaks of the FFT and IAA and the
parametric method (F-ESPRIT) with one assumed sinusoid. The ENF estimation
accuracy is improved using the adaptive method (IAA) because of improved spectral
resolution for several frames. Fig. 2-4 shows a comparison of the spectrum of one frame
of the Data2, where the poor frequency resolution of the FFT results in a relatively poor
estimate of the network frequency compared to the IAA algorithm.
−7
4
x 10
3.5
True frequency (119.952 Hz)
FFT
IAA
squared magnitude
3
2.5
2
1.5
1
0.5
0
119.8
119.85
119.9
119.95
120
120.05
frequency (Hz)
120.1
120.15
120.2
Figure 2-5. Power Spectrum of one Frame (Data2): strong interference signal
49
Figure 2-6. Extracted ENF via Frequency Tracking (Data2 - scaled to 60 Hz)
Fig. 2-7 shows this extracted ENF harmonic using the STFT and (2–7) matched
with the database. From this figure, there are several frames where the ENF is
estimated inaccurately, due to the fact that the frequency corresponding to the maximum
spectral peak for those frames do not correspond to the ENF. This can occur if there
is another signal present with frequency within the limits of the acceptable range of
the ENF as illustrated in Fig. 2-5. This figure shows that for both spectral estimation
techniques used (IAA, FFT) the ENF harmonic estimate using (2–7) will be 120 Hz,
whereas the true frequency is approximately 119.95 Hz.
This problem can be rectified using our dynamic programming based frequency
tracking algorithm presented above.
Fig. 2-6 shows the spectral peak locations computed using the TRIAA and the
corresponding ENF estimate using dynamic programming. The estimate of the network
frequency using this tracking algorithm is then matched to the database in Fig. 2-7,
which provides a better match when compared to using (2–7), which can also be seen in
this figure, Fig. 2-8 (absolute error) and also from Table 2-5.
A few important points to make are that the frequency tracking algorithm uses the
peak locations for each frame estimated either by the adaptive algorithm (IAA) or the
FFT. The results show that the estimated ENF is more accurate when the peak locations
50
Figure 2-7. Matching extracted ENF to Database (Data2 - scaled to 60 Hz)
Figure 2-8. Absolute error of Algorithms: STFT and TRIAA (Track)
of IAA are used. This is as a result of the inaccurate estimates in some frames caused
by the poor resolution of using FFT. Also, all the numbers presented can be improved
upon slightly by using the entire dataset (44.1kHz) for analysis. For example, the STFT
maximum correlation of 0.9125 will be improved to 0.9158 without re-sampling, which
may not be worth the increased computational complexity.
2.6
Conclusions
When it comes to digital audio verification, the reliability of the method used for
authentication cannot be overemphasized. This chapter demonstrates a reliable method
of extracting the network frequency from a digital recording when the ENF cannot be
51
extracted from some of the frames using the FFT based periodogram either because
of poor spectral resolution or a stronger interference signal within said frame. These
problems were solved by using an iterative adaptive method (IAA), which provides better
spectral resolution than the FFT based approach. Also a frequency tracking method
based on dynamic programming was used for accurate extraction of the ENF even in the
presence of a strong interference signals within ENF limits.
From the results presented, the FFT gives slightly better estimates of the network
frequency when the signal-to-interference-plus-ratio is very high as is the case with the
first dataset. However, in most digital recordings, there will be significant interferences
from the recorded speech signals and other surrounding sounds that could lead to poor
estimation performance using the FFT due to its poor resolution and high side lobe
problems. As the results have shown, the adaptive techniques and frequency tracking
method should be adopted for ENF estimation, especially in challenging environments.
52
CHAPTER 3
DATA-ADAPTIVE RELAX FOR RFI SUPPRESSION FOR THE SYNCHRONOUS
IMPULSE RECONSTRUCTION (SIRE) RADAR
3.1 Chapter Summary
This next two chapters focus on remote sensing applications, specifically on the
Synchronous impulse reconstruction radar (SIRE) UWB radar built by the Army research
laboratory for landmine detection. This chapter focuses on suppression of Radio
Frequency Interference (RFI) for ultra-wideband (UWB) radar signals, sampled using
this synchronous impulse reconstruction (SIRE) time equivalent sampling scheme. This
equivalent sampling scheme is based on the Army Research Lab’s (ARL) efforts to
build an ultra-wideband (UWB) radar in forward looking mode that samples returned
radar signals using low rate and inexpensive analog-to-digital (A/D) converters. The
cost effectiveness of this SIRE UWB radar makes it plausible for actual ground missions
for detecting buried explosive devices. However, the equivalent time sampling scheme
complicates RFI suppression as the RFI samples are aliased and irregularly sampled
in real time. In this chapter, the data-dependent RELAX and multi-snapshot RELAX
algorithms are presented as an intermediate step to the previously proposed averaging
scheme by the Army Research Laboratory, in order to enhance RFI suppression for
this sampling scheme. A direct application of the RELAX algorithm is computationally
intensive so an efficient method for generating the spectrum of this equivalently sampled
data is proposed in this chapter that provides a factor of 10 improvement in computation.
The proposed suppression technique involves modelling the narrowband RFI signals as
a sum of sinusoids and applying the aforementioned algorithms. The RELAX algorithm
improves the RFI suppression performance without altering the target signatures
compared to AR modelling. The multi-snapshot RELAX algorithm which provides a more
accurate sinusoidal model than the RELAX algorithm, improves on the RELAX algorithm
in terms of suppression. However, the target signatures are suppressed as the number
of sinusoids increases. The analysis of the algorithms is performed using sniff (passive)
53
data collected using the SIRE radar in addition to simulated wide-band echo signals
(point-target signatures).
3.2 Introduction
Ultra-wideband (UWB) radar is a commonly used tool for various remote sensing
applications. Such applications include but are not limited to the use of low frequency,
high bandwidth pulses for detecting improvised explosive devices (IEDs) and land mine
targets. The effective detection of land mines and other IEDs could lead to increased
safety for various ground related missions [52].
The use of low frequencies in UWB radar is necessary for foliage or ground
penetration, whereas the use of wideband pulses are necessary for good resolution
(ability to detect targets from clutter) [53]. However, because of these requirements, the
data (target returns) collected by the UWB radar will be corrupted by signals in the radio
frequency spectrum (specifically the UHF/VHF bands). These signals include FM Radio,
TV broadcasts and other narrowband and wideband communication signals. The ability
to effectively detect targets is reduced by the presence of these radio frequency signals.
General methods of suppressing RFI and their limitations are discussed in detail in [53]
for conventional UWB radar, which is the case when the returned signals are sampled
regularly at or above the Nyquist rate.
Due to the large bandwidth of the returned radar signals, conventional sampling
will require high rate analog-to-digital (A/D) converters to digitize the returned signals.
These high speed A/D converters are expensive to build and makes practical applications
improbable. In other to improve on the cost of UWB radars, the Army Research
Laboratory (ARL) is currently working on an equivalent time sampling UWB radar in
forward looking mode, referred to as the Synchronous Impulse Reconstruction (SIRE)
radar [54]. This radar uses low rate (inexpensive and commercially available) A/D
converters to sample the returned signals (approximately 3GHz bandwidth), which
makes the radar more feasible for adoption in practice. This equivalent time sampling
54
scheme takes advantage of the fact that the scene is not changing with time, hence
aliasing of the returned target signals can be prevented. However, this is not the case
with the radio frequency signals which are changing with time. Aliasing and the irregular
sampling caused by the time-equivalent scheme becomes an issue when it comes to the
subject of RFI suppression as discussed in the next two sections.
This chapter focuses on suppression of RFI signals for this equivalent time-sampling
scheme. The goal is to model the narrowband interference as a sum of sinusoids in real
time and estimate and subtract the sinusoids before averaging to achieve further
suppression.
A cyclic optimization algorithm known as RELAX [31] is proposed for estimating the
parameters of the sinusoids (in an iterative manner). This algorithm is an asymptotic
maximum likelihood approach [55] and is computationally and conceptually simple. It
has been applied to problems like non-contact vital sign detection for more accurate
estimates of respiratory rates and heart rates [56]. It has also been shown to estimate
the parameters of sinusoids accurately even in the presence of colored noise [55]. The
multi-snapshot RELAX [57] algorithm, which is an extension of the RELAX algorithm, will
be used to provide a more accurate sinusoidal model for the SIRE sampling scheme.
The RELAX algorithm or multi-snapshot RELAX are implemented as an intermediate
step to the already proposed averaging method [58] to achieve further suppression.
In Section 3.3, the time equivalent SIRE sampling scheme is described. Section
3.4 briefly describes the limitations of some conventional methods to the SIRE sampled
data for RFI suppression. The averaging method proposed in [58] for RFI suppression
of SIRE sampled data is also discussed in this section, along with its performance.
In Section 3.5, the RELAX algorithm, along with a fast computation of the spectrum
of irregularly sampled SIRE data for this algorithm is presented; the multi-snapshot
RELAX algorithm is also described in this section. The results are presented in Section
3.7, starting with simulations that show how the RELAX algorithm suppresses aliased
55
sinusoids for simulated data. The sniff (passive data collected using the SIRE UWB
radar) is then used to test the effectiveness of the proposed algorithms, which are
compared to AR modelling of the interference based on this sampling scheme (see
Appendix). Finally, the conclusions of this chapter are presented in Section 3.8.
3.3 SIRE Equivalent Sampling Scheme
In this section, the Synchronous Impulse Reconstruction (SIRE) equivalent time
sampling technique as detailed in [54] is briefly described. This time equivalent sampling
scheme poses some challenges on identifying and hence suppressing RFI sources, due
to the fact that the RFI sources are changing with time as will be discussed.
The SIRE sampling scheme involves sampling the returned radar signals from a
scene at a significantly lower sampling rate fs , (with corresponding sampling period
s), than the Nyquist rate, which leads to aliased samples. N aliased samples are
collected per pulse repetition interval (PRI) or fast time, and for each subsequent PRI,
N more samples are collected with the range profile shifted by e (in time). After K
pulse repetition intervals (PRIs) or slow time, a total of K × N aliased samples are
collected. These samples are interleaved as shown in Fig. 3-1, which gives an effective
sampling rate of fe
= 1/e that is equal to, or greater than the Nyquist rate. Because the
scene of interest in not changing with time, the returned samples from a given range bin
theoretically should also remain unchanged in time. Therefore, the interleaved samples
are theoretically effectively sampled above the Nyquist rate and should be unaliased.
The measurements from each range profile are typically repeated M times and
added coherently to improve the signal-to-noise ratio (SNR). Fig. 3-1 shows the special
case of M
= 1. Table 1 summarizes the parameters used by ARL in the SIRE radar
pertaining to RFI suppression [54].
The RFI signals, which are collected in addition to the desired target returns, on the
other hand, are changing with time. Therefore, when the collected data are interleaved,
they do not represent the true time samples of the RFI signals.
56
Table 3-1. ARL Parameters for Synchronous Reconstruction Radar.
Radar A/D sampling rate
= 40 MHz
s = 25 ns
PRF = 1 MHz
PRI = 1 µs
N =7
K = 193
K × N = 1351
e = 129.53ps
fe = 7.72 GHz
fs
Radar A/D sampling period
Pulse repetition frequency
Pulse repetition interval
Number of range profiles (per slow time)
Interleaving factor
Total number of range profiles
Effective sampling period
Effective sampling rate
For instance, consider a complex sinusoid sampled at fe (see Tab. 3-1), with time
samples h[n]
=
ejωo n . The periodogram estimate of the spectrum of h[n] is given by
ϕ(ω ) = (1/L)|H (ω )|2 , where
H (ω ) =
L
∑
ej (ωo −ω)n
i=1
(3–1)
is the discrete-time Fourier transform (DTFT) of h[n] and L
= K × N is the total number
of samples. If this complex sinusoid is sampled using the SIRE technique (M = 1), the
time samples of the interleaved signal will be given by:



h[l(T + 1)]
for l = 0, 1, . . . , K − 1






for l = K, . . . , 2K − 1
~h[l] = h[(l − K )(T + 1) + K ]


