AN ABSTRACT OF THE DISSERTATION OF
Eduardo Xavier Alban for the degree of Doctor of Philosophy in
Electrical and Computer Engineering presented on May 26, 2011.
Title: Immune Radio Architecture for Platform Interference
Abstract approved:
Mario E. Magaña
The decrease of switching times accompanied by the corresponding increase of
clock speeds and data rates, contributes to improve the overall system
computational performance. At the same time, they also affect wireless
communications due to an increment in the emissions of electromagnetic
radiation on the radio bands.
In this dissertation we have studied the sources and the performance in terms of
the bit error rate (BER) noise generated at the receiver by the electromagnetic
emissions of computing platforms.
Statistical analysis of platform noise measurements done on the 2.4 GHz band
have shown that this type of noise is Non-Gaussian, namely, it is K-distributed.
Motivated by this fact, a statistical model that is consistent with the physical
characteristics of the process rather than only on data fitting has been derived.
The accuracy of the noise model is of paramount importance, as it allows us to
design radios that are immune to it. Also, we have derived a new method for
estimating the parameters of the K-distribution when a limited number of
samples are available. The method is based on an approximation of the Bessel
function of the second kind that reduces the complexity of the estimation
formulas in comparison with the ones based on the maximum likelihood criterion.
The method presented is shown to have better performance in comparison with
existing methods of the same complexity yielding a lower mean squared error
when the number of samples used for the estimation is relatively low.
Finally, the mitigation of the platform noise has also been accomplished.
c
Copyright by Eduardo Xavier Alban
May 26, 2011
All Rights Reserved
Immune Radio Architecture for Platform Interference
by
Eduardo Xavier Alban
A DISSERTATION
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented May 26, 2011
Commencement June 2012
Doctor of Philosophy dissertation of Eduardo Xavier Alban presented on
May 26, 2011.
APPROVED:
Major Professor, representing Electrical and Computer Engineering
Director of the School of Electrical Engineering and Computer Science
Dean of the Graduate School
I understand that my dissertation will become part of the permanent collection
of Oregon State University libraries. My signature below authorizes release of my
dissertation to any reader upon request.
Eduardo Xavier Alban, Author
ACKNOWLEDGEMENTS
One of the pleasure of working in my Ph.D. is meeting wonderful people,
knowing them, sharing memories, learning from, and especially getting support
from them. I particularly grateful to Dr. Mario Magaña for his patience,
motivation, support and from whom I have received guidance during all my time
at Oregon State University. I would also like to thank Harry Skinner, Kevin
Slattery, and Intel for their support on this research.
During my time at Corvallis, I have met a lot of wonderful people through school
and as a member of ALAS, that I cannot named one by one, to whom I am
gratefully thankful for their friendship and all the great moments. Especially, my
gratitude goes to Panupat Poocharoen for his friendship and advice.
For me family is always important, I am greatly thankful to Calvin Hughes and
his wonderful family, who always make me feel welcome and part of their family.
And then there is my family, whom I can never thank enough; my parents:
Eduardo and Marianna, my sister: Gabi, and my brother in law: Dario. Even
though they have been far, their support and love have contribute greatly to the
completion of this thesis.
Finally there is Anne, whom without her support this thesis would have never
been a reality.
TABLE OF CONTENTS
Page
1 Introduction
1
2 Platform Noise Sources
12
2.1
Broadband Signals . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Spectrum for Sequences with even rise and fall times . . . .
2.1.2 Spectrum for sequences with uneven rise and fall times . . .
18
20
22
2.2
Narrowband Signals . . . . . . . . . . . . . . . . . . . .
2.2.1 Clocks . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Periodic Signals of Short Duration . . . . . . . .
2.2.3 Spread Spectrum Clock . . . . . . . . . . . . . .
2.2.4 Spread Spectrum Clock (SSC) with a Triangular
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42
47
2.3
Platform Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
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Profile
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3 Statistical Analysis of Broadband Noise
58
3.1
Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4 Platform Interference in OFDM
83
4.1
Communication Model . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.2
OFDM System Description . . . . . . . . . . . . . . . . . . . . . . .
84
4.3
Platform Interference in OFDM . . . . . . . . . . . . . . . . . . . .
4.3.1 Analysis of the impact of SSC on an OFDM radio . . . . . .
4.3.2 Platform Broadband Noise . . . . . . . . . . . . . . . . . . .
89
91
95
4.4
Bit Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.1 Narrowband Interference Bit Error Rate . . . . . . . . . . . 100
4.4.2 Broadband Noise Bit Error Rate . . . . . . . . . . . . . . . . 103
5 Noise Immune Radio Architecture
112
5.1
The Narrowband Case . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2
The Broadband Case . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.1 Parameter Estimation Methods . . . . . . . . . . . . . . . . 122
TABLE OF CONTENTS (Continued)
Page
5.2.2
5.2.3
5.2.4
5.2.5
5.2.6
5.2.7
Maximum Likelihood Estimation
Cramer-Rao lower bound . . . . .
New estimation method . . . . . .
Estimator bias . . . . . . . . . . .
Simulation Results . . . . . . . .
Broadband Noise Mitigation . . .
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127
130
131
134
135
144
6 Conclusions
155
Appendices
158
A
Pseudo Random Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B
Distribution of N (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
C
Estimation of the Bit Error Rate for OFDM under Narrowband Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
C.1
BER in BPSK Modulation . . . . . . . . . . . . . . . . . . . . . . . 173
C.2
BER in QAM Modulation . . . . . . . . . . . . . . . . . . . . . . . 175
C.3
Narrowband Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Bibliography
179
LIST OF FIGURES
Figure
Page
1.1
EM emissions from electronic devices . . . . . . . . . . . . . . . . .
4
2.1
Electromagnetic emissions . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Platform Layout
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Platform Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4
Random data spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.5
Broadband Noise Spectrum for raise time distinct than raise time .
28
2.6
M-stage shift register . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.7
Autocorrelation of a Pseudo Noise Random Sequence . . . . . . . .
31
2.8
Pseudo Noise Random Sequence . . . . . . . . . . . . . . . . . . . .
32
2.9
PseudoNoise Random Sequence Spectrum . . . . . . . . . . . . . .
33
2.10 PseudoNoise Random Sequence Spectrum overlapped by a sinc2
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.11 One period of a clock signal . . . . . . . . . . . . . . . . . . . . . .
35
2.12 100 MHz Clock, 50% duty cycle . . . . . . . . . . . . . . . . . . . .
37
2.13 100 MHz Clock, 50% duty cycle . . . . . . . . . . . . . . . . . . . .
37
2.14 Simulated display signal . . . . . . . . . . . . . . . . . . . . . . . .
38
2.15 Spectrum of the stream signal [1 1 1 0 1 0 0 1 0 1]
. . . . . . . . .
39
2.16 Spread Spectrum Frequency Profile . . . . . . . . . . . . . . . . . .
43
2.17 Spread Spectrum Frequency Kiss Shape Profile
. . . . . . . . . . .
44
2.18 Clock 100 MHz 50% duty . . . . . . . . . . . . . . . . . . . . . . .
45
2.19 Spread Spectrum Clocking spectrum . . . . . . . . . . . . . . . . .
45
2.20 Spectrum of a SSC with triangular profile . . . . . . . . . . . . . .
52
2.21 SSC spectrogram using Blackman-Tukey . . . . . . . . . . . . . . .
53
LIST OF FIGURES (Continued)
Figure
Page
2.22 Comparison of the instantaneous spectrum of a SSC signal with a
regular clock signal . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.23 Quasi-Band Interference Model . . . . . . . . . . . . . . . . . . . .
55
2.24 Platform noise
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.1
Platform noise spectrum on a 20 MHz channel on the ISM band . .
61
3.2
Horn antenna set up . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.3
Horn antenna gain response (source: ETS-LINDGREN Horn antenna Model 3164-03) . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.4
FCC class B digital devices limits at a distance of 3 meters . . . . .
63
3.5
Narrowband antenna set up . . . . . . . . . . . . . . . . . . . . . .
64
3.6
Time domain data from platform running 3DMark06 . . . . . . . .
65
3.7
Time domain baseband data from unit under test . . . . . . . . . .
65
3.8
Histogram plots of the noise IQ components . . . . . . . . . . . . .
66
3.9
CDF plots of the noise IQ components . . . . . . . . . . . . . . . .
67
3.10 Laptop Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.11 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.12 Fitting of the empirical CDF of the noise with K-distribution CDF
80
4.1
Communication Model . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.2
OFDM baseband spectrum . . . . . . . . . . . . . . . . . . . . . . .
85
4.3
OFDM Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.4
OFDM Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.5
Platform Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.6
Optional caption for list of figures . . . . . . . . . . . . . . . . . . .
90
4.7
OFDM Spectrum with Narrowband Interference . . . . . . . . . . .
91
LIST OF FIGURES (Continued)
Figure
Page
4.8
OFDM Spectrum with Broadband Interference . . . . . . . . . . . .
92
4.9
Optional caption for list of figures . . . . . . . . . . . . . . . . . . .
93
4.10 BER comparison for broadband noise with σ 2 = 1 × 10−6 , ν = 1
and b = 0.0005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.11 BER comparison for broadband noise with σ 2 = 4 × 10−6 , ν = 1
and b = 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.12 BER comparison for broadband noise with σ 2 = 6 × 10−6 , ν = 1.5
and b = 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.13 BER comparison for broadband noise with σ 2 = 8 × 10−6 , ν = 0.5
and b = 0.002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.14 BER of 4-QAM OFDM with narrow band interference . . . . . . . 104
4.15 Bit error rate for BPSK and 16-QAM modulation with K-distributed
noise ν = −0.99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.16 Bit error rate for BPSK and 16-QAM modulation with K-distributed
noise ν = −0.45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1
OFDM avoidance strategy . . . . . . . . . . . . . . . . . . . . . . . 113
5.2
OFDM receiver architecture with platform interference mitigation . 115
5.3
Platform interference mitigation algorithm block diagram . . . . . . 116
5.4
Adaptive NLMS filter . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.5
Mitigation of a narrowband interferer, SIR = −6 dB, using threshold decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.6
Mitigation of a narrowband interferer, SIR = −12 dB, using threshold decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7
Results using threshold decision with 2 interferers at SIR = −3.5
dB each . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.8
Parameter estimation (MoM) ν = 0.5 b = 0.02 . . . . . . . . . . . . 125
LIST OF FIGURES (Continued)
Figure
Page
5.9
MSE estimators comparison: ν = 0.5 b = 0.02 . . . . . . . . . . . . 137
5.10 MSE estimators comparison: ν = −0.2 b = 0.8
5.11 MSE estimator comparison: N = 32 b = 0.02
. . . . . . . . . . . 138
. . . . . . . . . . . . 140
5.12 MSE estimator comparison: N = 64, b = 0.02 . . . . . . . . . . . . 141
5.13 MSE estimator comparison: N = 128, b = 0.02
. . . . . . . . . . . 142
5.14 MSE estimator comparison: N = 256, b = 0.02
. . . . . . . . . . . 143
5.15 Generated noise with K-distribution: ν = 1 b = 0.0005 . . . . . . . 148
5.16 Generated noise with K-distribution: ν = 1 b = 0.001 . . . . . . . . 148
5.17 Generated noise with K-distribution: ν = 0.5 b = 0.1 . . . . . . . . 149
5.18 K-noise mitigation in 4-QAM OFDM systems ν = 2.5 and b = 1 . . 153
5.19 K-noise mitigation in BPSK OFDM systems ν = 2.5 and b = 1 . . . 153
5.20 K-noise mitigation in BPSK OFDM systems ν = 1.5 and b = 2 . . . 154
LIST OF TABLES
Table
Page
1.1
FCC emission for Class B digital devices at 3 meters . . . . . . . .
4
3.1
χ2 goodness-of-fit test for normality . . . . . . . . . . . . . . . . . .
68
3.2
Kolmogorov-Smirnov test results for α = 0.05 . . . . . . . . . . . .
82
5.1
OFDM parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
DEDICATION
I want to dedicate this thesis to my parents, Eduardo and Marianita, and to
Anne for all their love, patience and support in reaching my goals.
Chapter 1 – Introduction
Electronic devices with wireless capability have become popular and are now part
of the daily activities of people. In fact, most of today’s personal electronic devices
come with wireless communications capability as a built-in feature. Laptops, smart
phones, blackberrys, PDAs, iPads, iPhones, e-books readers and tablets such as the
Kindle of Amazon or the Apple’s iPad come with wireless capability. These devices
not only have the capability to work with one of the current wireless standards,
but some also have incorporated the capability to work with more than one. This
latter assertion has become the tendency of newer devices that come with more
computational capabilities. Protocol standards such as 802.11(g,n), Bluetooth
or Wimax have become an integral part of these devices. The importance and
wide-spread use of wireless devices on the daily lives of Americans is reflected on
a survey made by the Pew Research Centers Internet & American Life Project,
which established on the report [1] that 56% of Americans connect wireless to
internet. Another interesting datum, also on the same report, is the increase on
the use of mobile devices from 24% on 2007 to 32% on 2009, which is a clear sign
of the growing trend that exists on the adoption of these devices.
The demand of services has not only increased in terms of quantity, but also
in terms of quality. Users demand lighter devices with higher computational capabilities and longer battery life, among others. This is translated into a search of
2
higher efficiencies on the entire system. Consequently, there is a continuous effort
by engineers for redesigning computing devices to make more and better services
available on them. The focus of this dissertation is on the radio, as the component that actually makes wireless communications possible, where the redesign has
to consider architectures that provide greater sensitivity, lower operation power,
higher efficiency power amplifiers, and noise immunity among others. In dealing with the new specifications and their corresponding requirements, there is the
natural progression of improving current technologies by optimizing and adapting
existent architectures, or the development of new architectures.
The main purpose of this dissertation is the characterization of the interference at the radio caused by the electromagnetic emissions from the elements of a
computing device and the design of mitigation techniques that improves the radio communications. In a radio, the limitations could be of physical nature, e.g.
frequencies of operation or power consumption, or more specific for us they could
appear as a consequence of noise that was not entirely considered before due to
its negligible effect on the device performance. The conditions under which the
radio operates now have changed and those issues that were neglected before have
started to affect the radio communications and it is now imperative to design new
methods and architectures that fulfill the requirements that present and future
wireless devices have.
Electrical and electronic devices generate electromagnetic radiation that can
interfere with the well functioning of any nearby electronic device, known as Electromagnetic Interference (EMI). Because of this, regulatory institutions around
3
the world have imposed strict radiation emissions limits to all digital devices on
the different frequency bands that they operate, e.g. Code of Federal Regulations
title 47 Part 15, 47CFR15 [2] in the US, that designers need to follow. A scheme of
different devices emitting electromagnetic radiation is shown in figure 1.1, where
the electromagnetic fields are represented by dotted lines. As it is shown, the coexistence and normal operation of the two laptops and the mobile device, separated
by a relatively small distance, are possible because the electromagnetic power emanating from each of them does not affect the others. Thanks to the regulations,
different platforms can coexist together and work as they were designed to function. But the regulations were established for addressing the interference that a
device could cause to other devices but not to itself. For a long time, this was not a
problem since the emitting radiation of the electronic devices components did not
affect the radio operation frequencies or its impact fell within the assumptions of
radio designers. But that is not the case anymore, multiple electronic components
have been constantly increasing their frequency of operation and now the range
of the radiation frequency at their operating frequencies and their harmonics have
started to become harmful for the radio. For example, in the United States, the
Federal Communications Commission (FCC) requires that Class B digital devices
such as personal computers (for a more detailed description of this class see [2])
which electronics operate over 9 kilohertz (kHz) must not have, at a distance of 3
meters, radiated emissions greater than the ones shown on table 1.1. These limiting levels protect devices located at distances farther than 3 meters, but nowadays
radios are completely integrated with other radiating electronic devices such as
4
Frequency of emission
(Mhz)
30-88
88-216
216-960
Above 960
Field strength
(µV/m)
100
150
200
500
Field strength
(dBµV/m)
40
43.5
46
54
Table 1.1: FCC emission for Class B digital devices at 3 meters
clocks, high speed interconnects, graphic cards, CPUs which maximum dimension
does not exceed 30 centimeters, e.g. iPods, Pda’s, smart Phones, etc. It becomes
evident that an integrated radio receiver that is not perfectly shielded will suffer
the interference from a noise source placed one meter or less away from a noise
source when we compare the noise level indicated on table 1.1 with the sensitivities
of a radio, around 80 dBm (complying with the IEEE802.11-2007 standard at 2.4
GHz and 1 Mb/s [3]) or 27 dBµV.
Figure 1.1: EM emissions from electronic devices
5
By platform we refer to all the elements, such as clocks, graphic cards, hard
drives, memories, that are part of electronic devices having computing capabilities,
e.g. laptops. Then, it follows that we refer to the term platform interference as the
interference caused at the radio by the noise generated by the different platform elements. Even though, shielding of the radiation sources could be considered as the
natural solution to mitigate the interference, it is expensive and physically impossible to have a perfect enclosure of the radiating elements, because the connections
that each element has cause leakage that cannot be avoided.
Previous work has contributed to the characterization of potential sources of
electromagnetic emissions from elements of a computer platform, [4] [5] [6] [7],
and in finding different solutions to the interference that they can cause to other
systems. The most common signals found on a computing platform are periodic signals, e.g. clocks and synchronization sequences, and two-state random
sequences, e.g. signals produced by hard drive activity. The difficulty in characterizing this noise lies in the fact that inside a computing platform we not only
have one radiating source for each of these signals but multiples ones with different parameters (base clocks running at 100 MHz and 133.33 MHz, high speed
interconnects working a different transmission rates) radiating from different positions across the entire platform over different periods of time. This presents a
challenge for the radio since it does not see each individual signal separately but
as a combination (which in the best of cases would be linear) of them. Deterministic signals such as clocks can easily be distinguished since their spectra are
well defined, e.g. periodic signals consist of frequency components at multiples
6
of the operating frequency. For random sequences, the combination of streams
at different rates traveling across the platform gives a spectrum that in the radio
band of interest could not have any distinguishable feature, moreover it changes
with time. Nevertheless, we have to be careful to make oversimplifications of the
problem by only using a distribution that fits the data without a connection to
the physical process or by applying theoretical concepts that result in a distribution that does not match the actual data. The use of the central limit theorem
is an example of the latter one, where the considerable number of emitters could
lead to the wrong conclusion that the distribution seen by the radio is Gaussian.
Experimental results ([6, 8, 9]) show the disagreement with this assertion.
The behavior of periodic signals such as clocks and their spectra is a well known
result that can be derived using Fourier analysis. However, pressed by the regulations on reduction of electromagnetic emissions, researchers at Lexmark developed
a technique where the clock is modulated in order to reduce the instantaneous
amplitude of it. The technique, known as Spread Spectrum Clock (SSC), was
made public by Hardin et. al. in [10] and has become a standard technique used
nowadays by designers of computing platforms. The potential of interference that
the SSC can cause was already recognized by the same authors in [11]. This is
confirmed by Aoki et al. in [12], where the authors use an anechoic chamber to
measure the effects of SSC in the wireless LAN frequency bands using the throughput as a metric of performance. The book by Slattery et al. ([6]) is a comprehensive
study of the interference caused by different elements in a platform. Hardin and
other authors conducted a study of SSC interference on digital TV receivers in [13].
7
Measurements of the impact of Spread Spectrum Clock in the WLAN frequency
band (2.4 Ghz) are found in a paper by Ogata et al. in [14]. Also, Shimizu et.
al. in [15] do a numerical evaluation of the Bit Error Rate for the OFDM-based
standard IEEE802.11a without error correction under SSC interference. In [16]
the authors present a theoretical evaluation of the BER and extend their study
by including the effect of convolutional coding on the numerical evaluations of the
performance. A more analytical study of the Spread Spectrum Clock technique
has been done by Matsumoto et al. in [17] and in [18]. SSC interference in OFDMbased communications systems has been studied by Matsumoto et al. in [19] for
WLANs and in [20], [21] for Ultra Wide Band (UWB) systems.
Signals modulated by random sequences have been already characterized by
Titsworth et. al. in [22], where analytical expressions of their spectra are derived.
Their potential as sources of interference is pointed out by Clayton in an updated
version of its landmark book [5], also Slattery et. al. in [6]. The little attention
that this type of interference has received for radio designers is reflected in the
lack of literature available. This could be explained either by the fact that before
it affected none of the operating frequencies of radios or by making the wrong
assumption of being of Gaussian Nature, or both. The most notable attempts to
establish a statistical model for the emissions of random sequences that matches the
physical characteristics of noise are the works of Middleton [23] and Jakeman [24],
with the former one being the most widely known. A recent work by Bhatti et. al.
in [25] characterizes the noise using Middleton models. Also, non-Gaussian stable
distributions have been proposed to model the noise [26]. Nassar et al. in [8] use the
8
alpha-stable distribution, as an approximation of the Middleton models, to model
the noise taken from lab measurements. Some authors have seen as a disadvantage
the fact that stable distributions have infinite variance and used different models
that describe the experimental data. That is the case of the authors in [9] who have
tried to model the experimental data using nonlinear dynamical models based on
Chaos theory. Also, models based on Bernoulli-Gaussian, Poisson-Gaussian and
Hidden Markov chains are good approaches to describe the noise [27].
It has been established that there are three approaches to mitigate electromagnetic interference, as it is point out by Clayton in [5]:
• Mitigation at the emissions source
• Making the coupling path inefficient
• Making the receiver immune
The third item is the focus of this dissertation. It is important to point out that radios have been designed as independent systems separate from the platform where
they reside and not as a whole. Therefore, designers have to consider platform interference on their designs, specially with the advent of smaller devices with higher
levels of integration.
Efficient wireless communications have been possible thanks to the good statistical characterization of the noise generated at the radio receiver. This noise has
been well modeled because of its stationarity and Gaussian nature, but platform
interference does not fit this characterization. Platform noise is of non Gaussian
nature and is non-stationary [6]. Consequently, the radio architectures that are
9
now employed in electronic devices do not perform as efficiently as they do in a
standalone manner because they have not considered that platform interference
could occur at all. The consequence of interference is a reduction of radio performance as it is manifested in the increase of the bit error rate (BER) and the
decrease in throughput. A new radio design has to be designed that takes into
account not only the noise generated by the receiver RF front-end but also those
produced by the platform. Specifically, we focus our attention on the design of
radio receivers of communications systems that implement a multi-carrier modulation technique known as OFDM, where OFDM stands for Orthogonal FrequencyDivision Multiplexing, that we have chosen in this dissertation because of its robustness and specially its wide spread adoption in current wireless protocols such
as IEEE802.11a/n/g [28], IEEE802.16 (aka WIMAX), digital television known as
DVB-T and future cellular technology such as Long Term Evolution (LTE).
Different techniques have been proposed to mitigate platform interference. Mitigation techniques can be applied either in the transmitter, in the receiver, at the
noise sources, or in all of them. The effectiveness of them depends on the correct
identification of the platform noise that affects the radio. In [29], the authors use
sensing windows to evaluate if the communication is affected by interference. The
testing is performed in the physical layer or media access control layer by comparing received signal strength with a threshold or identifying an incorrect preamble.
The authors in [21] propose the use of the Viterbi decoding scheme using Channel
State Information (CSI). Skinner et al. in [30] propose a modification of the modulation parameters of the clock to avoid interference. Guo et al. in [31] present the
10
use of per-tone estimation of the noise. The obtained information is used for soft
decoding in the Viterbi decoder and, in the case when MIMO is used, for the selection of an antenna that is not severely affected by the noise. Nassar et al. in [8]
evaluate the performance of Wiener filtering, coherent Bayes detection and myriad
filtering for noise following an alpha stable distribution. In spite of the many solutions that have been proposed, there are still open questions on how to deal with
platform noise sources in the radio which leaves a wide room for improvement.
In most cases, people in the electromagnetic compatibility and the radio design
communities have been pursuing their goals separately rather than in a complementary fashion to effectively address platform noise. The major contribution of
this work is the development of a new methodology to characterize platform noise
with mathematical rigor. The impact of the interference on the performance of the
radio is also analyzed. Moreover, the noise characteristics are exploited to apply
different mitigation techniques.
We can summarize the main contributions of this dissertation as follows. A rigorous characterization of the noise that affects the radio communications generated
by computing elements is presented. It is shown that the random components of
the noise follow the K-distribution on its double-sided version. Also, a new method
for estimating the parameters of the K-distribution is presented. The method performs better than other methods of the same complexity when a limited number
of samples are available. The estimation of the parameters is the first step in
characterizing the noise, however it does not give a complete characterization over
time. Therefore, we derive a state-space model of the broadband noise. Also the
11
impact of K-noise in radio communications is quantified using the Bit Error Rate
metric. Finally, we developed mitigation algorithms for narrowband and broadband platform noise in OFDM systems by means of the NLMS and an extended
Kalman filter, respectively.
This dissertation is organized as follows: in chapter 2 we present an analysis
of some of the main signals found on the different interconnects of the platform
that constitute the main sources of platform noise interference. The analysis is
presented in both time and frequency domains. In chapter 3 we present an analysis
of measurement data of noise collected from laptops. A characterization of the
noise in terms of its statistical behavior is also presented. In chapter 4, the impact
of platform interference on the performance of the system in terms of the Bit Error
Rate (BER) for Orthogonal Frequency-Division Multiplexing (OFDM) systems is
presented. In chapter 5 some radio architectures and techniques to estimate and
mitigate the interference in OFDM are presented. Finally, chapter 6 presents some
conclusions and proposes some future work.
12
Chapter 2 – Platform Noise Sources
The purpose of this chapter and the following is to establish a theoretical framework
that leads us to characterize the platform noise that affects the radio communications.
All the components that are found in electronic devices produce some sorts of
electromagnetic emissions (figure 2.1) as a consequence of charges moving through
the elements of a circuitry that turn into radiators under certain conditions [32].
Although shielding, absorbers and other electromagnetic mitigation techniques
are used across the platform to comply with the FCC emission power regulations,
they are mainly used to limit the emissions of the equipment to avoid causing
interference to nearby devices and to protect it from emissions of external sources
but not to protect all the internal components from the electromagnetic emissions
of each other. The emissions can affect the normal operation of them due to the
interference that they could cause. This type of interference is known as Platform
Interference, [6], and in this chapter we focus on an analytical and numerical
characterization of it.
Figure 2.2 shows a diagram representation of an INTEL architecture motherboard [33] used in a typical nowadays computing platform. From the diagram it is
easy to identify the different elements of the motherboard which can be grouped
together according to the core chip where they are directly connected. The first
13
Figure 2.1: Electromagnetic emissions
one correspond to the central procesing unit (CPU) alone. The second group corresponds to the Northbridge chip and the elements connected to it such as the
DRAM modules (in this case a DDR2 technology as it is shown on the diagram)
and the graphics card. The third group is composed by the elements connected to
the Southbridge chip such as USB ports, serial ports, LAN ports, the power management and clock chips among others. The interconnections between the different
components are represented by lines with a double arrow representing the two-way
communication. The organization and location of the different elements on the
platform are not only important for good performance of a computer system but
also for the identification of the possible sources of interference, such as high speed
interconnections (PCIe), clocks and others.
The increase of processing capacity of computing platforms has been translated
into higher speeds of operation of the digital systems inside them. Consequently,
digital signals traveling across the platform at higher speeds than previous tech-
14
Figure 2.2: Platform Layout
15
nologies which in turn are the main sources of electromagnetic emissions [5]. The
emissions could affect directly the radio receiver on the platform either because of
its main frequency of operation or its harmonics or both overlap with the operating frequency bands of the radio. From all the elements that can be identified
on figure 2.2, the ones that are the main sources of the electromagnetic emissions
that could affect the radio, as it is noted in [32] [5] [6], are
• Clocks
• High-speed interconnects
• Cables
• Power delivery networks
• LCD panels
• Memory subsystems
• Processing components
We point out that it is of our interest to study the signals that are generated and
used by the components previously listed and not the physical characteristics of
each component, owing to the fact that in this dissertation we address the effect
of the interference at the radio receiver where a new architecture that mitigates
the noise needs to be found. We notice that a change in the fabrication process or
the emergence of newer techniques, such as shielding, and materials, such a change
16
from copper to other material for the interconnects, could reduce interference considerably. But, to some extent we want a robust design and hence consider the
characteristics of the signals for a worst case scenario.
Figure 2.3 presents a snapshot of the noise that we want to characterize. The
noise was generated by a laptop running the benchmark software 3DMark and it
was captured using an oscilloscope with the laptop placed inside a shielding room.
Platform Noise Spectrum
−60
−70
−80
Amplitude dBs
−90
−100
−110
−120
−130
−140
−150
1.5
2
2.5
3
Frequency (GHz)
Figure 2.3: Platform Noise
Now, according to the spectral characteristics of the signals interfering with
the radio receiver and the bandwidth that they occupy in comparison to the radio
receiver bandwidth, Slattery et. al in [6] have classified the interference in three
categories:
1. Narrowband
17
2. Quasi-band
3. Broadband
Clocks, synchronization signals and modulated clocks are examples of the first
two categories, with the third one being the only signal that is considered to be a
Quasi-band interference. And since a modulated clock cannot be analyzed without
doing it first for a regular clock, both are analyzed under the Narrowband category.
In a later chapter it will be made evident the usefulness of this approach because
some mitigation techniques can be used for the interference cause by both.
The elements in these categories not only share the characteristic of being
generated on the platform but also of being of non-Gaussian nature. Thus, we
define the set of signals generated from electromagnetic emissions at the platform
to be I ⊂ Ω, where Ω is the universe of all the physical signals that are present in
the transmission of data. In other words, the elements that cause interference to
the radio that we are analyzing here are the elements of the set
I= i:i∈
/ N (u, σ 2 ), i cause interference at the receiver
where N (u, σ) refers to a Gaussian distribution with any mean u and variance σ 2 .
In the following sections we present the analysis of the most common signals
that are found on computing platforms. The signals have been grouped together
according to their frequency domain characteristics. This will become clear as we
move forward through this chapter. First, signals that generate a broadband spectrum are analyzed. Under this classification fall signals that are random whose
18
spectra occupy an entire range of frequencies that expands beyond the radio spectrum. We also analyze the signals whose spectra are considered to be narrow
since they do not have spectral components over the entire frequency range, with
periodic signals falling in this category.
For the rest of the chapter we assume that the bandwidth of a radio receiver,
to which the bandwidth of the interfering signals are compared, is given by B.
2.1 Broadband Signals
Broadband Interference comprises of all the signals whose bandwidths are wider
than the bandwidth of the radio receiver and cause interference to it. That is, for
interfering signals with bandwidth B ′ , those which are considered broadband have
B < B′.
A typical signal with a spectrum signature that causes broadband noise is a
digital data signal. Data signals are the majority of signals present on computing
platforms and they usually travel through the high-speed interconnects, such as
PCI express or DDR buses, which connect the different components of the computer. These signals are primarily found at the processor, the graphics card, and
the hard drives. These signals are of random nature and are generated all over
the platform. They are mainly two-state signals where the difference between its
rise and fall times is small enough as for both times to be considered equal in the
19
analysis. The power spectra of random digital signals can be computed using the
results in [22], also found in [34]. For convenience we follow the notation found
in the latter one. Then, the power spectrum of a sequence produced from values
from one of the a symbols {E1 , ...Ea } that are modulated by a set of waveforms
si (t) is given by [34]
a
∞
n 2 n
1 X X
p i Si
Φ(f ) = 2
δ f−
T n=−∞ i=1
T T
a
a X
a
X
2
1X
′
′
2
pi |Si (f )| + Re
pi Si ∗ (f )Sk′ (f )Pik (f )
+
T i=1
T
i=1 k=1
!
(2.1)
where the first part constitutes the discrete part of the spectrum and the rest is
the continuous part of the spectrum. Si′ (f ) is the Fourier transform of s′i (t) defined
as
s′i (t)
= si (t) −
a
X
pk sk (t),
(2.2)
k=1
and Pik (f ) is obtained as
Pik (f ) =
∞
X
pik (n)e−j2πnf T ,
(2.3)
n=1
where pik (n) is the (i,k)th element of the state transition matrix P n . Then, it
follows that pik (n) is the probability that the k-signal is transmitted after the
i-signal at the nth step.
First, we compute the power power spectrum of sequences with even rise and
fall times and then for sequences with uneven ones.
20
2.1.1 Spectrum for Sequences with even rise and fall times
Let us assume we have a digital sequence whose only possible values are A and
−A with even rise and fall times, then the power spectrum of a random sequence
is computed as follows:
Since the rise and fall times are even, we notice that there exist only two modulating sequences which are the negative of each other s1 (t) = −s2 (t). Consequently,
S1 (f ) = −S2 (f ). Now, whenever s1 (t) is a rectangular waveform we have that
s1 (t) =



