High-Resolution Time-Frequency Methods' Performance
Analysis
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Shafi, Imran et al. “High-Resolution Time-Frequency Methods'
Performance Analysis.” EURASIP Journal on Advances in Signal
Processing 2010 (2010): 1-7.
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http://dx.doi.org/10.1155/2010/806043
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European Association for Signal Processing / Hindawi
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Final published version
Accessed
Thu May 26 06:22:55 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/61007
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Hindawi Publishing Corporation
EURASIP Journal on Advances in Signal Processing
Volume 2010, Article ID 806043, 7 pages
doi:10.1155/2010/806043
Research Article
High-Resolution Time-Frequency Methods’ Performance Analysis
Imran Shafi,1 Jamil Ahmad,1 Syed Ismail Shah,1 Ataul Aziz Ikram,1 Adnan Ahmad Khan,2
Sajid Bashir,3 and Faisal Mahmood Kashif4
1 Information
and Computing Department, Iqra University Islamabad Campus, Sector H-9, Islamabad 44000, Pakistan
of Telecommunication Engineering, NUST, Islamabad 44000, Pakistan
3 Computer Engineering Department, Centre for Advanced Studies in Engineering, Islamabad 44000, Pakistan
4 Laboratory for Electromagnetic and Electronic Systems (LEES), MIT Cambridge, Cambridge, MA 02139-4307, USA
2 College
Correspondence should be addressed to Imran Shafi, imran.shafi@gmail.com
Received 31 December 2009; Accepted 6 July 2010
Academic Editor: L. F. Chaparro
Copyright © 2010 Imran Shafi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This work evaluates the performance of high-resolution quadratic time-frequency distributions (TFDs) including the ones
obtained by the reassignment method, the optimal radially Gaussian kernel method, the t-f autoregressive moving-average spectral
estimation method and the neural network-based method. The approaches are rigorously compared to each other using several
objective measures. Experimental results show that the neural network-based TFDs are better in concentration and resolution
performance based on various examples.
1. Introduction
The nonstationary signals are very common in nature
or are generated synthetically for practical applications
like analysis, filtering, modeling, suppression, cancellation,
equalization, modulation, detection, estimation, coding,
and synchronization. The study of the varying spectral
content of such signals is possible through two-dimensional
functions of TFDs that depict the temporal and spectral
contents simultaneously [1]. Different types of TFDs are
limited in scope due to multiple reasons, for example, low
concentration along the individual components, blurring of
autocomponents, cross terms (CTs) appearance in between
autocomponents, and poor resolution. These shortcomings
result into inaccurate analysis of nonstationary signals.
Half way in this decade, there is an enormous amount
of work towards achieving high concentration along the
individual components and to enhance the ease of identifying the closely spaced components in the TFDs. The
aim is to correctly interpret the fundamental nature of the
nonstationary signals under analysis in the time-frequency
(TF) domain [2]. There are three open trends that make this
task inherently more complex, that is, (i) concentration and
resolution tradeoff, (ii) application-specific environment,
and (iii) objective assessment of TFDs [1–3]. Tradeoff
between concentration and CTs’ removal is a classical problem. The concepts of concentration and resolution are used
synonymously in literature whereas for multicomponent
signals this is not necessarily the case, and a difference
is required to be established. High signal concentration is
desired but in the analysis of multicomponent signals resolution is more important. Moreover, different applications
have different preferences and requirements to the TFDs.
In general, the choice of a TFD in a particular situation
depends on many factors such as the relevance of properties
satisfied by TFDs, the computational cost and speed of the
TFD, and the tradeoff in using the TFD. Also selection
of the most suited TFD to analyze the given signal is not
straightforward. Generally the common practice have been
the visual comparison of all plots with the choice of most
appealing one. However, this selection is generally difficult
and subjective.
The estimation of signal information and complexity
in the TF plane is quite challenging. The themes which
inspire new measures for estimation of signal information
and complexity in the TF plane, include the CTs’ suppression,
concentration and resolution of autocomponents, and the
ability to correctly distinguish closely spaced components.
2
Efficient concentration and resolution measurement can
provide a quantitative criterion to evaluate performances of
different distributions. They conform closely to the notion of
complexity that is used when visually inspecting TF images
[1, 3].