···





 h[(l − 6K )(T + 1) + 6K ] for l = 6K, . . . , 7K − 1
= (fe/PRF) (the other variables are described in Tab. 3-1). The corresponding
~ (ω)|2 , where H~ (ω) is the DTFT of ~h[l] and is
periodogram is given by ϕ~(ω ) = (1/L)|H
where T
57
given by:
K−
∑1
~ (ω)=
H
e
l=0
··· +
jωo l(T +1) −jωl
e
7∑
K−1
+
K−
∑1
ejωo ((l−K )(T +1)+K ) e−jωl
l=K
ejωo ((l−6K )(T +1)+6K ) e−jωl
(3–2)
l=6K
Therefore,
~ (ω) =
H
( 6
∑
ej (ωo −ω)sK
) (K−1
∑
s=0
)
ej (ωo (T +1)−ω)r
(3–3)
r=0
Fig. 3-2 shows the periodogram spectral estimate of the regularly sampled sinusoid
ϕ(ω ) and the interleaved SIRE sampled signal ϕ~(ω ). The spectrum of the complex
~ (ω) can
sinusoid is not only distorted, but it peaks at a different frequency. Note that H
be re-written as:
~ (ω) =
H
( 6
∑
ej (ωo −ω)sK
) (K−1
∑
)
ej (ωo −ω)r ejωo T r
r=0
s=0
(3–4)
Figure 3-1. Synchronous Impulse Reconstruction (SIRE) equivalent time sampling.
= 2πm/T , where m ∈ Z, then H~ (ω) reduces to H (ω). This condition
implies that the frequency (in Hz) of the complex sinusoid f = fe ωo /2π = m × PRI,
Therefore, if ωo
is an integer multiple of the pulse repetition frequency. Unless this condition is true,
interleaving will lead to distortion of the complex sinusoid.
58
0
SIRE (interleaved)
Regularly sampled
−10
−20
dB
−30
−40
−50
−60
−70
0
0.1
0.2
0.3
0.4
Normalized frequency (cycles/sample)
0.5
Amplitude
Figure 3-2. Spectrum of SIRE sampled complex sinusoid sampled after interleaving
compared to the spectrum of sampled regularly above the Nyquist rate.
1
0.5
Magnitude
0
0
0.2
0.4
0.6
time (s)
0.8
1
−6
x 10
1
0.5
0
0
20
40
60
80
frequency (MHz)
100
120
Figure 3-3. Spectrum SIRE sampling pattern: One fast time pulse (N
= 7 samples).
A single complex sinusoid, sampled regularly below the Nyquist rate (fs ), should
consist of a single peak at an ambiguous frequency in the frequency domain (in a
bandwidth of fs ). However, due to the irregular sampling pattern of the SIRE sampling
technique, a single sinusoid will be seen as multiple peaks within this bandwidth.
Fig. 3-3 shows the SIRE sampling pattern (in real time) and its corresponding
spectrum for a single fast time pulse (N samples). As expected, this will result in a
sinc like function every 40 MHz (fs ) in the frequency domain. However repeating this
sampling pattern K times will correspond to sampling in the frequency domain as
seen in Fig. 3-4. Therefore, the spectrum of a single sinusoid sampled using the SIRE
59
sampling scheme will correspond to convolving the spectrum of this sampling scheme
with that of a sinusoid resulting in multiple peaks.
In the next section, we discuss some of existing algorithms for RFI suppression as
Amplitude
well as the limitations posed by this sampling scheme, based on the analysis above.
1
0.5
Magnitude
0
0
0.5
1
1.5
time (s)
2
2.5
−6
x 10
1
0.5
0
0
20
40
60
80
frequency (MHz)
Figure 3-4. Spectrum SIRE sampling pattern (N × K
100
120
= 1351 samples).
3.4 Existing RFI Suppression Methods
One popular technique for RFI suppression based on conventional sampling
involves the use of notch filters. This method involves estimating the spectrum of the
corrupted signal and removing the spikes in this spectrum using a notch filter. This
method works well for narrowband interference sources. However, it will introduce
sidelobes in the time-domain [59–61]. Filtering techniques in general, suffer from filter
transients and reduced data length. The notch filtering problem is even more severe
because of the ambiguity in frequency for the SIRE sampling scheme based on the
analysis in the previous section and Fig. 3-2 for the interleaved signals. Also, if the
analysis of the corrupted signal is performed in real-time (before interleaving), one
interference source will appear to have multiple peaks in the spectrum due to irregular
sampling as seen in Fig. 3-4, which makes this method not applicable.
Modelling the RFI using AR models can also be used for suppression (see
Appendix). The irregular sampling of the SIRE data makes this endeavour challenging.
60
However, the SIRE sampled data is sampled regularly in fast time and slow time, and
this can be exploited for AR modelling. There are only N
= 7 samples sampled regularly
= 193 regularly sampled samples in slow time. These
slow time samples can be used for AR modelling with N = 7 snapshots allowing for
in fast time, whereas there are K
more freedom in the choice of the AR model order (see Appendix for AR modelling of
SIRE sampled data).
Modelling the narrowband RFI as a sum of sinusoids, and estimating their
parameters has been shown to be effective for suppressing RFI with little signal
distortion [59],[62]. This method involves estimating the amplitude, frequency and
phase of each interfering sinusoid and subtracting the resulting sinusoid from the
corrupted data. The effectiveness depends on how accurate these parameters are
estimated, and is reduced if the sinusoidal model for the RFI signals starts to breakdown
[62]. This occurs when the duration of data is greater than the modulation time (inverse
of modulation bandwidth) of the RFI signals. For instance, a 3 kHz narrowband voice
channel will have a modulation time of approximately 0.3 ms, whereas wideband TV
signals with bandwidth of several kHz will have a much smaller modulation time [62]. If
the duration of the processed data is greater than this modulation time, the estimated
parameters will change during the acquisition time, leading to less effective suppression.
These methods are also computationally expensive when many interference sources are
estimated.
When the RFI signals are sampled using the SIRE equivalent sampling scheme,
estimation of these parameters becomes even more challenging due to the irregular
sampling pattern and aliasing introduced, even if the model is accurate.
Another technique for RFI suppression is using passive data to adaptively suppress
RFI from active radar data by projecting the measured active data to a signal subspace
created by the passive data. This method assumes orthogonality between the desired
target signatures and the RFI, and has been shown to be effective for RFI suppression
61
in [63] for conventionally sampled data. However, as noted in [54], this method is
inadequate for suppression of SIRE sampled data, due to the irregular sampling pattern
and aliased samples of the RFI. These challenges have prompted the need for new RFI
suppression techniques for the SIRE sampling scheme.
The averaging method proposed in [58] and also detailed therein, has been shown
to suppress wideband and narrowband interferers. The method is based on repeating
the measurements from the same range profile M times and averaging the repeated
measurements. The averaged samples are then interleaved and used for generating
SAR images. Fig. 3-2 shows the amount of suppression as a function of the number of
repeated measurements based on simulated RFI sources. A similar plot can be seen
in [58]. An important point to note is that this method of suppression does not take
into account any properties of the RFI signal, which is the motivation for improving the
performance.
−10
Suppression (dB)
−15
−20
−25
−30
−35
0
200
400
600
800
Number of Pulses Averaged (M)
1000
Figure 3-5. RFI Suppression (dB): Averaging method (M realizations) for simulated
SIRE sampled RFI signals.
In this chapter, the averaging method is improved by analyzing the data in ’real-time’
(before interleaving). The aliased samples of the data in ’real-time’ are modelled as a
sum of sinusoids, in other to achieve further suppression with little signal distortion. The
parameters of the sinusoids are estimated and the resulting sinusoids are subtracted
62
from the data using the RELAX algorithms [31],[57] (to provide accurate estimates of the
parameters) before averaging.
Based on the analysis in the previous section, a single sinusoid appears as multiple
peaks due to the irregular SIRE sampling pattern in ’real-time’. However, in theory,
estimating the parameters of a single sinusoid from the maximum peak location of the
spectrum, and subtracting this from the data, will correspond to its removal from the
spectrum. This will eliminate all the multiple aliased peaks (Fig. 3-4). This analysis will
be shown on simulated sinusoids in the results section.
The RELAX algorithm and its multi-snapshot counterpart are described in the next
section and the steps for RFI suppression are also presented.
3.5 Proposed RFI Suppression Method: RELAX and Averaging
3.5.1 Modelling of RFI
The proposed suppression method, entails modelling RFI signals of length L
collected in real time (before interleaving) as a sum of P complex-valued aliased
sinusoids as described in Eq. (3–5):
z=
P
∑
p=1
αp a(fp )
(3–5)
where αp and fp are the complex amplitude and frequency of the pth sinusoid and
[
a(fp ) = 1
ej 2πfp
···
ej 2π(L−1)fp
The received measurement signal can be written as y
]T
= z + s + n, where z, s, and
n, are the RFI signal, desired target returns, and receiver noise, respectively. The target
returns have a wide bandwidth relative to the RFI signals and can be modelled as white
noise [53] ,[64]. RFI suppression, then, becomes a case of estimating the parameters
of multiple sinusoids in the presence of white noise. The non-linear least squares
(NLS) approach (an asymptotic Maximum Likelihood approach [55],[2]) estimates these
63
parameters by minimizing the following non-linear least squares cost function in Eq.
(3–6),
}P
{
α
^p, f^p p=1
2
P
∑
= argmin y − αpa(fp)
P
p=1
{α^p ,f^p }p=1
(3–6)
where P is the number of sinusoids, which can be estimated using a model-order
selection tool like the Bayesian Information Criterion (BIC) [65]. This method can
approach the Cramer-Rao bound in performance, but it involves a multi-dimensional
search and hence involves complex computations for the case of multiple sinusoids.
It can also be sensitive to initializations [2],[66]. The RELAX algorithm can be used
for solving the problem in an iterative manner reducing the computational complexity
significantly [31]. This conceptually and computationally simple algorithm was shown to
estimate sinusoidal parameters accurately and robustly even in the presence of colored
noise [55]. The parameters are estimated for the above non-linear least squares fitting
problem in an iterative manner as described below.
3.5.2 RELAX Algorithm
The RELAX algorithm estimates the parameters as follows: Let
yp , y −
P
∑
i=1,p̸=i
α
^ia(f^i)
(3–7)
The frequency and complex amplitude estimates of the pth sinusoid are, respectively,
estimated by:
f^p
= argmax |aH (fp)yp|2
(3–8)
H
= a (Lfp)yp |f =f^
(3–9)
fp
and
α
^p
|
{z
}
DTFT of yp
64
p
p
The RELAX algorithm steps are given by:
= 1. Estimate f^1 and α^1 from y.
•
Step 1: Assume P
•
Step 2: Assume P = 2. Compute y2 based on estimates from the previous
step and estimate f^2 and α
^2 . Compute y1 and re-estimate f^1 and α^1 . Re-iterate
previous steps until practical convergence.
•
•
Step 3: Assume P = 3. Compute y3 and estimate f^3 and α
^3 . Re-compute y1 and
^
^
^
re-estimate f1 and α
^1 from f2 , α^2 , f3 ,α^3 . Re-iterate until convergence or a fixed
number of iterations.
Remaining Steps: Continue until P
= P^, which is an estimated or desired number.
Note that, the frequencies and complex amplitudes in (3–8) and (3–9), respectively,
are estimated using the DTFT of the signals yp . This can be efficiently computed using
the FFT and zero-padding for conventionally (regularly) sampled data.
Based on Fig. 3-3 and the analysis leading to Eq. (3–4), as previously discussed,
the interleaving process of a SIRE sampled sinusoid leads to a distortion of that signal
except for a specific case, being that the frequency of the sinusoid is an integer multiple
of the PRI. The analysis of the RFI using the RELAX algorithm will, therefore, be
performed on the data in real-time (before the interleaving process). As will be shown
in the results section, the estimated complex amplitudes and frequencies (although
possibly ambiguous), can be used to accurately reconstruct the aliased RFI samples
and yield effective RFI suppression using the RELAX algorithm.
The RELAX algorithm requires the computation of the spectrum of the received
samples. For irregularly sampled SIRE data, this spectrum can be computed using an
FFT after re-sampling the data (interpolating with zeros). Re-sampling this data to give a
regularly sampled data with effective sampling frequency of fe , will lead to a significantly
long data sequence with most of the samples being zero. For instance, one realization
= 1) of a SIRE sampled data, sampled at fs = 40 MHz , contains N = 7 aliased
samples per PRI. After re-sampling to an effective rate of fe = 7.72 GHz, each PRI will
consist of T = fe × PRI = 7720 samples. Hence a total of T × K = 7720 × 193 ≈ 1.5
(M
65
million samples per realization. Therefore, applying a direct FFT (with zero-padding)
to this re-sampled data to estimate the frequencies and complex amplitudes becomes
computationally intensive with a computational complexity of O(T KlogT K ).
Note that similar to Fig. 3-4, a single sinusoid sampled using the SIRE sampling
technique and re-sampled as discussed above to give an effective sampling rate
of fe
= 7.72 GHz will repeat itself approximately every 40MHz (A/D rate) in the
frequency domain, due to aliasing. In order to reduce the computational complexity
of this re-sampling scheme, the regular sampling of the data in both fast and slow time
can be exploited and the spectrum can be computed only on a 40 MHz bandwidth to
save on computations. The analysis is performed as follows:
The spectral estimate for SIRE sampled data in real time (before interleaving) based
on parameters in Tab. 3-1 is given by:
X (f ) =
∑∑
n
xm,n e−j 2πf (mm +nn )
(3–10)
m
where
n
= 0, 1, 2, · · · , N − 1 (N = 7)
m
= 0, 1, 2, · · · , K − 1 (K = 193)
n = s = 25 ns, (ADC sampling rate),
m =
PRI + e .
A direct computation of the spectrum in Eq. (3–10) is obviously computationally
intensive, especially for a fine grid size in frequency. Assuming
f
= k1 + k2 f
m
66
(3–11)
where
k1
= 0 , 1, 2, · · · , K 1 − 1
K1
= T = 7720
k2
= 0 , 1, 2, · · · , K 2 − 1
K2
= 1/(mf )
f =
fixed grid size (in Hz)
Eq. (3–11) is the frequency grid (in Hz) on which the spectrum in Eq. (3–10) will be
computed. Note that the choice of K2 determines the grid spacing f and the choice
of the k1 values determines the portion of the bandwidth in which the spectrum is to be
estimated. For instance, k1
= 0, 1 · · · T = 7720 computes the spectrum over the entire
7.72 GHz (effective sampling rate) bandwidth.
For the frequency grid specified in (3–11), the spectrum in (3–10) can be re-written
as follows:
X (f ) =
∑∑
n
xm,n e−j 2π( m +k2 )(mm +nn )
k1
(3–12)
m
which simplifies to:
X (k1 , k2 )
=
∑
e−j 2π( m +k2 f )nn
k1
n
X (k1 , k2 ) =
∑
−j 2π K2 m
k
xm,n e
2
m
∑
e−j 2π( m +k2 f )nn Xn (k2 )
k1
(3–13)
n
From Eq. (3–13), the spectrum is computed by summing up multiple FFTs. Also
because the signal is aliased, the spectrum needs only to be computed over a small
portion (40 MHz - A/D sampling rate) of the entire bandwidth. The computational
complexity of this algorithm is O(N K2 logK2
+ K1 K2 N ). Note that the bottle neck of
this algorithm is in the second term. When the spectrum is computed over the entire
frequency grid, K1
= T = 7720, the computational complexity is on the same order as
re-sampling and applying an FFT. However, when the spectrum is computed over a 40
67
= 40), this algorithm drastically improves on the computation. For
example, for a frequency grid with spacing (f ) of approximately 2 kHz, the spectrum in
MHz bandwidth (K1
Fig. 3-4 was computed in 0.26 secs when the SIRE sampled data was re-sampled and
the FFT was applied directly using the MATLAB software. However, the spectrum based
on Eq. (3–13) was computed in 0.035 secs on a 40 MHz grid.
Table 3-2. Suppression Algorithm: RELAX + Averaging
Step 1:
RELAX (P sinusoids estimated).
- Compute the DTFT of the measured data y from (3–13) and estimate f^1
and α
^1 , using (3–8), (3–9) and (3–13).
- Compute y2 using (3–7) and its DTFT using (3–13). Estimate f^2 and α
^2 .
Re-estimate f^1 and α
^1 from y1 and iterate. Continue for yp , f^p and
α
^p (3 ≤ p ≤ P ) (Section IV.B).
Step 2:
Reconstruct aliased RFI samples using {f^i }Pi=1 and {α
^i}Pi=1 . Subtract
from each realization (y).
Step 3:
Average residue from each realization and interleave.
This spectrum is used in the RELAX algorithm (3–8) and (3–9), to estimate the
parameters of the sinusoids present. The RELAX algorithm is applied here to one
realization (M
= 1) of SIRE sampled data (which correspond to a data with an
acquisition time of 0.193 ms based on Tab. 3-1). Therefore a narrowband interference
source with a modulation bandwidth of 5 kHz or less can be accurately approximated as
a single tone, whereas multiple sinusoids are needed to model an interference source
with wider bandwidth. The sinusoidal model begins to break down for very wideband
interferers. In the next subsection we propose the multi-snapshot RELAX algorithm
for SIRE sampled data that provides a more accurate sinusoidal model for the RFI
signals by using fewer samples (smaller modulation time), for suppression. The overall
proposed RFI suppression algorithm can be summarized in the following steps as shown
in Tab. 3-2 for RELAX.
68
3.5.3 Multi-snapshot RELAX Algorithm
The multi-snapshot RELAX algorithm [57] uses N
= 7 samples (150 ns acquisition
time) for RFI suppression. Interference sources with modulation bandwidth of 6.7 MHz
or less can be accurately modelled as sinusoids, which includes wideband interferers
like TV broadcasts etc.
The multi-snapshot RELAX algorithm (M-RELAX for short) proposed for angle and
waveform estimation in [57] is a modification of the originally proposed RELAX algorithm
[31]. The algorithm estimates the angle of arrival (using multiple snapshots of the data)
and the corresponding waveform for each snapshot.
This algorithm is proposed here for RFI suppression of SIRE sampled data to
=
provide a more accurate sinusoidal model for the RFI signals. Here, each set of N
7 fast time samples is treated as a snapshot. The data is split into K = 193 total
snapshots based on the parameters in Tab. 3-1. The frequency of a single tone is
estimated by averaging the periodogram of each snapshot and finding the frequency
that maximizes the average. The complex amplitudes of each snapshot is estimated by
finding the complex value of the spectrum of each snapshot at the estimated frequency.
Note that for a single complex sinusoid, K
= 193 complex amplitudes are estimated
from each snapshot, whereas only one frequency is estimated. The parameters are
estimated as given in Eq. (3–14) (modification of NLS for the multi-snapshot case [57]):
{
}P
α
^ p, f^p p=1
2
K P
∑
∑
= argmin
αp (k )a(fp )
x(m) −
{αp fp }P
p=1
p=1 m=1
(3–14)
= [αp(1), αp(2), . . . , αp(K )] contains the estimated complex amplitudes
of the pth sinusoid for each of the K snapshots, f^p is the estimated frequency of the
where αp
pth sinusoid for all snapshots and x(m) is the mth snapshot. These parameters are
estimated as follows. Let
xp (m) , x(m) −
P
∑
i=1,p̸=i
69
α
^i(m)a(f^i )
(3–15)
The estimates described above are given by:
f^p = argmax
fp
K
∑
m=1
|aH (fp )xp (m)|2
(3–16)
and
α
^p(m) =
aH (fp )xp (m)
|
|fp =f^p ,
L
{z
}
DTFT of xp (m)
m = 1 , 2, . . . , K
(3–17)
The multi-snapshot RELAX algorithm steps are as follows:
= 1. Estimate f^1 and α^1 (m) from x(m), for m = 1, 2, . . . , K.
•
Step 1: Assume P
•
Step 2: Assume P = 2. Compute x2 (m) based on the estimates from the previous
step and estimate f^2 and α
^2 (m), for m = 1, 2, . . . , K. Compute x1 (m) and
^
re-estimate f1 and α
^1 (m), for m = 1, 2, . . . , K. Re-iterate previous steps until
practical convergence.
•
•
Step 3: Assume P = 3. Compute x3 (m) using {α
^ p, f^p}2p=1 and estimate f^3 and
α
^3 (m), for m = 1, 2, . . . , K. Re-compute x1 (m) and re-estimate f^1 and α^1 (m) from
{α
^ p, f^p}3p=2 , for m = 1, 2, . . . , K. Then re-compute x2 (m) and re-estimate f^2 and
α
^2 (m) from {α^ p, f^p}p=1,3 , for m = 1, 2, . . . , K. Re-iterate until convergence.
Remaining Steps: Continue until P
= P^, which is an estimated or desired number.
For the SIRE sampled data, the spectrum (DTFT) of each snapshot can be
described as follows. Let x(m)
=
−1=7
{dm (n)}N
correspond to the mth snapshot,
n=0
where dm (n) denotes the nth sample of the mth fast time pulse (m
= 1, 2, . . . K). Note
that each snapshot is regularly sampled (at the A/D rate). The spectral estimate can
therefore be computed using an FFT multiplied by a corresponding phase shift (over a
40 MHz bandwidth). The spectrum of each snapshot is given by:
Xm (f ) =
where (f ∈
6
∑
n=0
dm (n)e−j 2πf (nn +mm )
(3–18)
(0, fs)) is the frequency (in Hz), n = 1/fs and m are the sampling period
and the time difference from one snapshot to the next, respectively. Equation (3–18)
70
above can be simplified as follows:
Xm (f ) = e−j 2πf (mm )
6
∑
n=0
dm (n)e−j 2π fs n
(3–19)
dm (n)e−j 2π R n
(3–20)
f
which simplifies to:
Xm (r) = e−j 2π R (
r
for a discrete frequency grid r
mm
n
)
6
∑
n=0
r
= 0, 1 . . . R − 1. It is important to note that parameter
identifiability (maximum number of sinusoids that can be uniquely identified) [67–69],
becomes an issue with this approach. Given N
= 7 real valued samples, only up
= 2 sinusoids (amplitude, frequency, and phase), can be uniquely identified.
Estimating more than P = 2 sinusoids will significantly distort the target signatures.
to P
The overall proposed RFI suppression algorithm can be summarized in the steps as
shown in Tab. 3-3 for multi-snapshot RELAX.
We also consider Auto-regressive(AR) modelling of the RFI data for suppression,
however due to desired signal distortion, this approach is not effective. AR for RFI
suppression for the SIRE radar is described in the next section.
Table 3-3. Suppression Algorithm: M-RELAX + Averaging
Step 1:
M-RELAX (P sinusoids estimated)
- Compute the DTFT of the mth snapshot x(m) of the measured data y
from (3–20) and estimate f^1 and α
^1 (m), using (3–16), (3–17).
- Compute x2 (m) using (3–15) for each snapshot and its DTFT using
(3–20). Estimate f^2 and α
^2 (m). Re-estimate f^1 and α^1 (m) and iterate.
^p (3 ≤ p ≤ P ). (Section IV.C).
Continue for yp , f^p and α
Step 2:
^i(m)}Pi=1 for each
Reconstruct aliased RFI samples using {f^i }Pi=1 and {α
snapshot. Subtract from each realization (y).
Step 3:
Average residue from each realization and interleave.
71
3.6
Autoregressive (AR) Modelling
Auto-regressive (AR) models, which is commonly used for modelling narrowband
(”peaky”) signals, can be used for estimating and suppressing RFI signals. The
measured signal (RFI, desired target returns, and thermal noise) is modelled as an
AR process [62]. The AR modelling (linear prediction modelling) equation is written as:
y [ tn ] = −
q
∑
i=1
a[i]y [tn − i] + u[tn ]
(3–21)
where, y [tn ] is the measured data sequence, u[tn ] corresponds to the white noise term
at a time instant tn and q corresponds to the AR order, which is determined by the
number of spectral peaks and their widths. The assumption is that the first term on the
right hand side of Eq. (3–21) corresponds to the RFI signal. The suppression process
therefore involves estimating {a[i]}qi=1 and using the coefficients to suppress the RFI
signals. Note that Eq. (3–21) can be re-written as:
y [tn ] = H (z )u[tn ]
(3–22)
where H (z ) = 1/A(z ) = 1/(1 + a[1]z −1 + . . . + a[q ]z −q ), with z −1 being the delay operator.
The RFI suppression process involves passing the measured data through the inverse
filter 1/H (z ) = A^(z ) (from the estimated AR coefficients {^
a[i]}qi=1 ).
The well-known methods for solving for the AR coefficients in (3–21) include the
Yule-Walker (YW) method, Prony method and the modified Prony method [2]. The YW
and Prony methods give similar results for large data samples. However, for smaller data
records the Prony method tends to give gives more accurate AR estimates [2].
If both sides of the forward linear prediction equation (3–21) are multiplied by
y [tn − tm ], and the expectation is taken, the well-known Yule-Walker equations are
obtained.
72