A 0 ≤ t < T


0
,
otherwise
where its corresponding Fourier transform is given by
S1 (f ) = AT e−jπf T
sin(πf T )
.
πf T
(2.4)
Since we have only two possible symbols and we consider that the sequence does
not contain memory, that is the current outcome in a sequence is independent of
the previous outcome, then we define the probability that s1 (t) occurs as p. It
follows that the probability that s2 (t) occurs is given by 1 − p. Therefore, the
transition matrix is given by


p 1 − p
P =
.
p 1−p
21
Then, by substituting equation (2.1) with the expressions of S1 (f ), S2 (f ) and the
entries of P we have that
Φ(f ) =
∞
n
n 2 n
1 X pS
δ
f
−
−
(1
−
p)S
1
1
T 2 n=−∞
T
T
T
2
2
p(1 − p)|S1 (f )|2 + p(1 − p)|S1 (f )|2
T
T
∞
(2p − 1)2 X n 2 n 4p(1 − p)
=
|S1 (f )|2 .
+
δ f−
S1
2
T
T
T
T
n=−∞
+
(2.5)
Now, if the outcomes of the sequence are equiprobable, p = 1/2, the power spectrum of a two-level random sequence is given by
1
|S1 (f )|2 ,
T
2
sin(πf T )
2
,
=A T
πf t
Φ(f ) =
= A2 T sinc2 (f T ),
(2.6)
where 1/T = fbr corresponds to the data rate of the random signals in bits per
second (bps), A2 T ∈ R+ is the power spectrum density amplitude and the sinc
function is given by
sinc(x) =



1,
x=0
.


 sin(πx) , x 6= 0
πx
The normalized power spectrum calculated for any T is shown in figure 2.4. The
spectrum has nulls at the frequencies corresponding to the data rate of the signal
and its multiples.
22
FH f L dBs
f
1
2
3
4
5
6
-10
-20
-30
-40
-50
Figure 2.4: Random data spectrum
2.1.2 Spectrum for sequences with uneven rise and fall times
Whenever a sequence has transition times that differ from one another, it produces
a distinctive spectrum signature. The power spectrum of such signals is computed
here.
A two-state signal has two transition times. The one between the low state to
the high state is known as rise time, tr , and the transition from the high state to
the low state is known as fall time, tf .
The power spectrum calculation of a random waveform with states A and −A
and with transition times tr and tf now ensues. The waveforms that constitute
the building blocks of two-state random signals with transient times tr and tf are
23
described by
s1 (t) =



−A +


A,
2A
t,
tr
0 < tr
,
tr < t < T
s2 (t) = +A
0 ≤ t ≤ T,



A − 2A t, 0 < tf
tf
,
s3 (t) =


−A,
tf < t < T
s4 (t) = −A
0 ≤ t ≤ T.
It clearly follows that s2 (t) = −s4 (t), then S2 (f ) = −S4 (f ). Since we already
computed the Fourier transform of s2 (t) on the previous derivation, we only have
to compute the Fourier transforms of s1 (t) and s3 (t).
Then, the Fourier transforms of s1 (t) and s3 (t) are given by
tr
Z T
2A
−i2πf t
−A +
S1 (f ) =
t e
dt +
Ae−i2πf t dt,
tr
0
tr
A −1 + e−2if πtr + if πtr + ie−2if πT f πtr
,
=
2f 2 π 2 tr
Z
and
tf
Z T
2A
−i2πf t
S3 (f ) =
A−
t e
dt −
Ae−i2πf t dt,
t
f
0
tf
−2if πtf
−2if πT
A 1−e
− if πtf − ie
f πtf
.
=
2f 2 π 2 tf
Z
24
It is clear that when tr = tf , S3 (f ) = −S1 (f ).
Now, in a random sequence, the signals just described occur for the following
states:
• s1 (t) appears when the pair (-A,A) occurs
• s2 (t) appears when the pair (A,A) occurs
• s3 (t) appears when the pair (A,-A) occurs
• s4 (t) appears when the pair (-A,-A) occurs
Thus, the probabilities for each of the transitions from one state to another are
given by
p1 = qp,
p2 = p2 ,
p3 = pq,
p4 = q 2 ,
where p is the probability that a high state, A, appears in a sequence and q is
the probability of a low state, −A. We can now proceed to compute the power
spectrum using equation (2.1) and the results just obtained. Then, the discrete
component of the spectrum whenever the probability of each state is the same is
25
given by
4
∞
n 2 n
1 X X
p i Si
,
Φd (f ) = 2
δ f−
T n=−∞ i=1
T T
∞
n
n
n 2 p2 X n n
= 2
+ S2
+ S3
+ S4
.
δ f−
S1
T n=−∞
T
T
T
T
T
Now, since S2
n
T
= −S4
n
T
and
A −1 + e−2if πtr + if πtr + ie−2if πT f πtr
S1 (f ) + S3 (f ) =
2f 2 π 2 tr
−2if πtf
A 1−e
− if πtf − ie−2if πT f πtf
,
+
2f 2 π 2 tf
A 1 − e−2if πtr (tr − tf )
,
=
2f 2 π 2 tr tf
we have that
2
n
∞
p2 X A 1 − e−2i T πtr (tr − tf ) n
Φd (f ) = 2
.
δ
f
−
2
T n=−∞ T
2 Tn π 2 tr tf
(2.7)
This result clearly shows the fact that for tr = tf the discrete spectrum vanishes,
so the discrete spectrum is significant only when there is a difference between the
rise and the fall times.
Now, in order to compute the continuous spectrum we need to find the equivalent signals given by
s′i (t) = si (t) −
4
X
k=1
pk sk (t),
26
where
4
X
k=1
pk sk (t) = pqs1 (t) + p2 s2 (t) + pqs3 (t) − q 2 s2 (t).
Hence,
Si′ (f )
= Si (f ) −
4
X
pk sk (t)
k=1
= Si (f ) − pqS1 (f ) − (p2 − q 2 )S2 (f ) − pqS3 (f ).
With equiprobable probabilities p = q, it follows that
Si′ (f )
Ap2 tf 1 − e−2if πtr − tr 1 − e−2if πtf
.
= Si (f ) −
2f 2 π 2 tf tr
The transition matrix P that characterizes the sequence for q = 1 − p is given by

0

0

P =
p


p
and since

 −(−1 + p)p

 −(−1 + p)p

P2 = 
 −(−1 + p)p


−(−1 + p)p
p 1−p

0 

p 1−p
0 

,
0
0
1 − p


0
0
1−p
p
2
p2
p2
p2
2

−(−1 + p)p (−1 + p) 

−(−1 + p)p (−1 + p)2 

,
2 
−(−1 + p)p (−1 + p) 

2
−(−1 + p)p (−1 + p)
27
it can easily be shown that P n = P 2 for n > 1. Hence, the Z transform which is
used here for notation convenience of the elements of the matrix can be expressed
as
Pkl (z) =
∞
X
pkl (n)z −1
n=1
(2)
= pkl z −1 + pik
z −2
,
1 − z −1
where the matrix P (z) is given by

(1−p)pz 2
1−z
pz +
p2 z 2
1−z
(1 − p)z +
(1−p)pz 2
1−z
(1−p)2 z 2
1−z




 (1−p)pz2
2
2 z2
2 z2
(1−p)pz
(1−p)
p


pz + 1−z (1 − p)z + 1−z


1−z
1−z
P (z) = 
,
2
2
2
2
2
2
pz + (1−p)pz
(1−p) z 
(1−p)pz
p z
(1
−
p)z
+

1−z
1−z
1−z
1−z 


2
2
2
2
(1−p)2 z 2
(1−p)pz
p z
pz + (1−p)pz
(1
−
p)z
+
1−z
1−z
1−z
1−z
where we notice that Pik (e−j2πf T ) = Pik (z)|z=e−j2πf T .
The continuous part of the spectrum is computed using equation (2.1) with the
corresponding values obtained from the previous expressions, i.e.
!
4
4 X
4
X
2
1X
′∗
′
2
′
pi |Si (f )| + Re
pi Si (f )Sk (f )Pik (f ) .
Φc (f ) =
T i=1
T
i=1 k=1
(2.8)
Figure 2.5 shows the power spectrum of a random signal with uneven transition
times for p = 1/2 that gives p1 = p2 = p3 = p4 = 1/4, A = 1 and T = 1. The
uneven times produce spikes at the fundamental and at the harmonics that do not
appear on signals that have equal transition times.
28
FH f L dBs
40
30
20
.
.
.
.
.
.
10
f
1
2
3
4
5
6
0
Figure 2.5: Broadband Noise Spectrum for raise time distinct than raise time
Random signals can also be well modeled by using pseudo random bit streams
(PRBS). PRBS are deterministic sequences generated using shift registers, as the
one shown in figure 2.6, that have characteristics of random signals. The use
of these signals is justified since, by being deterministic, they allow us to have
a uniform testing scenario for different platforms. Moreover, not all the signals
generated at the computing platform are completely random. Some are deterministic sequences that are used for synchronization between the different components.
Also, the generation of pseudorandom sequences at the software or hardware level
is done by using some kind of deterministic algorithm to generate pseudo random
sequences and more important, as it becomes clear later, this allows us to have a
spectrum generator in the radio that produces signals of similar characteristics to
the broadband interference to help to mitigate its effect.
Although PBRS sequences are signals that are generated in a deterministic way,
the algorithm used produces sequences with properties that approximate quite
29
output
1
2
3
4
...
N-1
N
Figure 2.6: M-stage shift register
well the ones with random numbers. They are binary sequences where each bit
has an equal probability of being either zero or one, regardless of the outcome
of the preceding bit. We investigate the characteristics of these signals by first
generating a sequence of ones and zeros using a random number generator. Then,
we generate the PBRS signal by using pulse amplitude modulation on the sequence.
We consider the case when the duration of the pulse in both states is the same but
we notice that the transition time between them could be different, as it happens
in practical devices where the rise and fall times differ. This produces a unique
spectrum signature with peaks at frequencies where only nulls exist, as it was
previously derived.
Now, in a maximum length PRBS sequence that takes on only two possible
values, one or zero, the length of the sequence is given by M = 2n − 1, with n
being the maximum number of shift registers. The number of ones in one period
are found to be
(M +1)
2
= 2n−1 and the number of zeros are
(M −1)
2
= 2n−1 − 1. Then,
30
the probability of obtaining a one at any given time is given by
2n−1
,
2n − 1
1
=
1 ,
2 − 2n−1
p=
but when n is sufficiently large it follows that the PRBS behaves more like a
random signal, i.e.
p≈
1
2
for n large.
Moreover, the PRBS autocorrelation is given by
N −1
1 X
R̂ibr (k, ∆) =
ibr (k)ibr (k − ∆).
N k=1
Then , it follows that (see Appendix A)
R̂ibr (∆) =



 a2 ,
∆ = 0, ±M, ±2M, . . .
.
(2.9)


