Compressed Sensing for Ultra Wideband (UWB) Systemsrqiu/publications/thesis... · 2015. 3. 14. · FOR ULTRA WIDEBAND (UWB) SYSTEMS Daniel Zahonero Inesta ... One of the latest breakthroughs

AN ABSTRACT OF A THESIS

COMPRESSED SENSINGFOR ULTRA WIDEBAND (UWB)

SYSTEMS

Daniel Zahonero Inesta

Master of Science in Electrical Engineering

The demanding characteristics of the UWB technology include ex-tremely high sampling rates in the receiver. These sampling rates requiresophisticated devices, sometimes out of the scope of the state-of-art tech-nology. Among the different methods to make the reception possible,Compressed Sensing seems to be the one that presents better perfor-mance. It consists basically of compressing the information while thisis sampled, avoiding processing a huge chunk of redundant data and low-ering the sampling rate. In order to reconstruct the data after the com-pression, different methods have come up presenting different favorablefeatures. However this methods also present a trade-off between sam-pling rate and processing time. Using real measurements of the channel,the performance in different environments has been analyzed for differentfrequencies. Thanks to the theoretical account and practical results thisstudy will help to understand better the Compressed Sensing techniquesapplied to a real communication, and specifically, to Ultra Wideband.

COMPRESSED SENSING

FOR ULTRA WIDEBAND (UWB)

SYSTEMS

A Thesis

Presented to

the Faculty of the Graduate School

Tennessee Technological University

by

Daniel Zahonero Inesta

In Partial Fulfillment

of the Requirements for the Degree

MASTER OF SCIENCE

Electrical Engineering

May 2009

Copyright c© Daniel Zahonero Inesta, 2009All rights reserved

ii

CERTIFICATE OF APPROVAL OF THESIS

COMPRESSED SENSING

FOR ULTRA WIDEBAND (UWB)

SYSTEMS

by

Daniel Zahonero Inesta

Graduate Advisory Committee:

Dr. Robert C. Qiu, Chairperson Date

Dr. Periasamy K. Rajan Date

Dr. Omar ElKeelany Date

Approved for the Faculty:

Francis OtuonyeAssociate Vice President forResearch and Graduate Studies

Date

iii

STATEMENT OF PERMISSION TO USE

In presenting this thesis in partial fulfillment of the requirements for a Master

of Science degree at Tennessee Technological University, I agree that the Univer-

sity Library shall make it available to borrowers under rules of the Library. Brief

quotations from this thesis are allowable without special permissions, provided that

accurate acknowledgment of the source is made.

Permission for extensive quotation from or reproduction of this thesis may by

granted by my major professor when the proposed use of the material is for scholarly

purposes. Any copying or use of the material in this thesis for financial gain shall not

be allowed without my written permission.

Signature

Date

iv

DEDICATION

This thesis is dedicated to my parents

who have given me invaluable educational opportunities.

v

ACKNOWLEDGMENTS

Appreciation is extended to Peng Zhang whose knowledge in the field and

support has been vital for the fulfillment of this thesis. To Dr. Robert C. Qiu whose

advices and recommendations have been tremendously helpful. I would also like to

thank all my lab mates for their help and support throughout my stay here.

Finally, I would also like to thank the Department of Electrical and Computer

Engineering, and Center for Manufacturing Research for its invaluable help, putting

at my disposal all the equipment and resources available.

vi

TABLE OF CONTENTS

Page

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Chapter

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . 3

2. UWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 UWB Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Pulse-Shape Distortion . . . . . . . . . . . . . . . . . . . . . 102.2.2 Multiple-Access Interference . . . . . . . . . . . . . . . . . . 112.2.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . 112.2.4 High Sampling Rate . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. COMPRESSED SENSING . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Compressive Sensing Theory . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 CS for UWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Communication Model . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Communication Model: Theoretical Approach . . . . . . . . 253.3.2 Communication Model: Practical Approach . . . . . . . . . . 27

vii

Table of Contents viii

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4. THE UWB CHANNEL . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Channel Measurement . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Hallway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Metalic Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1.3 Inter-Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Channel Sounding Results . . . . . . . . . . . . . . . . . . . . . . 354.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5. CS ALGORITHMS COMPARISON . . . . . . . . . . . . . . . . . . . 425.1 Basis Pursuit (BP) . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.1 Algorithm Definition . . . . . . . . . . . . . . . . . . . . . . 445.1.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Orthogonal Matrix Pursuit (OMP) . . . . . . . . . . . . . . . . . 495.2.1 Algorithm Definition . . . . . . . . . . . . . . . . . . . . . . 505.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Stagewise Orthogonal Matrix Pursuit (StOMP) . . . . . . . . . . 555.3.1 Algorithm Definition . . . . . . . . . . . . . . . . . . . . . . 565.3.2 Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.2.1 False Alarm Rate (FAR) . . . . . . . . . . . . . . . . . 585.3.2.2 False Discovery Rate (FDR) . . . . . . . . . . . . . . . 59