 r(1)
r[0] · · · r[−q + 1] a[1]
 . 

 . 
..
 ..  = −  ...
  .. 
.










r(n)
r [q ] · · ·
r[0]
a[q ]
(3–23)
which can be re-written as r = −Ra. Where r and R are the covariance vecotr and
matrix of the data. The AR coefficients (a) are estimated by solving Eq. (3–23). The
Yule-Walker method estimates the coefficients by replacing r with the standard biased
autocorrelation sequence (ACS) estimator [2]. The Prony method solves the forward
linear prediction equation (3–21) using least squares (LS). The problem reduces
to (3–23), with the covariance sequence estimated by the standard unbiased ACS
estimator [2].
The Modified covariance (Prony) method (which improves on the Prony method)
combines the forward linear prediction in (3–21) and the backward linear equation given
below in (3–24) to solve for the AR coefficients using least squares:
y [ tn ] = −
q
∑
i=1
ab [i]y [tn + i] + ub [tn ]
(3–24)
where ab [i] = a[i].
This Modified covariance method is applied to the SIRE sampled data which is
sampled regularly in fast-time and slow-time. N
samples (K
= 7 sets of the slow-time regular
= 193 samples per set) are used for AR modelling. The Modified covariance
73
equations can be written in matrix form as follows (for each set of slow time samples):




y (q )
y [0]



 y [q − 1] · · ·





..
..
..




.
.
.



 a[1]









 y (K − 1) 
y [K − 2] · · · y [K − q − 1] 
a[2]



 = −

(3–25)



  .. 


 y [1]
 . 
y
(0)
·
·
·
y
[
q
]










.
.
.



 a[q ]
..
..
..