− a2 , otherwise
M
Figure 2.7 shows the autocorrelation of a PRBS sequence with M = 10 and T = 1.
The autocorrelation consists of thin triangles functions centered at zero and at
multiples of M, which resembles a train of Kronecker delta functions.
31
RHΤL
1.5
1.0
0.5
Τ
-20
10
-10
20
-0.5
Figure 2.7: Autocorrelation of a Pseudo Noise Random Sequence
Now, for M large we have that
lim R̂ibr (∆) =
M →∞



 a2 , ∆ = 0


0,
,
otherwise
where it is clear that its envelope has a triangular shape. Then, from any standard
Fourier transform table we find that the power spectrum of the envelope is equal
to
2
Φ̂ibr (w) = a T
sin(wT /2)
wT /2
2
.
(2.10)
A result, it coincides with the one already computed before for a modulated random
sequence. This confirms the validity of using PRBS signals to study broadband
interference caused by data signals.
Figure 2.8 shows a simulated PBRS signal with an amplitude of 1 volt, pulse
32
duration T equal to 1 nanosecond, rise time tr of 100 picoseconds and a fall time
tf equal to 100 picoseconds. The total simulation time of the PBRS signal is
200 nanoseconds. The frequency characteristics of a PBRS signal is analyzed by
PRBS Signal
Rise Time = Fall Time = 100ps
1
Amplitude (V)
0.8
0.6
0.4
0.2
0
−0.2
0
50
100
Time (ns)
150
200
Figure 2.8: Pseudo Noise Random Sequence
computing the Fourier Transform of its autocorrelation function by means of the
Fast Fourier Transform (FFT). Figure 2.9 shows the computed spectrum for the
signal in figure 2.8.
Figure 2.10 shows the agreement between the analytical spectrum and the one
computed by means of the DFT from the simulation previously described. The
nulls are localized at the fundamental frequency and its harmonics, as it can be
inferred from the analytical result.
We clearly notice that the spectrum is spread over a wider range of frequencies
than the bandwidth of most of the wireless standards such as that of 802.11n, 40
MHz, and that of 802.16, which goes up to 20 MHz, which are small compared
33
Spectrum of PBRS signal
Rise Time = 100 ps Fall Time = 100ps
0
−10
Power in dbm
−20
−30
−40
−50
−60
−70
−80
0
0.5
1
1.5
2
2.5
3
Frequency (GHz)
3.5
4
4.5
5
Figure 2.9: PseudoNoise Random Sequence Spectrum
Spectrum of PBRS signal
Rise Time = 100 ps Fall Time = 100ps
0
PBRS spectrum
|Asinc(f)|
−10
Power in dbm
−20
−30
−40
−50
−60
−70
−80
0
0.5
1
1.5
2
2.5
3
Frequency (GHz)
3.5
4
4.5
5
Figure 2.10: PseudoNoise Random Sequence Spectrum overlapped by a sinc2 function
34
with the one shown in the figure whose signature spectrum is over their entire
frequency ranges. This would not be a problem if its power level were below the
noise floor level. However, since the radiated power of these signals could turn out
to be above the noise floor and on the sensitivity range of the radio, they directly
affect the incoming signals at the receiver decreasing the performance of it.
2.2 Narrowband Signals
Narrowband interference is produced by unwanted signals whose center frequencies
lie inside the frequency range of the radio of interest and with bandwidths that are
smaller than the one of the radio receiver. That is, signals of bandwidth B ∗ where
B ∗ << B,
and with center frequency fn ∈ FR , where FR is the set of operating frequencies of
the radio receiver.
In a platform, the narrowband interference signals come in the frequency domain as narrow sharp peaks that usually have amplitudes higher than the amplitudes of the broadband interference.
In this section, we analyze the main sources of this type of interference such as
clocks and display sequences whose spectra have narrow sharp peaks at multiples
of their fundamental frequency. These peaks are potential sources of interference
if they lie on the frequency bands where the wireless protocols of our interest are
35
implemented.
2.2.1 Clocks
The clock is one of the most important elements of any platform and it is used
to synchronize the different processes that run on a computing platform such as
writing/reading data to/from RAM, refreshing contents and performing calculations inside a computing platform. The clock has been increasing its speed from a
few megahertz to somewhere in the gigahertz range with a trend that is expected
to continue in the years to come. On the one hand, this rate increment has been
beneficial for computer performance. On the other hand, it turns out to be harmful
for the radio receiver due to its spectral characteristics as we describe next.
Amplitude
1.0
0.8
0.6
0.4
0.2
t1
t2
2. ´ 10-9
t3
4. ´ 10-9
t5
t4
6. ´ 10-9
8. ´ 10-9
timeHsL
1. ´ 10-8
Figure 2.11: One period of a clock signal
Figure 2.11 shows one cycle of a clock with a period of ten nanoseconds. In the
graph, the different transition times are marked with dashed lines and labeled as
36
t1 , t2 , t3 , t4 and t5 . Thus, the rise time is given by tr = t2 − t1 and the fall time by
tf = t4 − t3 . In close form, the clock signal over one period can be described as
v(t) =

t−t

− (t −t 1)/5


2
1
A
1
−
e
, t1 ≤ t ≤ t2







A,
t2 ≤ t ≤ t3
t−t

− (t −t 3)/5


4
3
),
A(e







0,
.
(2.11)
t3 ≤ t ≤ t4
otherwise
where A is the amplitude of the clock which in the case of figure 2.11 is equal to
one.
In order to investigate the spectrum that a clock signal produces, we apply the
FFT to a simulated clock signal sampled at 10 picoseconds. Figure 2.12 shows 100
nanoseconds of a 100 MHz clock with duty cycle of fifty percent, a rise time equal
to 500 picoseconds and fall time equal to 500 picoseconds. The computed spectrum
of the clock is shown in figure 2.13. The figure shows the signature spectrum of a
clock signal with sharp peaks at the fundamental frequency and the corresponding
harmonics as we expected due to its periodicity. We also notice that even the high
harmonics have high power peaks which can lie in the same frequency range of the
radio receiver disturbing its operation.
37
100 MHz Clock
50% duty cycle, rise time = fall time = 500ps
Amplitude (V)
1
0.8
0.6
0.4
0.2
0
−0.2
0
20
40
60
Time (ns)
80
100
Figure 2.12: 100 MHz Clock, 50% duty cycle
100 MHz Clock, 50% duty cycle
rise time: 500ps ; fall time: 500ps
0
Power in dBs
−10
−20
−30
−40
−50
−60
0
0.5
1
1.5
2
2.5
3
Frequency (GHz)
3.5
4
4.5
Figure 2.13: 100 MHz Clock, 50% duty cycle
5
38
2.2.2 Periodic Signals of Short Duration
Clocks are not the only periodic signals that can be found on a computing platform.
Other periodic signals are also used to synchronize and control a particular function
inside the computing platforms. These are deterministic signals composed of short
sequences of ones and zeros which are transmitted periodically. Examples of these
are display sequences which are used for power management to refresh the screen
and to send wake-up or sleep signals to it.
Display signals are streams of ones and zeros that are transmitted periodically
for control and synchronization of video digital displays. Given their short lengths
and periodicity, their behaviors in the frequency domain are similar to that of a
clock signal as we now verify.
Display Signal
1
Amplitude (V)
0.8
0.6
0.4
0.2
0
−0.2
0
1
2
3
4
5
Time (ns)
6
7
8
9
10
Figure 2.14: Simulated display signal
We simulate a ten symbol stream, [1 1 1 0 1 0 0 1 0 1], with a symbol duration
39
of 1 nanosecond (ns), fall time of 100 picoseconds (ps), rise time of 100 picoseconds
(ps) and a maximum amplitude of one volt, shown on figure 2.14. The spectrum
of the signal is obtained by means of the FFT and the result is shown on figure
2.15.
The spectrum signature is similar to that of a clock signal since the display
signal produces sharp peaks in the frequency domain but in this case there is a
greater number of harmonics, not only the odd harmonics.
Display signal spectrum
−10
Power in dBs
−20
−30
−40
−50
−60
0
0.5
1
1.5
2
2.5
3
Frequency (GHz)
3.5
4
4.5
5
Figure 2.15: Spectrum of the stream signal [1 1 1 0 1 0 0 1 0 1]
2.2.2.1 Baseband Representation
It is a standard practice in communication theory to use baseband representations
of the transmitted and received signals because of their mathematical convenience
in analyzing and simulating different communication systems. Therefore, we pro-
40
ceed to find a baseband representation of the noise generated by the elements of
the platform that affect the radio receiver. The derivation of this representation is
as follows:
Let ι(t) be the noise generated by the elements of the platform, fc be the center
frequency of the receiver radio and B be the operating bandwidth. Then, it follows
that the noise affecting the incoming signal is given by
ι′ (t) = ι(t) ∗ hB (t),
where hB (t) is a bandpass filter located at the receiver front-end. Without loss of
generality, we assume an ideal bandpass filter given by
HB (f ) =



1, fc − B/2 ≤ |f | ≤ fc + B/2
.


0, otherwise
2.2.2.2 Clock baseband representation
Let v(t) be a clock with levels 1 and -1 generated at the platform, and be the
T = 2L then it can be described by the following Fourier series
4
v(t) =
π
∞
X
1
sin
k
k=1,3,5,...
2kπt
T
.
The action of the filter v ′ (t) = v(t) ∗ hB (t) at the receiver from end will pass only
the elements whose frequencies are inside range of operation of the radio. Let F
41
be the range of operation of the radio receiver, then F = {f : |f − fc | ≤ B/2}.
Now, let the indicator function 1F be given by
1F (x) =



1, x ∈ F


0, x ∈
/ F,
then for any function v(t) with a Fourier series given by v(t) =
P∞
n=−∞ cn e
jnt
, we
have an equivalent representation given by v(t) ∗ HB (t) = v(t) × 1F (n). Thus,
4
v (t) =
π
′
=
4
π
=
4
π
=
4
π
∞
X
2kπt
× 1F (k/T )
T
∞
X
2kπt
1
Im exp j
× 1F (k/T )
k
T
k=1,3,5,...
∞
X
1
2kπt
exp (−j2πkfc t) exp (j2πkfc t) × 1F (k/T )
Im exp j
k
T
k=1,3,5,...
∞
X
1
2kπt
Im exp j
− j2πfc t exp (j2πfc t) × 1F (k/T )
k
T
k=1,3,5,...
1
sin
k
k=1,3,5,...
Now, the low-pass representation of a the bandpass signal is given by [34]
v ′ (t) = Re[v l (t)ej2πfc t ]
and since Im{c} = Re{−jc} the lowpass representation of v ′ (t) is given by
4
v (t) = −j
π
l
∞
X
1
k
exp j2π
− fc t × 1F (k/T )
k
T
k=1,3,5,...
(2.12)
42
where fc is the center frequency of the bandpass filter, B is its bandwidth and 1/T
is the rate of the clock.
2.2.3 Spread Spectrum Clock
In order to reduce the electromagnetic emissions of a clock, engineers have found
that modulating the clock reduces the peak amplitude of the emissions. This
technique, as we describe below, produces a reduction on the maximum emitted
power by spreading the signal over a wide range of frequencies. This could cause
an interfering signal with a bandwidth B ⋆ that turns out to be bigger, smaller or
equal than the bandwidth of the radio receiver.
B⋆ ≤ B
or
B ⋆ > B.
This explains why the authors of [6] have designated the interference that the clock
modulation can cause as Quasiband Interference.
The clock modulation method is better known as Spread Spectrum Clocking
(SSC) [10] since the spectrum of the modulated signal occupies a wider range of
frequencies than the original one. The modulation of the clock has been done using
a profile which can be of linear or non linear nature. In general, the frequency of
the clock can be modulated by a signal V (t) such that the instantaneous frequency
of the mth harmonic is given by [10]
f (t) = mfc + ρmfc V (t),
43
where fc is the nominal frequency of the clock, 0 < ρ << 1 is the maximum
frequency deviation and the modulation frequency is given by fm = ρmfc . Thus,
it follows that the phase is given by
θ = 2π
Z
t
f (τ )dτ,
−∞
= 2πmfc t + 2πmfc ρ
Z
t
V (τ )dτ.
−∞
Examples of a linear modulation profile is the triangular profile. This is described
analytically for one modulation period over time as
V (t) = 1 −
4|t|
Tssc
− Tssc ≤ t ≤ Tssc ,
(2.13)
where Tssc is the modulation period. Figure 2.16 shows a plot of the modulation
profile just described.
frequency
fc
H1 - ∆L fc
0.5
1
fm
fm
Figure 2.16: Spread Spectrum Frequency Profile
time
44
Another well-known modulation profile is the kiss profile, named after its shape
shown in figure 2.17, which is given by
M (t) =



0.45t3 + 0.55t,
for −1 ≤ t ≤ 1
(2.14)


−0.45(t − 2)3 − 0.55(t − 2), for 1 ≤ t ≤ 3.
Normalized frequency
1.0
0.5
1
-1
2
3
time
-0.5
-1.0
Figure 2.17: Spread Spectrum Frequency Kiss Shape Profile
We now proceed to compare the spectrum of an unmodulated clock with a
modulated one. On the one hand, we have figure 2.18 showing a 100 MHz clock
with a duty cycle of 50 percent, rise time of 20 picoseconds and fall time of 20
picoseconds. On the other hand, figure 2.19 shows the spectrum of the same clock
modulated with a modulation amount of 1 percent and modulation frequency of
30 MHz.
As we already saw in previous sections, the unmodulated clock presents sharp
peaks at the fundamental frequency an its harmonics. The SSC spectrum shows
45
Spectrum of a 100 MHz Clock
50% duty cycle, rise time = fall time = 20ps
15
5
Power in dbm
−5
−15
−25
−35
−45
−55
−65
0
0.05
0.1
0.15
0.2
0.25
0.3
Frequency (GHz)
0.35
0.4
0.45
0.5
Figure 2.18: Clock 100 MHz 50% duty
Spread Spectrum Clocking 100 MHz
50% duty cycle, rise time = fall time = 20ps
15
5
Power in dbm
−5
−15
−25
−35
−45
−55
−65
0
0.1
0.2
0.3
Frequency (GHz)
0.4
0.5
Figure 2.19: Spread Spectrum Clocking spectrum
46
a higher utilization of bandwidth but it is clear the reduction of power level over
the entire frequency band, an example of this is the reduction of the power at
the fundamental frequency of more than 10 dBs. It is evident the trade off that
exists when using the spread spectrum method between the power amplitude and
the bandwidth of the signal, where lower power levels require higher bandwidth
utilization and vice versa.
2.2.3.1 Modulated clock baseband representation
A clock modulated by a profile V (t) is given by,
4
v(t) =
π
∞
X
1
sin
k
k=1,3,5,...
2kπ
T
t+ρ
Z
t
V (τ )dτ
−∞
.
then, in analogy to the derivation of the unmodulated clock, it follows that its low
pass representation is given by
4
v (t) = −j
π
l
∞
X
Z t
1
2πk
t+ρ
V (τ )dτ − j2πfc t × 1F (k/T )
exp j
k
T
−∞
k=1,3,5,...
(2.15)
In the next subsection we derive the spectrum of a clock modulated by a triangular profile.
47
2.2.4 Spread Spectrum Clock (SSC) with a Triangular Profile
We perform an analysis of the spectrum of a clock modulated by a triangular signal
profile given by
V (t) = 1 −
4|t|
,
T
−
T
T
≤t≤ .
2
2
We have that,
Z
t
−∞
t
4|τ |
dτ
T
−T /2
t
2τ |τ | = τ−
T
−T /2
T
2t|t| 2T /2|T /2|
= t+ −
−
2
T
T
2t|t|
= t−
.
T
V (τ )dτ =
Z
1−
Now, we notice that the waveform of the modulating signal modulates the frequency from a low frequency given by fm − fm ρ to a high frequency fm + fm ρ and
then goes to the lowest frequency again, with this occurring in a periodic manner.
We also notice that the spectrum from high to low, is the same as the one from
low to high. Then the spectrum that the modulation produces is either from a low
frequency to a the high frequency or the other way around.
For a regular clock with levels 1 and -1 with T = 2L, its Fourier series is given
by:
4
v(t) =
π
∞
X
1
sin
k
k=1,3,5,...
2kπt
T
,
48
thus, its kth component is given by
vk (t) =
4
sin (2πkfc t)
kπ
where fc = 1/T . Then, it follows that the kth component of the modulated clock
(SSC) with profile V (t) is given by
Z t
4
vk (t) =
sin 2πkfc t + ρ
V (τ )dτ
.
kπ
−∞
Specifically, when the modulated profile is a triangular wave we have that
2t|t|
4
sin 2πkfc t + ρ t −
.
vk (t) =
kπ
T
(2.16)
The low pass representation of the kth component is given by [34] vk (t) =
Re[fkl (t)ej2πkfc t , but we know that
4
2t|t|
vk (t) =
.
Im exp j2πkfc t + ρ t −
kπ
T
Hence,
vkl (t)
2t|t|
4
exp j2πkfc ρ t −
.
= −j
kπ
T
49
2.2.4.1 Spectrum Derivation Using Fourier Series
We now proceed to compute the spectrum of the SSC. Since vkl (t) is periodic we
can find its Fourier series, i.e.
vkl (t) =
∞
X
Cn ej2πnt/T .
n=−∞
Then the magnitude and phase of the spectrum at frequencies ±n/T , where n =
0, 1, 2, . . . , are given by |Cn | and ∠Cn .
Now the Fourier series coefficients for the exponential term are computed as
Cn′
1
=
T
Z
T /2
ej2πkfc ρ(τ −
2τ |τ |
T
) e−j2πnτ /T dτ,
−T /2
where the argument of the exponentials can also be expressed for convenience as
j2πkfc ρτ − j2πkfc ρ
τ
n
2τ |τ |
2τ |τ |
− j2πn = j2π(kfc ρ − )τ − j2πkfc ρ
T
T
T
T
2
2
a
a
= aτ + bτ |τ | +
− ,
4b 4b
with a = j2π (kfc ρ − n/T ) and b = −j2πkfc ρ2/T , then whenever t < 0
a2 a2
aτ + bτ |τ | +
−
=−
4b 4b
√
a
√ − bτ
2 b
2
+
a2
,
4b
50
and when t ≥ 0 we have that
a2 a2
−
=
aτ + bτ |τ | +
4b 4b
√
a
√ + bτ
2 b
2
−
a2
.
4b
Then, the Fourier coefficients are given by
"Z
!
2
0
2
√
1
a
a
Cn′ =
dτ
exp − √ − bτ +
T
4b
2 b
−T /2
! #
2
Z T /2
√
a2
a
√ + bτ −
dτ
+
exp
4b
2 b
0
"
#
Z √a
Z √a +√bT /2
2 b
2 b
−a2
a2
1
2
2
=√
−e 4b
e−u du + e 4b
eu du
√
a
a
bT
√
√
+ bT /2
2 b
2 b
#
"
a
Z
Z √a +√bT /2
√
2
2
2 b
2 b
−a
a
1
2
2
eu du
−e 4b
e−u du + e 4b
=√
√
a
a
bT
√
√
+ bT /2
2 b
2 b
"
#
a
√a +√bT /2
√
2
2
−a
a
1
2 b
2 b
=√
−e 4b erf(u) a √
+ e 4b erfi(u) a
√ + bT /2
√
bT
2 b
2 b
"
√
a2
1
a
a
√ − erf
√ + bT /2
=√
− e 4b erf
bT
2 b
2 b
#
√
−a2
a
a
+ e 4b erfi √ + bT /2 − erfi √
,
2 b
2 b
51
replacing with the values of a and b we have that
"
n 2
πT 1
′
kfc ρ −
Cn = √
− exp −j
4kfc ρ
T
−j4πkfc ρT
s
s
!!
!
−jπT −jπT n
n p
erf
kfc ρ −
kfc ρ −
+ −jπkfc ρT
− erf
4kfc ρ
T
4kfc ρ
T
s
!
−jπT πT n 2
n p
+ exp j
kfc ρ −
kfc ρ −
+ −jπkfc ρT
erfi
4kfc ρ
T
4kfc ρ
T
s
!! #
n
−jπT − erfi
kfc ρ −
,
4kfc ρ
T
where erfi(x) = −j erf(jx). Then the k th component of the clock can be described
by a Fourier series as
vkl (t)
∞
4 X ′ −j2πnt/T
= −j
C e
,
kπ n=−∞ n
=
∞
X
n=−∞
−j
4 ′ −j2πnt/T
C e
,
kπ n
4
4
with an amplitude at n/T given by | − j kπ
Cn′ | and a phase given by ∠−j kπ
Cn′ .
Figure 2.20 shows the computed spectrum for one of the components of a
modulated clock where all the quantities have been normalized.
The spectra shown in figures 2.19 and 2.20 do not exactly correspond to what is
happening in the frequency domain at some instant of time since it just represents
the average spectrum calculated assuming time-invariance. However, the spectrum
of a SSC interfering signal is a periodic time-variant signal. In order to reflect what
it is really happening at each instant of time we make use of an adaptive spectrum
52
Amplitude dBs
0.1
0.01
0.001
10-4
f
-1
0
1
2
Figure 2.20: Spectrum of a SSC with triangular profile
estimation method that uses a Blackman-Tukey windowing method in order to
track the changing spectrum over time.
Figure 2.21 shows the spectrogram of the modulated clock where the horizontal
and vertical axes represent frequency and time respectively. The amplitude has
been normalized such that the maximum value corresponds to 0 dBs. The different
amplitude levels are represented by different colors and intensities, with the red
corresponding to the highest level and the blue to the lowest level. We have now a
clear visualization of the behavior of a modulated clock in the frequency domain.
The sharp peaks at the fundamental and its harmonics sweeps the corresponding
frequencies bands over time, shown in 2.19, each one going back and forth between
the minimum and the maximum frequency, established by the modulation profile.
We can quantify the total average bandwidth that each sharp peak occupies in a
53
down spreading scheme by the following formula
Bk = k · ρ · fnom
(2.17)
where n corresponds to the kth harmonic and Bk is the total bandwidth that it
occupies. Figure 2.21 shows the spectrum of a 133 Mhz clock with a spreading of
Spectogram: 133MHz Clock
SSC: 30 kHz, 1 % modulation
0
0
−5
10
−10
Time (µs)
20
−15
−20
30
−25
40
−30
50
−35
−40
60
100
200
300
400
500
600
Frequency (MHz)
700
800
900
1000
−45
Figure 2.21: SSC spectrogram using Blackman-Tukey
8 percent, then the bandwidth that we expect, for example, that the 3rd harmonic
occupies is given by
B3 = 3 × 0.08 × 133 = 31.92MHz,
a quantity that can be also estimated by analyzing the plot.
54
Figure 2.22: Comparison of the instantaneous spectrum of a SSC signal with a
regular clock signal
We can also analyze the instantaneous spectrum of a spread spectrum signal
by taking snapshots of the spectrum at different times. Figure 2.22 shows a representation of different snapshots of a SSC signal superimposed with the spectrum
of a normal clock over a range of frequencies that contains two harmonics. The
dotted arrowed lines represent the harmonics of a normal clock at frequencies f1
and f2 . The solid dotted lines represent the instantaneous frequency of the modulated clock. As we already mentioned, the amplitude of a SSC is lower than a
regular clock but the location of the impulses changes over time, moving over the
range of frequencies that each harmonic of a SSC signal covers.
Going further, we can analyze one of the frequency components of the SSC and
notice that if we sample the frequency and we associate each of the frequencies that
the component takes over time with a state, we have that its behavior resembles a
Markov chain. This is justified by the fact that at any given time, the frequency
at which the signal can be found only depends on the previous state, and since
either it advances or goes back to another frequency then at any state that is not
55
an edge one we have that
P (sm |sm±1 ) =
1
2
where sm represents the mth state.
f
i
f
i+1
. . .
f
f
Figure 2.23: Quasi-Band Interference Model
This becomes a very handy fact that can be be exploited by the receiver to
mitigate the interference that it causes. Figure 2.23 shows a Markov chain with N
states, where the initial state fi is the lowest frequency that the modulated clocks
takes and ff is the highest one. The probability of going from one state to another
is equal to 1/2, except on the edges where there is only one possible state.
However, it is clear that at any given instant of time, the spectrum of a quasiband interferer behaves similarly to a narrowband interferer. Therefore, a technique that is used to mitigate a narrowband interference could also be used to
mitigate a quasiband interference, as long as the method is modified to adapt to
the time varying nature of a quasiband signal spectrum.
2.3 Platform Noise
In this chapter we have analyzed separately the characteristics of the signals which
are primarily responsible of the electromagnetic emissions of the different platform
56
Platform Noise Spectrum
−60
Narrowband
−70
Broadband
−80
Amplitude dBs
−90
−100
−110
−120
−130
−140
−150
1.5
2
2.5
3
Frequency (GHz)
Figure 2.24: Platform noise
components. But, at the radio what we have is a combination of all of them.
Narrowband and broadband platform emissions are all received by the antenna as
whole.
An example of this is the spectrum shown in figure 2.24. The spectrum is
obtained by performing the Fourier transform of platform noise captured by an
oscilloscope connected to a radio antenna. In the plot the narrowband part and
the broadband part of the noise can be clearly identified. But, as it was pointed
out before, noise changes over time and even though the presence of a certain
narrowband noise contributor would be directly reflected on the spectrum with
the appearance or not of spikes, the presence of broadband interferers is not that
straight forward. Broadband noise presents its own difficulties since the main
57
contributors operate at different data rates, with different signal levels and not
always are on as clocks are.
From the radio perspective, it is an almost impossible task to separate broadband contributors to the platform noise as it can be done for the narrowband case
only by frequency analysis. Specially, given the narrowband bandwidth that the
radio antenna has in comparison with the bandwidth of the emissions. Therefore,
for broadband signals, it is more appropriate to analyze the data over time and
find a statistical model that characterizes the process. This model would not only
match its frequency characteristics but also its time-varying characteristics.
In the next chapter a model that fits the experimental data for broadband
interference is derived.
58
Chapter 3 – Statistical Analysis of Broadband Noise
In this chapter, a statistical characterization of the noise generated by the elements
of a computing platform is done.
Efficient wireless communications have been possible thanks to a fairly good
statistical characterization of the noise. This noise, generated at the radio receiver
RF front-end, has been well modeled because of its stationarity and Gaussian
nature. However, the interference generated by the platform does not fit this characterization [25], [8],[6] because of its non-Gaussian nature. Although, shielding
of the radiation sources could be considered as the natural solution to mitigate
the interference, it is expensive and a perfect enclosure of the radiating elements
is physically impossible, causing a leakage that cannot be avoided. Therefore,
a characterization of the noise is essential to quantify the impact on the radio
communications.
It has been a common practice to describe the noise as the result of a Gaussian
random process, though experimental data may suggest otherwise. Different NonGaussian models have been developed to overcome this but not always do these
models come from distributions with parameters that depend on the physical properties of the process. Most of these models are used because it is practical to fit the
data rather than representing the physical process that gives rise to them. A model
that overcomes this difficulty has been developed by analyzing several data sets
59
that were experimentally acquired using the setup explained later in this chapter.
The most notable attempts to establish a statistical model that matches the
physical characteristics of noise are the works of Middleton [23] and Jakeman et.
al [35], with the former one being the most widely used. A recent work by Bhatti
et. al. in [25] characterizes the noise from platforms using Middleton models.
Also, non-Gaussian stable distributions have been proposed to model the noise
[26]. Nassar et al. in [8] use the alpha-stable distribution, as an approximation of
the Middleton models [23], to model the noise taken from lab measurements. Some
authors have seen as an disadvantage the fact that stable distributions have infinite
variance which hardly can be linked to a physical process in a straight forward
manner. That is the case of the authors in [9] who have taken another approach
by modeling the experimental data using strange attractors (nonlinear dynamic
chaotic systems) as Lorenz, Chua and Rössler. Also, models based on BernoulliGaussian, Poisson-Gaussian and Hidden Markov chains have been identified to be
alternative approaches to describe the noise [27].
3.1 Noise Analysis
In the previous chapter we studied platform noise signals by grouping them according to their frequency characteristics as either narrowband or broadband. On
the one hand, narrowband signals are easier to identify in the frequency domain
due to the sharp peaks that occur at multiples of the fundamental frequency with
amplitudes that are usually above the average value of the spectrum. Most of
60
them are clock-like deterministic signals that are essential for the operation of
computing platforms. On the other hand, broadband signals come from numerous
heterogeneous sources carrying random data signals, transmitted at different data
rates over different periods of time, which are harder to identify. Their presence
can hardly be analyzed using spectral techniques because of the lack of recognizable unique features, which makes mitigation techniques based on their frequency
characteristics not practical, especially when not all the spectral characteristics of
the noise are available for analysis.
The difficulty in identifying specific features of broadband emissions becomes
even greater in radio receivers because the frequency content that is captured by
the narrow band antenna is just a fraction of the total frequency content of them.
This lack of resolution limits the ability of the receiver to find unique features in
the frequency domain that could be seen with broadband antennas, e.g. the nulls
that appear on data transmission at multiples of the data rate for random data
transmission, and in most of the cases the impact can only be quantified as the
increase on the noise floor as it is seen in figure 3.1. The figure shows the baseband
spectrum of the noise for a 20 MHz channel in the 2.4 GHz band. In our work
we removed all the narrowband signals by means of filtering since our purpose is
to analyze the broadband contributions to the noise. We can identify the effect
of platform emissions by comparing its intensity with the platform in idle state.
The figure clearly shows that there is an increase of about 10 dB of the noise floor,
without any other recognizable feature seen on the spectrum.
The works of [8] and [9] have used different approaches to model the noise
61
WIFI antenna over Idle Platform
WIFI antenna over Platform 3D Mark
−80
Amplitude (dBs)
Amplitude (dBs)
−80
−90
−100
−110
−120
−130
−90
−100
−110
−120
−130
−10
−5
0
5
Frequency (MHz)
10
(a) Idle state
−10
−5
0
5
Frequency (MHz)
10
(b) Running 3DMark
Figure 3.1: Platform noise spectrum on a 20 MHz channel on the ISM band
measurements taken using a similar configuration as the one shown in figure 3.2.
The set up shows a computing device located under a diagonal dual polarized horn
antenna that is connected to an oscilloscope where the IQ data corresponding
to the band of interest is acquired. The antenna used is a broadband antenna
with a frequency range from 400 MHz to 6 GHz with antenna gain characteristics
shown on figure 3.3. This technique is a useful method that allows to measure the
emissions of a platform over a wide range of frequencies [36] which later can be
compared with the spectral mask given by the regulatory bodies, e.g. FCC Class
B devices regulations shown in figure 3.4, to check the unit compliance with the
Electromagnetic Compatibility Regulations.
A drawback of this method is that it does not give and accurate measurement
of the noise as it is seen by the radio receiver since no compensation is performed to
match the horn antenna response with the narrowband receiver antenna. Therefore, new data was collected using the configuration shown in figure 3.5. The
62
Figure 3.2: Horn antenna set up
Figure 3.3: Horn antenna gain response (source: ETS-LINDGREN Horn antenna
Model 3164-03)
63
Electric Field
(dBµV/m)
FCC limits Class B digital devices
55
53
51
49
47
45
43
41
39
37
35
10
100
1000
Frequency (MHz)
Figure 3.4: FCC class B digital devices limits at a distance of 3 meters
experimental set up consists of an RF enclosure, with a shielding performance of
120 dB, inside which the platform under test was placed to avoid signals from
other electronic components that disrupt the measurements. The data were collected using an oscilloscope connected to the antenna of the platform through the
connections of the enclosure. IQ data were later obtained by further processing
them in the band of interest from the captured data. Different data sets were
acquired using the set up just described and analyzed here.
The unit under test was the motherboard of a laptop without any kind of enclosure. A WIFI antenna was placed fifteen centimeters above the unit with both
inside the enclosure box to avoid interference from the environment. The antenna,
MPCI08001 by Ethertronics, is an omni directional one designed to work at frequencies in the 2.4-2.49 GHz and 4.9-5.9 GHz range. Data were taken using the
oscilloscope while the platform was running the software 3DMark06. The software
is a commercial benchmark program that runs different tests to quantify the per-
64
Figure 3.5: Narrowband antenna set up
formance of the unit under test. The justification behind using the benchmark
is that it tests all the buses to their maximum usage, which from the radio perspective can be considered to be a worst case scenario. The disadvantage of using
the benchmark is that it does not mimic the conditions under which the radio
will operate for all the users. The behavior of the different platform components
depends on the applications run by each user, but since it is unrealistic to analyze
each of the cases, the radio has to be prepared to deal with the worst case scenario
as it is the one that 3DMark06 mimics.
Figure 3.6 shows different sets of time domain data captured using the oscilloscope. The data were further processed by passing them through a bandpass filter
with a bandwidth of 22 MHz centered at 2.437 GHz, corresponding to channel 6
on the WLAN channelization. Then the data were down converted to baseband
where the lowpass complex representation (IQ data) of the signal was obtained.
The real and imaginary data of one of the sets are shown on figure 3.7.
65
Amplitude
Platform noise from laptop running 3DMark
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
Amplitude
−0.1
−0.05
0
0.05
0.1
−0.1
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.1
−0.05
0
0.05
0.1
−0.1
−0.05
−0.05
time (ms)
0
0.05
0.1
0
0.05
0.1
time (ms)
Figure 3.6: Time domain data from platform running 3DMark06
Real Component
Amplitude (V)
0.01
0.005
0
−0.005
−0.01
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.04
0.06
0.08
0.1
Imaginary Component
Amplitude (V)
0.01
0.005
0
−0.005
−0.01
−0.1
−0.08
−0.06
−0.04
−0.02
0
time (ms)
0.02
Figure 3.7: Time domain baseband data from unit under test
66
In order to find the distribution that governs the process that generates the
data, a histogram of the data was obtained for both IQ components, figure 3.8
shows the normalized histogram of both components. Even though the histograms
give us an insight of the distribution they miss some of its subtleties. One of the
first assumptions that we are tempted to make about the data is that it follows a
Gaussian distribution. One of the most well known graphical approaches to find
out if the assumption is right is by using a probability plot.
Imaginary Component
1
0.9
0.9
0.8
0.8
0.7
0.7
Normalized Frequency
Normalized Frequency
Real Component
1
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
−0.01
−0.005
0
Amplitude
0.005
0.01
0
−0.01
−0.005
0
0.005
0.01
Amplitude
Figure 3.8: Histogram plots of the noise IQ components
In a probability plot, the axes are modified so that for a specific distribution
the cumulative distribution function (CDF) of the data that follow that probability
distribution appear in the plot as a straight line. Any deviation from the line is
an indication that the data do not belong to that distribution. This information is
fundamental in comparing different sets of data and also in characterizing them.
67
The empirical cumulative distribution functions of the IQ data obtained from the
platform running 3DMark were plotted using the normality plots shown in figure
3.9. In the plots, it can be seen that the assumption of the data coming from a
Gaussian distribution is not valid since there exits a prominent deviation of the
tails from the model. It is also interesting to note that both components follow a
0.999
0.997
0.999
0.997
0.99
0.98
0.99
0.98
0.95
0.90
0.95
0.90
0.75
0.75
Probability
Probability
fairly similar pattern.
0.50
0.25
0.10
0.05
0.25
0.10
0.05
0.02
0.01
0.02
0.01
Real (Skew = 0.938 Kurtosis= 22.9)
Normal Distribution
0.003
0.001
−0.01
0.50
−0.005
0
Data
0.005
Imaginary (Skew = 0.583 Kurtosis= 17.3)
Normal Distribution
0.003
0.001
0.01
−0.01
−0.005
0
0.005
0.01
Data
Figure 3.9: CDF plots of the noise IQ components
Even though the CDF plots provide a good graphical evidence that the data
is non-Gaussian, they are not truly statistical tests since we rely only on visual
inspection to reach conclusions. Therefore, a hypothesis test that validates and
reinforces the conclusions reached is been used. To this end, consider the following
simple hypotheses:



H0 : Data comes from a normal Distribution: X ∼ N (µ, σ 2 )


H1 : Data do not come from a normal Distribution: X ≁ N (µ, σ 2 )
,
68
Data Set
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
p-value
1.75 × 10−08
2.04 × 10−09
0.000110741
1.99 × 10−08
5.10 × 10−24
8.71 × 10−25
0.000202552
1.88 × 10−14
4.19 × 10−11
5.67 × 10−08
0.003284874
6.54 × 10−14
0.034974958
1.43 × 10−07
9.56 × 10−05
3.96 × 10−13
1.47 × 10−99
9.85 × 10−208
Conclusion
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Reject H0
Table 3.1: χ2 goodness-of-fit test for normality
where X represents the data under test. We want to either accept or reject the
null hypothesis H0 , which corresponds, respectively, to reject or accept H1 , for the
baseband components of the data. For this purpose, we used the well known χ2
goodness-of-fit test with a significance level of 5%. The results for different sets
of data are presented in table 3.1. After analyzing the data, it becomes clear that
data do not follow a Gaussian distribution, which confirms the conclusion reached
through the normality plots.
It is well known that radio receivers are optimized to demodulate signals that
are corrupted with additive white Gaussian noise (AWGN), therefore any emission
69
generated by a computing platform that follows a Gaussian distribution and is
white would be part of the noise model that is already taken care by the receiver.
Therefore, any mitigation attempt would not improve the performance under the
same modulation and coding scheme since its design is already optimum. But if
the emissions do not follow a Gaussian distribution, as it was confirmed here, the
receiver would lose its optimality. Therefore, we need a model of the noise able to
match the statistics of the experimental data and to explain the physical process
that generates it. The next sections are devoted to find and test that model.
3.2 Noise Model
The previous analysis bring us to the question of what kind of distribution the
noise follows if it is not Gaussian.
The different elements on a computing platform are potential sources of radiation that contributes to the noise that affects radio communications. Because
their behaviors can be considered, to some extent, to be independent of each other,
there is the tendency to use the central limit theorem (CLT) to assert that the
statistics of the noise are the ones for the Gaussian distribution. Experimental results have shown that the latter assertion does not always holds. These finding do
not contradict the CLT but rather show the form in which the theorem is loosely
applied.
The CLT assumes besides the independence of the random variables that each
of them has finite mean and variance, and applies to the sum of a large number of
70
random variables, i.e. limn→∞
Pn
i=1
Xi . It is clear that if any of those conditions
fails the theorem cannot be applied. Gaussian mixture, generalized Gaussian and
stable distributions have been proposed to model non-Gaussian processes [37]. The
most appealing of them is the set of stable distributions because of its versatility
and its inclusion of distributions with infinite variance. The latter one prompted
the development of the generalized central limit theorem (GCLT) where the CLT
is a special case, in few words it establishes that the sum of stable random variables
gives a stable distribution.
A model of the platform noise that has parameters related to its physical characteristics prevents us to use any of both theorems. On the one hand, experimental
evidence shows the disagreement with the CLT. On the other hand, the use of the
GCLT can lead us to distributions with infinite variance, which as mathematical
models can be appealing but not as models with parameters that describe the
physical process. This leads us to find a model with parameters based on the
characteristics process.
In constructing the model, the first thing to notice is that the number of radiated sources on a platform varies with time. In a computer, there are some
processes that are fundamental regarding of its use but there are others that are
user specific, application-specific and even platform-specific. Hence, different platform elements are used at every instant of time.
Then, it follows that the number of radiated sources is time dependent, N (t).
A condition that differs with the one assumed by the CLT and the GCLT which
assume that the number of random variables that generates the process tends to
71
be large at all times, N → ∞, not just in average. Therefore, we need to find the
distribution that better describes the statistics of n(t), a realization of the process
N (t), in order to derive the model for platform noise.
In a computing platform, every process that runs on it demands a different
amount of resources that depends on the application. It is evident that this translates into the number of radiating sources, N (t), which varies over time. Thus,
N (t) is a random process that can be described as a random walk that takes on
positive integer values, i.e. N (t) ∈ Z+ . Now, if we assume that the process has
the Markovian property with unitary increments or decrements, we get a birth and
death process [38], i.e.
P (N (t + h) − N (t) = k|N (t) = n) =




λn h + o(h)




, when k = 1
µn h + o(h)
, when k = −1 , (3.1)





(1 − λ − µ )h + o(h) , when k = 0

n
n
where λn is the birth rate and µn is the death rate when n sources are present.
Then, for the special case of a birth-death process with immigration, we have that
the distribution of N (t) is given by [39] (see the Appendix B for the details)
P (N (t) = n) =
n+r−1
n
N̄
r
n r
r + N̄
n+r
.
(3.2)
72
Figure 3.10: Laptop Emissions
Figure 3.10 shows a scheme of the setup that we are analyzing. A radio antenna
is fixed at an almost equidistant position from all the main radiators in a platform
that runs a benchmark software that tests the performance of the unit. This creates
the perfect scenario to test the maximum level of emissions from the platform.
Then, the emission captured by the antenna from the ith element is given by:
zi = ai ejθi ,
where ai is the amplitude and θi is the phase. We notice that there is a line
of sight between the receiving antenna and the radiating elements. Thus, we
only expect that the amplitude of each individual interference source to suffer
attenuation manly due to free-space path loss and since the signals used in a
computing platform have predetermined levels representing the high and low states,
we do not expect a high variation on the received amplitude between realizations
that come from the same source. Therefore, ai would be the maximum received
amplitude from the radiation of the ith component and the random phase θi is
responsible of the random fluctuations in the signal.
73
Moreover, since the antenna is fixed it is not unrealistic to assume that, on
average, the maximum emissions from each of the sources can be considered to be
the same, with the differences controlled by the phase. Thus, we can assume that
the amplitude of each of the components is given by
ai = ā i = 0, 1, . . .
where ā is the average maximum amplitude of the emissions.
Then, the baseband equivalent noise generated by a platform at the receiver,
the sum of all the contributions of the different radiators, is given by
N (t)
1 X jΘi (t)
Z(t) = √
āe
,
N̄ i=1
(3.3)
√
where the scaling factor 1/ N̄ is due to energy conservation, since the electromagnetic emission from any computing platform is finite. N (t) is the number of noise
sources at time t, ai and θi (t) are the amplitude and the angle of the i component.
N̄ is the average number of radiated components. Thus, the real component of the
noise is given by
X(t) = Re(Z(t)).
Now, assuming that the elements in the emissions sources are independent we
74
have that the conditional characteristic function of X is given by
ϕX(t)|N (t)=n (w) = E[ejwX(t) ],
= E[e
=
n
Y
jw √1
E[e
N̄
Pn
i=1
ā cos(θi (t))
jw √1 ā cos(θi (t))
N̄
],
].
i=1
And with θi (t) being uniformly distributed between 0 and 2π we obtain the well
known result
2π
1
1
ϕX(t)|N (t)=n (w) =
exp jw √ ā cos(θ) dθ,
2π
N̄
0
n
āw
.
= J0 √
N̄
Z
Moreover, N (t) is negative binomial distributed with probability mass function
75
given by equation (3.2). Then,
n āw
,
ϕX(t) (w) = E J0 √
N̄
n
n ∞ X
N̄
āw
n+r−1
=
J0 √
n
r
N̄
n=0
n+r
r
,
r + N̄
−r

āw
r
N̄
J
0 √N̄
r
1 −
 ,
=
N̄ + r
N̄ + r
−r


N̄ J0 √āwN̄
N̄ 
 ,

1−
1+
=
r
N̄ + r
−r
N̄
āw
N̄
N̄
1+
J0 √
−
= 1+
,
r
r
N̄ + r
N̄
−r
āw
N̄
1 − J0 √
.
= 1+
r
N̄
(3.4)
Now, the asymptotic behavior for a large average number of sources is given
by [24]
lim ϕX(t) (w) =
N̄ →∞
ā2 w2
1+
4r
−r
,
which can also be expressed as
ϕX (w) =
1
,
(1 + b2 w2 )r
(3.5)
with b2 = ā2 /4r. Thus, the characteristic function in (3.5) results in the K-
76
distribution given by
1
pX (x) = √
πbΓ(r)
|x|
2b
r−1/2
Kr−1/2
|x|
b
,
(3.6)
with r > 0, b > 0. The same derivation can be done for the distribution of the
imaginary component, which will lead to a similar result, realizing that sin(θ) =
cos π2 − θ .
Now, we can express 3.6 in the more conventional fashion by substituting r by
ν + 1, then it follows that the K-distribution is given by [40]
1
pX (x) = √
πbΓ(ν + 1)
|x|
2b
ν+1/2
Kν+1/2
|x|
b
|x|
b
,
(3.7)
with ν > −1, b > 0.
The moments are computed as follows:
∞
1
E[X ] =
x √
πbΓ(ν + 1)
−∞
k
Z
k
|x|
2b
ν+1/2
Kν+1/2
dx
which gives the following result:
2k−1 (−b)k + bk Γ 1+k
Γ
2
√
E[X ] =
πΓ(ν + 1)
k
k
2
+ν+1
Now, whenever k is an odd integer
E[X k ] = 0
k = 1, 3, 5, . . . ,
.
(3.8)
77
and if k is a even integer
k
E[X ] =
2k bk Γ
1+k
2
Γ k2 + ν + 1
√
πΓ(ν + 1)
k = 2, 4, 6, . . . .
The calculation of the moments is important because they are the link to the
physical characteristics of the process since through them we can compute the
parameters of the distribution [41], [42].
The second moment for a zero-mean distribution is equal to the variance.
Hence,
22 b2 Γ
1+2
2
Γ (1 + ν + 1)
,
πΓ(ν + 1)
√
22 b2 π(ν + 1)Γ(ν + 1)
√
=
,
2 πΓ(ν + 1)
var(X) =
√
= 2b2 (ν + 1).
(3.9)
Now in terms of ā, we have that
var(X) =
ā2
,
2
(3.10)
78
the fourth moment is given by the following
24 b4 Γ 52 Γ (2 + ν + 1)
√
,
E[X ] =
πΓ(ν + 1)
√
24 b4 3 π(ν + 2)(ν + 1)Γ (ν + 1)
√
=
,
4 πΓ(ν + 1)
4
= 12b4 (ν + 2)(ν + 1),
from which the kurtosis, β2 , can be computed as
E[X 4 ]
,
E[X 2 ]2
3(ν + 2)
.
=
ν+1
β2 ≡
Therefore
ν=
3
− 1,
β2 − 3
(3.11)
and
b=
r
var(X) (β2 − 3)
.
6
(3.12)
3.3 Experimental Results
In order to test the suitability of the derived model, we performed two fitting tests,
one graphical and the other analytical. We use the data sets that were collected
using measurement set up in figure 3.5, as it was explained in a previous section.
Figure 3.11 shows the normalized histograms of the I-Q components of the
79
Real Component
1
0.9
0.8
Histogram
Gaussian
K
Normalized Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.01
−0.008
−0.006
−0.004
−0.002
0
Amplitude
0.002
0.004
0.006
0.008
0.01
0.008
0.01
(a) Real component
Imaginary Component
1
0.9
0.8
Histogram
Gaussian
K
Normalized Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.01
−0.008
−0.006
−0.004
−0.002
0
Amplitude
0.002
0.004
(b) Imaginary component
Figure 3.11: Histograms
0.006
80
Real Component
Imaginary Component
1
1
0.9
0.9
0.8
0.8
0.7
0.6
0.6
Probability
Probability
K CDF
Empirical CDF
0.7
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
−0.01
K CDF
Empirical CDF
0.1
−0.008
−0.006
−0.004
−0.002
0
Data
0.002
0.004
(a) Real component
0.006
0.008
0.01
0
−0.01
−0.008
−0.006
−0.004
−0.002
0
Data
0.002
0.004
0.006
0.008
0.01
(b) Imaginary component
Figure 3.12: Fitting of the empirical CDF of the noise with K-distribution CDF
noise captured from a laptop motherboard. Both histograms show very symmetric
distributions around the mean with a high concentration of values around it. This
causes the distribution to have a sharp peak that decays rapidly which results in
heavy tails. We estimated the best fitting parameters to the data for the Gaussian
distribution and the K-distribution by using the method of moments (MOM), using
the formulas in equations (5.5) and (3.11) for the K-distribution. The obtained
distributions are plotted along with the histograms on figure 3.11. As expected the
Gaussian distribution does not fit very well the data, which graphically confirms
the analytical results. On the other hand, the K-distribution appears to be a valid
model for the experimental data. We verified the assumption that the noise comes
from the K-distribution by first computing the empirical cumulative distributions
of the I-Q components.
Figure 3.12 shows the cumulative distributions of the measured noise and the
K-distribution. The graphs show small discrepancies between the lower and upper
81
curvature of the empirical and the K CDF. But for most of the values each CDF
coincides with each other. The last assertion can be also validated by analytical
methods which lead us to a stronger conclusion than the graphical evidence. We
test the goodness-of-fit of the measured data with the distribution using the well
known One-Sample Kolmogorov-Smirnov test for the following hypotheses:



H0 : Data come from the K-distribution
,


H1 : Data do not come from the K-distribution
The hypotheses were tested for real and imaginary components of the experimental
data with a significance level of 5%. Table 3.2 summarizes the results that were
obtained for the IQ components of several data sets. In the table, we do not make
a distinction of both components since we assumed both to be independent and
that is how they were treated when the parameters were computed. These results
confirm that most of the components of the data sets are fitted well with the Kdistribution and therefore confirming the validity of the model described in the
previous section.
In the next chapter, we investigate the impact of platform noise described here
and in the previous chapter on a OFDM radio architecture.
82
Data Set
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
p-value
0.915302727
0.062464117
0.423125619
0.671242784
0.339459155
0.537040638
0.204022971
0.071819388
1.09605E-10
0.652270317
9.43378E-09
0.680015823
0.735575096
0.972053516
0.450313111
0.302436539
0.002872387
0.0384982
b
0.000519
0.000648
0.000430
0.000521
0.002461
0.002550
0.000202
0.000329
0.003697
0.000611
0.003376
0.000788
0.000780
0.000647
0.000720
0.000795
0.002234
0.002950
ν
0.66
0.20
0.48
0.14
-0.38
-0.45
8.73
2.55
-0.70
-0.11
-0.67
0.03
0.02
1.07
0.88
-0.02
-0.40
-0.52
Conclusion
Accept
Accept
Accept
Accept
Accept
Accept
Accept
Accept
Reject
Accept
Reject
Accept
Accept
Accept
Accept
Accept
Reject
Reject
Table 3.2: Kolmogorov-Smirnov test results for α = 0.05
83
Chapter 4 – Platform Interference in OFDM
In this chapter we describe the architecture of an OFDM communication system.
This system has many advantages over others and it has become the standard
transmission scheme for the actual high data rate protocols and the next generation
ones such as WIMAX, 802.11n or Long Term Evolution (LTE). We also describe
the impact that the different platform noises previously studied have on the system.
4.1 Communication Model
Every communication system consists of three main parts: the transmitter, the
propagation medium and the receiver. In particular, a wireless communication
model may be represented by the simplified model shown in figure 4.1 where x(t),
y(t), n(t) and h(t; τ ) represent the transmitted signal, the received signal, the
noise and the channel. Briefly, what the diagram represents is the transmission
of x(t) through a time-varying channel described by h(t; τ ). At the receiver the
signal is assumed to be affected by Gaussian noise with zero mean and variance
σ 2 , n(t) ∼ N (0, σ 2 ). The received signal y(t) is down converted, sampled and
further processed thorough an architecture that performs the necessary operations
to obtain an output stream of bits that closely resembles the original message with
minimum possible errors.
84
Figure 4.1: Communication Model
In general, the receiver architecture has been designed and optimized taking
into account the impulse response of the channel and the statistics of the noise n(t)
which is assumed to be white and Gaussian. Moreover, the radio designers have
never taken into consideration the noise generated by the platform or they have
assumed that it can be incorporated into the Gaussian noise. But, as we analyzed
in the previous chapters, platform interference has started to become harmful to
the radio receiver and since it is of non-Gaussian nature the receiver architecture
employed up to now is not optimum anymore.
4.2 OFDM System Description
Orthogonal Frequency Division Multiplexing (OFDM) is a multi carrier modulation technique that is employed in frequency selective channels. The purpose of
OFDM is to divide the transmission into orthogonal subchannels such that each
85
of them experiences a flat fading instead of a frequency selective one [43].
OFDM can be implemented by means of the Discrete Fourier Transform (DFT)
and its inverse (IDFT) which creates the orthogonal subcarriers that are needed
for transmission. An implementation of an N-FFT OFDM transmitter is shown
in figure 4.2 which we describe next. The input bits are first modulated using a
linear modulator, in this case QAM, and then they are passed to a Serial-to-Parallel
converter. The output is an N-collection of QAM symbols, which are the input of
an IFFT block that produces the symbols to be transmitted. Before transmission,
a cyclic prefix is added to induce circular convolution between the channel and the
transmitted data and to eliminate inter-symbol interference (ISI) [43]. Then the
stream of symbols is passed through a digital to analog converter and up converted
to the transmission frequency.
Figure 4.2: OFDM baseband spectrum
Mathematically, an OFDM system with QAM modulation can be described as
follows: The input bits si are first mapped to a L-QAM symbol X[k] as
Z/2Z → C,
{s1 , . . . , sl } 7→ Xk ,
86
where X[k] ∈ C is a symbol from an alphabet A of QAM symbols. The modulated
signal is further processed by an IFFT block, then the baseband transmitted signal
before passing through the digital to analog converter (D/A) is given by
N −1
1 X
X[k]ej2πnk/N .
x[n] = √
N k=0
Then, an OFDM symbol is given by x = [x[0], x[1], ..., x[N − 1]]T . We can also
express the IFFT operation in vector notation as
X = Qx,
where Q is the IFFT matrix given by


1
1
···
1
1



N −1 
2
1
W
W
.
.
.
W
N
1 
N
N


Q = √ .
,
.
.
.
.


N  ..
..
..
..
..



2(N −1)
(N −1)2
N −1
1 WN
WN
. . . WN
with WN = e−j2π/N . Since Q−1 = QH ,
x = Q−1 X = QH X.
A cyclic prefix is then added to the signal for mitigating the Intersymbol Interfer-
87
ence (ISI), thus the transmitted OFDM-symbol is given by
xcp = [x[N − m], ..., x[N − 1], x[0], x[1], ..., x[N − 1]].
OFDM Symbol Spectrum
Normalized Power in dBs
0
−20
−40
−60
−80
−100
−120
−10
−5
0
Frequency (MHz)
5
10
Figure 4.3: OFDM Transmitter
Figure 4.3 shows the spectrum of the baseband representation of a OFDM
symbol ready to be transmitted with a bandwidth of 20 MHz.
Figure 4.4 shows the OFDM receiver which performs the inverse operations
that were carried out at the transmitter. It first down converts the received signal
into baseband, then it converts the stream from analog to digital. The cyclic prefix
is removed and the FFT is applied to the data that has been previously converted
from serial to parallel. Finally, the output bits are obtained from the FFT symbols
by demodulating them using a QAM demodulator.
We can describe the demodulation process mathematically by describing the
received OFDM symbol after the down conversion, the analog-to-digital operation
88
Figure 4.4: OFDM Receiver
and its cyclic prefix removed as
y = [y[0], y[1], ..., y[N − 1]] .
The symbol is later passed through the FFT block where the obtained kth value is
given by
N −1
1 X
Y [k] = √
y[n]e−j2πnk/N .
N n=0
(4.1)
Finally, the output bits are obtained by a decision function D such as
D:
C → Z/2Z
Y [k] 7→ {ŝ1 , . . . , ŝl }
(4.2)
(4.3)
In the next section, we describe the effects of Platform Interference on a communications system before heading to the next chapter, where new radio architectures
are explored in order to mitigate the interference.
89
4.3 Platform Interference in OFDM
In this section we analyze the impact of the noise generated by a computing platform. Figure 4.5 shows a communication model that includes the Platform Interference. The model now incorporates the platform interference where the received
signal can be expressed analytically as
y(t) = x(t) ∗ h(k; τ ) + n(t) + i(t),
(4.4)
where i(t) ∈ I is the result of the emissions on the platform; n(t) ∼ N (0, σ 2 ) and
h(t; τ ) is the time varying channel. The interference component is the result of all
the electromagnetic emissions, either narrowband, broadband or both, from the
platform, so it can be separated in its constituent members as
i(t) = inb (t) + ibb (t),
(4.5)
where inb (t) corresponds to the narrowband noise and ibb (t) corresponds to the
broadband noise.
We analyze first the effect of narrowband interference in a communication system. To do this, we analyze a regular clock and a modulated one as a sources of
interferece. Now, OFDM is a multicarrier technique and without lost of generality
we can assume that, on the one hand, each of the components of regular clocks
affects only one sub band. On the other hand, modulated clocks could affect more
that one sub band over a symbol time. Figure 4.6 shows the different effects in the
90
Figure 4.5: Platform Model
frequency domain of a clock and a modulated clock interferer on an OFDM signal.
In the first case, the spectrum of the interferer is at a frequency fi that overlaps
one of the OFDM sub bands, f4 . In the second case, the interferer affects more
than one sub band but at different instants of time.

Frequency
f1
f2
f3
f4 fi
f5
f6
(a) Narrowband Noise
Frequency
f1
f2
f3
f4 fi
f5
f6
(b) Quasi-band Noise
Figure 4.6: OFDM narrowband interference
Figure 4.7 shows the baseband spectrum of an OFDM symbol with a bandwidth
of 10 Mhz and a narrowband interferer. The interferer has a sharp peak located
at 1 Mhz and sidelobes levels starting at -80 dBs less from the peak. In most of
the cases, the effect of the sidelobes has a minor effect at the reception and can be
91
OFDM Symbol Spectrum
20
Narrowband Interference
OFDM Symbol
Normalized Power in dBs
0
−20
−40
−60
−80
−100
−6
−4
−2
0
Frequency (MHz)
2
4
6
Figure 4.7: OFDM Spectrum with Narrowband Interference
ignored, but if the level of interference is too high the sidelobes effect cannot be
neglected.
Broadband noise, on the other hand, could affect the entire receiver bandwidth.
As it is shown in figure 4.8, the broadband interference spectrum occupies almost
the entire range of frequencies except for the nulls affecting all the sub bands.
4.3.1 Analysis of the impact of SSC on an OFDM radio
The impact of a modulated clock on an OFDM radio is now quantified. The
spectrum of a harmonic of a modulated clock, as the one calculated before for a
triangular profile, gives the representation of the signal in the frequency domain
over the entire period of the modulating profile. However, the real impact inside a
92
OFDM Symbol Spectrum
0
Normalized Power in dBs
−20
−40
−60
−80
−100
−120
−6
Broadband Interference
OFDM Symbol
−4
−2
0
Frequency (MHz)
2
4
6
Figure 4.8: OFDM Spectrum with Broadband Interference
radio band depends on the relation between the modulation time of the clock, 1/fm ,
and the duration of the received symbol, Tsym , of the communication protocol used
by the radio. Figure 4.9 shows a comparison between the baseband spectrum of a
modulated clock harmonic for different symbol times. The spectrum is shown on
the left side and the part of the profile corresponding to each spectrum is shown
on the right side. It is clear from the figure that there is a noticeable difference
between the range of frequencies that are affected in each case, which in OFDM
corresponds to a different number of affected sub carriers. This information allows
us to use narrow band mitigation techniques in each affected sub carrier.
The number of affected sub carriers can be estimated as follows: Let a clock
being modulated by a triangular profile with modulation frequency fm and a modulation amount ρ. Now, let Tsym be the symbol duration of an OFDM radio trans-
93
Amplitude
1.00
Frequency
0.50
1.0
Symbol Time
0.20
0.5
0.10
0.05
Time
-0.4
0.2
-0.2
0.4
0.02
-0.5
0.01
Frequency
-3
-2
0
-1
1
2
-1.0
3
(a) Spectrum for symbol time: 40% of period
(b) Symbol time: 40% of period
Frequency
Amplitude
1.00
1.0
0.50
0.5
0.20
0.10
Time
-0.4
0.05
0.2
-0.2
0.02
0.4
-0.5
0.01
Frequency
-3
-2
-1
0
1
2
-1.0
3
(c) Spectrum for symbol time: 80% of period
(d) Symbol time: 80% of period
Frequency
Amplitude
1.00
1.0
0.50
0.5
0.20
0.10
Time
-0.4
0.05
0.02
0.2
-0.2
0.4
-0.5
0.01
Frequency
-3
-2
-1
0
1
2
3
(e) Spectrum for symbol time: 10% of period
-1.0
(f) Symbol time: 10% of period
Figure 4.9: Normalized spectra of modulated clocks for different symbol times
94
mission and ∆fc be the sub carrier spacing. Then, we have that at any given time
ti the relationship between it and the frequency at which the clock is modulated
is given by
ti =
1
fi+c ,
4ρfc fmod
then the frequency range in hertz that is covered by a SSC clock from time t1 to
t2 either in the raising or the falling side of the profile, as in 4.9(b) and 4.9(f), is
obtained as follows:
1
1
f2 −
f1 ,
4ρfc fmod
4ρfc fmod
1
∆f,
=
4ρfc fmod
t2 − t1 =
thus,
∆f = 4ρfc fmod ∆t.
Now, if Tsym <
1
2fm
and it falls on either side of the profile, we have that ∆t = Tsym .
Then the estimated number of affected sub carriers is given by,
Nsub
On the other hand, if Tsym ≥
∆f
=
,
∆fc
4ρfc fmod Tsym
.
=
∆fc
1
2fm
and the reception of the symbol starts is syn-
chronized with the clock modulation profile, then ∆t =
1
,
2fm
anything different
95
from that would result in a smaller number of affected sub carriers, is given by
Nsub
2ρfc
.
=
∆fc
Therefore, the percentage of affected sub carriers is given by
N% =