This paper presents the performance evaluation of high
resolution TFDs that include well-known quadratic TFDs
and other established and proven high resolution and interesting TF techniques like the reassignment method (RAM)
[4], the optimal radially Gaussian kernel method (OKM) [5],
the TF autoregressive moving-average spectral estimation
method (TSE) [6], and the neural network-based method
(NTFD) [7, 8]. The methods are rigorously compared to each
other using several objective measures discussed in literature
complementing the initial results reported in [9].
2. Experimental Results and Discussion
Various objective criteria are used for objective evaluation
that include the ratio of norms-based measures [10], Shannon & Rényi entropy measures [11, 12], normalized Rényi
entropy measure [13], Stankovi measure [14], and Boashash
and Sucic performance measures [15]. Both real life and
synthetic signals are considered to validate the experimental
results.
2.1. Bat Echolocation Chirps Signal. The spectrogram of bat
echolocation chirp sound is shown in Figure 1(a), that is
blurred and difficult to interpret. The results are obtained
using the TSE, RAM, OKM, and NTFD, shown in Figure 1.
The TF autoregressive moving-average estimation models for nonstationary random processes are shown to be a
TF symmetric reformulation of time-varying autoregressive
moving-average models using a Fourier basis [6]. This
reformulation is physically intuitive because it uses time
delays and frequency shifts to model the nonstationary
dynamics of a process. The TSE models are parsimonious
for the practically relevant class of processes with a limited
TF correlation structure. The simulation result depicted in
Figure 1(c) demonstrates that the TSE is able to improve on
the Wigner Distribution (WVD) in terms of resolution and
absence of CTs; on the other hand, the TF localization of the
components deviates slightly from that in the WVD.
The reassignment method enhances the resolution in
time and frequency of the classical spectrogram by assigning
to each data point a new TF coordinate that better reflects
the distribution of energy in the analyzed signal [4]. The
reassigned spectrogram for the bat echolocation chirps signal
is shown in Figure 1(d). The evaluation by various objective
criteria is presented in graphical form at Figure 6 criterions
comparative graphs. The analysis indicates that the results
of the reassignment and the neural network-based methods
are proportionate. However, the NTFD’s performance is
superior based on Ljubisa measure.
On the other hand, the optimal radially Gaussian
kernel TFD method proposes a signal-dependent kernel
that changes shape for each signal to offer improved TF
representation for a large class of signals based on quantitative optimization criteria [5]. The result by this method
EURASIP Journal on Advances in Signal Processing
is depicted in Figure 1(e) that does not recover all the
components missing useful information about the signal.
Also the objective assessment by various criteria does not
point to much significance in achieving energy concentration
along the individual components.
2.2. Synthetic Signals. Four synthetic signals of different
natures are used to identify the best TFD and evaluate their
performance. The first test case consists of two intersecting
sinusoidal frequency modulated (FM) components, given as
x1 (n) = e− jπ((5/2)−0.1 sin(2πn/N))n + e jπ((5/2)−0.1 sin(2πn/N))n . (1)
The spectrogram of the signal is shown in Figure 2(a),
referred to as test image 1 (TI 1).
The second synthetic signal contains two sets of nonparallel, nonintersecting chirps, expressed as
x2 (n) = e jπ(n/6N)n + e jπ(1+(n/6N))n + e− jπ(n/6N)n
(2)
+ e− jπ(1+(n/6N))n .
The spectrogram of the signal is shown in Figure 3(a),
referred to as test image 2 (TI 2).
The third one is a three-component signal containing a
sinusoidal FM component intersecting two crossing chirps,
given as
x3 (n) = e jπ((5/2)−0.1 sin(2πn/N))n + e jπ(n/6N)n + e jπ((1/3)−(n/6N))n .
(3)
The spectrogram of the signal is shown in Figure 4(a),
referred as test image 3 (TI 3). The frequency separation
is low enough and just avoids intersection between the
two components (sinusoidal FM and chirp components) in
between 150–200 Hz near 0.5 sec. This is an ideal signal to
confirm the TFDs’ effectiveness in deblurring closely spaced
components and check its performance at the intersections.