5.4 Simulation in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . 605.4.1 Measuring Matrix . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 635.5.1 Accuracy on the Recovery . . . . . . . . . . . . . . . . . . . 635.5.2 Number of Measurements . . . . . . . . . . . . . . . . . . . . 715.5.3 Processing Time . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.1 CS applied to UWB . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Reconstruction Algorithms . . . . . . . . . . . . . . . . . . . . . 826.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

LIST OF TABLES

Table Page

3.1 Bit-Rate due to the number of pulses . . . . . . . . . . . . . . . . . . . . 293.2 Sampling Rate at the Receiver . . . . . . . . . . . . . . . . . . . . . . . 295.1 Sampling BP Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

ix

LIST OF FIGURES

Figure Page

2.1 UWB Receiver Block Diagram . . . . . . . . . . . . . . . . . . . . . . . 133.1 Symbol. Pulse Position Modulation (PPM) . . . . . . . . . . . . . . . . 263.2 UWB Communication System Based on Compressed Sensing . . . . . . . 274.1 Vector Network Analyzer (VNA) . . . . . . . . . . . . . . . . . . . . . . 324.2 3.1-10 GHz UWB Antenna . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Hallway Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Metallic Box Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Inter-Car Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.6 Hallway Channel in Frequency Domain . . . . . . . . . . . . . . . . . . . 374.7 Hallway Channel Impulse Response . . . . . . . . . . . . . . . . . . . . . 374.8 Metallic Box Channel in Frequency Domain . . . . . . . . . . . . . . . . 384.9 Metallic Box Channel Impulse Response . . . . . . . . . . . . . . . . . . 394.10 Inter-Car Channel in Frequency Domain . . . . . . . . . . . . . . . . . . 404.11 Inter-Car Channel Impulse Response . . . . . . . . . . . . . . . . . . . . 415.1 Heuristic approximation to the minimization problem . . . . . . . . . . . 445.2 Atoms related to non-zero elements (OMP) . . . . . . . . . . . . . . . . 505.3 Quasi-Toeplitz Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 Error Rate of BP Recovery for different levels of sparsity . . . . . . . . . 645.5 Error Rate of BP Recovery (6-8 GHz) . . . . . . . . . . . . . . . . . . . 655.6 Error Rate BP Recovery Metallic Box (4-6 GHz) . . . . . . . . . . . . . 655.7 Error Rate BP Recovery Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . 665.8 Error Rate OMP Recovery (4-6 GHz) . . . . . . . . . . . . . . . . . . . . 675.9 Error Rate OMP Recovery (6-8 GHz) . . . . . . . . . . . . . . . . . . . . 675.10 Error Rate OMP Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . . . . . 685.11 Error Rate OMP Metallic Box (4-6 GHz) . . . . . . . . . . . . . . . . . 685.12 Error Rate StOMP Hallway (4-6 GHz) . . . . . . . . . . . . . . . . . . . 695.13 Error Rate StOMP Hallway (6-8 GHz) . . . . . . . . . . . . . . . . . . . 705.14 Error Rate StOMP Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . . . . 715.15 Error Rate StOMP Metallic Box (4-6 GHz) . . . . . . . . . . . . . . . . 715.16 Error Rate OMP Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . . . . . 735.17 Error Rate OMP Metallic Box (4-6 GHz) . . . . . . . . . . . . . . . . . 735.18 Error Rate OMP Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . . . . . 735.19 Error Rate OMP Metallic Box (4-6 GHz) . . . . . . . . . . . . . . . . . 735.20 Error Rate OMP Inter-Car (4-6 GHz) . . . . . . . . . . . . . . . . . . . 735.21 Error Rate OMP Metallic Box (4-6 GHz) . . . . . . . . . . . . . . . . . 73

x

List of Figures xi

5.22 Sampling Rate (High Sparsity) . . . . . . . . . . . . . . . . . . . . . . . 745.23 Sampling Rate (Low Sparsity) . . . . . . . . . . . . . . . . . . . . . . . . 745.24 Processing Time of BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.25 Processing Time of OMP . . . . . . . . . . . . . . . . . . . . . . . . . . 775.26 Processing Time of Stagewise OMP . . . . . . . . . . . . . . . . . . . . . 785.27 Processing Time (High Sparsity) . . . . . . . . . . . . . . . . . . . . . . 785.28 Processing Time (Low Sparsity) . . . . . . . . . . . . . . . . . . . . . . . 79