l
m

 Nsub × 100%

if Nsub < Nt
Nt


100%
,
(4.6)
otherwise
where Nt is the total number of the subcarriers that OFDM uses.
4.3.2 Platform Broadband Noise
In chapter 3 we show that broadband noise generated by the platform is Kdistributed. It was argued there that the assumption that people make of broadband noise being Gaussian does not match the experimental data. Even though
the last assumption is convenient for radio designers, we show here through simulations that this conservative approach not only differs from experimental data
but it also translates into more costly designs than if the K-distribution is used.
Monte Carlo simulations were conducted for an OFDM system with a FFT
size of 1024 with 720 of them being data carriers. The modulation used in the
simulations is binary phase-shift keying (BPSK) and the bandwidth is 10 MHz.
The average energy signal is equal to 1 × 10−6 . The bit error rate (BER) was
computed for communications affected by platform broadband noise that is K-
96
0
10
No interference
K−noise
Gaussian
−1
10
−2
Error Rate
10
−3
10
−4
10
−5
10
−6
10
−4
−2
0
2
4
SNR
6
8
10
12
Figure 4.10: BER comparison for broadband noise with σ 2 = 1 × 10−6 , ν = 1 and
b = 0.0005
97
distributed and Gaussian distributed with variance σ 2 . The performances of both
scenarios are compared for different parameters of the K-distribution.
0
10
No interference
K−noise
Gaussian
−1
10
−2
Error Rate
10
−3
10
−4
10
−5
10
−6
10
−4
−2
0
2
4
SNR
6
8
10
12
Figure 4.11: BER comparison for broadband noise with σ 2 = 4 × 10−6 , ν = 1 and
b = 0.001
Figures 4.10 shows the BER for radio communications without interference and
affected by broadband noise with variance σ 2 = 4 × 10−6 . Broadband noise was
generated from the Gaussian and the K distributions where the parameters of the
K-distribution are ν = 1 and b = 0.0005. From the graph, we see that at low SNR
the BER is comparable for all the cases. At high SNR, the difference of assuming
the platform noise to be Gaussian distributed is of up to 2 dB.
Figures 4.11 and 4.12 show the performance of the system for a platform broad-
98
0
10
No interference
K−noise
Gaussian
−1
10
−2
Error Rate
10
−3
10
−4
10
−5
10
−4
−2
0
2
4
SNR
6
8
10
12
Figure 4.12: BER comparison for broadband noise with σ 2 = 6 × 10−6 , ν = 1.5
and b = 0.001
99
band noise parameters σ 2 = 6×10−6 , ν = 1.5, b = 0.001 and σ 2 = 8×10−6 , ν = 0.5
and b = 0.002 respectively. The cost of assuming the wrong model of platform interference is clearly seen in both figures.
0
10
No interference
K−noise
Gaussian
−1
Error Rate
10
−2
10
−3
10
−4
10
−4
−2
0
2
4
SNR
6
8
10
12
Figure 4.13: BER comparison for broadband noise with σ 2 = 8 × 10−6 , ν = 0.5
and b = 0.002
Finally, figure 4.13 shows the BER for parameters σ 2 = 8 × 10−6 , ν = 0.5 and
b = 0.002 where even for low SNR there is cost for assuming the interference is
Gaussian.
100
4.4 Bit Error Rate
In this section we quantify the performance of a communication system when
platform noise is present. The metric used for this purpose is the Bit Error Rate
(BER). We present first the BER for narrowband interference and then for the
broadband noise derived on the previous chapter.
4.4.1 Narrowband Interference Bit Error Rate
The performance of OFDM systems in narrowband interference, clocks or any
periodic signal, has already been quantified by many authors. Its derivation follows
basic principles found in any standard communications book as in [34]. The BER
of spread spectrum clocks (SSCs) interference in OFDM systems has also been
derived by other authors in [16, 19]. For completeness, we present here some of
those results whose derivation can be found in Appendix C.
In particular, we consider here that the narrowband or quasi-band interference
can be described as the sum of L sinusoidal interfering signals, even with modulated
clocks the approximation is valid under some conditions as it was shown on the
previous section.
Now, the sampled received signal in the baseband, assuming an ideal channel
and perfect synchronization, is given by
N −1
1 X
√
X(k)ej2πnk/N + ν[n] + ina [n],
y[n] =
N k=0
(4.7)
101
then, at the receiver the DFT is performed on the incoming signal resulting on the
kth sample being
Y [k] = X[k] + N [k] + Ina [k],
where N [k] and Ina [k] are, respectively, the kth sample of the Fourier transforms
of the Noise and the Interference. Now, ν is Gaussian with variance value of σ 2 ,
then N is also Gaussian with variance σ 2 [34]. X is the kth data symbol from some
constellation.
The interference is given by the sum of L narrowband inteferers such that
ina (n) =
L
X
ina (n, m),
m=1
where we obviously consider the case when |fm | < fs /2. Then, each ina (n, m) is
described as,
ina (n, m) = bm exp (j2πfm nTs + φm ) .
Now, the DFT of the interference is given by
Ina (k) =
L
X
Ina (k, m),
m=1
where each component has a DFT representation given by
k
Ina (k, m) = bm ej(π( N −fm T )(N −1)+φ)
sin(N π( Nk − fm T ))
,
sin π( Nk − fm T )
102
where we define
k
Ψ(k, m) = ej(π( N −fm T )(N −1)+φm )
sin(N π( Nk − fm T ))
,
sin π( Nk − fm T )
γs = A2 /2σ 2 and γm = b2m /2σ 2 . Then, the conditional Bit Error Rate for BPSK
modulation, as derived in Appendix C, is given by
!
"
N
−1
L
X
X
1
1
√
√
erfc
PeBPSK (γs , γi |φ, f ) =
γs +
γm Re{Ψ(k, m)}
N k=0 4
m=1
,
!#
L
X
1
√
√
+ erfc
γs −
γm Re{Ψ(k, m)}
4
m=1
(4.8)
and the conditional BER for M-QAM modulation is given by
PeQAM (γs , γi |φ, f )
√
!
X√
M −1
√
=1− 1− √
2 + erfc
γs −
γm Re{Ψ(k, m)}
2 M
m
.
! !!2
X√
√
− erfc − γs −
γm Re{Ψ(k, m)}
m
(4.9)
Both error probabilities were computed under the premise that the phase and
the frequency could be random. In that case the knowledge of their distributions,
p(φ) and p(f ), would allow us to compute the actual probabilities but it does not
guaranteed that closed-form expression can be found. Then, the bit error rate can
103
be obtained from the conditional probabilities as
Pe (γs , γi ) =
Z Z
φ
P e(γs , γi |φ, f )p(φ)p(f )dφdf.
(4.10)
f
In order to investigate the effect of narrowband interference in an Additive
White Gaussian Noise (AWGN) environment, an OFDM system was simulated
using 4-QAM. Finding a mathematical expression for the case when more than
one tonal signals interfere turns out to be a burden, so it is better to perform
simulation to quantified the impact on the performance. Figure 4.14 shows the Bit
Error Rate (BER) of the simulated system without interference and with different
levels of a narrowband interferer. The figure shows a clear decrease in performance
due to the interference. We also notice that when the energy of the interference is
large, an error floor occurs.
4.4.2 Broadband Noise Bit Error Rate
We have derived a model for the broadband emissions generated by a platform and
now we want to compute the impact of this noise on the radio communications.
We compute the BER for a digital system that uses either BPSK or M-QAM
modulation affected by noise that is K-distributed. Let the received signal be
Y [k] = X[k] + I[k].
104
OFDM 4−QAM over AWGN channel
with different Narrowband Interferers
0
10
−1
10
Bit Error Rate
−2
10
−3
10
−4
10
−5
10
Theoretic without Interferer
Interferer 0dB ratio
Interferer 6dBs ratio
Interferer 12dBs ratio
2 Interferers 3.5 dBs ratio
−6
10
0
2
4
6
8
10
12
14
Eb/No
Figure 4.14: BER of 4-QAM OFDM with narrow band interference
where X is the transmitted signal with Eb energy and I is a K-distributed platform
noise with parameters b and r.
4.4.2.1 BPSK
The probability of error for a BPSK modulated signal affected by K-noise is computed as follows:
p
p
p
P e = P (I < − Eb |X = Eb )P (X = Eb )
p
p
p
+ P (I > Eb |X = − Eb )P (X = − Eb ),
105
since the distribution is symmetric around 0 and we are analyzing an equiprobable
BPSK signaling, we have that
p
p
p
p
1
1
P e = P (I < − Eb |X = Eb ) + P (I < − Eb |X = Eb ),
2
2
p
p
= P (I < − Eb |X = Eb ),
and since the interference is K-distributed we have that:
Pe =
Z
√
− Eb
−∞
1
√
πbΓ(ν + 1)
|x|
2b
ν+ 21
Kν+ 1
2
|x|
b
dx.
(4.11)
Solving equation (4.11) using the formulas from [44] and after some further simplifications we have that
√
√
1
(ν + 1) Eb
Eb
1
1 3 1
Pe = √
π(ν + 1)Γ(ν + 1) −
Γ
+ ν 1 F2
; , − ν; 2
b
2
2 2 2
4b
2 π(ν + 1)Γ(ν + 1)
√ 2ν+2 Eb
1
3
Eb
Γ − − ν 1 F2 ν + 1; + ν, 2 + ν; 2
,
−
2b
2
2
4b
(4.12)
where p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z) is the generalized hyper geometric function.
The expression just derived is cumbersome and in some cases it is more convenient to derive bounds that make the solution of a problem more tractable. We first
106
find the moment generating function from the characteristic function as follows:
MX (t) = ϕX (−it),
=
1
.
(1 − b2 t2 )ν+1
Then, we can find a bound on the bit error rate using the Chernoff bound [45], i.e.
Pe ≤ e−t
≤ e−t
√
√
Eb
Eb
MX (t),
∀t > 0,
1
.
(1 − b2 t2 )ν+1
Now, the t that minimizes the last expression can be found by performing a simple
minimization, i.e.
√
√
e−t Eb
(1 − b2 t2 )ν+1
e−t Eb
∂
=0
∂t (1 − b2 t2 )ν+1
p
2b2 (ν + 1)t
− Eb +
=0
1 − b2 t 2
from which we have that
−
p
p
Eb + Eb b2 t2 + 2b2 (ν + 1)t = 0,
then
t=
−2b2 (ν + 1) ±
p
4b4 (ν + 1)2 + 4Eb b2
√ 2
,
2 Eb b
107
and since t > 0 and ν > −1
ν+1
t=− √ +
Eb
s
1
(ν + 1)2
+ 2.
Eb
b
Therefore,
Pe ≤ e
1−
b2
ν+1−
√
+
− ν+1
Eb
q
E
r 2 + 2b
b
q
(ν+1)2
Eb
+
2 ν+1
1
b2
q
Eb
2
exp ν + 1 − (ν + 1) + b2
=
q
ν+1 .
2b2 (ν+1)
Eb
1 + (ν+1)2 b2 − 1
Eb
(4.13)
We can also find a better approximation of the probability of error by noticing
that for large
|x|
b
≫ |(ν + 1/2)2 − 1/4|, namely [46]
Kν+1/2
|x|
b
s
πb − |x|
e b,
2|x|
ν+1/2
≈
and
Pe =
Z
√
− Eb
1
√
πbΓ(ν + 1)
|x|
2b
|x|
b
dx,
Kν+1/2
ν+1/2 s
1
πb − |x|
|x|
√
e b dx
≈
2b
2|x|
πbΓ(ν
+
1)
−∞
√ 2−ν−1 Γ ν + 1, bEb
p
, for Eb ≫ b|(ν + 1/2)2 − 1/4|
=
Γ(µ + 1)
−∞
Z − √E b
(4.14)
108
where Γ(s, x) is the incomplete gamma function given by
Γ(s, x) =
Z
∞
ts−1 e−t dt.
x
√
Eb < b|(ν + 1)2 − ν − 1|, we still can find a lower bound for the
√
probability by choosing a value p > Eb such that p ≫ b|(ν + 1)2 − ν − 1|, the
Now, if
lower the value of p the tighter the bound then
ν+1/2
|x|
|x|
dx
Pe =
Kν+1/2
2b
b
−∞
ν+1/2
Z − √E b
|x|
|x|
1
√
+
dx,
Kν+1/2
b
πbΓ(ν + 1) 2b
−p
Z − √E
ν+1/2
b
2−ν−1 Γ ν + 1, pb
|x|
|x|
1
√
≈
+
dx
Kν+1/2
Γ(ν + 1)
b
πbΓ(ν + 1) 2b
−p
2−ν−1 Γ ν + 1, pb
.
(4.15)
≥
Γ(ν + 1)
Z
−p
1
√
πbΓ(ν + 1)
4.4.2.2 Rectangular QAM
The BER for the case of a rectangular QAM modulation can be derived from the
√
previous results. For PAM with M levels and one-half the average signal power
[34] the probability of symbol error is given by
P√
M
√
M −1
√
P (| Re{I(k)}| > A),
M
√
M −1
= √
(P (Re{I(k)} > A) + P (Re{I(k)} < −A)) .
M
=
109
Since the distribution is symmetric,
P√
where A =
q
3Es
.
M −1
M
=
√
M −1
√
(2P (Re{I(k)} < −A)) ,
M
We can also find an expression in terms of the average bit
energy, Eb , rather than in terms of the average symbol energy, Es , using the relationship Es = Eb log2 (M ). Then, we have that
P√
M
=
√
√
A(ν + 1)
1
1 3 1
1
M −1
√
√
Γ
+ ν 1 F2
; , −
π(ν+1)Γ(ν+1)−
b
2
2 2 2
π(ν + 1)Γ(ν + 1)
M
!
2ν+2 1
3
A2
A
Γ − − ν 1 F2 ν + 1; + ν, 2 + ν; 2
. (4.16)
−
2b
2
2
4b
Thus, for a QAM signal with M levels, the probability is given by [34]
Pe = 1 − (1 − P√M )2 .
We can approximate the distribution whenever A ≫ b|(ν + 1/2)2 − 1/4|, i.e.
P√M ≈
√
M −1
√
M
2−ν Γ ν + 1, Ab
Γ(ν + 1)
!
.
Then, the BER is approximated by
Pe ≈ 1 −
1−
√
M −1
√
M
2−ν Γ ν + 1, Ab
Γ(ν + 1)
!!2
.
(4.17)
110
0
10
Probability of bit error
−1
10
−2
10
BPSK Exact
BPSK Approximation
BPSK Chernoff Bound
16−QAM Exact
16−QAM Approximation
−3
10
0
2
4
6
8
10
12
SNR per bit
14
16
18
20
Figure 4.15: Bit error rate for BPSK and 16-QAM modulation with K-distributed
noise ν = −0.99
Figures 4.15 and 4.16 show the bit error rate for communication systems affected by K-distributed noise with ν = −0.99 and ν = −0.45, respectively. Specifically, the exact and the approximate BER for a BPSK and 16-QAM modulation
are shown as means of comparison. Also the Chernoff bound for BPSK is also
shown.
In the next chapter, we present different architectures designed to improve the
performance of a communication system under interfering signals such as the ones
studied in this dissertation.
111
0
10
−1
10
Probability of bit error
−2
10
−3
10
−4
10
−5
10
BPSK Exact
BPSK Approximation
BPSK Chernoff Bound
16−QAM Exact
16−QAM Approximation
−6
10
0
2
4
6
8
10
12
SNR per bit
14
16
18
20
Figure 4.16: Bit error rate for BPSK and 16-QAM modulation with K-distributed
noise ν = −0.45
112
Chapter 5 – Noise Immune Radio Architecture
In the previous chapters, we described the challenges that a computing platform
radio receiver faces since it was not designed to deal with the interference caused by
the platform itself. The presence of platform noise results in higher power consumption and a reduction of the throughput [6] in comparison to an interference-free
radio at some given SNR. This noise causes a loss of sensitivity in today’s radios
and as a consequence a higher Signal-to-Noise Ratio (SNR) needs to be used to
accomplish the BER requirements for a given application.
The design of new architectures and methods to address the interference problem can fall into one of the following categories:
• Interference Avoidance
• Interference Cancellation
In the interference avoidance category, the transmitter needs to know before
hand the characteristics of the interference signal so that a suitable modulation,
error correction code, pulse shape or any other technique can be used to transmit
an interference-protected signal. For example, OFDM as a multiple carrier system
gives us the flexibility to decide what is sent on each subcarrier, then we can
avoid a narrowband interferer by sending no usable data in the corresponding
subcarrier, whose frequency has to be known at the transmitter, as shown in figure
113
5.1. Another implementation for narrowband interference is the one proposed by
InterferenceAvoidance on OFDM
Frequency
f1
f2
f3
fi
f5
f6
Figure 5.1: OFDM avoidance strategy
Wu in [47]. It consists of using spreading codes over all bands such that the
interference is spread out over all subcarriers. This minimizes the effect of the
interference making its effects almost negligible.
In this dissertation, we are mostly interested in improving the performance
of the receiver. Therefore, our radio design lies within the cancellation category,
where the receiver has to deal with the presence of interference and the transmitter
does not have any knowledge of the conditions of either the channel, the receiver
or both. Conceptually, our noise immune receiver architecture needs to do two
things: estimate the interference and cancel it. Both of these operations can be
either in time, in frequency or in both domains.
In this chapter, we first present an architecture for the narrowband scenario
that consists of an estimation part and a mitigation part. We exploit the fact that
the noise consists mostly of tonal signals produced by clocks and periodic signals
114
on the platform. Then, we present an estimation technique for the broadband
scenario and a mitigation using an extended Kalman filter based on a derived
state-space model.
On the one hand, tonal noise has been widely studied with a considerable quantity of mitigation architectures and methods available on the literature, [48, 49, 50]
among the newest ones. Consequently, we have focused our research on the performance of OFDM under tonal interference and in finding a modified architecture
adapted to our needs.
On the other hand, K-distributed noise has not been studied in the context
that has been presented here. The estimation of the parameters of the noise
distribution is essential before any attempt to mitigate it could be applied. But,
we encounter the difficulty that only a limited number of samples is available
for this task. OFDM implementations, e.g. LTE, are not the exception and as we
mentioned, only a limited number of samples, known as pilots symbols, are used for
synchronization, channel estimation or as in our case, noise estimation. Different
low complexity estimation techniques based mostly on the method of moments,
have been developed by different researchers but they do not perform well with
a small number of samples. Consequently, we have derived here a new estimator
that retains the simplicity and computational requirements of other methods but
has a better performance than them when a small number of samples is available.
115
5.1 The Narrowband Case
Narrowband noise is produced by the harmonics of clocks and periodic signal as
seen in a previous chapter. In this section we present an architecture that estimates
and mitigates the noise produced by them.
The receiver architecture is based of the one described in [50]. This is shown in
figure 5.2, where the noise shaping and mitigation blocks have been adapted to the
case studied here. Since the radio receiver is implemented in the same platform
that is responsible for the interference, it exploits this fact to obtain information
to generate the reference signal that is used to mitigate the noise.
Figure 5.2: OFDM receiver architecture with platform interference mitigation
Figure 5.3 shows a scheme of the architecture that is used to estimate narrowband noise. The estimation is performed under the assumption that the peak
amplitude of the interferer is greater than the sub band component that it is affecting. The OFDM signal is demodulated, modulated and substracted from the
original one, this leaves mostly the noise when the SNR is high to be used to estimate the interference. A normalized linear mean square (NLMS) is then used
to remove the noise from the received signal. Mathematically we can express the
process as follows:
116
Signal is Received
Demodulate Signal
Modulate signal
-
+
Generate Tonal
Signal
Hard Decoding
NLMS
FFT
Set to 0 values
less than threshod
Demodulate Signal
Figure 5.3: Platform interference mitigation algorithm block diagram
First, the received signal goes through the FFT block Y = Qy. The result is
then demodulated and subtracted from the signal before demodulation. This has
the objective of estimating the interference in the frequency domain. The resulting signal is then passed through a noise shaping filter which is designed according
to the characteristics of the expected platform interference. Then, the estimated
interference is transformed back into the time domain and the interference cancellation is then performed. In other words, the received signal is mapped to a QAM
symbol and a hard decision is made,
Y 7→ s̃
this eliminates some of the errors, specially at high SNR. Only those severely
affected by noise interference remain in error. The bits are then modulated again,
117
i.e.
s̃ 7→ Z.
Then, the interference is estimated as
D = Y − Z ≈ N + I.
The estimate of the interference is obtained by passing the signal through a shaping
filter that finds the most accurate ĩ ∈ I such that the estimated interference
becomes the input for the adaptive LMS filter. Namely,
î = IF F T (D̃),
where D̃ is the output of the shaping filter whose input is D.
Figure 5.4: Adaptive NLMS filter
118
The idea behind this approach is to estimate the interference so it can be
subtracted from the received signal. This is done using the Normalized Linear
Mean Square (NLMS), where the coefficients of the filter are computed by
wn+1 = wn + β
î∗ (n)
ǫ+ k î(n) k2
e(n),
where ǫ is a relatively small positive constant that guarantees the convergence of
the algorithm. Figure 5.4 shows a scheme of the NLMS filter that is used.
Figure 5.5: Mitigation of a narrowband interferer, SIR = −6 dB, using threshold
decision
We test the suitability of the architecture just described by simulating the
OFDM system under AWGN and narrow band interference with a 4-QAM modulation. First, we simulate a narrowband interferer with a signal to interference
ratio (SIR) of −6 dB. The noise shaping is implemented using a threshold scheme,
119
where everything below is considered to be noise and discarded. The filter shapes
the result with the characteristics of the interferer that are known. Figure 5.5
shows the results of the mitigation architecture with different number of taps for
the filter.
Figure 5.6 shows the results of the mitigation architecture for a narrowband
interferer with a SIR of −12 dB with respect to the average signal amplitude. The
interference is so severe that there exists an error floor that an increase in the
number of tabs of the filter barely improves the BER.
Figure 5.6: Mitigation of a narrowband interferer, SIR = −12 dB, using threshold
decision
Moreover, figure 5.7 shows the results of the mitigation architecture for two
narrowband interferers with each of them having a SIR of -3.5 dB. The error floor
that can be seen in all the plots are due to the spectral leakage of the interfering
120
signals. The spectral leakage occurs due to the N-FFT that is applied to the
incoming signal.
Figure 5.7: Results using threshold decision with 2 interferers at SIR = −3.5 dB
each
5.2 The Broadband Case
We have shown that the broadband noise produced by the platform can be well
described by the two-sided K-distribution. In this section, we derive a new method
for estimating the parameters of the distribution when a limited number of samples is available. The derivation is done under the premise that in any OFDM
architecture only a limited number of pilot symbols is available for estimation.
The estimation of the parameters of a distribution with a limited number of
samples available is usually a challenging task. A reduced number of samples
121
constrains the use of the method of moments (MoM) which despite having low
complexity has relatively low performance due to its high dependency on the sample size. Estimators with smaller variance based also on moments are usually
preferred, though the one based on the maximum likelihood (ML) method turns
out to be the method of choice. Despite the fact that ML estimators are optimal,
in some cases they require either the evaluation of uncommon functions or the
solution of non-linear equations when no closed-form expressions of them exist.
Different types of estimators for the K-distribution have been proposed that
try to overcome the high variance of the MoM and the difficulty in finding ML
estimates. Iskander et al. in [51] propose the use of fractional moments which
they show produce estimates with lower variance than the MoM. The authors
in [52] and [53] propose the use of logarithmic estimators. ML estimates for a
limited range of ν were presented by Raghavan in [41] based on an approximation
of the K-distribution using the Gamma distribution. Iskander and Zoubir in [54]
introduce a generalized version of the distribution that the authors named as the
generalized Bessel K distribution (GBK) thanks to which in [51] they were able to
find a ML closed-form expression for b of the K-distribution parameters in terms of
ν. The distribution results from a generalized Gamma random variable with scale
parameter that is also generalized Gamma distributed. The probability density
function (pdf) is given by
fXGBK (x) =
2c
2c (α1 +α2 )−1
x
β
βΓ(α1 )Γ(α2 )
Kα2 −α1
2c !
x
2
,
β
(5.1)
122
with c, α1 , α2 , β ∈ R+ . It is understood that fXGBK (x) ≡ fXGBK (x; α1 , α2 , β, c). The
double-sided K-distribution can be derived from the GBK as
1
fX (x) = fXGBK (|x|; 1/2, ν + 1, 2b, 2).
2
(5.2)
Other methods rely heavily on numerical methods to find the parameters estimates. They make use of iterative methods such as the Expectation-Maximization
in [55], 2-D maximizations as in [56], neural networks as in [52] or non-linear techniques, among others. These methods are robust albeit computationally expensive,
which make real time applications infeasible.
5.2.1 Parameter Estimation Methods
We give brief descriptions of some of the estimators that have been proposed for
the K distribution which will be compared with our proposed method. The use of
any of these methods depends on the samples and the computational resources that
are available. Specifically, we proceed to find estimators for the two parameters ν
and b that define the double-sided distribution.
5.2.1.1 Method of Moments
The method of moments (MoM) computes the parameters of a distribution by
finding expressions in terms of its moments. The moments are then replaced by
their corresponding empirical ones computed from the sample set.
123
The moments of a K-distributed random variable X are defined as µk ≡ E[X k ]
and they are computed as follows:
µk =
=
Z
∞
−∞
k−1
2
xk fX (x),
Γ
(−b)k + bk Γ 1+k
2
√
πΓ(ν + 1)
k
2
+ν+1
.
(5.3)
We notice that whenever k is an odd integer E[X k ] = 0 and when k is an even
integer the moments are given by
µk =
2k bk Γ
Γ k2 + ν + 1
√
,
πΓ(ν + 1)
1+k
2
k = 2, 4, 6, . . .
Then, it follows that the second moment (the variance for zero-mean distributions)
is given by
µ2 =
22 b2 Γ
1+2
2
√
Γ (1 + ν + 1)
,
πΓ(ν + 1)
= 2b2 (ν + 1),
and the fourth moment by
24 b4 Γ 52 Γ (2 + ν + 1)
√
,
µ4 =
πΓ(ν + 1)
= 12b4 (ν + 1 + 1)(ν + 1).
(5.4)
124
Thus, the kurtosis which is defined as β2 ≡ µ4 /µ22 is given by
β2 =
3(ν + 1 + 1)
.
ν+1
It follows that the estimators of ν and b are found to be
ν̂ =
3
β̂2 − 3
− 1 and b̂ =
v u
u µ̂ β̂ − 3
t 2 2
6
,
(5.5)
respectively, where µ̂2 is the second empirical moment of the data and β̂ is the
empirical kurtosis.
The estimators are computationally inexpensive and easy to implement but
they depend heavily on the number of samples available. A small sample set
causes the variance of the estimator to increase to levels that make the estimation
unreliable as it is seen on Fig. 5.8. The figure shows MoM estimates and their
variance for parameters ν = 0.5 and b = 0.02 over 5000 independent trials as a
function of the sample size.
5.2.1.2 Fractional Moments
Iskander in [42] noticed that fractional moments produce estimates with lower
variances. His method proposes the use of the ratio
αp,1 =
µp+2
µp µ2
0 < p ≤ 2,
(5.6)
125
0.03
b
0.02
Exact
MoM
0.01
0
−0.01 1
10
2
3
10
4
10
10
50
ν
Exact
MoM
0
−50 1
10
2
3
10
4
10
10
N
Figure 5.8: Parameter estimation (MoM) ν = 0.5 b = 0.02
which in the case of the type of K-distribution studied here comes with the restriction of p 6= 1. Replacing the expression with moments formula given in (5.3),
applying the properties of the gamma function and after some simplifications, we
find that the expression for the double-sided K-distribution turns out to be
αp,1 =
=
√
1+p+2
Γ p+2
+ ν + 1 πΓ(ν +
2
2
1+p
p
1+2
Γ
Γ 22 + ν
+
ν
+
1
Γ
2
2
2
+ p) p2 + ν + 1
Γ
Γ
(1
(ν + 1)
,
1)
,
+1
(5.7)
126
which is independent of b. Then, the parameter ν can be easily estimated using
the corresponding empirical ratio, namely
ν̂ =
(p + 1)(p/2 + 1) − α̂p,1
α̂p,1 − (p + 1)
0 < p ≤ 2, p 6= 1.
(5.8)
The value of b can be estimated using the empirical second moment or any
other moment since they establish a relation between b and ν.
5.2.1.3
X r log(X) estimation
Blacknell and Tough in [53] proposed an estimator of ν based on X r log(X) which
gives a comparable accuracy with the fractional moments estimators among others.
They noticed that setting r = 1 leads to simple expressions for the estimator of ν
of the one-sided K distribution. We proceed with the derivation of the estimate
that corresponds to the double-sided K distribution. This derivation follows the
one given in [53], though in this case it is easily seen that setting r = 2 gives
an estimate that does not have ν as an argument of any ψ(·), Γ(·) or any other
exotic function. We also notice that the distribution is zero-mean and symmetric
around the mean, therefore it makes sense to work with the absolute values since
the logarithm of negatives values is not defined.
127
Now,
)Γ( 2r + ν + 1)
2r br Γ( 1+r
2
√
E[|X| ] =
,
πΓ(ν + 1)
ψ(1) ψ(ν + 1)
E[log(|X|r )] =
+
+ log(b),
2
2
1+r
r
1
) + ψ(1 + + ν)).
E[|X|r log(|X|r )] =E[|X|r ] (log(4b2 ) + ψ(
2
2
2
r
(5.9)
Now, setting r = 2 on the previous results we evaluate the following expression:
ψ( 32 ) ψ(1 + 1 + ν) ψ(1) ψ(ν + 1)
E[|X|2 log(|X|)]
−
E[log(|X|)]
=
log(2)
+
+
−
−
.
E[|X|2 ]
2
2
2
2
(5.10)
1
Since ψ(ν + 1 + 1) = ψ(ν + 1) + ν+1
and ψ(3/2) = ψ(1) − 2 log(2) + 2, we have that
ν=
1
2 log(|X|)]
2 E[|X|E[|X|
2]
− 2E[log(|X|)] − 2
− 1.
(5.11)
The estimate ν̂ is obtained by replacing the expected values by their corresponding
empirical estimates.
5.2.2 Maximum Likelihood Estimation
Maximum Likelihood (ML) estimation is one of the most reliable estimator even
when only a limited number of samples exist. Also, the ML estimator is asymptotically unbiased and it attains the Cramer-Rao lower bound asymptotically better
than any other unbiased estimator. Though the ML estimator performs better than
other methods, its high complexity prevents its use when limited computational
128
capabilities are available.
The ML estimator results from the maximization of the likelihood or the loglikelihood function, whichever gives a more tractable expression. In the case of the
K-distribution the log-likelihood l is preferred and it is given by
l = l(x1 , . . . , xN |b, ν),
N
N
X
|xi |
1 X
1
.
log(|xi |) +
log Kν+1/2
+ ν+
= N log √
2 i=1
b
πbΓ(ν + 1)(2b)ν+1/2
i=1
(5.12)
The parameters estimators are obtained by maximizing the log-likelihood function,
but it is evident from its partial derivatives
∂l
(ν + 3/2)
= −N
+
∂b
b
N
X
|xi |
i=1
2b2