Yet another test case is adopted from Boashash [15] to
compare the TFDs’ performance at the middle of the signal
duration interval by the Boashash’s performance measures.
The authors in [15] have found the modified B distribution
(β = 0.01) as the best performing TFD for this particular
signal at the middle. The signal is defined as
x4 (n) = cos 2π 0.15t + 0.0004t 2
+ cos 2π 0.2t + 0.0004t 2 .
(4)
The spectrogram of the signal is shown in Figure 5(a),
referred to as test image 4 (TI 4).
The synthetic test TFDs are processed by the neural
network-based method and the results are shown in Figures
2(b)–5(b), which demonstrate high resolution and good
concentration along the IFs of individual components.
However, instead of relying solely on the visual inspection of
the TF plots, it is mandatory to quantify the quality of TFDs
by the objective methods. The quantitative comparison can
be drawn from Figure 6 (in Figure 6, the abbreviations not
mentioned earlier are the spectrogram (spec), Zhao-AtlasMarks distribution (ZAMD), Margenau-Hill distribution
EURASIP Journal on Advances in Signal Processing
3
NTFD
Spectrogram
160
180
140
160
140
100
120
80
100
80
60
60
40
40
20
20
50
100
150
200
Frequency (Hz)
250
50
300
100
150
200
250
Frequency (Hz)
(a)
300
(b)
TFD obtained by the TSE
Reassigned spectrogram
(c)
(d)
TFD obtained by the OKM
175
150
Time (s)
Time (s)
120
125
100
75
50
25
0
50
100
150
200
250
Frequency (Hz)
300
350
400
(e)
Figure 1: TFDs of the multicomponent bat echolocation chirp signal by various high resolution t-f methods.
350
4
EURASIP Journal on Advances in Signal Processing
Resultant image after processing through NN
2.5
2.5
2
2
Time (s)
Time (s)
Test spectrogram TED for synthetic test signal
1.5
1.5
1
1
0.5
0.5
0
0
100
200
300
400 500 600
Frequency (Hz)
700
800
0
900
0
100
200
300
(a)
400 500 600
Frequency (Hz)
700
800
900
(b)
Figure 2: TFDs of a synthetic signal consisting of two sinusoidal FM components intersecting each other. (a) Spectrogram (TI 2) [Hamm,
L = 90], and (b) NTFD.
Resultant image after processing through NN
1.2
1
1
0.8
0.8
Time (s)
Time (s)
Test spectrogram TED for synthetic test signal
1.2
0.6
0.6
0.4
0.4
0.2
0.2
0
0
100
200
300
400 500 600
Frequency (Hz)
700
800
900
(a)
0
0
100
200
300
400 500 600
Frequency (Hz)
700
800
900
(b)
Figure 3: TFDs of a synthetic signal consisting of two-sets of non-parallel, non-intersecting chirps. (a) Spectrogram (TI 3) [Hamm, L = 90],
and (b) NTFD.
(MHD), and Choi-Williams distribution (CWD)), where
these measures are plotted individually for all the test
images. On scrutinizing these comparative graphs, the NTFD
qualifies the best quality TFD for different measures.
Boashash’s performance measures for concentration
and resolution are computationally expensive because they
require calculations at various time instants. We take a
slice at t = 64 of the signal and compute the normalized
instantaneous resolution and concentration performance
measures Ri (64) and Cn (64). A TFD that, at a given time
instant, has the largest positive value (close to 1) of the
measure Ri is the TFD with the best resolution performance
at that time instant for the signal under consideration. The
NTFD gives the largest value of Ri at time t = 64 in Figure 7
and hence is selected as the best performing TFD of this
signal at t = 64.
On similar lines, we have compared the TFDs’ concentration performance at the middle of signal duration interval.
EURASIP Journal on Advances in Signal Processing
5
Resultant image after processing through NN
2.5
2.5
2
2
Time (s)
Time (s)
Test spectrogram
1.5
1.5
1
1
0.5
0.5
0
0
100
200
300
400 500 600
Frequency (Hz)
700
800
0
900
0
100
200
(a)
300
400 500 600
Frequency (Hz)
700
800
900
(b)
Figure 4: TFDs of a synthetic signal consisting of crossing chirps and a sinusoidal FM component. (a) Spectrogram (TI 4) [Hamm, L = 90],
and (b) NTFD.