CHAPTER 1

INTRODUCTION

1.1 Motivation

One of the latest breakthroughs made in wireless communications has been

the Ultra Wideband Technology. The growing demand on wireless services has over-

crowded the RF Spectrum over the last decade and new services face a problem of

allocation. Since in 2002 the FCC authorized UWB operations over the top of other

bands, it has become one of the most promising alternatives. However, its demanding

features in terms of sampling and processing information require new methods and

ways to set the transmission.

It has been demonstrated theoretically that Compressive Sensing technique is

a perfect suitable candidate to implement UWB. Following this premise, this study

presents as a goal to demonstrate that CS can implement UWB, supporting the

idea not only with theoretical background but with practical results. The study also

analyzes its limitations dealing with frequency and different channels. The channels

used in the study will vary from favorable line-of-sight indoor to hostile outdoor

environments. The frequency will be modified in order to study the influence in the

performance of the CS.

1

2

Finally a comparison of the main algorithms that perform CS will be made.

The key features, advantages and disadvantages will be discussed easing the election

of them according to the conditions and constraints.

1.2 Literature Survey

Since 2002, there has been a noticeable increase in literature published about

UWB. Many books have been written introducing the priciples[1][2] and fundamentals[3][4]

. Some other came up presenting the applications[5], and finally a radio model sys-

tem was introduced [6] [7][8][9]. In the last years, matematicians came up with

a new revolucionary solution that applied Compressed Sensing to communications

[10][11], and more specifically to UWB[12][13]. Numerous studies have been made

about the CS alogrithms applied to communication, describing them in detail: Basis

Pursuit[14][15][16][17], Orthogonal Matching Pursuit[18][19] and Stagewise Oothogo-

nal Matching Pursuit[20]. Even some of them are compared in a given situation[21]

[22]. However there is no study that goes that far and compare the three of them in

different environments and frequency bands.

1.3 Research Approach

Using Matlab, the CS algorithms will be applied to simulate a transmission

using different channel data from real measurements. To implement the communica-

3

tion model, VNA measurements of the channel are made for two Frequency Bands.

The frequency data is transformed into time domain using IFFT to obtain the im-

pulse response. The surroundings where the channel is measured are a Hallway, a

Metallic Box and an Inter-Vehicle environment. The two 2 GHz Bandwidth Chan-

nel will be centered in 5 GHz and 7 GHz. With the obtained data, three different

featured algorithms are used for the same environments in order to achieve a reliable

comparison.

1.4 Organization of the Thesis

Chapter 2 presents a briefly introduction and the fundamentals of UWB focus-

ing mainly in the Channel, the PPM modulation and the Challenges UWB entails.

The Chapter 3 introduces the Compressed Sensing concept. The CS theoretical

background is explained and its application to communication is discussed. Finally

a communication model for UWB is proposed.

In the Chapter 4 the Channel is examined for the three environments after

explaining the measuring method. The Channel souning results are commented later

on in the chapter.

Chapter 5 focuses in three algorithms used for CS. The algorithms are ex-

plained in detail pointing out the strengths and weaknesses. The simulation of the

4

earlier proposed model is set out. Eventually, the results of the simulations are ex-

plained and critically analyzed.

Chapter 6 gathers all the final conclusions and contributions for the thesis as

well as points out the future work.

CHAPTER 2

UWB

The principle of UWB is based on transmitting in a short-range with large

bandwidth using low energy levels. This way the power is spread all over the spectrum

and becomes immune to frequency flat fading[1]. The UWB Technology also offers a

robust performance under multipath environments, but its most interesting feature

is the possibility to reuse the frequencies. Spreading the power over the spectrum

allows transmitting with low power levels making the signal noise-like so that it causes

minimal or no interference to other signals that may be transmitted in the same

frequency. Hence UWB is able to coexist with current radio systems and no need to

be allocated in the spectrum. UWB technology was designed for military applications

but new U.S. Federal regulations and the demand for higher data rates at short-range

opened up UWB for commercial applications [23]. The centimeter accuracy in ranging

and communications provides unique solutions to applications, including logistics,

security applications, medical applications, control of home appliances, search-and-

rescue, family communications and supervision of children[2].