y (K − q − 1)
y [K − q ] · · ·
y [K − 1]
where q is the AR order. The equation can be re-written as
yn = −Yn a,
for n = 1, 2, . . . N
with N
=7
(3–26)
The least-squares solution of this overdetermined linear system of equations is given by:
a = −(YnT Yn )−1 YnT yn
for n = 1, 2, . . . N
with N
=7
(3–27)
where (YnT Yn )−1 estimates the covaraince matix and YnT yn estimates the ACS in (3–23).
A more accurate estimate of the covariance matrix in (3–27) is derived by averaging the
N
= 7 snapshots.
3.7
Experimental Results
3.7.1 Simulations
In this section, a signal consisting of three sinusoids in white noise (SNR = 10dB)
is simulated and sampled using the SIRE equivalent scheme based on the parameters
= 1). The sinusoids have frequencies
= 650.255 MHz, all with amplitudes of 1.
in Tab. 3-1, with no repeated measurements (M
f1
= 111.111 MHz, f2 = 300 MHz and f3
Note that the samples obtained will also correspond to a signal containing sinusoids with
= f1 mod fs, fa2 = f2 mod fs and fa3 = f3 mod fs, as well as a signal
containing sinusoids fa1 + kfs , fa2 + kfs and fa3 + kfs (where k ∈ Z and fs is the A/D
frequencies fa1
rate). This ambiguity in frequency is caused by aliasing due to the low A/D rate of the
radar. The RELAX algorithm can be used to accurately estimate the complex amplitudes
74
of these sinusoids as well as the ambiguous frequencies. This is achieved using the
spectrum in (3–13) estimated only on a 40 MHz bandwidth to save on computations.
The estimated parameters are then used to reconstruct the aliased samples, in order to
suppress the sinusoids through subtraction.
Original Signal
Amplitude
Amplitude
0
−2
1200
1250
1300
Samples
Spectrum
1350
0.4
0.2
0
0
10
20
Frequency (MHz)
30
2
0
−2
1200
Magnitude
Magnitude
Residue
2
1250
0.2
0
40
0
10
20
Frequency (MHz)
Magnitude
Amplitude
−2
1300
Samples
Spectrum
1350
0.4
0.2
0
20
Frequency (MHz)
30
2
0
−2
1200
Magnitude
Amplitude
0
10
40
Residue
2
0
30
B
Residue
1250
1350
0.4
A
1200
1300
Samples
Spectrum
40
1300
Samples
Spectrum
1350
0.4
0.2
0
0
C
1250
10
20
Frequency (MHz)
30
40
D
Figure 3-6. RFI suppression (SIRE sampling) - Signal and spectrum of simulated data
containing 3 real-valued sinusoids in white noise after suppression using
RELAX with P (real-valued) sinusoids estimated (A) Original data, (B) P = 1,
(C) P = 2, (D) P = 3.
Fig. 3-6, shows the original signal, its spectrum and the progression of suppression
as the number of estimated aliased parameters increases. From Fig. 3-6, we observe
that the spectrum of the three sinusoids contains multiple peaks, due to the irregular
sampling as described previously. By estimating the ambiguous frequency and complex
amplitudes of each of the sinusoids (on only a 40 MHz bandwidth), multiple aliased
peaks are suppressed. The purpose of the above simulations is to show the ability of the
75
RELAX algorithm to estimate the ambiguous frequency and complex amplitudes of the
sinusoids on a small bandwidth correctly, to effectively suppress the sinusoids (including
the multiple aliased peaks).
In the next subsection the sniff dataset collected using ARL’s SIRE UWB radar is
analyzed and the RFI is suppressed using both the RELAX and multi-snapshot RELAX
algorithms. Comparison with AR modelling of the RFI is also provided.
3.7.2 Sniff Experimental Data
The sniff data to be analyzed, was collected by ARL using the SIRE UWB radar
in passive mode based on the parameters in Tab. 3-1. Each set of data consists of
L
= K × N = 1351 samples. In this subsection this RFI data will be analyzed using
the proposed algorithms. Two sets of RFI data with different energy levels are analyzed.
For simplicity they will be referred to as File1 and File2. Each set of the data, consists of
M
= 88 realizations. The amount of suppression achieved are presented in Table3-4.
A wideband echo signal which represents a return from a single point target is
simulated. This signal is added to each realizations (M
= 88) of the sniff data, in a way
that the echo signal adds up coherently. The goal is to show how much distortion is
introduced to the desired signals after the application of the RFI suppression algorithms.
Fig. 3-7 shows the amount of suppression achieved when the RELAX algorithm
with P real-valued sinusoids are suppressed for each realization and the residues are
averaged (File1). These results are compared to straightforward averaging, also in this
figure. A similar analysis is performed for the multi-snapshot RELAX algorithm and the
amount of suppression can be seen in Table 3-4 and 3-5. The average power of the
signals {s^(i)}Li=1 , are computed using (3–28):
( L
)
∑
10 log10
|s^(i)|2 /L
i=1
(3–28)
From Table 3-4 and 3-5, it is clear that the amount of suppression increases as the
number of real-valued sinusoids increases for the RELAX algorithm. This improvement
76
Table 3-4. RFI Suppression (dB): File 1 (P~
Avg.
RELAX
= 1)
26.61 (P = 4)
27.79 (P = 7)
28.06 (P = 10)
20.89 23.46 (P
M-RELAX
= 1)
*PI (P = 4)
*PI (P = 7)
*PI (P = 10)
24.19 (P
Table 3-5. RFI Suppression (dB): File2 (P~
Avg.
= 1)
= 1)
RELAX
M-RELAX
= 1)
21.55 (P = 4)
22.30 (P = 7)
22.61 (P = 10)
= 1)
*PI (P = 4)
*PI (P = 7)
*PI (P = 10)
18.49 20.02 (P
20.22 (P
RELAX/M-RELAX (P~ ) AR (q)
= 1)
29.11 (P = 4)
30.38 (P = 7)
31.27 (P = 10)
27.04 (P
23.21 (2)
24.52 (20)
RELAX/M-RELAX(P~ ) AR (q)
= 1)
23.47 (P = 4)
23.65 (P = 7)
25.97 (P = 10)
22.40 (P
20.03 (2)
20.37 (20)
*PI - Parameter identifiability not met.
comes at a cost of increased computational complexity. However, the target signatures
are left basically unaltered as can be seen in Fig. 3-8.
The multi-snapshot RELAX algorithm shows a significant amount of suppression
of the data as the number of sinusoids increases. Due to the issue of parameter
identifiability discussed in the previous section, estimating more than P
sinusoids (in theory), using only N
= 2 real-valued
= 7 real-valued samples per-snapshot will effectively
suppress all the samples to zero. This leads to the suppression of the target energy, as
can also be seen in Fig. 3-8.
The multi-snapshot RELAX algorithm can be seen to improve on the suppression
with little target distortion for P
= 1 based on the real RFI data collected using the SIRE
radar. This algorithm is combined with the RELAX algorithm to effectively suppress both
wideband and narrowband interferers. This improvement is seen in Table 3-4 and 3-5
and Fig. 3-9 shows the reconstructed echo after suppression.
A similar analysis is performed for AR modelling. The AR modelling improves on the
suppression compared to averaging as can be seen in Fig. 3-10. However, this inverse
77
30
30
Averaging
RELAX (P = 10) and averaging
Averaging
RELAX (P = 1) and averaging
10
Amplitude
10
Amplitude
20
20
0
0
−10
−10
−20
−20
−30
0
200
400
600
800
Samples
1000
1200
−30
0
1400
200
400
600
800
Samples
A
1000
1200
1400
B
Figure 3-7. RFI suppression - RELAX algorithms with P (real-valued) sinusoids
estimated and suppressed from sniff data (File1) compared to averaging. (A)
P = 1, and (B) P = 10.
Echo signal
RELAX
M−RELAX
60
40
Amplitude
Amplitude
40
20
20
0
0
−20
−20
−40
650
Echo signal
RELAX
M−RELAX
60
660
670
680
690
−40
650
700
660
670
680
Samples
Samples
A
B
690
700
Echo signal
RELAX
M−RELAX
60
Amplitude
40
20
0
−20
−40
650
660
670
680
690
700
Samples
C
Figure 3-8. Echo retrieval (File1) - RELAX with P (real-valued) sinusoids compared to
ideal echo signal. (A) P = 1, (B) P = 2, and (C) P = 10.
78
Echo signal
RELAX/M−RELAX
60
40
Amplitude
Amplitude
40
20
20
0
0
−20
−20
−40
650
Echo signal
RELAX/M−RELAX
60
660
670
680
690
−40
650
700
660
670
680
Samples
Samples
A
B
690
700
Echo signal
RELAX/M−RELAX
60
Amplitude
40
20
0
−20
−40
650
660
670
680
Samples
690
700
C
Figure 3-9. Echo retrieval (File1) - RELAX with P (real) sinusoids combined with
M-RELAX with P~ = 1 real sinusoid, compared to ideal echo signal. (A)
P = 1, (B) P = 2, and (C) P = 10.
filtering technique leaves the desired signal distorted. This distortion is increased as
the model order increases (due to filtering transients) as seen in Fig. 3-9. Hence the
combined RELAX and multi-snapshot RELAX outperforms the AR approach in terms of
both RFI suppression and desired target echo preservation.
3.8
Conclusions
In this chapter, we have proposed a method for RFI suppression for the SIRE
UWB radar, which is a cost efficient system of sampling returned radar signals used for
detecting land mines and IEDs developed by ARL. The low sampling rate and irregular
sampling pattern of this radar poses a challenge for Radio Frequency Interference
(RFI) suppression as the measured RFI signals will be severely aliased. In this chapter,
we have discussed the challenges of RFI suppression for this radar and proposed
79
30
30
Averaging
AR−2
10
0
10
0
−10
−10
−20
−20
−30
0
200
400
600
800
Samples
1000
1200
Averaging
AR−20
20
Amplitude
Amplitude
20
−30
0
1400
200
A
400
600
800
Samples
1000
1200
1400
B
Figure 3-10. RFI suppression - AR modelling with order q compared to averaging for
sniff data (File1). (A) q = 2, and (B) q = 20.
Echo
AR−2
60
40
Amplitude
Amplitude
40
20
0
−20
−40
650
Echo
AR−20
60
20
0
−20
660
670
680
Samples
690
−40
650
700
A
660
670
680
Samples
690
700
B
Figure 3-11. Echo retrieval (File1) - AR modelling with order q, compared to ideal echo
signal. (A) q = 2, and (B) q = 20.
using the RELAX algorithm and its multi-snapshot counterpart as an intermediate
step to the already proposed averaging scheme for RFI mitigation, for the SIRE UWB
radar. The results show that the RELAX algorithm can suppress RFI further than just
averaging without altering desired target echo signals. The RELAX algorithms are easy
to implement since they just involve FFTs. They have been shown to outperform AR
modelling of the RFI singals.
The multi-snapshot RELAX uses a shorter time-duration (and fewer samples)
for suppression, which yields a more accurate wideband model of the RFI as sum of
sinusoids compared to the RELAX algorithm. However, this algorithm significantly
80
suppresses target signatures as the number of sinusoids increases and is limited to
estimating only one sinusoid. Combining this algorithm assuming just one sinusoid, with
the RELAX algorithm increases the suppression performance with little signal distortion.
81
CHAPTER 4
DATA-ADAPTIVE, SPARSE SUPER-RESOLUTION IMAGING FOR THE SIRE FLGPR
RADAR
4.1 Chapter Summary
In the previous chapter, the problem of RFI suppression was presented for the
Forward Looking Ground Penetrating Radar (FLGPR) known as the SIRE radar built by
the Army Research Laboratory. FLGPR has multiple applications, one of which includes
its use for detecting landmines and other buried improvised explosive devices (IEDs). In
this chapter, we focus on data-adaptive high resolution imaging for this SIRE FLGPR.
The standard method for generating SAR images for this radar is the back-projection
algorithm, which is limited by poor resolution and high side-lobes problems. In this
chapter, we consider using the Sparse Iterative Covariance-based Estimation (SPICE)
and the Sparse Learning via Iterative Minimization (SLIM) algorithms for generating
sparse high-resolution images for FLGPR. The pre-processing involves an orthogonal
projection of the received measurements to a subspace related to the region of
interest for data and clutter reduction. These user-parameter free algorithms are
capable of providing sparse results as well as improved resolution synthetic aperture
radar (SAR) images. We also examine the well-known CLEAN approach based on
a signal model in the time domain for imaging. We show using simulated data that
the SPICE/SLIM algorithms provide higher resolution than CLEAN and standard
backprojection algorithm. Imaging using real data collected via the Synchronous
Impulse Reconstruction (SIRE) radar, a multiple-input multiple-output (MIMO) FLGPR
radar developed by the Army Research Laboratory (ARL)) is also used for analysis.
4.2 Introduction
The global problem of landmines and other buried improvised explosive devices
(IEDs) is affecting both military and civilians alike [70–74], and effective as well as
efficient methods for detecting these devices is very important in the world today.
Methods for detecting landmines include but are not limited to the use of metal
82
detectors, infrared sensing of the land surface [75], biological sensors such as animals
(dogs) [76] and more recently detecting fumes of the mines using lasers to ionize the air
[74]. Radar is an excellent tool for remote sensing applications [77]. Ground penetrating
radar (GPR) which transmits an electromagnetic wave into the ground and examines
the back-scattered returns to determine buried objects has become a useful tool for
effectively detecting landmines and IEDs [78–80].
By operating in forward looking mode, GPR can be applied to the problem of
landmine detection as it inspects the ground surface with a safe stand-off range as
can be seen in Fig 4-1. Impulse based forward looking ground penetrating radar
(FLGPR) typically transmits a mono-cycle pulse with typical operating frequency
range spanning the UHF, and L bands [54, 79]. The low frequency of GPR provides
the necessary ground penetrating properties of the radar and the large bandwidth
provides the necessary down-range resolution. The cross-range resolution on the other
hand is limited by the antenna beamwidth [81]. Increasing the antenna physical size
can improve cross-range resolution. However, this is limited by physical antenna size
constraints. Side-looking synthetic aperture radar techniques improve cross-range
resolution by synthesizing a virtual aperture much larger than the physical aperture
[11, 82–84]. However, in forward looking mode, the cross-range resolution is limited
by the physical radar size. A multi-input multi-output (MIMO) radar [85] can be used to
enhance this resolution [86][87]. For example, Fig. 4-1 shows the FLGPR for landmine
detection built by the Army Research Laboratory (ARL) known as the synchronous
impulse reconstruction (SIRE) radar [54]. This radar consists of 2 transmitters and 16
receivers and exploits waveform diversity [54, 86, 88] to enhance cross-range resolution
by alternatively transmitting between its two transmitters.
The well-known conventional method for imaging for this type of radar is the
standard back-projection method [89, 90]. This approach also known as the delay-and-sum
(DAS) approach, suffers from high sidelobe problems and is limited by poor resolution.
83
Figure 4-1. Forward looking ground penetrating radar [54]
High resolution imaging is important for separating closely spaced targets as well as
distinguishing targets from clutter and such imaging techniques should be investigated.
In this chapter, we focus on sparse high-resolution imaging for impulse based
FLGPR. A signal model in the time domain is established since the transmitted impulse
is well localized in time. Based on this model, the well-known CLEAN [91], [92] approach
is analyzed for imaging compared to the standard backprojection algorithm. This
technique eliminates side-lobes, but it provides no improvement in imaging resolution
over the standard backprojection algorithm.
Two recently proposed, user-parameter free and data-adaptive methods are
considered here for imaging. The Sparse Learning via Iterative Minimization (SLIM)
[29] and the SParse Iterative Covariance-based Estimation (SPICE) [93] methods are
capable of providing sparse, as well as high resolution imaging results. These methods
are applied to the data of significantly lower dimension for impulsed based FLGPR
to achieve sparse high resolution imaging results. The data-reduction is achieved via
a pre-processing technique, which involves an orthogonal projection of the received
data to the subspace spanned by the dominant singular vectors of a steering matrix
corresponding to the imaging ROI. An efficient decomposition of the steering matrix is
performed using an eigenvalue decomposition of a matrix of much reduced dimension.
84
A conjugate gradient SPICE (CG-SPICE) algorithm is also introduced in this paper
similar to the CG-SLIM algorithm [29] to speed up the computation of SPICE.
The improvement in imaging results over the standard backprojection method
are shown via simulated results and also real measured experimental data. For real
experimental data, the SIRE radar developed by ARL [54, 94] is used for analysis. This
radar was also presented as a MIMO radar in [88].
The remaining sections of this chapter are organized as follows. In Section 4.3, a
proposed data model is presented for forward looking GPR based on the ARL’s SIRE
radar. This MIMO radar is also briefly described therein. In Section 4.4, the standard
back-projection method for imaging is analyzed as well ARL’s Recursive Sidelobe
Minimization (RSM) algorithm which is based on the BP algorithm and iteratively and
effectively suppresses sidelobes [54, 95]. Based on the proposed model we also show
that the CLEAN approach can be used for sparse imaging for impulse based FLGPR.
In Section 4.5 we present the sparse and adaptive methods for improved resolution as
well as the pre-processing step of orthogonal projection for clutter and data reduction.
Section 4.6 contains the numerical results based on simulated and real data. Finally the
conclusions are drawn in Section 4.7.
4.3 Data Model: SIRE Impulse Based FLGPR
For impulse based FLGPR, we consider the SIRE radar which is designed by ARL
and mounted on an SUV for landmine detection [54]. The radar geometry as seen in
Fig. 4-1 consists of two transmitters and sixteen receivers. Each transmitter transmits
an impulse with a frequency range of 0.3-3.0 GHz, which determines the downrange
resolution. The cross range resolution is determined by the physical 2 m aperture of
this radar. This radar can be described as a practical example of a MIMO radar which
exploits waveform diversity by transmitting orthogonal waveforms from the two transmit
antennas located at the edges of the receive array [88]. These orthogonal waveforms
are achieved by alternatively transmitting narrow pulses (in ”ping-pong” mode [96]) from
85
Figure 4-2. SIRE FLGPR: 2D aperture for SAR imaging
each transmitter. This creates a virtual aperture which is effectively almost double the
physical 2 m aperture of the radar hence improving cross-range resolution [97].
The 2D-aperture of received measurements shown in Fig 4-2 is used for image
= 1 · · · K receive measurements for a desired imaging area
consisting of i = 1 · · · L targets (pixels). Let rk (t) denote the kth receive measurement.
formation where there are k
This measurement can described by the following equation:
rk (t) =
where αk,i
target, τk,i
=
=
1
1
Rt (k,i) Rr (k,i)
Rt (k,i)+Rr (k,i)
c
L∑
+M
i=1
αk,i zi s(t − τk,i ) + nk (t)
(4–1)
is the propagation path-loss, zi is the reflectivity of the ith
is the trip time delay from transmitter to target to receiver,
where Rt (k, i), Rt (k, i) are the distances of transmitter to target and the target to
receiver, respectively. The speed of propagation is given by c and nk (t) is thermal noise
associated with kth measurement. Note that the model of the received measurement
in (4–1) takes into account the contribution of the M scatterers outside the ROI, i.e.,
86
desired imaging area which consists of L pixels. The model in (4–1) can be simplified
into the following linear equation:
y = Cz
 + n 