NNTFD for synthetic signal 4
120
120
100
100
80
80
Time (s)
Time (s)
Test spectrogram for closely spaced components
60
60
40
40
20
20
0.05 0.1
0.15 0.2 0.25 0.3 0.35
Frequency (Hz)
0.4
0.45
0.5
0.05 0.1
(a)
0.15 0.2 0.25 0.3 0.35
Frequency (Hz)
0.4
0.45
0.5
(b)
Figure 5: TFDs of a signal consisting of two linear FM components with frequencies increasing from 0.15 to 0.25 Hz and 0.2 to 0.3 Hz,
respectively. (a) Spectrogram and (b) NTFD.
A TFD is considered to have the best energy concentration
for a given multicomponent signal if for each signal component, it yields the smallest instantaneous bandwidth relative
to component IF (Vi (t)/ fi (t)) and the smallest side lobe
magnitude relative to main lobe magnitude (AS (t)/AM (t)).
The results plotted in Figure 7 comparative graphs for
Boashash concentration resolution measure indicate that
the NTFD gives the smallest values of C1,2 (t) at t = 64
and hence is selected as the best concentrated TFD at time
t = 64.
3. Conclusion
The objective criteria provide a quantitative framework
for TFDs’ goodness instead of relying solely on the visual
measure of goodness of their plots. Experimental results
6
EURASIP Journal on Advances in Signal Processing
Shannon entropy measure
16
14
12
10
Type of TFDs
Type of TFDs
RAM
OKM
NTFD
TSE
OKM
RAM
NTFD
TSE
Spec
10−5
OKM
RAM
NTFD
TSE
CWD
MHD
ZAMD
8
10−4
CWD
10
MHD
12
10−3
ZAMD
Numerical value of the measure
14
WVD
Ratio of norm based measure
10−2
WVD
Volume normalised Renyi entropy measure
Spec
Numerical value of the measure
(b)
16
6
CWD
Type of TFDs
(a)
18
MHD
Spec
ZAMD
8
6
OKM
RAM
NTFD
TSE
CWD
MHD
ZAMD
WVD
101
18
WVD
Numerical value of the measure
102
100
Renyi entropy measure
20
Spec
Numerical value of the measure
103
Type of TFDs
(c)
(d)
Ljubisa measure
106
105
104
OKM
RAM
NTFD
TSE
CWD
MHD
ZAMD
102
WVD
103
Spec
Numerical value of the measure
107
Type of TFDs
TI 3
TI 4
TI 1
TI 2
(e)
Figure 6: Comparison plots, numerical values of criterion versus method employed, for the test images 1–4, (a) The Shannon entropy
measure, (b) Rényi entropy measure, (c) Volume normalized Rényi entropy measure, (d) Ratio of norm based measure, and (e) Ljubisa
measure.
EURASIP Journal on Advances in Signal Processing
0.9
7
Modified Boashash concentration measure (Cn )
Numerical value of the measure
0.8
0.7
0.6
0.5
0.4
0.3
0.2
NTFD
Modified B
BJD
CWD
ZAMD
WVD
0
Spec
0.1
Type of TFDs
C1
C3
(a)
1
Boashash normalised instantaneous resolution measure (Ri )
Numerical value of the measure
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
NTFD
Modified B
BJD
CWD
ZAMD
WVD
0.5
Spec
0.55
Type of TFDs
(b)
Figure 7: Comparison plots for the Boashash’s TFD performance
measures versus different types of TFDs, (a) The proposed modified
Boashash’s concentration measure (Cn (64)), and (b) The Boashash’s
normalized instantaneous resolution measure (Ri ).
demonstrate the effectiveness of the neural network-based
approach against well-known and established high resolution TF methods including some popular distributions
known for their high CTs suppression and energy concentration in the TF domain.
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[3] B. Boashash, Time-Frequency Signal Analysis and Processing: A
Comprehensive Reference, Elsevier, Oxford, UK, 2003.
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