5

6

2.1 UWB Technology

2.1.1 Definition

To be considered UWB, the signal bandwidth has to be greater than 500 MHz

or having a fractional bandwidth1 of larger than 20% and the radiated power cannot

exceed -41.3 dBm/MHz according to the Federal Communication Commission(FCC)[24].

UWB transmissions transmit information by generating radio energy at specific time

instants with the shape of a pulse. These pulses are on the order of nano-seconds

and are used as the elementary pulse shaping to carry the information[3]. The basic

pulse used for UWB is a Gaussian monocycle in which the width determines the

center frequency and the bandwidth. A typical UWB pulse is between 0.2 and 0.5

nanoseconds width. The monocycle itself contains no data, so that a long sequence

of monocycles with data modulation is used for communication.

2.1.2 Antennas

In UWB technology, power level is much more important than in standard

narrowband system. Thus an effective UWB antenna is a critical part of an overall

UWB system design. A wide variety of antennas is suitable for UWB applications.

1

2(fH − fC)(fH + fC)

7

They can be classified as directional and non-directional. High-gain antennas concen-

trate energy into a narrower solid angle than omnidirectional antennas. An isotropic

antenna radiates equally in every direction. Differences between using a directional

or non-directional is a tradeoff between Gain, the Field of view and the size of the

antenna. Regulatory constraints limit the power to the same peak radiated emission

limit, so basically what is reached by using the directional antenna is to reduce the

emissions in undesired directions. These kind of antennas can be implemented in

relatively compact planar designs. The design of the receiving antenna can affect

dramatically to the link performance[5]. UWB antennas can be modeled as the front-

end pulse shaping filters that affect the baseband detection[2]. Antennas act as a

filter for the generated UWB signal, and only allows those signal components that

radiate to be passed. The basic effect of antennas is that they produce the derivative

of the transmitted or received pulse waveform. This also has the effect of extending

the duration of the transmitted and received pulse. This extension of pulse duration

decreases the time resolution of the system[25].

2.1.3 Channel

Propagation through the medium will attenuate and distort the incident pulse-

based signals, but the most important feature is that when a short UWB pulse prop-

agates through the channel, multiple pulses are received via multipath. These pulses

8

have shapes different from the incident short pulse[2]. If the channel is well charac-

terized, the effect of disturbances and other sources of perturbation can be reduced

by proper design of the transmitter and receiver. Detailed characterization of UWB

radio propagation is required for successful design of UWB communication systems[4]

and necessary for the algorithms to reconstruct the original signal from data sampled.

The channel can be measured in the Frequency Domain (FD) using a frequency

sweeping technique. Using a set of narrow-band signals a wide frequency band is

swept thanks to a vector network analyzer (VNA). This measurement corresponds to

S21 parameter measurement set-up, where the device under test (DUT) is the radio

channel. There is another way to measure the channel in Time Domain (TD) using

channel sounders that are based on impulse transmission or direct sequence spread

spectrum signaling[25]. During the sweep the channel must be static to maintain

the conditions during the soundings. For fast changing channels, other sounding

techniques are needed. Another possible errors is the frequency shift caused by the

propagation delay when long cables are used, or when the flight time of the sounding

signal is long[25].

Signal Analysis using IFFT will provide the Time Domain signal from the

Frequency Domain signal. This processing is possible since the receiver has a down-

conversion stage with a mixer device. To convert the signal into the time domain

Hermitian Processing is used[25]. This method is based in reflecting the negative

9

conjugate to the negative frequencies. The result is transformed into time domain

by using the IFFT[25]. Parameters related to penetration, reflection, path loss, and

many other effects should be considered frequency-variant and investigated more

carefully.[4]

2.1.4 Modulations

The different techniques to encode the information in UWB include amplitude,

polarity and position. In order to choose a modulation, data rate, transceiver com-

plexity, spectral characteristics, robustness against narrowband interference, inter-

symbol interference and error performance must be taken in account. They can be

grouped as:

• On-Off Keying (OOK)

• Pulse-Amplitude Modulation (PAM)

• Pulse-Position (PPM)

• Biphase Modulation (BPM)

• Transmitted-Reference Modulation (TRM)

In this thesis, the communication model will emulate a K-sparse PPM encoding

transmission. Each symbol is K-sparse, meaning that for N possible positions there

10

will be just K (K<<N) pulses. These possible positions of the pulses will determine

the symbol and provide the information.

2.2 Challenges

Although it has been demonstrated that UWB is a great alternative for short-

distance low-power wireless applications[26], it has its weak points. These issues must

be minimized in order to make UWB a reliable technology. The main disadvantages

are based on the large frequency band that makes the pulses deform through prop-

agation and the low-level power that makes multiple-user communication hard to

implement as well as stress the distortion.