β
+n
γ
= A B 
(4–2)
= [r1 (0), . . . , r1 (T − 1), . . . , rK (0), . . . , rK (T − 1)]T , which is a vector of received
measurements stacked together; β = {βi }Li=1 are the pixel values in the desired imaging
grid to be estimated, γ = {γi }M
i=1 corresponds to the pixel values outside the ROI and n
is the noise vector. The matrix A of dimensions (T K ) × L consists of delayed and scaled
where y
versions of the transmitted signal, given by:


 α1,1 s(τ1,1 ) · · · α1,I s(τ1,I ) 


..
..
...

A = 
.
.



αK,1 s(τK,1 ) · · · αK,I s(τK,I )
where the vector s(τk,i )
(4–3)
= {s(t − τk,i )}Tt=0−1 is the transmitted impulse delayed by τk,i.
This data model is used for FLGPR SAR imaging in this paper. In the next section
the backprojection algorithm as well as the CLEAN algorithm are described for SAR
imaging.
4.4 Back-projection/Delay-and-sum (DAS) Based Methods
The standard backprojection (BP) algorithm is a well-known and widely used
algorithm for FLGPR SAR imaging (also known as the delay-and-sum (DAS) algorithm).
This algorithm is limited in downrange resolution by the bandwidth of the transmitted
impulse and in cross-range resolution by the physical (or virtual) aperture of the radar.
One other limitation of this algorithm is that it produces images with high sidelobes.
A Recursive Sidelobe Minimization (RSM) algorithm based on the BP algorithm was
proposed in [54], [95] for effectively suppressing sidelobes. The CLEAN approach
[91] (which is also based on this BP/DAS algorithm) can also be used for eliminating
87
sidelobes [92] as well as accurately estimating weak targets by iteratively subtracting
the contributions of stronger targets from the received data based on the proposed data
model. In this section, we describe these algorithms as well as analyze the CLEAN
approach based on the proposed data model for impulse based FLGPR SAR imaging.
4.4.1 Back-projection/DAS
Based on Fig 4-2, the backprojection algorithm is described as follows: Consider
the ith pixel in this figure with location (xi , yi , zi ) relative to a predefined reference point
or origin. For a specific transmit-receive pair, let (xr,k , yrk , zrk ) and (xt,k , ytk , ztk ) denote
the transmitter and receiver locations in this coordinate system for k
= 1, . . . , K. The
delay due to the transmitted EM pulse from the transmitter corresponding to the kth
receive measurement to the ith pixel back to the corresponding receiver is given as:
τk,i
√
= τacq + 1c ( (xt,k − xi)2 + (yt,k − yi)2 + (zt,k − zi )2
√
+ (xr,k − xi)2 + (yt,k − yi)2 + (zt,k − zi)2 )
(4–4)
where τacq is the acquisition time delay associated with the radar system. The estimate
of the reflection coefficient at the ith pixel given by βi is given as:
1
β^i =
K
∑
wk rk (t − τk,i )
(4–5)
K k=1
This estimate is simply a summation of delayed receive measurements with the
propagation loss compensated by a weighting factor wk . This is referred to as coherent
processing of the received measurements which improves SNR by a factor K compared
to using a single received measurement.
The backprojection/DAS algorithm is limited in resolution and suffers from poor
sidelobes. The recursive sidelobe minimization (RSM) algorithm proposed by ARL
for sidelobe suppression effectively suppresses sidelobes by generating multiple DAS
images with apertures with randomly missing measurements and selecting the minimum
value across all images [54]. In the next subsection we analyze a CLEAN approach
88
based on the data model in (4–1) for sparse imaging. These DAS based algorithms
do not improve imaging resolution compared to the standard DAS algorithm. We then
present new approaches to imaging using the SLIM and SPICE algorithms for improved
resolution in imaging after preprocessing via an orthogonal projection of the data.
4.4.2 Sparse: CLEAN Method
The data-dependent CLEAN algorithm [91] (also known as matching pursuit) for
image formation based on the data model in (4–1) is briefly described and analyzed
here for imaging. This technique was introduced to produce ’CLEANer’ images
(where prior knowledge of the point spread function was required) [92]. The standard
DAS algorithm suffers high sidelobe problems. CLEAN can be used to eliminate
the side-lobes of strong returns so that weak targets can be revealed by eliminating
contributions of strong targets from the receive measurements.
The CLEAN algorithm can therefore be used iteratively to find the pixel location
of the strongest target and then subtract all the contributions of that target from the
data. The next strongest point is then computed more accurately based on the updated
measurements. The RELAX algorithm [31] can therefore be used to get even more
accurate estimates as well as improved imaging resolution. The RELAX approach
involves estimating the strongest target, subtracting the contributions of this target and
estimating the next strongest target. The initial pixel value of the strongest target is
then re-estimated based on the new estimate of the next strongest target. These two
estimates are then iterated back and forth to achieve more accurate estimates. The
process is then repeated for all the targets in the scene of interest.
Due to the exponential increase in computation as the number of targets increases,
this process proves to be too computationally intensive for practical purposes when
there are many scatterers in the ROI.
The much faster CLEAN approach is described in the following steps based on the
data model in (4–1).
89
•
Step 1: Determine the brightest estimate P (io ) and corresponding location io from
the backprojection image (based on 4–5).
•
Step 2: Subtract the contribution of brightest point from the received signals
(update receive measurements).
–
–
rk up = rk − λ(io )s(^τk,i0 )
(i ) × 1
λ(io ) = P K
α
k
= 1···K
o
k,io
•
Step 3: Generate a new image by filling in the io th pixel with P (i0 ).
•
Step 4: Use updated received signals to regenerate back-projection image.
•
Iteration: Repeat previous steps with regenerated image until reaching a
predefined threshold. η > Pσ(i2o ) .
–
–
η threshold (typically chosen as 1).
σ 2 Noise variance.
This CLEAN approach, although effective for accurate and sparse imaging, is
limited in resolution, which is similar to the standard backprojection/DAS algorithm. We
therefore, investigate super-resolution methods for FLGPR SAR imaging.
4.5
Super-resolution Methods
In this section, a new approach for high-resolution imaging is presented for impulse
based FLGPR. The 2D aperture in Fig. 4-2 for imaging of the specified grid will result
in a data vector in (4–1) y ∈ RT K×1 that is large (on the order of 106 ), making practical
applications of adaptive techniques infeasible. Also the availability of a single data
vector makes well known high resolution data adaptive approaches, such as the
CAPON, APES, as well as subspace based methods [2], not directly applicable. Another
challenge is that the data vector will contain clutter reflections from regions outside the
imaging ROI, which need to be effectively suppressed prior to imaging.
We propose a data filtering and reduction approach via time gating and orthogonal
projection to reduce interference from scatterers outside the ROI. This approach
involves a singular value decomposition of the steering matrix and a projection of the
90
Figure 4-3. Time gating
data using the dominant singular vectors, which effectively reduces the data dimension
to practical levels. This projection also filters the data to effectively utilize received
energy from only the ROI. A computationally efficient method of this projection is
performed via an eigenvalue decomposition to obtain an updated data model.
Using the updated model, two recently proposed algorithms (SPICE and SLIM) are
used for imaging to produce sparse, accurate and high resolution imaging results, even
with a single data vector. The process is described in the next subsections.
4.5.1 Orthogonal Projection and Time Gating
−1
For time gating, consider the kth receive measurement {rk (t)}Tt=0
. Based on
(4–4), the delays of the pixels in the ROI imaging grid, τ k
= [τk,1 , τk,2 , . . . , τk,I ], to
the corresponding transmit-receive pair for this measurement can be computed. The
minimum and maximum delays between this transmit-receive pair and the pixels in the
= min(τ k ) and τk = max(τ k ), respectively. The kth
τ
by discarding data outside
receive measurement can then be updated to {rk (t)}t=τ
ROI imaging grid are given by τkmin
max
kmax
kmin
these computed delays.
Without loss of generality, this procedure is shown in Fig. 4-3 for a colocated
transmit-receive pair centered below (with a pre-specified standoff-range) the imaging
ROI grid. Interference from the regions below the minimum delay line and above the
maximum delay line are discarded. The data vector in (4–1) can then be updated to
91
yg = [r1 (τ1 ), . . . , r1 (τ1 ), . . . , rK (τK ), . . . , rK (τK )]T , and the updated data model is:
min
max
max
min
yg = Ag β + Bg γ + ng .
(4–6)
with Ag ∈ RQ×L and Bg ∈ RQ×M (Q < T K).
Note from Fig. 4-3, interference from scatterers outside the ROI still contribute
to the receive measurements. In this paper, an orthogonal projection of the the data
to relevant singular vectors of the data matrix is performed to reduce the effects of
the scatterers outside this grid as well as to reduce the data dimension significantly
(by a factor of 103 in practical applications), allowing for practical applications of high
resolution imaging techniques. This interference reduction via orthogonal projection
is performed in a computationally efficient manner via an eigen-decomposition and is
motivated and described as follows.
Consider the following penalized least squares optimization problem used for
square-root LASSO [98] (for notational simplicity, we eliminate the subscript g from
(4–6)).
argmin ||y − Aβ − Bγ||2 +λ||β||1
β ,γ
(4–7)
Note here that the sparsity promoting ℓ1 constraint is placed on β (the desired
estimates), whereas no constraint is placed of γ due to the fact that even if γ is sparse,
the subsequent transformation eliminates this sparsity. Let Bγ
= UB B VB T γ
,
UB γ~
via a singular value decomposition of B. The optimization problem in (4–7) for γ
~
(which is no longer sparse) given β is unconstrained and the solution is given as
^~
γ
= UB T (y − Aβ). Given γ^~ , the constrained optimization problem in (4–7) is now given
as follows:
^
β
= argmin ||(I − UBUB T )(y − Aβ)||2 +λ||β||1
β
Since B and hence UB is unknown, we assume that UB and UA , where UA are the
singular vectors of A are orthogonally compliment to each other. Then Eq. (4–8)
92
(4–8)
becomes:
^
β
= argmin ||UA UA T (y − Aβ)||2 + λ||β||1
(4–9a)
β
^ = argmin ||UA T (y − Aβ)||2 + λ||β||1
β
(4–9b)
β
Consider the following singular value decomposition of the steering matrix A ∈
RQ×L , with Q > L in practice1 :
T
A = UV

= UA


 A 
U~ A   
0
[ ]
V
T
(4–10)
= UA AVT
~ A ] are the left singular vectors of A and the columns of
where the columns of [UA , U
V are their right counterparts, and the singular values of A are on the diagonal of the
diagonal matrix A .
Due to the find grid used for the ROI, some of the singular values of A in A
are quite small. By discarding the small singular values of A, we approximate A as
A ≈ Us s Vs T . Then the optimization problem in Eq. (4–9b) becomes:
^
β
= argmin ||UsT (y − Aβ)||2 +λ||β||1
(4–11)
β
Then using Us for orthogonal projection yields:
Us T y = Us T Aβ + ϵ,
~y = A~ β + ϵ
(4–12a)
(4–12b)
Via a series of simulations, we found that the number of columns in Us is not sensitive
to the image formation performance. We choose the dimension of Us by analyzing the
1
Note that since the radar illuminates a large area M > Q, but we do not consider M
as the contributions of the clutter (Bγ) are eliminated based on the decomposition
93
metric ||A − Us s Vs T ||F /||A||F . Note here that ~
y ∈ Rs×1 and A~ ∈ Rs×1 where s << Q (a
factor of 103 reduction in practical applications).
The projection described in (4–12b) can be obtained in a computationally efficient
way via an eigenvalue decomposition in lieu of an SVD. Note that in (4–10) UA ∈
RQ×L , which makes the SVD decomposition in this equation computationally intensive
(O(Q2 L)) in practical scenarios.
Consider the following eigenvalue decomposition2
AT A = VVT
where V ∈ RL×L and = 2A . Then UA
(4–13)
= AV−A 1 , where the diagonal matrix −A 1 can be
written as:

−1
−A 1 =  s
0

0 

−1
n
(4–14)
1
with the diagonal of the sub-matrix −
s consisting of the inverse of the dominant
singular values of A. Then Us
= AVs −s 1 , with Vs corresponding to the s dominant
eigenvectors of AT A.
The updated data vector in (4–12b) is given as:
~y = UsT y = (−s 1 )Vs T AT y
(4–15)
whereas the updated steering matrix is given as:
A~ = UTs A
= −s 1 Vs T AT A
= −s 1 Vs T VVT
= s Vs T
2
Complexity of eigenvalue decomposition - O(L3 )
94
(4–16)
The updated data model in (4–12b), obtained using Us can be obtained using the
decomposition in (4–13). This decomposition can be performed offline as long as prior
knowledge of the imaging grid is known, which is typically the case.
A different approach for orthogonal projection that improves on computation is
described next. Consider the steering matrix G which corresponds to the imaging
ROI with a much coarser grid. The matrix G can then be generated by selecting
the appropriate columns of A. The matrix G is used to approximate Us . However,
unlike Us this matrix is not semi-unitary. The updated data vector is then given as
~y = (GT G)−1 GT y and the updated steering matrix is given as A~ = (GT G)−1 GT A. This
new projection approach, skips the computation AT A and its subsequent decomposition,
significantly improving computation at cost of less interference suppression..
Based on the updated model in (4–12b), we present below two recently proposed,
data-adaptive and iterative approaches for high resolution FLGPR SAR imaging. They
are the Sparse Learning via Iterative Minimization (SLIM) [29] and the SParse Iterative
Covariance-based Estimation (SPICE) [93], which is equivalent to square-root LASSO
with λ
= 1 [99]. These two approaches are user-parameter free and are capable
of producing sparse and high resolution estimates when only a single data vector is
available for imaging; they are described next.
4.5.2 SLIM
The SLIM method [29] is a maximum-aposteriori (MAP) approach for sparse
signal recovery. The sparse recovery problem is can be solved by optimizing the ℓ1
optimization cost function in (4–17) based on the linear model in (4–12):
^
β
= argmin ||~y − A~ β||2 +λ||β||1
(4–17)
β
The SLIM algorithm can be considered as an ℓq norm approach norm for 0 < q ≤ 1, that
considers the following hierarchial Bayesian model (with a sparsity promoting prior) for
95
Table 4-1. SLIM Algorithm
Initialization:
^ (0) with the DAS algorithm and σ^(0)
Obtain initial estimate β
^ (0) and (4–20)
based on β
SLIM (nth) Iteration: Repeat the following steps until convergence
^ (n+1) = A~ diag(|β(n)|2−q )A~ T
Step 1: Compute: R
= A~ P(n) A~ T + σ(n) I.
^ (n) = P(n) A~ T R^ −(n1)~y.
Step 2: Compute: β
Step 3: Update: σ
^(n) = L1 ||~y − A~ β^ (n)||22 .
estimation [29].
~ β, σI)
~y|β, σ ∼∏
CN (A
2
e− (|β | −1)
f (β ) ∝
q
f (σ ) ∝ 1
i
+ σ(n) I
(4–18a)
q
(4–18b)
i
(4–18c)
SLIM estimates the desired sparse vector β, and the noise variance σ, iteratively by
minimizing the negative logarithm cost of the posterior density given by:
cq (β, σ ) = L log σ +
The choice of q
1 ||~y − A~ β||2 + ∑ 2 (|β |q −1)
2
σ
i
q
i
(4–19)
= 1 simplifies this cost function to the well-known ℓ1 norm constraint
for sparse estimation [29]. Minimizing the cost function in (4–19) yields the following
estimates:
where P
^
β
= PAT (A~ PA~ T + σI)−1~y
= PA~ T R−1~y
σ
^ = L1 ||~y − A~ β||22
(4–20a)
(4–20b)
= diag(p) and p = |β|2−q . These estimates are obtained in an iterative manner
based on the steps in Tab. 4-1.
4.5.3 SPICE
The SPICE method for parameter estimation [30, 93] modifies the model in (4–12)
as follows:
~y = A~ β + ϵ
= Dx
96
(4–21)
Table 4-2. CG SPICE Algorithm
Obtain initial estimate x
^j(0)
Initialization:
j
= 1, . . . , L + s.
= dTj ~y/dTj dj , and pj(0) = |x^j(0) |/ωj
SPICE (nth) Iteration: Repeat the following steps until convergence
^ (n) = Ddiag(p(n) )DT = DP(n)DT .
Step 1: Compute R
1 y.
Step 2: Compute (using CG): s(n) = R−
( n) ~
CG Initialization: s(n)(0)
CG Iterations (mth):
= 0; r(0) = q(0) = ~y
= (r(Tm) r(m) )/(q(Tm) q(m) )
- s(n)(m+1) = s(n)(m) + α(m) q(m)
^ (n)q(m)
- r(m+1) = r(m) − α(m) R
- q(m+1) = q(m) + (r(Tm+1) r(m+1) )/(r(Tm) r(m) )
Step 3: Update ^
x(n+1) = P(n) DT R−(n1)~y. j = 1, . . . , L + s.
Step 4: Update pj (n+1) = x
^j(n+1) /ωj j = 1, . . . , L + s.
- α(m)
where D
= [A~ , I] and x = [βT , ϵT ]T . This method is a covariance fitting approach to
parameter estimation that minimizes the following covariance fitting cost function:
||R1/2 (~
y~yT − R)||F
(4–22)
where R = E{~
y~yT } = DPDT is the covariance matrix of the data and the diagonal matrix
P is now given as:


Ps 0 
P = 

0 Pϵ
with Ps
(4–23)
= diag(ps ) and Pϵ = diag(pϵ) being the diagonal matrices containing the power
estimates of β and the noise and interference residue ϵ, respectively.
The criterion in (4–22) simplifies to the following minimization problem to estimate
both x and p = [ps T , pTϵ ]T .
{^
x, p^} = argmin x
x,p
where ωj
T
−1
P x+
L+s
∑
j =1
ωj pj
s.t.
Dx = ~y
(4–24)
= ||dj ||/||~y|| and ||dj || is the jth column of D. The estimates are given as:
97
^x = PDT R−1~y
|βj |
pj =
,
j = 1, . . . , L + s
(4–25a)
(4–25b)
ωj
+s and p = {p }L+s .
which are solved iteratively till convergence [30]; with x = {xj }Lj =1
j j =1
To improve on the computationally efficiency of the SPICE algorithm for FLGPR
SAR imaging, a conjugate gradient based SPICE algorithm (CG-SPICE) is presented in
this paper. This approach is similar to the conjugate gradient SLIM algorithm described
in [29], [100]. The steps of this CG-SPICE algorithm are described in Tab 4-2.
Based on [101], the SPICE optimization problem can be re-written as:
L+s
∑
argmin ||x||1 =
|xj | s.t.
x^
j =1
Dx = ~y
For real-valued data, let ~
hj , max(xj , 0) and hj , −min(xj , 0). Note that xj
and |xj |= ~
hj
(4–26)
= ~hj − hj
+ hj . The optimization problem in Eq. (4–26) can then be augmented to
[102]:
argmin uT h
h^
where h
s.t.
D h = ~y
and h ≥ 0
(4–27)
= [~h1 , . . . ~hL+s , h1 . . . hL+s], D = [D, −D] and u = [1, 1 . . . , 1]T . The linear
program in (4–27) can be solved efficiently to provide sparse estimates for FLGPR
SAR imaging. This approach is on the same order (computationally) as the cyclic
optimization approach in Tab 4-2 implemented using the conjugate gradient, and faster
(approximately 2 times) without CG based on numerical simulations.
Note that the estimates in SLIM and SPICE have the same form with the difference
lying in the estimation of the noise and interference residue. SPICE estimates the
reflection coefficients and the noise and interference residue simultaneously with the
noise and interference residue variance of each elements not necessarily being equal
unlike the case in SLIM. However, simulation results show similar performance of the
two algorithms with SPICE being less susceptible to the noise and interference residue.
This algorithms are robust and effective for generating sparse results.
98
Table 4-3. Subspace approximation
No. of singular vectors - s
||A − Us s Vs T ||F /||A||F
15
0.902
100
0.512
300
0.182
Unlike most well-known adaptive algorithms, SLIM can dynamically estimate the
user parameter (which in this case, corresponds to the noise power estimate) of the
original LASSO cost function (for sparse parameter estimation) [103] which is sensitive
to the choice of this parameter (λ). The SPICE criteria can also be reduced to the
criteria in (4–17) with λ
= 1 [99] (a special case of the square-root lasso [98] which is
insensitive to the choice of the user-parameter λ).
These robust user parameter free algorithms are applied here to problem of
FLGPR SAR imaging. Analysis is performed on both simulated and real experimentally
measured SIRE FLGPR data for imaging, and the results are presented in the next
section.
4.6 Numerical and Experimental Results
In this section, we perform sparse high resolution imaging for FLGPR using
orthogonal projection (using Us ) for clutter and data reduction. The SPICE and
SLIM algorithms are considered for high resolution imaging. We also analyze the
well-known CLEAN approach for imaging based on the proposed data model and show
the ability of this well-known algorithm to yield sparse and accurate results. The CLEAN
approach, however, is limited in imaging resolution and does not improve resolution
over the standard BP algorithm. The coarse grid approach using G is also analyzed for
projection.
For FLGPR SAR imaging analysis, we use the SIRE radar designed by ARL for
imaging based on the setup in Figs. 4-1 and 4-2. For simulations, we consider an
imaging area which has a range swath of 4 m and a cross-range swath of 5 m. A
99
0
35
Eigenvalues
Threshold = 15
Threshold = 100
Threshold = 300
cross−range (meters)
30
2
Eigenvalues
25
20
15
10
5
0
0
−5
−10
1
−15
0
−20
−25
−1
−30
−35
−2
200
400
600
Eigenvalue number
800
−2
1000
A
−1
0
1
downrange (meters)
2
B
0
0
2
−5
−10
1
−15
0
−20
−25
−1
−30
cross−range (meters)
cross−range (meters)
2
−35
−2
−2
−1
0
1
downrange (meters)
−40
2
−40
C
−5
−10
1
−15
0
−20
−25
−1
−30
−35
−2
−2
−1
0
1
downrange (meters)
2
−40
D
Figure 4-4. Subspace dimension (s) for high resolution imaging (A) Eigenvalues of AT A,
(B) s = 15, (C) s = 100, and (D) s = 300
minimum standoff distance of 8 m is used for simulations, with a maximum standoff
distance of 14 m. Simulation was run with three targets placed at various [x, y, z ]
locations in meters marked by the symbol ’X’. The imaging area consists of L
= 10000
pixels.
An analysis of the subspace dimension, (i.e., the number of dominant singular
values of A) is performed first. Targets at locations [0,0,0], [-0.3, 1, 0], and [1.5, 1.5,
0] are simulated. Fig. 4-4 shows the SPICE algorithm applied with various thresholds.
From Tab. 4-3, thresholds as large 0.5 based on the criterion ||A − Us s Vs T ||F /||A||F
100
0
0
2
cross−range (meters)
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
−1
0
1
downrange (meters)
A
2
B
0
0
2
cross−range (meters)
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−40
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
−1
0
1
downrange (meters)
C
2
−40
D
0
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−1
0
1
downrange (meters)
2
−40
E
Figure 4-5. FLGPR SAR Imaging - detection of weak target (A) Back-projection, (B)
RSM, (C) CLEAN, (D) SLIM, and (E) SPICE
(normalized scale of 0 to 1) yield desirable results with all the targets detected as can be
seen in Fig. 4-4.
The next analysis involves detecting weak targets buried by the sidelobes of much
stronger targets. Three targets are again simulated at locations [0,0,0], [-0.1, 0.8, 0],
101
0
0
2
cross−range (meters)
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
−1
0
1
downrange (meters)
A
2
B
0
0
2
cross−range (meters)
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−40
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
−1
0
1
downrange (meters)
C
2
−40
D
0
cross−range (meters)
2
−10
1
0
−20
−1
−30
−2
−2
−1
0
1
downrange (meters)
2
−40
E
Figure 4-6. FLGPR Imaging - resolution improvement (A) Back-projection, (B) RSM, (C)
CLEAN, (D) SLIM, and (E) SPICE
and [1.5, 1.5, 0], with the strong targets 10 times stronger than the weak target as
shown in Fig. 4-5. From this figure we can see that the weak target is buried by the
sidelobes of the stronger target using the standard backprojection algorithm. The
CLEAN approach, which iteratively subtracts out the contributions of the strongest target
102
0
0
2
−10
1
0
−20
−1
−30
cross−range (meters)
cross−range (meters)
2
−2
−2
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
A
−1
0
1
downrange (meters)
2
B
0
0
2
−10
1
0
−20
−1
−30
cross−range (meters)
cross−range (meters)
2
−2
−2
−40
−10
1
0
−20
−1
−30
−2
−1
0
1
downrange (meters)
2
−40
−2
C
−1
0
1
downrange (meters)
2
−40
D
Figure 4-7. Orthogonal projection using (A) Us (High SNR), (B) G (High SNR), (C) Us
(Low SNR), and (D) G (Low SNR)
from the receive measurements, can effectively and accurately detect this weak target.
The SPICE and SLIM algorithms are also applied post orthogonal projection (with
threshold ||A − Us s Vs T ||F /||A||F = 0.2) and the weak target is revealed.
The CLEAN approach, although effective in providing sparse and accurate results,
is limited in resolution and has no improvement over the standard backprojection
algorithm. The high resolution imaging methods for FLGPR, provide improvement in
resolution over the backprojection-based algorithms (BP, RSM and CLEAN). Fig. 4-6
shows three targets at locations [0,0,0], [0, 0.75, 0], and [1.5, 1.5, 0]. Two of these
targets are ’closely’ spaced and are clearly resolved by SLIM and SPICE. The SPICE
and SLIM algorithms provide almost a factor of 2 improvement in imaging resolution.
103
220
0
220
210
cross−range (meters)
cross−range (meters)
210
0
−10
200
190
−20
180
−30
170
160
−10
200
190
−20
180
−30
170
160
−10
0
10
downrange (meters)
−40
−10
0
10
downrange (meters)
A
−40
B
220
0
cross−range (meters)
210
−10
200
190
−20
180
−30
170
160
−10
0
10
downrange (meters)
−40
C
Figure 4-8. Real data - SIRE FLGPR SAR Imaging: (A) Back-projection, (B) RSM, and
(C) SPICE
104
Receiver Operating Characteristics (ROC)
Probability of Detection (PD)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Backprojection
RSM
SPICE
0.2
0.1
0
0
1
2
3
4 5 6 7 8 9 10 11 12 13
Number of False Alarms
Figure 4-9. Receiver Operating Curve (ROC) comparison: FLGPR SAR Imaging
This experiment is repeated using G for orthogonal projection. The results are
compared to using the semi-unitary matrix Us for projection and are shown in Fig. 4-9. A
gain in computation is achieved at a cost of less interference suppression.
The proposed approach is verified using real experimentally measured data.
Results based on real SIRE data provided by the Army research lab can be seen in
Fig. 4-8. In this figure a subimages 2m in range are continuously formed based on
overlapping 2D apertures to generate the entire image [54]. Based on this figure,
significant interference reduction can be seen by the high resolution SPICE compared to
the standard backprojection algorithm, with some of the targets of interest marked in red
oval circles.
A quantitative numerical analysis is performed to show the effectiveness of the
high-resolution SPICE approach for FLGPR SAR imaging. Several targets with varying
strengths are simulated and the receiver operating characteristics curve (ROC) is shown
in Fig. 4-9. This curve which shows the probability of detection versus the number
of false alarms is generated by using a simple threshold detector. The image under
analysis is segmented into regions, and the maximum pixel value in each region is
retained. The threshold is incremented in steps and for each threshold, the number
105
of alarms are recorded. The alarms that fall outside the regions with targets present
(based on prior knowledge) are considered false alarms. As shown in Fig. 4-9, when
there is full detection, the SPICE approach has less false alarms the the BP algorithms.
4.7
Conclusions
In this chapter, we have considered new approaches to imaging for forward looking
ground penetrating radar. The pre-processing involves a proposition of a data model
in the time domain, which takes into account the contributions of clutter outside the
imaging area. An orthogonal projection of the measured data to a subspace spanned by
the steering matrix corresponding to the imaging ROI is then used for clutter reduction
as well as significant data reduction, making it feasible for practical applications of high
resolution methods. The steering matrix decomposition is performed efficiently and
depends only on the prior knowledge of the desired imaging area and hence can be
performed offline. Two recently proposed, data-adaptive approaches, SPICE and SLIM
are used for FLGPR SAR imaging. They are user parameter free algorithms and have
the ability to provide sparse and high resolution images using a single data vector, unlike
other well-known high resolution methods. The results using simulated data show that
SLIM and SPICE provide improvement in resolution close to a factor of two compared
to the backprojection based algorithms including BP, RSM, CLEAN. A new conjugate
gradient based SPICE algorithm is also introduced in this paper for more efficient
computations of the estimates.
106
CHAPTER 5
CONCLUDING REMARKS AND FUTURE WORK
Due to limitations of data independent methods for spectral estimation, data-adaptive
methods are currently being investigated for improved performance. In this dissertation,
we focused on efficient and effective applications of data-adaptive methods to real world
sensing problems.
In Chapter 2, the basic problem of harmonic retrieval is investigated. The problem
pertains to digital audio forensics. The contribution we make to this problem involves
coming up with a more reliable and accurate way of estimating the network frequency
buried in an audio recording using data adaptive techniques. The proposed approach
involves spectral analysis using a robust high resolution algorithm and tracking the
network frequency via a dynamic programming approach. The approach yields
significant improvement in the estimation of the embedded network frequency when
this signal is weak compared to the audio recording (a major challenge for this problem).
Chapters 3 and 4 are the focus of this dissertation. In these chapters, the
Synchronous Impulse Reconstruction (SIRE) radar, which is a remote sensing tool for
landmine detection is analyzed and studied. In Chapter 3, we propose a new approach
of Radio Frequency Interference suppression for this radar. This new approach can
provide an improvement of close to 7 dB in RFI suppression without distorting the
desired target signatures. This approach is implemented in an efficient way by exploiting
the equivalent sampling technique of this radar.
Chapter 4 focuses on sparse high resolution imaging for this SIRE Forward Looking
Ground Penetrating radar (FLGPR). In this chapter, we establish a signal model in the
time domain since the transmitted impulse is well localized in time. This data model
takes into account the contributions of clutter outside the imaging region of interest
(ROI). We propose a pre-processing step of orthogonal projection to mitigate the effects
of clutter outside the ROI which is present in the collected data. Recently proposed
107
robust, sparse high resolution algorithms for imaging are then applied to provide
improved imaging resolution. We achieve close to a factor of 2 improvement in imaging
resolution compared to the standard methods currently used for SAR imaging for this
radar.
Our current and future work focuses on more efficient ways to implement the
pre-processing step of orthogonal projection. In lieu of a decomposition of the steering
matrix corresponding to imaging ROI and then projection of the data, we propose a
direct projection of the data to the steering matrix corresponding to the ROI with a much
coarser grid to improve computation. This matrix approximates the set of orthogonal
vectors that spans the subspace of the matrix corresponding to the ROI. Results show
significant improvement in computation at cost of less interference suppression.
108
REFERENCES
[1] S. Kay, Modern Spectral Estimation: Theory and Applications.
NJ: Prentice-Hall, 1988.
[2] P. Stoica and R. L. Moses, Spectral Analysis of Signals.
Prentice-Hall, 2005.
[3] S. Marple, Digital Spectral Analysis with Applications.
Prentice-Hall, 1987.
Englewood Cliffs,
Upper Saddle River, NJ:
Englewood Cliffs, NJ:
[4] P. Stoica and R. L. Moses, Introduction to Spectral Analysis, Upper Saddle River,
NJ, 1997, vol. 1.
[5] G. M. Jenkins and D. G. Watts, Spectral analysis.
Holden-Day, 1968.
San Francisco, CA:
[6] A. Schuster, “On the investigation of hidden periodicities with application to a
supposed 26 day period of meteorological phenomena,” Terrestrial Magnetism,
vol. 3, no. 1, pp. 13–41, 1898.
[7] J. Allen, “Short term spectral analysis, synthesis, and modification by discrete
Fourier transform,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 25, no. 3, pp. 235–238, Jun 1977.
[8] L. Du, J. Li, P. Stoica, H. Ling, and S. Ram, “Doppler spectrogram analysis of
human gait via iterative adaptive approach,” Electronics Letters, vol. 45, no. 3, pp.
186–188, 29 2009.
[9] S. Ram and H. Ling, “Analysis of microDopplers from human gait using
reassigned joint time-frequency transform,” Electronics Letters, vol. 43, no. 23,
November 2007.
[10] S. Haykin, Advances in spectrum analysis and array processing Volume 3.
Prentice Hall, 1991.
[11] C. Jakowatz, D. Wahl, P. Eichel, D. Ghiglia, and P. Thompson, Spotlight-Mode
Synthetic Aperture Radar: A signal processing approach. Norwell, MA: Kluwer
Academic Publishers, 1996.
[12] E. Gudmundson, P. Stoica, J. Li, A. Jakobsson, and M. Rowe, “Spectral estimation
of irregularly sampled exponentially decaying signals with applications to RF
spectroscopy,” J. Magn. Resonance, vol. 201, no. 1, pp. 167–176, March 2010.
[13] B. Edde, RADAR: Principles, Technology and Applications.
NJ: Prentice-Hall, 1993.
109
Upper Saddle River,
[14] R. Armitage, C. Landis, and R. H. et al, “Power spectral analysis of sleep EEG