Kν−1/2

|xi |
b
+ Kν+3/2
Kν+1/2
|xi |
b
|xi |
b

(5.13)
(5.14)
,
and
∂l
= −N (log(2b) + ψ(ν + 1)) +
∂ν
N
X
i=1
|xi | +
N
X
i=1
∂
K 3
∂ν ν+ 2
Kν+ 1
2
|xi |
b
|xi |
b
,
that finding closed-form solutions for both parameters is a difficult task. In fact,
closed-form expressions cannot be directly obtained from them [41, 51, 56, 55, 53].
Iskander et al. in [51] derived a closed-form expression for one of the parameters
by first finding an expression for the parameter β from the generalized Bessel K
129
distribution:
!
N
1 X
ψ(α1 ) + ψ(α2 )
+
log(xi ) ,
β = exp −
2
N i=1
(5.15)
where ψ(α) is the digamma function. The parameter b of the double-sided Kdistribution can easily be obtained following the relationship given in (5.2). Replacing the corresponding values (α1 , α2 , β, c) = (1/2, ν + 1, 2b, 2), it follows that
!
N
1
X
ψ(
)
+
ψ(ν
+
1)
1
1
log(|xi |) ,
+
b = exp − 2
2
2
N i=1
(5.16)
which turns out to be a non-linear function of ν. The authors also found, from the
maximization of the likelihood function that
N
X
i=1
where
K(xi ) log
xi
β
N
N
X
1 X
xi
=
+ N,
K(xi )
log
N i=1
β
i=1
c c 2
xi 2
+ Kα2 −α1 +1 2 xβi
Kα2 −α1 −1 2 β
xi
,
K(xi ) =
c/2
β
xi
Kα2 −α1 2 β
(5.17)
(5.18)
for the generalized Gamma distribution. The maximum likelihood estimates can
then be found using the expressions just presented and the equivalence in (5.2) for
the double-sided K-distribution. Although there exists a closed-form expression for
one the parameters, we are still required to use computationally intensive methods
to find both parameters estimates.
In [51] the maximum likelihood estimates are found using a cubic spline interpolation. In [55] the iterative method known as the Expectation-Maximization
130
is employed. Finally in [52] and [57] neural networks are used. This has lead us
to investigate a new estimate that retains the simplicity of the estimators previously presented which yields accurate estimates when we only have access to small
sample sets of the process.
5.2.3 Cramer-Rao lower bound
The Cramer-Rao lower bound (CRLB) gives a bound on the performance of an
estimator. Specifically, it tells us about the minimum value that the variance of
an unbiased estimator can achieve. In other words,
CRB(θ) ≤ var(θ),
(5.19)
CRB(θ) = E[(∂θ l(x1 , x2 , . . . , xN ; θ))2 ],
(5.20)
where θ is the estimator.
The CRLB is given by
where l(·) is the log-likelihood function. Deriving a closed-form expression of the
CRLB for the K-distribution would be a daunting task but for specific parameters
of the distribution it can be computed numerically. Kay and Hu in [58] have also
proposed a method to compute the bound using the characteristic equation. For
the purpose of comparing the different estimation methods we proceed to evaluate
the CRLB numerically.
131
5.2.4 New estimation method
We propose an approximate method that leads us to a more tractable expression
for the estimation of the K-distribution parameters. This approximation turns out
to have better performance than others for a low number of samples.
We now proceed to derive estimators that have as a starting point an expression
found in [51] that results from the maximization of the log-likelihood of the GBK
distribution. They found that maximizing the log-likelihood gives the following
expression:
N
X
i=1
where
K(xi ) log
xi
β
N
N
X
1 X
xi
=
+ N,
K(xi )
log
N i=1
β
i=1
c c 2
xi 2
+ Kα2 −α1 +1 2 xβi
Kα2 −α1 −1 2 β
xi
.
K(xi ) =
c/2
β
xi
Kα2 −α1 2 β
(5.21)
(5.22)
Since standalone expressions derived from the previous equations are not known,
using numerical methods as in [51] are the only methods to find the estimators.
We notice that the presence of the modified Bessel function of the second kind,
Kg (·) adds complexity to the derivation of estimators. Thus, it is more convenient
to express Kg (·) in terms of well known functions since it eases the burden of
finding the estimators. A suitable approximation will do the trick for a certain
range of values.
Now, it follows that for a large argument the modified Bessel function K can
132
be approximated by [46]
Kλ
|xi |
b
≈
s
πb − |xi |
e b ,
2|xi |
|xi | ≫ b
(5.23)
where the approximation corresponds to an equality regardless of its argument
when λ = ± 21 .
We proceed to obtain an expression for the estimator of b for the double-sided
distribution by first using the equivalence defined in (5.2) and the approximation
(5.23) in (5.22). It follows that
K(|xi |) =
Kν−1/2
|xi |
b
+ Kν+3/2
|xi |
b
|xi |
,
2b
Kν+1/2 |xbi |
q
q
|x |
|x |
πb − bi
πb − bi
+
e
e
2|xi |
2|xi |
|xi |
q
≈
,
|xi |
πb − b
2b
e
2|xi |
=
Now, let
PN
i=1 K(|xi |)f
N
X
i=1
|xi |
b
|xi |
.
b
(5.24)
then
K(|xi |)f
|xi |
b
N
X
|xi |
|xi |
≈
,
f
b
b
i=1
(5.25)
holds whenever f (·) is a monotonic function. Then, assuming that only a small
fraction of the data does not satisfy the condition |xi | ≫ b and using (5.24) on
133
(5.21) we have that
!
N
1 X
|xi |
= N,
−
log
K(xi ) log
N i=1
2b
i=1
!
N
N
X
1 X
|xi |
|xi |
|xi |
log
−
log
≈ N,
b
2b
N i=1
2b
i=1
N
X
|xi |
2b
which holds since the log function is monotonic.
It is now easy to show in a few steps that the estimator of b is given by
!
N
N
1 X
1 X
|xi | log(|xi |) −
log(|xl |) .
b̂ =
N i=1
N l=1
(5.26)
The expression for b̂ just derived does not depend on ν, it is computationally
inexpensive and it can be implemented using a fairly simple architecture.
The parameter ν can be easily estimated from any of the moments and the
estimate of b just derived. One of the options is to use the second moment as
follows:
ν̂ =
µ̂2
b̂2
− 1.
(5.27)
The estimators assume that the |xi |’s values that are larger than b outnumber
those that are not, because this ensures that the actual value of the K function is
mostly dominated by the approximation (5.23). On the one hand, if b is infinitesimally small the condition is easily met since almost all |xi | are larger than it. On
the other hand, if b is large, our estimators still perform well whenever N is small
since only a fraction of the |xi |’s would be smaller than b and the approximation
134
holds.
5.2.5 Estimator bias
The bias of the estimator is given by bias(b̂) = E[b̂ − b]. We compute first E[b̂], it
follows that
!#
N
N
1 X
1 X
|xi | log(|xi |) −
log(|xl |)
E[b̂] = E
N i=1
N l=1
(
)
N
N
1 X
1 X
=
E [|xi | log(|xi |)] −
E [|xi | log(|xl |)]
N i=1
N l=1
"
N −1
1
= E [|xi | log(|xi |)] −
E [|xi | log(|xl |)] − E [|xi | log(|xi |)]
N
N
1
= 1−
E [|xi | log (|xi |)] − E [|xi |] E [log (|xl |)]
N
1
1
2
E[|xi |] log(4b ) + ψ (1) + ψ (ν + 3/2) − E [|xi |] E [log (|xl |)]
= 1−
N
2
ψ(1) ψ (ν + 1)
1
1
2
= 1−
E [|xi |]
log(4b ) + ψ (1) + ψ (ν + 3/2) −
−
− log(b)
N
2
2
2
1 2bΓ(ν + 3/2)
ψ (ν + 3/2) − ψ (ν + 1)
√
= 1−
log(2) +
N
2
πΓ(ν + 1)
Then, the bias is given by
bias(b̂) = b
1
1−
N
Γ(ν + 3/2)
√
2 log(2) + ψ (ν + 3/2) − ψ (ν + 1) − 1 .
πΓ(ν + 1)
(5.28)
135
The asymptotic bias can be computed from the previous expression as
Γ(ν + 3/2)
2 log(2) + ψ (ν + 3/2) − ψ (ν + 1) − 1 , (5.29)
lim bias(b̂) = b √
N →∞
πΓ(ν + 1)
where it can be easily seen that limN →∞ bias(b̂) = 0 only when ν = 0. Therefore,
the estimator just derived is not unbiased or asymptotically unbiased, except for
ν = 0.
The accuracy of the estimator can be improved by subtracting the bias from
the estimator, but by doing so we ended up in the same situation as before, even
though without the modified Bessel function of the second kind, since the bias of b
depends on the value of ν. Therefore, in order to find the estimates we will need an
iterative process to find the values which increases the computational requirements
of the estimators.
The performance of this estimator and the ones described before are quantified
next.
5.2.6 Simulation Results
A series of Monte Carlo simulations were carried out to evaluate the performance
of the estimators. The estimation methods were applied over 5000 realizations of
a K-distributed process and their performances were analyzed using as a measure
the mean-square error. For comparison, a maximum likelihood was computed
using a one-dimensional search for the parameter ν with (5.16) substituting the
136
corresponding value in (5.18). Also, the Cramer-Rao lower bound (CRLB) was
evaluated numerically to compare it with the others estimators.
We first analyze the performance of the estimators in terms of the number of
samples available. The analyses are constrained to take on parameters with small
values, ν < 3 and b < 10 which does not limit its applicability since most of the
known processes fall within those ranges. Fig. 5.9 and Fig. 5.10 show some of
the results for a K-random random process with parameters ν = 0.5, b = 0.02
and for ν = −0.2, b = 0.8. Comparing with the other methods, our estimator
outperforms them whenever N is small, N < 500 for the cases shown here. We
notice that when N is large, the performance of our estimator does not improve
as the other methods do, being outperformed by them. This is not an unexpected
result since our estimator is based on an approximation that does not guarantee the
asymptotic unbiasedness of the estimator for all values of ν. We also notice that
for small N our estimator performs similar to the maximum likelihood estimator
with some values giving a slightly better performance due to the inaccuracies of
the one-dimensional search that depends on the choices of the grid spacing.
The behavior of the estimators in terms of ν for the range −1 to 1.5 is also analyzed. We present here simulation results with b = 0.02 and N = 32, 64, 128, 256, 512.
Figs. 5.11, 5.12 and 5.13 show the estimators when N = 32, 64, 128. The results
confirm that our estimators are more consistent and perform better than the others
except for ν < −0.5, where the fractional MoM and the |x| log |x| performs slightly
better. The simulations show that our method performs quite well for parameters
b < 10 and ν < 3. This does not limit the scope of the estimator since there is
137
b
−3
10
MoM
fractional MoM 1/10
|x|2 log |x|
New estimator
CRLB
ML
−4
Mean−Square Error
10
−5
10
−6
10
−7
10
−8
10
1
10
2
3
10
4
10
10
N
(a) Estimators of b
ν
8
10
MoM
fractional MoM 1/10
|x|2 log |x|
New estimator
CRLB
ML
6
Mean−Square Error
10
4
10
2
10
0
10
−2
10
−4
10
1
10
2
3
10
10
4
10
N
(b) Estimators of ν
Figure 5.9: MSE estimators comparison: ν = 0.5 b = 0.02
138
b
0
10
MoM
fractional MoM 1/10
2
|x| log |x|
New estimator
CRLB
ML
−1
Mean−Square Error
10
−2
10
−3
10
−4
10
1
10
2
3
10
10
4
10
N
(a) Estimators of b
ν
6
10
MoM
fractional MoM 1/10
2
|x| log |x|
New estimator
CRLB
ML
4
Mean−Square Error
10
2
10
0
10
−2
10
−4
10
1
10
2
3
10
10
4
10
N
(b) Estimators of ν
Figure 5.10: MSE estimators comparison: ν = −0.2 b = 0.8
139
a wide range of K-distribution processes with parameters inside that range. The
proposed estimator together with the maximum likelihood estimator (MLE) are
the ones whose mean-squared error is closer to the CRLB when N is small. In
general, the MLE is closer to the CRLB for all values of N .
The performance of the other estimators improves as the number of samples
available for the estimation increases, as it in seen on Fig. 5.14 for N = 256.
Specially the |X| log |X| outperforms the rest but for some values in which our
estimators still outperform them.
In [53], it was argued that the estimator based on |x|r log |x| comes as a natural
limit of the fractional moments estimator of [42], this turns out to be true when
the number of samples N is large, where how large N should be depends on the
parameters ν and b, but not when N is small as it is shown in the figures. The
results confirm that the proposed estimator outperforms other estimators that are
comparable in terms of complexity and computational requirements.
We have derived a new estimation method for the K-distribution. The method
provides improved performance over existing techniques when only a limited number of samples is available. It has been shown through Monte Carlo simulations
that the method produces estimates with smaller variance than others while maintaining their simplicity and computational requirements low.
140
N = 32 b = 0.02
−2
10
MoM
fractional MoM 1/10
|x|2 log |x|
New estimator
CRLB
−3
Mean−Square Error
10
−4
10
−5
10
−6
10
−1
−0.5
0
0.5
1
1.5
1
1.5
(a) Estimators of b
8
10
MoM
fractional MoM 1/10
6
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
4
10
2
10
0
10
−2
10
−4
10
−1
−0.5
0
ν
0.5
(b) Estimators of ν
Figure 5.11: MSE estimator comparison: N = 32 b = 0.02
141
N = 64 b = 0.02
−2
10
MoM
fractional MoM 1/10
−3
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
−4
10
−5
10
−6
10
−1
−0.5
0
0.5
1
1.5
1
1.5
(a) Estimators of b
8
10
MoM
fractional MoM 1/10
6
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
4
10
2
10
0
10
−2
10
−4
10
−1
−0.5
0
ν
0.5
(b) Estimators of ν
Figure 5.12: MSE estimator comparison: N = 64, b = 0.02
142
N = 128 b = 0.02
−2
10
MoM
fractional MoM 1/10
−3
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
−4
10
−5
10
−6
10
−1
−0.5
0
0.5
1
1.5
1
1.5
(a) Estimators of b
6
10
MoM
fractional MoM 1/10
4
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
2
10
0
10
−2
10
−4
10
−6
10
−1
−0.5
0
ν
0.5
(b) Estimators of ν
Figure 5.13: MSE estimator comparison: N = 128, b = 0.02
143
N = 256 b = 0.02
−3
10
MoM
fractional MoM 1/10
Mean−Square Error
|x|2 log |x|
New estimator
CRLB
MLE
−4
10
−5
10
−6
10
−1
−0.5
0
0.5
1
1.5
1
1.5
(a) Estimators of b
6
10
MoM
fractional MoM 1/10
4
Mean−Square Error
10
|x|2 log |x|
New estimator
CRLB
MLE
2
10
0
10
−2
10
−4
10
−6
10
−1
−0.5
0
ν
0.5
(b) Estimators of ν
Figure 5.14: MSE estimator comparison: N = 256, b = 0.02
144
5.2.7 Broadband Noise Mitigation
We have shown that the broadband emissions of platform elements follow the
K-distribution. We have also proposed a method to estimate the parameters of
the distribution when a limited number of samples, pilot symbols in OFDM, is
available. We now present a method to mitigate the interference using an extended
Kalman filter based on the assumption that the noise can be described by a statespace model that produces a K-distribution.
5.2.7.1 Broadband noise state-space model
We notice that a K-distributed random variable can be generated by the multiplication of a Gaussian random variable and a Gamma distributed one [40], both
independent of each other, i.e.
Z=
√
XY,
where Y ∼ N (0, b2 ) and X ∼ Γ(ν + 1, 2) is given by
fX (x) =
1
2ν+1 Γ(ν
+ 1)
xν e−x/2 ,
x ≥ 0, ν > −1.
(5.30)
Then, it follows that the evolution in time of the K-distributed random variable
1/2
Z can be expressed as Zt = Xt Yt , where, on the one hand, Yt can be described
as an Ornstein-Uhlenbeck process [59] associated with the stochastic differential
145
equation (SDE) given by
dYt = −Yt dt +
(y)
where dWt
√
(y)
2b2 dWt ,
(5.31)
is a 1-dimensional white noise process [60], and, on the other hand, Xt
has a stationary distribution given by (5.30) with a stochastic differential equation
described by [60]
(x)
dXt = (2ν + 2 − Xt )dt + (4Xt )1/2 dWt .
(5.32)
We now derive the SDE of Zt , but first we present an important result from
stochastic theory known as Ito’s formula. Let Ut be an Ito process, then if Vt =
g(t, Ut ), we have that [60]
dVt =
∂g
∂g
1 ∂ 2g
(t, Ut )dt +
(t, Ut )dUt +
(t, Ut ) · (dUt )2 .
∂t
∂u
2 ∂u2
1/2
We define Rt ≡ Xt
dRt =
=
−
dXt
1/2
−
1/2
h
2Xt
1
2Xt
1
3/2
8Xt
h
and using Ito’s formula we find that
(dXt )2
3/2
8Xt
(5.33)
,
(x)
(2ν + 2 − Xt )dt + (4Xt )1/2 dWt
(x)
i
,
(x)
(2ν + 2 − Xt )2 dtdt + 4Xt dWt dWt
(x)
+ 2(2ν + 2 − Xt )(4Xt )1/2 dtdWt
(5.34)
i
146
and since dtdt = dtdWt = dW dt = 0 and dW dW = dt [60], then
dRt =
2ν + 2 − Xt − 1
2Rt
(x)
dt + dWt .
(5.35)
Finally, using the Ito product with the previous results it follows that
dZt = Rt dYt + Yt dRt + dRt dYt ,
√
2ν + 1 − Xt
(y)
1/2
(x)
2
+ Yt
= Xt
−Yt dt + 2b dWt
dt + dWt
,
2Rt
p
2ν
+
1
−
X
t
(y)
(x)
= −Zt dt + 2b2 Xt dWt + Zt
dt + Zt dWt .
(5.36)
2Xt
We can express the obtained results in vector form as





1/2


(x)
 dWt 
0
dXt   (2ν + 2 − Xt ) 
 2(Xt )
 dt +  1/2 √

= 
,
(y)
3
2ν+1
1
2
Zt − 2 + 2Xt
2b Xt
Zt Xt
dZt
dWt
(x)
where ν and b are the parameters of the K-distribution and dWt
(5.37)
(y)
and dWt
are
independent white noise processes.
As was mentioned before, we are interested in mitigating platform noise of
OFDM systems. Specifically, we focus on the received signal after it is down
converted and sampled. Therefore, there is the need to obtain the equivalent
difference equations form the ones just obtained. The difference equations that are
147
obtained using the Euler’s method with step size Ts are given by


 

1/2


0
x[k + 1]  x[k] + (2ν + 2 − x[k])Ts  2(x[k]Ts )
 w1 [k]
+  2
1/2 p

=


z [x]
2ν+1
3
2
T
z[k] + z[k] − 2 + 2x[k] Ts
2b x[k]Ts
z[k + 1]
w2 [k]
x[k] s
(5.38)
where w1 and w2 are independent, zero mean, unit variance Gaussian random
variables and Ts is the sampling period.
We test the accuracy of the model by generating time series for different parameters of ν and b using (5.38). The histograms of the time series are calculated and
compared with the probability density function of the K-distribution that results
from the corresponding parameters ν and b.
Figures 5.15, 5.16, 5.17 show time series and their respective histograms for
different parameters ν and b. The corresponding K-distribution probability density
function is also plotted on the histograms. The figures clearly show the agreement
between the generated data distribution and the K-distribution.
Now, following the notation used in the previous chapter, at the receiver we
have that the sampled signal that has been down converted is expressed as
y[k] = x[k] ∗ h[k; τ ] + n[k] + ibb [k].
(5.39)
Let us define z[k] = ibb [k] and v[k] = x[k] ∗ h[k; τ ] + n[k], then the received signal
is given by
y[k] = z[k] + v[k].
(5.40)
148
Generated K−noise: ν = 1 b = 0.0005
Generated K−noise: ν = 1 b = 0.0005
1
Histogram generated noise
K−distribution
0.01
0.9
0.008
0.8
0.006
0.7
Normalized Bins
Amplitude
0.004
0.002
0
−0.002
−0.004
0.6
0.5
0.4
0.3
−0.006
0.2
−0.008
0.1
−0.01
0
500
1000
0
−0.01 −0.008 −0.006 −0.004 −0.002
1500
units of time
(a) Time Series
0
0.002
Amplitude
0.004
0.006
0.008
0.01
(b) Histogram
Figure 5.15: Generated noise with K-distribution: ν = 1 b = 0.0005
Generated K−noise: ν = 1 b = 0.001
Generated K−noise: ν = 1 b = 0.001
1
0.02
Histogram generated noise
K−distribution
0.9
0.015
0.8
0.01
0.7
Normalized Bins
Amplitude
0.005
0
−0.005
0.6
0.5
0.4
0.3
−0.01
0.2
−0.015
0.1
−0.02
200
400
600
800
units of time
1000
(a) Time Series
1200
1400
0
−0.01 −0.008 −0.006 −0.004 −0.002
0
0.002
Amplitude
0.004
0.006
0.008
0.01
(b) Histogram
Figure 5.16: Generated noise with K-distribution: ν = 1 b = 0.001
Finally, we can find the state-space equations of the platform broadband noise by
expressing the previous equation and the ones in equation (5.38) in vector notation
149
Generated K−noise: ν = 0.5 b = 0.1
Generated K−noise: ν = 0.5 b = 0.1
1
2
Histogram generated noise
K−distribution
0.9
1.5
0.8
1
0.7
Normalized Bins
Amplitude
0.5
0
−0.5
0.6
0.5
0.4
0.3
−1
0.2
−1.5
0.1
−2
200
400
600
800
units of time
1000
1200
0
−1
1400
−0.8
−0.6
(a) Time Series
−0.4
−0.2
0
0.2
Amplitude
0.4
0.6
0.8
1
(b) Histogram
Figure 5.17: Generated noise with K-distribution: ν = 0.5 b = 0.1
as
x[k + 1] = F (k; x[k], w[k]) ,
y[k] = C(k, x[k]) + v[k],
where x[k] = x[k]
z[k] , w[k] = w1 [k]
(5.41)
(5.42)
w2 [k] , C(k, xk ) = z[x] and F (·) is
a nonlinear function given by the right side expression of equation (5.38).
5.2.7.2 Extended Kalman filter
In order to estimate the broadband noise from the received signal, we make use
of the extended Kalman filter with the state-space model just derived. The filter
needs the linearization of the model by constructing the following matrices: A =
∂F ∂F ∂C ,
B
=
and
C
=
. The computed matrices are
∂x
∂w
∂x x=x̂[k|yk ]
x=x̂[k|yk ]
x=x̂[k|yk ]
150
given by,


A=

1 − Ts +
2ν+1
−Ts 2x
2 [k]
C= 0 1 .
Ts
w
x[k] 1
(z 2 [k]Ts )1/2
w1
2x3/2 [k]
+
1/2
2 (x[k]Ts )
B =  2
1/2
z [k]Ts
x[k]
q
+
0
q
2b2 Ts
w
2x[k] 2
1 + − 23 +
2ν+1
2x[k]
Ts +