2.2.1 Pulse-Shape Distortion

Unlike narrow-band communications, UWB uses a large portion of the Spec-

trum and the attenuation due to the propagation does not affect all the frequencies of

the band the same way. The received power from higher frequencies of the band will

be lower than the received from the lower frequencies originating a distortion of the

pulse. Moreover the low-level power of the signal makes this distortion much more

noticeable. To face these problems matched filters were used to correlate the pulses

with predefined templates. Because of the changing conditions of the channel they

are doomed to fail, so an equivalent distorted signal is used as a reference signal in

11

the matched filter.

2.2.2 Multiple-Access Interference

In the case of the presence of several users transmitting over the same chan-

nel, pulses originating in other transmission links may collide with pulses belonging

to a reference transmission, giving rise to an interference noise called Multi-User

Interference[7]. Fortunately as the channel can be modeled as pseudo-random(PN)

noise code, by shifting each monocycle at a pseudo-random time interval, the pulses

appear to be white background noise to users with a different PN code[8]. Hence

UWB technology allows several users to transmit at the same time, using the same

frequency range.

2.2.3 Channel Estimation

The characteristic low level of energy transmitted in UWB limits the range of

the transmission and makes it hard to implement in an outdoor environment. However

UWB is very powerful when it comes to indoor scenario due to the rich-multipath. It

only occurs if the channel is known since in UWB it is assumed stationary or quasi-

stationary[2]. Unlike other technologies UWB channel needs to be considered in time

domain rather than frequency domain due to its unique features. In opposition to

narrowband communications, the signals are huge-bandwidth pulses instead of sine

12

waves so that selective fading is no longer a problem. The overlapping superposi-

tion of unresolved multipaths hardly affects the received signal so the time-resolvable

multipath will carry great part of the energy of the transmission. The channel is con-

sidered quasi-static so the collected total energy is almost constant in each instant[9].

Due to this a close estimation of the channel can be obtained. Some studies consider

propagation as the single most important issue in the success of UWB technology [4].

Thus, a well characterized channel is fundamental for a successful UWB communica-

tion. The attenuations, interferences and delays affect the design of the transmitter

and receiver so having all the information in advance will result in a more efficient

communication. Also, as shown later on, knowing the channel time-response will be

essential for the recovering in compressed sensing. In the receiver, the transmitted

signal will be correlated with a template signal which will reflect the effect of the

channel so that it will be a reliable reference for the received signals to compare. The

attenuations and propagation delays must be estimated to predict the shape of the

template signal that matches the received signal[5]. However the characteristics of

the UWB make the pulse vulnerable to distortion and hard to interpret the channel.

The huge bandwidth of the pulse makes different frequency components attenuate un-

evenly causing distortion in an already low-power pulse. To deal with this drawback

training sequences known by the receiver are used[27]. These predefined training se-

quences are transmitted and compared with the original, the difference between them

13

Figure 2.1: UWB Receiver Block Diagram

will help to depict the channel response in terms of attenuation, resolvable multipaths

and delay spread.

2.2.4 High Sampling Rate

One of the most important challenges UWB has to face is how to manage the

high speed transmissions, being the transformation from Analog domain to Digital

domain one of the key parts of the UWB receiver model (Figure 2.1). The obtained

analog signal is sampled at a 1/Ts, where Ts is the Time of the Symbol, in order

to get a sequence of digital values which be able to work with. This sampler will

limit the speed of the transmission, the information unable to sample due to higher

speed than the sampler can handle will be lost. Finally the sampled sequence will be

processed afterward by a Maximum Likelihood Sequential Estimator (MLSE) that

will interpret the symbol detected.

As in any other Telecommunication area, sampling is always a challenge. This

task in UWB technology, due to its features, is even more complicated. Accurate

14

data detection for pulsed UWB systems is crucial to achieve good Bit Error Rate

(BER), performance, capacity, throughput and network flexibility[2]. Most of the

UWB communication applications are targeting between 100 Mbps and 500 Mbps

transmissions, remarkably faster compared with current wireless standards [4]. The

ultra-short pulse widths go from tens of picoseconds to few nanoseconds[28][29] re-

quiring extremely high sampling rate in the receiver to collect them. In a UWB

communication, every symbol is transmitted with a low duty cycle over a large num-

ber of frames gathering adequate symbol energy while maintaining low power density