in twins discordant for chronic fatigue syndrome.” J. Psychosomatic Research,
vol. 66, pp. 51–57, 2009.
[15] J. Mukai, “Spectral analysis of all-night sleep EEG in narcoleptic patients and
normal subjects,” J Sleep Res., pp. 63–71, 2003.
[16] C. Grigoras, “Digital audio recording analysis: The electric network frequency
criterion,” The International Journal of Speech Language and the Law, vol. 12,
no. 1, pp. 63–76, 2005.
[17] L. Xu, J. Li, and P. Stoica, “Adaptive Techniques for MIMO Radar,” in Sensor Array
and Multichannel Processing, 2006. Fourth IEEE Workshop on, July 2006, pp.
258–262.
[18] J. Proakis and D. Manolakis, Digital Signal Processing: Principles, Alogrithms and
Applications. Upper Saddle River, NJ: Prentice-Hall, 2007.
[19] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proceedings of the IEEE, vol. 57, no. 8, pp. 1408–1418, Aug. 1969.
[20] P. P. Stoica, Z. Wang, and J. Li, “Robust Capon beamforming,” IEEE Signal
Processing Letters, vol. 10, no. 6, pp. 172–175, June 2003.
[21] E. Larsson, P. Stoica, and J. Li, “Spectral estimation via adaptive filterbank
methods: a unified analysis and a new algorithm,” Signal Processing, vol. 82, pp.
1991–2001, March 2002.
[22] J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR
imaging,” IEEE Transactions on Signal Processing, vol. 44, no. 6, pp. 1469–1484,
Jun 1996.
[23] P. Stoica, H. Li, and J. Li, “A new derivation of the APES filter,” IEEE Signal
Processing Letters, vol. 6, no. 8, pp. 205–206, Aug. 1999.
[24] Y. Wang, J. Li, and P. Stoica, Spectral Analysis of Signals: The Missing Data
Case. Morgan and Claypool, 2005.
[25] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Source localization and
sensing: A nonparametric iterative adaptive approach based on weighted least
squares,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 1,
pp. 425–443, Jan. 2010.
[26] W. Roberts, P. Stoica, J. Li, T. Yardibi, and F. Sadjadi, “Iterative Adaptive
Approaches to MIMO Radar Imaging,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 1, pp. 5–20, Feb. 2010.
110
[27] P. Stoica, J. Li, and H. He, “Spectral Analysis of Nonuniformly Sampled Data: A
New Approach Versus the Periodogram,” IEEE Transactions on Signal Processing,
vol. 57, no. 3, pp. 843 –858, March 2009.
[28] P. Stoica, J. Li, and J. Ling, “Missing Data Recovery Via a Nonparametric Iterative
Adaptive Approach,” Signal Processing Letters, IEEE, vol. 16, no. 4, pp. 241–244,
April 2009.
[29] T. Xing, W. Roberts, J. Li, and P. Stoica, “Sparse Learning via Iterative
Minimization with Application to MIMO Radar Imaging,” IEEE Transactions on
Signal Processing, vol. 59, no. 3, pp. 1088–1101, March 2011.
[30] P. Stoica, P. Babu, and J. Li, “SPICE: A Sparse Covariance-Based Estimation
Method for Array Processing,” IEEE Transactions on Signal Processing, vol. 59,
no. 2, pp. 629–638, 2011.
[31] J. Li and P. Stoica, “Efficient Mixed-Spectrum Estimation with Applications to
Target Feature Extraction,” IEEE Transactions on Signal Processing, vol. 44, no. 2,
pp. 281–295, Feb 1996.
[32] R. C. Maher, “Audio Forensic Examination - authenticity, enhancemement, and
interpretation,” IEEE Signal Processing Magazine, vol. 26, no. 2, pp. 84–94, Mar.
2009.
[33] E. B. Brixen, “Techniques for the Authentication of Digital Audio Recordings,” in
Audio Engineering Society Convention 122, May 2007.
[34] C. Grigoras, “Applications of ENF criterion in forensic audio, video and
telecommunications analysis,” Forensic Science International, vol. 167, pp.
136–176, 2007.
[35] E. B. Brixen, “ENF - Quantification of the Magnetic Field,” in Proc. AES 33rd Conf.,
Audio Forensics - Theory and Practice, Denver, CO, Jun. 2008.
[36] A. J. Cooper, “The Electric Network Frequency (ENF) as an aid to authenticating
forensic digital audio recordings - an automated approach,” in Proc. AES 33rd
Conf., Audio Forensics - Theory and Practice, Denver, CO, Jun. 2008, pp. 1–6.
[37] U. Jönsson, C. Trygger, and P. Ögren, “Lecture notes on Optimal Control:
Optimization and System Theory,” unpublished.
[38] D. P. N. Rodrı́guez, J. Apolinário, and L. W. P. Biscainho, “Audio authenticity:
Detecting ENF discontinuity with high precision phase analysis,” IEEE Transactions on Information Forensics and Security, vol. 5, no. 3, pp. 534–543, Sept.
2010.
[39] M. Kajstura, A. Trawinska, and J. Hebenstreit, “Application of the Electrical
Network Frequency (ENF) Criterion - A case of a digital recording,” Forensic
Science International, vol. 155, pp. 165–171, 2005.
111
[40] Y. Liu, “A US-Wide Power Systems Frequency Monitoring Network,” in IEEE
Power Systems Conference and Exposition, Oct. 29 - Nov. 1 2006, pp. 159–166.
[41] N. G. Hingorani, “High-voltage DC Transmission - A power electronics workhorse,”
IEEE Spectrum, vol. 33, no. 4, pp. 63–72, Apr. 1996.
[42] Liu, Z. Yuan, P. N. Markham, R. Conners, and Y. Liu, “Wide-area frequency as
a criterion for digital audio recording authentication,” in IEEE Power and Energy
Society General Meeting, Jul. 2011, pp. 1–7.
[43] J. O. Smith and X. Serra, “PARSHL An Analysis/Synthesis program for
non-Harmonic Sounds Based on Sinusoidal Representation,” in International
Computer Music Conference, San Francisco, CA, 2004.
[44] G. Glentis and A. Jakobsson, “Time-Recursive IAA Spectral Estimation,” IEEE
Signal Processing Letters, vol. 18, no. 2, pp. 111–114, Feb. 2011.
[45] W. Roberts, P. Stoica, J. Li, T. Yardibi, and F. Sadjadi, “Iterative Adaptive
Approaches to MIMO Radar Imaging,” Selected Topics in Signal Processing,
IEEE Journal of, vol. 4, no. 1, pp. 5–20, Feb. 2010.
[46] M. Xue, L. Xu, and J. Li, “IAA Spectral Estimation: Fast Implementation Using
the Gohberg-Semencul Factorization,” IEEE Transactions on Signal Processing,
vol. 59, no. 7, pp. 3251–3261, Jul. 2011.
[47] G. Glentis and A. Jakobsson, “Efficient implementation of iterative adaptive
approach spectral estimation techniques,” IEEE Transactions on Signal Processing, vol. 59, no. 9, pp. 4154–4167, Sept. 2011.
[48] G. O. Glentis and A. Jakobsson, “Superfast Approximative Implementation of the
IAA Spectral Estimate,” IEEE Transactions on Signal Processing, vol. 60, no. 1,
pp. 472–478, 2012.
[49] P. Stoica, J. Li, and H. He, “Spectral Analysis of Nonuniformly Sampled Data: A
New Approach Versus the Periodogram,” IEEE Transactions on Signal Processing, vol. 57, no. 3, pp. 843–858, March 2009.
[50] M. Huijbregtse and Z. Geradts, “Using the ENF Criterion for Determining the Time
of Recording for Short Digital Audio Recordings,” in Proc. of the 3rd International
Workshop on Computational Forensics, ser. IWCF ’09, vol. 1, 2009, pp. 116–124.
[51] J. Gunarsson and T. McKelvey, “High SNR Performance Analysis of F-ESPRIT,”
in Conference Record of the 38th Asilomar Conference on Signals, Systems and
Computers, vol. 1, Nov. 2004, pp. 1003–1007.
[52] R. Miles, A. Dogariu, and J. Michael, “Bringing bombs to light,” IEEE Spectrum,
vol. 49, no. 2, pp. 38–43, February 2012.
112
[53] T. Miller, L. Potter, and J. McCorkle, “RFI Suppression for Ultra Wideband Radar,”
IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1142
–1156, Oct. 1997.
[54] L. Nguyen, “Signal and Image Processing Algorithms for the U.S. Army Research
Laboratory Ultra-wideband (UWB) Synchronous Impulse Reconstruction (SIRE)
Radar.” ARL Technical Report, ARL-TR-4784, April 2009.
[55] P. Stoica, A. Jakobsson, and J. Li, “Cisoid parameter estimation in the colored
noise case: asymptotic Cramer-Rao bound, maximum likelihood, and nonlinear
least-squares,” IEEE Transactions on Signal Processing, vol. 45, no. 8, pp.
2048–2059, Aug 1997.
[56] C. Li, J. Ling, J. Li, and J. Lin, “Accurate Doppler Radar Noncontact Vital Sign
Detection Using the RELAX Algorithm,” IEEE Transactions on Instrumentation and
Measurement, vol. 59, no. 3, pp. 687–695, March 2010.
[57] J. Li, D. Zheng, and P. Stoica, “Angle and waveform estimation via RELAX,”
IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 3, pp.
1077–1087, July 1997.
[58] L. Nguyen and M. Soumekh, “Suppression of Radio Frequency Interference (RFI)
for Synchronous Impulse Reconstruction Ultra-Wideband Radar,” ser. Proc. SPIE,
vol. 5808, 2005, pp. 178–184.
[59] R. Lord and M. Inggs, “Approaches to RF Interference Suppression for VHF/UHF
Synthetic Aperture Radar,” ser. Proc. COMSIG, 1998, pp. 95–100.
[60] D. Vu, L. Xu, M. Xue, and J. Li, “Nonparametric Missing Sample Spectral Analysis
and Its Applications to Interrupted SAR,” IEEE Journal of Selected Topics in Signal
Processing, vol. 6, no. 1, pp. 1–14, Feb 2012.
[61] L. Nguyen and R.Innocenti, “Suppression of sidelobes and noise in airborne
SAR imagery using the Recursive Sidelobe Minimization technique,” in Radar
Conference, 2010 IEEE, May 2010, pp. 522–525.
[62] M. Braunstein, J. Ralston, and D. Sparrow, “Signal processing approaches to radio
frequency interference (RFI) suppression,” in, Algorithms for Synthetic Aperture
Radar Imagery, A. Giglio, eds., Orlando, FL, USA, vol. 2230, pp. 190–208, April
1994.
[63] L. Nguyen, M. Ressler, D. Wong, and M. Soumekh, “Adaptive Coherent
Suppression of Multiple Wide-Bandwidth RFI Sources in SAR,” ser. Proc. SPIE,
vol. 5427, 2004, pp. 1–16.
[64] C. Yu, Y.Zhang, Z. Dong, and D. Liang, “Eigen-Decomposition Method for RFI
Suppression Applied to SAR Data,” in Multimedia Technology (ICMT), 2010
International Conference on, oct. 2010, pp. 1–4.
113
[65] P. Stoica and Y. Selén, “Model-order selection: a review of information criterion
rules,” IEEE Signal Processing Magazine, vol. 21, no. 4, pp. 36–47, July 2004.
[66] X. Huang and D. Liang, “Gradual RELAX algorithm for RFI suppression in
UWB-SAR,” Electronics Letters, vol. 35, no. 22, pp. 1916–1917, Oct 1999.
[67] M. Wax and I. Ziskind, “On unique localization of multiple sources by passive
sensor arrays ,” IEEE Transactions on Acoustics, Speech and Signal Processing,
vol. 37, no. 7, pp. 996–1000, Jul 1989.
[68] A. Nehorai, D. Starer, and P. Stoica, “Direction-of-arrival estimation in applications
with multipath and few snapshots,” Circuits, Syst., Signal Process., vol. 10, pp.
327–342, 1991.
[69] J. Li, P. Stoica, L. Xu, and W. Roberts, “On Parameter Identifiability of MIMO
Radar,” IEEE Signal Processing Letters, vol. 14, no. 12, pp. 968–971, Dec. 2007.
[70] S. M. Bird and C. B. Fairweather, “Military fatality rates (by cause) in Afghanistan
and Iraq: a measure of hostilities,” vol. 36, no. 4, pp. 841–846, 2007.
[71] R. Gunaratna, “Suicide terrorism: a global threat,” Janes Intelligence Review,
vol. 12, no. 4, pp. 52–55, 2000.
[72] H. Killers, “The global landmine crisis,” in 1998 report to the US Congress, US
Department of State, Bureau of Political-Military Affairs, US GPO, 1998.
[73] A. Soroush, F. Falahati, M. Zargar, M. Soroush, S. Khateri, A. Khaji et al.,
“Amputations due to landmine and unexploded ordnances in post-war Iran,”
Arch Iran Med, vol. 11, no. 6, pp. 595–7, 2008.
[74] R. Miles, A. Dogariu, and J. Michael, “Bringing bombs to light,” IEEE Spectrum,
vol. 49, no. 2, pp. 38–43, 2012.
[75] B. N. Nelsen, “A forward looking infrared sensor for landmine detection that
incorporates a novel method of identifying regions of interest and a fuzzy
inference system,” in Fuzzy Information Processing Society, 1999. NAFIPS.
18th International Conference of the North American, 1999, pp. 408–412.
[76] J. A. MacDonald and J. Lockwood, Alternatives for landmine detection.
corporation, 2003, no. 1608.
[77] M. Skolnik, Introduction to radar systems.
2001.
RAND
New York, NY: McGraw-hill New York,
[78] D. J. Daniels, Ground penetrating radar. London, UK: IET, 2004.
[79] S. Vitebskiy, L. Carin, M. Ressler, and F. Le, “Ultra-wideband, short-pulse
ground-penetrating radar: simulation and measurement,” IEEE Transactions
on Geoscience and Remote Sensing, vol. 35, no. 3, pp. 762–772, 1997.
114
[80] L. Peters, J. Daniels, and J. Young, “Ground penetrating radar as a subsurface
environmental sensing tool,” Proceedings of the IEEE, vol. 82, no. 12, pp.
1802–1822, 1994.
[81] E. Byron, Radar Principles, Technology, Applications.
Jersey: PTR Prentice hall, 1993.
Englewood Cliffs, New
[82] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight synthetic aperture
radar: signal processing algorithms. Norwood, MA: Artech House Inc., 1995.
[83] M. Soumekh, Synthetic aperture radar signal processing.
Sons, 1999.
[84] M. A. Richards, Fundamentals of radar signal processing.
Education, 2005.
New york, NY: Wiley &
Tata McGraw-Hill
[85] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela,
“MIMO radar: an idea whose time has come,” in Proceedings of the IEEE, Radar
Conference, 2004., April 2004, pp. 71–78.
[86] J. Li and P. Stoica, “MIMO Radar with Colocated Antennas,” IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 106–114, Sept. 2007.
[87] D. Bliss and K. Forsythe, “Multiple-input multiple-output (MIMO) radar and
imaging: degrees of freedom and resolution,” in Conference Record of the
Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003.,
vol. 1, Nov. 2003, pp. 54 – 59.
[88] O. Ojowu, Y. Wu, J. Li, and L. Nguyen, “SIRE: a MIMO radar for landmine/IED
detection,” in SPIE Defense, Security, and Sensing. International Society for
Optics and Photonics, 2013.
[89] H. V. Trees, Detection, Estimation, and Modulation Theory, Optimum Array
Processing. New York, NY: John Wiley & Sons, 2004.
[90] L. A. Gorham and L. J. Moore, “SAR image formation toolbox for MATLAB,” in
SPIE Defense, Security, and Sensing. International Society for Optics and
Photonics, 2010.
[91] J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer
baselines,” Astophys, J. Suppl., pp. 417–426, Aug. 1974.
[92] E. Karpat, “CLEAN technique to classify and detect objects in subsurface
imaging,” International Journal of Antennas and Propagation, 2012.
[93] P. Stoica, P. Babu, and J. Li, “New Method of Sparse Parameter Estimation in
Separable Models and Its Use for Spectral Analysis of Irregularly Sampled Data,”
IEEE Transactions on Signal Processing,, vol. 59, no. 1, pp. 35–47, Jan. 2011.
115
[94] M. Ressler, L. Nguyen, F. Koenig, D. Wong, and G. Smith, “The Army Research
Laboratory (ARL) Synchronous Impulse Reconstruction (SIRE) Forward-Looking
Radar.” ser. Proc. SPIE, Unmanned Systems Technology IX, vol. 6561, 2007.
[95] L. Nguyen and R. Innocenti, “Suppression of sidelobes and noise in airborne
SAR imagery using the Recursive Sidelobe Minimization technique,” in Radar
Conference, 2010 IEEE, May 2010, pp. 522 –525.
[96] J. Ender and J. Klare, “System architectures and algorithms for radar imaging by
MIMO-SAR,” in IEEE Radar Conference, May 2009, pp. 1 –6.
[97] J. Li and P. Stoica, MIMO Radar Signal Processing.
Sons, 2009.
Hoboken, NJ: John Wiley &
[98] A. Belloni, V. Chernozhukov, and L. Wang, “Square-root lasso: pivotal recovery of
sparse signals via conic programming,” Biometrika, vol. 98, no. 4, pp. 791–806,
2011.
[99] P. Babu and P. Stoica, “Connection between SPICE and Square-Root LASSO for
sparse parameter estimation,” Signal Processing, vol. 95, pp. 10–14, February
2014.
[100] J. Ling, J. Li, P. Stoica, and M. Datum, “Probing waveforms and adaptive receivers
for active sonar,” The Journal of the Acoustical Society of America, vol. 129, pp.
3640–3651, 2011.
[101] P. Babu and P. Stoica, “A combined linear programming-maximum likelihood
approach to radial velocity data analysis for extrasolar planet detection,” in IEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP),
2011, pp. 4352–4355.
[102] P. Stoica and P. Babu, “Algebraic Derivation of Elfving Theorem on Optimal
Experiment Design and Some Connections With Sparse Estimation,” IEEE Signal
Processing Letters, vol. 17, no. 8, pp. 743–745, 2010.
[103] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the
Royal Statistical Society. Series B (Methodological), pp. 267–288, 1996.
116
BIOGRAPHICAL SKETCH
Ode Ojowu Jr. was born in Zaria, Nigeria. He came to the United States in 2001
to pursue an academic career. He received a Bachelor of Arts in physics from Grinnell
College in 2005, as well as a Bachelor of Science and Master of Science in electrical
engineering from Washington University in 2007. He is currently with the Spectral
Analysis Lab (SAL) supervised by Prof. Jian Li at the University of Florida. He will
receive a Doctor of Philosophy in electrical engineering from the University of Florida in
the Fall of 2013.
His general research interest lies in the field of signals and systems with a focus on
data-adaptive spectral estimation techniques, array signal processing and radar signal
processing.
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