,

1/2
(2b2 x[k]) Ts x=x̂[k|yk ]
0
q
Ts
w
x 1



,
x=x̂[k|yk ]
(5.43)
(5.44)
(5.45)
Now, we have that Q1 [k], Q2 [k], are, respectively, the correlation matrix of the
process noise and the correlation matrix of the measurement noise. Then, Kalman
filter algorithm is given by
Initial Conditions:
x̂[1|y0 ] = E[x[1]],
R[1, 0] = E[(x[1] − E[x[1]])(x[1] − E[x[1]])H ],
For each k = 1, 2, . . . ,
We compute: A[k + 1, k], B[k + 1, k] and C[k + 1, k] as given by equations
(5.43), (5.44) and (5.45).
151
−1
G[k] = Re [k, k − 1]CH [k] C[k]Re [k, k − 1]CH [k] + Q2 [k]
(5.46)
α[k] = y[k] − C[k]x̂[k|yk−1 ]
(5.47)
x̂[k|yk ] = x̂[k|yk−1 ] + G[k]α[k]
(5.48)
x̂[k + 1|yk ] = F[k, x̂[k|yk ], 0]
R[k] = [I − G[k]C[k]] R[k, k − 1]
R[k + 1, k] = A[k + 1, k]R[k]AH [k + 1, k] + B[k]Q1 [k]BH [k].
(5.49)
(5.50)
(5.51)
5.2.7.3 Mitigation
Having developed a state-space model of the broadband noise we proceed to estimate it using the extended Kalman filter. The estimated noise is then subtracted from the received signal and the resulting signal is finally demodulated.
We notice that mitigation first requires the estimation of the parameters of the
K-distribution. Therefore, the estimation of the parameters is performed using
the method described in the previous section when a limited number of samples is
available as it is the case of OFDM systems where only a limited number of pilot
symbols is available.
We test the performance of the mitigation technique by means of Monte Carlo
simulations. We simulate an OFDM system with BPSK and 4-QAM modulation
where we have assumed a perfect channel, h(t; τ ) = 1. Gaussian noise along with
K-noise are added to the modulated signal according to the simulated signal-to-
152
FFT size
Number of data carriers
Number of pilot symbols
Prefix size
Bandwidth
1024
720
120
128
10 MHz
Table 5.1: OFDM parameters
noise ratio and the parameters of the K-noise. The bandwidth of the transmitted
signal is 10 MHz with a FFT size of 1024 and a unit symbol energy, Es = 1.
The pilot symbols were used to estimated the parameters of the noise. Table 5.1
summarizes the parameters of the OFDM system used in the simulation.
Figures 5.18, 5.19 and 5.20 show the plots of the bit error rate (BER) versus the
signal-to-noise ratio (SNR) of the simulated OFDM systems. The first plot shows
the results for a 4-QAM modulation for a K-noise with parameters ν = 2.5 and
b = 1. At low SNR the gain of the mitigation is minimum but with an increasing
SNR the gains go from 2 to 4 dBs. On the second plot, we observe the same gains
for a system with BPSK modulation for K-noise with the same parameters as the
first plot. The third plot shows a system with BPSK modulation but with noise
parameters ν = 1.5 and b = 2. In this case, the gains go to more than 6 dBs.
We notice that the effectiveness of the method depends directly on the parameters of the K-noise. We also notice that the discretization of the state-space model
was done using Euler’s method which is an approximate method and that a more
accurate model could provide a better mitigation performance.
153
K−noise ν = 2.5 b = 1
0
10
Error Rate
No mitigation
Mitigation
−1
10
−2
10
−4
−2
0
2
4
6
8
10
12
14
SNR
Figure 5.18: K-noise mitigation in 4-QAM OFDM systems ν = 2.5 and b = 1
K−noise ν = 2.5 b = 1
0
10
Error Rate
No mitigation
Mitigation
−1
10
−2
10
−4
−2
0
2
4
6
8
10
12
14
SNR
Figure 5.19: K-noise mitigation in BPSK OFDM systems ν = 2.5 and b = 1
154
K−noise ν = 1.5 b = 2
0
10
Error Rate
No mitigation
Mitigation
−1
10
−2
10
−4
−2
0
2
4
6
8
10
12
14
SNR
Figure 5.20: K-noise mitigation in BPSK OFDM systems ν = 1.5 and b = 2
155
Chapter 6 – Conclusions
In this dissertation we have studied the interference noise generated by computing
platforms. We have specially focused on all those emissions which are potential
sources of interference for computing platform radio receivers. The signals have
been studied by grouping them according to their spectra and bandwidth of the
radio receiver as narrowband and broadband signals.
Periodic signals, such as clocks and display signals, are the main contributors to
narrowband noise, while random data signals are the main contributors to broadband noise. On the one hand, narrowband noise has been well characterized and
some of the important results are presented here. On the the hand, broadband
noise has been considered to be Gaussian, but experimental data show that the
noise is Non-Gaussian so a new characterization of the noise is needed.
One of the main contributions of this work is the finding that the K-distribution
is an accurate model for the broadband noise that affects the radio receiver. The
K-distribution has been used in modeling signal scattering in sonar and radar and
recently for modeling fading on wireless channels, but we have shown that it also
models the platform interference on the radio bands. The validity of the model
was proven with experimental data from the captured emissions of a computing
platform.
Also, closed-form expressions for the impact on the performance in terms of the
156
bit error rate (BER) were derived. These expressions were described in terms of
uncommon functions for the broadband case and some bounds expressed as simple
functions were also derived.
Moreover, we have derived a new estimation method for the K-distribution. The
method provides an improved performance over existing techniques when only a
limited number of samples is available. It has been shown through Monte Carlo
simulations that the method produces estimates with smaller variance than others while maintaining their simplicity and computational requirements low. The
method is important because in OFDM systems only a small number of samples,
pilots symbols, are available for estimation of the noise.
The mitigation of narrowband interferers was accomplished by first estimating
the interference noise and then using an adaptive NLMS algorithm for OFDM
systems. For the case of broadband interference, the mitigation was performed by
first estimating the noise with an extended Kalman filter. The state-space model
that is used has been derived from stochastic differential equations. In both cases,
the proposed mitigation techniques improved the performance of OFDM systems.
The effectiveness of the mitigation techniques depend directly on the parameters
of the K-noise.
A natural extension of the work presented here will be the study of interference
avoidance techniques that can be employed when feedback between the receiver and
the transmitter exists. There are many mitigation techniques that have not been
considered here that could perform better than the ones presented here to mitigate
narrowband and broadband interference. Moreover, newer adaptive signal process-
157
ing algorithms, such as the ones presented in [61], could lead to better solutions
in dealing with existing and new types of interference. Also, newer manufacturing
and design technologies on computing platforms could make other sources of noise
not considered here to become relevant. This will open a field for studying which
mitigation techniques should be employed.
Finally, the models presented here as others available in the literature are derived based on assumptions that, while making the derivation more mathematically
tractable, try to stay as close as possible to the physical reality. Therefore, there
is still room for improving the existing models and/or the mathematic tools that
have been employed, thus far.
158
APPENDICES
159
Appendix A – Pseudo Random Noise
We present here the derivation of the spectrum of a Pseudo Random Binary Sequence (PRBS) following the one found in [62].
Without loss of generality we assume a binary sequence u(t) with a maximum
amplitude equal to one. Now, u(t) is a deterministic M-periodic signal,i.e. u(t) =
u(t − M ),
∀t. Then, the mean value, assuming ergodicity, is given by
N
M
1 X
1 X
u(t) =
u(t).
m , lim
N →∞ N
M t=1
t=1
Now, if u(t) is a maximum length PRBS the number of high states in the sequence
is found to be
(M +1)
2
= 2n−1 and the number of zeros is
(M −1)
2
(M − 1)
(M + 1)
1
+0·
1·
m=
M
2
2
1
1
= +
.
2 2M
= 2n−1 −1. Therefore
(A.1)
We can now evaluate the autocorrelation. We compute first the autocorrelation
160
when τ = 0 is given by
M
1 X
Ribr (0) =
[u(t)]2 ,
M t=1
M
1 X 2
=
u (t),
M t=1
M
1 X
=
u(t),
M t=1
=
1
1
+
.
2 2M
For τ = 1, . . . , M − 1 we have that
M
1 X
[u(t + τ )][u(t)],
Ribr (τ ) =
M t=1
M
1 X1
[u(t + τ ) + u(t) − (u(t + τ ) ⊕ u(t))],
Ribr (τ ) =
M t=1 2
"M
#
M
M
X
X
1 X
u(t + τ ) +
=
u(t) −
u(t + τ − l) ,
2M t=1
t=1
t=1
1
[mM + mM − mM ],
2M
m
= .
2
=
Using the result of equation A.1, we have
=
1
1
+
.
4 4M
161
Then
Ribr (τ ) =



1 +
2


1 +
4
1
2M
, τ = 0, M, 2M, . . .
1
4M
, otherwise
.
Then, the power spectrum can be computed by means of the relationship between the DFT and the DTFT. Since the autocorrelation of a periodic signal is
periodic we have that the power spectrum of the signal is given by
M −1
2π X
Φ(ω) =
Ck δ (ω − kω0 ) ,
M k=0
M −1
2πk
2π X
,
Ck δ ω −
=
M τ =0
M
where Ck are the Fourier Series coefficients (DFT coefficients) of the autocorrelation, thus
Ck =
M
−1
X
u=0
e−j2πku/M R(u),
162
and
C0 =
=
=
=
=
=
M
−1
X
R(u),
u=0
1
1
1
1
,
+
+ (M − 1)
+
2 2M
4 4M
M +1
M +1
+ (M − 1)
,
2M
4M
2(M + 1) + (M − 1)(M + 1)
,
4M
(M + 1)(2 + M − 1)
,
4M
(M + 1)2
.
4M
For k > 0,
M
−1
X
1
1
1
1
−j2πku/M
Ck = +
,
+
+
e
2 2M
4 4M
u=1
!
M −1
1
1
1 X −i2πkj/M
=
e
+
1+
,
2 2M
2 j=1
1
1 e−i2πk/M − e−i2πkM/M
1
,
+
1+
=
2 2M
2
1 − e−i2πk/M
1
1
1 e−i2πk/M − 1
=
,
+
1+
2 2M
2 1 − e−i2πk/M
M +1
=
.
4M
163
Using the previous results we have that the power spectrum is given by
#
2π
k
Φ(ω) =
,
C0 δ (ω) +
Ck δ ω − 2π
M
M
k=1
"
#
M
−1 X
2πk
2π(M + 1)
(M + 1)δ(ω) +
δ ω−
.
=
4M 2
M
k=1
"
M
−1
X
(A.2)
We can extrapolate the result just obtained for a digital signal with signal levels
−a and a. We can compute its power spectrum by noticing that any sequence of
ones and zeros can be mapped to a sequence y(t) of l’s and k’s through the following
formula: y(t) = l + (k − l)u(t). Thus, for levels −a and a we have that
y(t) = −a + 2au(t).
Then,
my = −a + 2amu ,
1
1
= −a + 2a + 2a
,
2
2M
a
=
.
M
164
Now, the autocorrelation of y(t) is given by
M
1 X
y(t + τ )y(t),
R(τ ) =
M t=1
M
1 X
=
[−a + 2au(t + τ )][−a + 2au(t)],
M t=1
M
1 X 2
[a − 2a2 u(t) − 2a2 u(t + τ ) + 4a2 u(t + τ )u(t)],
=
M t=1
=
a2
[M − 2mM − 2mM + 4M Ru (τ )],
M
= a2 [1 − 4m + 4Ru (τ )].
Then for τ = 0
1
1
1
1
R(0) = a2 [1 − 4[ +
)] + 4[ +
],
2 2M
2 2M
= a2 ,
and for τ > 0
1
1
1
1
R(τ ) = a2 [1 − 4[ +
)] + 4[ +
],
2 2M
4 4M
1
1
)],
= a2 [1 − 4( +
4 4M
a2
=− .
M
165
Therefore,
Ribr (τ ) =



 a2
, τ = 0, M, 2M, . . .


 − a2
.
, otherwise
M
We can follow the same procedure used for u(t) to obtain the power spectrum
for a PRBS with levels a and −a. The coefficients are computed as follows:
C0 =
M
−1
X
R(u),
u=0
= a2 − (M − 1)
=
a2
,
M
a2
.
M
Now, for k > 0,
2
Ck = a −
a2
=
M
M
−1
X
e
−j2πku/M
u=1
M−
M
−1
X
j=1
e
a2
,
M
−i2πkj/M
!
,
e−i2πk/M − e−i2πkM/M
a2
,
M−
=
M
1 − e−i2πk/M
a2
e−i2πk/M − 1
=
,
M−
M
1 − e−i2πk/M
a2
=
(M + 1).
M
166
Then the power spectrum becomes
"
M
−1
X
#
2π
k
C0 δ (ω) +
Ck δ ω − 2π
M
M
τ =1
"
#
M
−1
X
2πa2
2πk
=
.
δ(ω) + (M + 1)
δ ω−
M2
M
k=1
Φ(ω) =
(A.3)
167
Appendix B – Distribution of N (t)
We consider the process N (t) as the number of radiation sources that contribute
to the noise on the radio receiver at time t. Now, if we assume that the number
of sources depends only on the previous state and the increase or decrease of it
depends mostly on the present number then we can model N (t) as a continuous
Markov chain and specifically as a birth and death process. Following the works
of [39] and [35], we derived the distribution of N (t) again for the sake of completeness with the results presented here, as the way to link the parameters of the
distribution with the physical process under consideration and as an update of [35]
to the notation and definitions that is used nowadays, such as it is the case of the
generating function.
The process can be modeled using a linear growth model were the rate of new
sources added to the process is given by
λn = nλ + α,
with λ being the individual source rate and α being the rate of new events that
does not depend on the number of noise contributors. The rate at which sources
stop radiating is given by
µn = nµ,
168
then, we have that for small increments of time h the probability is given by [38]
P (N (t + h) − N (t) = k|N (t) = n) =




λn h + o(h)




, when k = 1
µn h + o(h)
, when k = −1 . (B.1)






(1 − λn − µn )h + o(h) , when k = 0
Now, the probability P (N (t + h) = n) can be found as
P (N (t + h) = n) =
1
X
k=−1
P (N (t + h) = N (t) + k|N (t) = n − k)P (N (t) = n − k),
(B.2)
it follows that
P (N (t + h) = n) =µn+1 hP (N (t) = n + 1) + λn−1 hP (N (t) = n − 1)
+ (1 − λn − µn ) hP (N (t) = n) + o(h).
× (P (N (t) = n + 1) + P (N (t) = n − 1) + P (N (t) = n)).
(B.3)
Dividing the previous expression by h with PN (n) = PN (N (t) = n) and h → 0,
we obtain
∂PN (n)
= µn+1 PN (n + 1) + λn−1 PN (n − 1) + (1 − λn − µn ) PN (n).
∂t
(B.4)
169
Now, the generating function [63] is given by
GN (z, t) =
∞
X
z n PN (n),
n=0
and its derivative by
∞
∂GN (z, t) X n−1
=
nz PN (n).
∂z
n=0
Using equation (B.4) and after some simplifications, it follows that
∞
∂GN (z, t) X n ∂
=
PN (n),
z
∂t
∂t
n=0
∂GN (z)
∂GN (z, t)
= (λz − µ)(z − 1)
+ α(z − 1)GN (z, t).
∂t
∂z
The solution of the differential equation just derived can be found using the method
of characteristics with the initial condition given by GN (z, 0) = z M as in [39],
resulting in
GN (z, t) =
µ−λ
µ − λz − λ(1 − z)e−(µ−λ)t
α/λ µ − λz − µ(1 − z)e−(µ−λ)t
µ − λz − λ(1 − z)e−(µ−λ)t
M
(B.5)
then the asymptotic behavior can be found by letting t → ∞ with µ > λ as
lim GN (z, t) =
t→∞
1 − λ/µ
1 − λ/µz
α/λ
,
which turns out to be the characteristic function of a negative binomial random
170
variable [45] given by
P (N (t) = n) =
n + α/λ − 1
(1 − λ/µ)α/λ (λ/µ)n .
n
(B.6)
Thus, N ∼ N B(α/λ, λ/µ) with mean
α λ/µ
,
λ 1 − λ/µ
α
.
=
µ−λ
N̄ =
Now, it is more convenient to express the distribution in terms of the mean and
the parameter r as in [24] by noticing the following:
α/λ
r
,
=
α/λ + α/(u − λ)
r + N̄
λ(u − λ)
=
,
λ(u − λ + λ)
u−λ
.
=
u
Then, the probability can be computed as
n+r−1
(1 − λ/µ)r (λ/µ)n ,
P (N (t) = n) =
n
n n
α λ
n+r−1
µ−λα
r
n
=
(1 − λ/µ) (λ/µ)
,
n
µ−λα
α λ
n
n
µ−λ
N̄
n+r−1
r
(1 − λ/µ)
=
r
µ
n
171
Therefore
P (N (t) = n) =
n+r−1
n
N̄
r
n r
r + N̄
n+r
.
(B.7)
172
Appendix C – Estimation of the Bit Error Rate for OFDM under
Narrowband Interference
In this section the BER for an OFMD system affected by narrowband interference
is computed.
Let, the received signal be given by
y(t) = h(t) ∗ x(t) + ν(t) + i(t),
where i ∈ I, h(t) is the channel and ν(t) is a zero-mean Gaussian noise. Now,
assuming an ideal channel and perfect synchronization, the down converted signal
sampled at Ts is given by
N −1
1 X
y[n] = √
X(k)ej2πnk/N + ν[n] + i[n].
N k=0
(C.1)
At the receiver the DFT is performed on the incoming signal with the kth sample
173
given by
N −1
1 X
√
Y [k] =
y[n]e−j2πnk/N ,
N n=0
N −1
1 X
=√
N n=0
!
N −1
1 X
j2πnl/N
√
X(k)e
+ ν[n] + i[n] e−j2πnk/N ,
N l=0
= X[k] + N [k] + I[k],
where N [k] and I[k] are, respectively, the kth sample of the Fourier transforms of
the Noise and the Interference, and X is the kth data symbol from some constellation. Since ν is Gaussian with variance σ 2 , then N is also Gaussian with variance
σ 2 [34].
C.1 BER in BPSK Modulation
Having a BPSK constellation consisting of two symbols {s1 , s2 }, means that the
conditional PDF’s of the a signal r, which correspond to the DFT of the received
signal, are given by
p(r|s1 , k, φ, f ) = √
1
2
e−(r−(A+Re{I(k)}))/2σ ,
2πσ
and
p(r|s2 , k, φ, f ) = √
1
2
e−(r−(−A+Re{I(k)}))/2σ ,
2πσ
174
where I(k) = f (k, φ, f ) with φ and f being either random variables or deterministic
values. Now,
Pe (γs , γi |s1 , k, φ, f ) =
=
=
=
=
Z
0
p(r|s1 , k, φ, f )dr
Z 0
1
2
2
√
e−(r−(A+Re{I(k)})) /2σ dr
2πσ −∞
Z − A+Re{I(k)}
√
2σ
1
2
√
e−u du
π −∞
Z ∞
1
2
√
e−u du
π A+Re{I(k)}
√
2σ
1
A + Re{I(k)}
√
,
erfc
2
2σ
−∞
and
Pe (γs , γi |s2 , k, φ, f ) =
Z
∞
p(r|s2 , k, φ, f )dr
Z ∞
1
=√
e−(r−(−A+Re{I(k)})) dr
2πσ 0
Z ∞
1
2
e−u/2σ du
=√
π A−Re{I(k)}
√
2σ
1
A − Re{I(k)}
√
= erfc
.
2
2σ
0
Then, the conditional probability of error for a BPSK modulation is given by
1
1
P e(γs , γi |k, φ, f ) = P (e|s1) + P (e|s2)
2
2
1
A + Re{I(k)}
A − Re{I(k)}
1
√
√
+ erfc
.
= erfc
4
4
2σ
2σ
175
C.2 BER in QAM Modulation
For a M-ary QAM, we have that the probability of error is the same as the probability of error of two PAM signals [34]. So we compute the BER for a PAM signal
first.
The amplitude values for the in-phase component for a rectangular M-QAM is
√
given by Am = (2m − 1 − M )d and d = 2A is the distance between symbols.
The probability of error of a PAM signal is the probability that the noise and the
interference do not exceed half the distance between symbols in either way. Then,
√
with M levels, the probability of error is given by
P√
M
=
=
=
=
=
√
M −1
√
P (| Re{N (k) + I(k)}| > A),
M
√
M −1
√
P (Re{N (k) + I(k)} > A) + P (Re{N (k) + I(k} < −A),
M
√
Z ∞
Z −A
(r−Re{I(k)})2
(r−Re{I(k)})2
M −1 1
−
−
2
2
2σ
2σ
√
√
e
dr +
dr ,
e
2πσ
M
−∞
A
!
√
Z ∞
Z ∞
1
1
M −1
2
2
√
√
e−u du + 1 − √
e−u du ,
A−Re{I(k)}
−A−Re{I(k)}
π
π
M
√
√
2σ
2σ
√
M −1
A − Re{I(k)}
−A − Re{I(k)}
√
√
√
2 + erfc
− erfc
.
2σ
2σ
2 M
176
Thus, for a QAM signal with M levels, the probability is given by
P e(γs , γi |k, φ, f ) = 1 − (1 − P√M )2
√
M −1
=1− 1− √
2 M
2
−A − Re{I(k)}
A − Re{I(k)}
√
√
− erfc
.
× 2 + erfc
2σ
2σ
C.3 Narrowband Noise
Let a sum of L narrowband inteferers be denoted by
ina (n) =
L
X
ina n, ml .
l=1
Then, its DFT is
Ina (k) =
L
X
Ina (k, ml ),
m=1
where a component at frequency fm = mfc is described as
ina (n, m) = bm exp (j2πfm nT + φm ) .
Now, each component has a DFT representation given by
Ina (k, m) =
k
k
j(π( N
−fm T )(N −1)+φ) sin(N π( N − fm T ))
,
bm e
sin π( Nk − fm T )
177
from where it clearly follows that
k
Ψ(k, m) = ej(π( N −fm T )(N −1)+φm )
sin(N π( Nk − fm T ))
.
sin π( Nk − fm T )
The real component of Ψ(k, m) is then given by
Re{Ina } =
L
X
m=1
L
X
bm Re {Ψ(k, m)} ,
sin(N π( Nk − fm T ))
k
.
− fm T (N − 1) + φm
=
bm cos π
N
sin π( Nk − fm T )
m=1
Now by defining γs = A2 /2σ 2 and γm = b2m /2σ 2 , we have that for BPSK the
mean probability of error across an OFDM symbol for a narrowband Ina is given
by
PeBPSK (γs , γi |φ, f )
N −1
1 X
pe (γs γi |k, φ, f ),
=
N k=0
!
"
L
N −1
X
1 X 1
√
√
=
γs +
γm Re{Ψ(k, m)}
erfc
N k=0 4
m=1
!#
L
X
1
√
√
,
+ erfc
γs −
γm Re{Ψ(k, m)}
4
m=1
(C.2)
178
and for QAM is given by
√
L
X
√
!
M −1
√
√
γs −
γm Re{Ψ(k, m)}
2 + erfc
2 M
m=1
.
! !!2
L
X
√
√
− erfc − γs −
γm Re{Ψ(k, m)}
PeQAM (γs , γi |φ, f ) = 1 −
1−
m=1
(C.3)
179
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