[28]. This low duty cycle makes it hard to calculate the Time Gating, time which

the receiver and the detector have to remain turned on in order to match the signal

as well as when they have to be turned on[6]. The duration time must be just the

expected duration of the signal, if they remain open too long, excess energy noise will

be collected affecting the signal-to-noise ratio. On the other hand, if the pulse is sam-

pled less time than needed, since there is not enough energy some information may

be lost. Another problem for synchronization is the dense multipath. Although it

entails large diversity and can enhance the energy capture, this is challenging during

synchronization phase because the channel and time information, not being avail-

able, are hard to figure out. However, the main challenge is to reduce the acquisition

time which calls for a more sophisticated ADC, sometimes out of the scope of the

state-of-art technology[30][31]. This limitation led to leave aside classical conception

15

of sampling (Nyquist) and seek for new techniques that allow more information rate

using less sample requirements. In order to address this challenge, a revolutionary

sampling theory appeared: Compressed Sensing. It permits to reconstruct a trans-

mitted signal using just a few percentage of the original number of samples, opening

a new range of possibilities in UWB. In broad strokes this technique is based on

compress while sampling, freeing the receiver from having to sample in a high rate

and to process a huge chunk of data due to the speed of the data. Thanks to this,

impressive low sampling rates can be achieved related to the degrees of freedom of

the information instead of to the Bandwidth of the signal. This thesis focuses its goal

on giving a general idea of how Compressed Sensing is applied to UWB and analyze

different reconstruction algorithms.

2.3 Summary

This chapter introduces the fundamentals and applications of UWB technol-

ogy. The advantages are pointed out, but also the challenges that UWB involves

emphasizing the challenge of dealing with great high-frequency. Since UWB has very

demanding sampling requirements, sometimes out of the scope of the state-of-art

technology, the A/D converter will be the bottleneck. To address this challenge, the

Compressed Sensing is proposed.

CHAPTER 3

COMPRESSED SENSING

Classical compression methods throw away the undesired information after

data acquisition. For few data this is not a problem, but when it comes to com-

press massive data, many resources are wasted acquiring and processing a chunk

of unnecessary data. As an answer for this inefficiency appeared the Compressed

Sensing (CS) which allows to compress the data while is sampled. In other words

Compressed Sensing suggest ways to economically translate analog data into already

compressed digital form [32][10]. This new way to sample achieves sampling rates

below Nyquist rate. It originates from the idea that it is not necessary to invest a lot

of power into observing the entries of a sparse signal in all coordinates when most of

them are zero anyway. Rather it should be possible to collect only a small number

of measurements that still allow for reconstruction[33]. CS requires a compressible

signal with certain sparsity, considering sparse a signal that can be written either

exactly or accurately as a superposition of a small number of vectors in some fixed

basis. The procedure then, gets the most of the properties of some signals in which a

small number of non-adaptive samples carries sufficient information to approximate

the signal properly.

This is potentially useful in applications where one cannot afford to collect or

16

17

transmit a lot of measurements[33] such as UWB is. Among the attractive features of

CS is the ability to reconstruct any sparse (or nearly sparse) signal from a relatively

small number of samples, even when the observations are corrupted by additive noise.

Nevertheless, the potential of CS in other signal processing applications is still not

fully known[12].

CS methods provide a robust framework for reducing the number of measure-

ments required to summarize sparse signals. For this reason CS methods are useful

in areas where analog-to-digital conversion costs are high.[34]

Research in this area has two major components[35]:

• Sampling: How many samples are necessary to reconstruct signals to a spec-

ified precision? What type of samples? How can these sampling schemes be

implemented in practice?

• Reconstruction: Given the compressive samples, what algorithms can efficiently

construct a signal approximation?

This thesis will address both areas presenting a study for different algorithms

and different sampling rates analyzing which one fits better for each situation, envi-

ronment and requirements.

18

3.1 Compressive Sensing Theory

The fundamental of Compressive Sensing is basically reconstruct a signal x

from a downsampled one y. In order to demonstrate the mathematical background,

the signal x and y will be consider as a superposition of spikes so that can be repre-

sented as vectors of n and m elements respectively, being m<<n. Mathematically:

y = Φx

Where Φ represents the downsampling matrix from the n elements into m.

The number of samplings will indicate the degree of compression of the signal and

will be given by the sampling rate. Thus instead of depending on the number of

samples intended to transmit or the bandwidth of the signal, the sampling rate will

depend on the degrees of freedom leaving the possibility to achieve surprisingly

low sub-Nyquist sampling rates.

To achieve the sparse vector required for the CS, a basis that provide a k-

sparse representation is needed. Hence x will be a linear combination of K vector

chosen from the basis. Considering α as the k-sparse vector and Ψ the basis:

x =∑N−1

n=0 ψnαn =∑K

l=1 ψlαl

Then the former problem can we rewritten in terms of a sparse vector:

y = Φx = ΦΨα

19

Leading to a l1 − norm optimization problem:

α̂ = argmin ‖α‖1 so that y = ΦΨα

Linear programming techniques like Basis Pursuit (BP) or greedy algorithms

such Orthogonal Matching Pursuit(OMP and StOMP) can be used to solve this

problem[13]. Throughout the thesis these algorithms will be compared in order to

find out which one suits better in each situation depending on the sampling rate, the

channel and the sparsity of the signal.

3.1.1 Isometry

To succeed in this reconstruction, the Matrix used as downsampling basis must

meet certain isometry requirements. Several studies about the isometry properties

of the sensing matrix (Φ) [11][14][36], revealed that a matrix that follows the Re-

stricted Isometry Hypotesis-also called Restricted Isometry Propierties (RIP)- can

ensure better recovery results.

To meet the Restricted Isometry Hypotesis is necessary that every set of

columns with a number of columns less than K behaves approximately like an orthog-

onal system. Hence if the columns of the sensing matrix (Φ) are approximately or-

thogonal, then the exact recovery phenomenon occurs. In the same studies[14][36],

the Uniform Uncertainty Principle is defined to set the isometry conditions that the

20

matrix has to meet. If a signal is just nearly sparse, keeping the largest coefficients

and setting the rest to zero, is possible to achieve an accurate reconstruction if the

sensing matrix undergoes the UUP (at K level). Or putted in another words, if the

matrix meets certain characteristics, being x a vector in <N and xK its best K-sparse

approximation, the recovery error will not be much worse than ‖x− xK‖l2 .

At this point, find a matrix that obeys the UUP is essential. This matrix should

be designed as collection of N vectors in M dimensions so that any subset of columns

of size about K be roughly orthogonal. Although it might be difficult to exhibit a

matrix which probably obeys the UUP for very large values of K, is well known that

randomized constructions can achieve so with high probability. The reason why this

holds may be explained by some sort of ”blessing of high-dimensionality.” Because

the high-dimensional sphere is mostly empty, it is possible to pack many vectors while

maintaining approximate orthogonality[11].

Being Φ a measurement matrix, it will obey the Uniform Uncertainty Prin-

ciple with oversampling factor λ if for all subsets T such that T ≤M/λ the following

is true [36]:

12MN≤ λmin (Φ∗TΦT ) ≤ λmax (Φ∗TΦT ) ≤ 3

2MN

What this theory basically tries to ensure is the condition that the geometry

of sparse signals should be preserved under the action of the sampling matrix. In

21

this case to build the sampling matrix a pre-coding matrix and a matrix representing

the channel will be combined. For the pre-coding matrix a Gaussian ensemble can

be picked:

F (i, j) := 1√NXi,j, Xi,ji.i.d.N (0, 1)

It will be combined with another matrix representing the channel. The resul-

tant matrix will be consider as the measurement ensemble and will have the rows

corresponding with the number of measurements and the number of columns match

the number of elements of the sparse vector. To keep meeting the properties of isom-

etry and orthogonality both the random pre-coding matrix and the channel matrix

should be incoherent. Since the channel is completely unpredictable, it is hard to

ensure such a thing although some properties of the channel have been demonstrated

empirically as necessaries to reach a good recovery. Among them a rich-multipath

environment has been proved as a necessary condition due to the fact that preserves

the orthogonality of the pre-coding matrix.

3.1.2 Measurements

In the theory of Compressed Sensing, sample is to apply a linear function

to a signal, being the process of collecting multiple samples the fact of applying a

sampling matrix to the signal[35]. Hence down-sample a signal X of length N into

22

M samples demands a sampling matrix of N rows and M columns. Obviously the

smaller the matrix the less measurements meaning slower sampling rate. Knowing in

advance the minimum number of measurements will result in an improvement of the

efficiency. Unfortunately although many accurate approximations can be made, is

impossible to predict the exact minimum necessary number of measurements needed

for reconstruction.

Without taking in account the sampling matrix characteristics a broadly re-

strictions can be made. Several studies pointed out[37][38] some restrictions for the

sparsity for a given number on measurements:

K ≤ C(M/logN)

being C a positive constant. Therefore the number of measurements can be

limited to:

M ≥ CKlog(N)

However the sampling matrix and its characteristics do affect the number

of measurements so they must be taken into account. There are few examples of

random matrices in terms of their behavior but this study will focus on simulations

made with Gaussian random matrices due to their general features. Considering this,

the minimum number of measurements can be restricted to:

23

M ≥ CKlog(NK )

ε2

δK ≥ ε

with epsilon being the Restricted Isometry Constant from the Uniform Un-

certain Principle, also written as:

(1− δK) ‖x‖22 ≤ ‖Φx‖

22 ≤ (1 + δK) ‖x‖2

2

where δK := sup‖Λ‖=Kδ (ΦΛ). When δK ≤ 1 imply that each collection of

r columns from Φ is nonsingular, which is the minimum requirement for acquiring

sparse signals.

The last but not least consideration that may limit the number of measure-

ments will be the algorithm used for reconstruction. These algorithms are very effec-

tive but some need more measurements than others to meet the same accuracy being

the first the less complex and fastest. This study will analyze this last dependency

and find out the reasons of the behavior of the algorithms and since channel affects

the measuring matrix and the reconstruction process, different environments will do

so.

24

3.2 CS for UWB

In this thesis the architecture used to represent the CS recovering for UWB

communication will be filter-based using a Finite Impulse Response (FIR) filter. To

put in practice this idea is necessary to assume a linear time-invariant system in which

the channel, once estimated will remain fixed during the transmission process. It will

allow to fairly compare the results for signals with different sparsity levels, something

impossible if the channel would be changing.

Back to the main equation:

y = Φx

For communications, y will be the sampled signal while x the transmitted.

The channel and the downsampling will be represented by a FIR filter so that the

equation can be rewritten as:

y (mTS) = h (mTS) ∗ x (mTS)

where TS is the sampling period and m the number of samples collected. Con-

sidering that h is a FIR filter:

h (t) =∑L−1

i=0 hiδ (t− iTh)

25

to implement the system a Toeplitz matrix will be needed. However as the

matrix has also to downsample will be necessary to define a quasi-Toeplitz in which

a each row is the row above, shifted the relation between Sampling Period and Tap

Period of the filter q = TS/Th. In other words a Toeplitz matrix in which the rows

that does not correspond to q multiples are removed.

3.3 Communication Model

In this study Compressed Sensing will be applied to a UWB communication.

The model will implement UWB transmission system including UWB transmitter,

the channel and a low rate receiver.

3.3.1 Communication Model: Theoretical Approach

The proposed architecture is based in a UWB series of pulses generated from

a sparse bit sequence that represent the information as shown in Figure 3.1. It will

be modulated with K-Pulse Position Modulation so there will be K pulses distributed

along N positions (K<<N). In this study the position of the pulses will be randomized

and the data obtained will be averaged for different combinations in order to get

reliable results.

These pulses pass through a FIR filter before being transmitted. This Pre-

coding filter will be combined with the channel -another FIR filter- giving as a result

26

Figure 3.1: Symbol. Pulse Position Modulation (PPM)

the φ matrix. It will be implemented by a PN sequence to ensure the matrix meet

all the necessary conditions to make possible the recovering of the signal.

In the receiver, the downsampling will be performed by deleting the undesired

rows of the matrix. The number of remaining rows will represent the sampling rate

or number of measurements per symbol. Thus the remaining matrix will have M rows

by N columns. Along this study different number of measurements will be tested for

different degrees of sparsity.

After analyzing the theory, an empiric study will help to understand the prac-

tical problems and limitations of the methods proposed. As in every real-environment

simulations, variations from theory due to irregularities of the channel are expected.

Also new challenges can arise from real channel data and its combination with the

pre-coding matrix. The results will help to understand the scope and limitations of

real implementation of the Compressed Sensing methods applied to UWB.

27

3.3.2 Communication Model: Practical Approach

The Model proposed to apply Compressed Sensing to UWB, as shown in Figure

3.2, is based on a measuring matrix (Φ) that combines the effect of the Pre-coding

filter and the Channel. The number of measurements will determine the Sampling

Rate at the A/D and will define the number of rows of the Matrix. The information

will be applied to a UWB Pulse generator and Modulated with a Pulse Position

Modulation (PPM). The pulses generated will have a 2 GHz bandwidth and will be

centered in the frequencies 5 GHz and 7 GHz to study the bands 4-6 GHz and 6-8

GHz.

Figure 3.2: UWB Communication System Based on Compressed Sensing

This model intends to transmit a symbol of 256 different positions for the

non-zero elements as portrait in Figure 3.1. Considering Pulse Position Modulation,

the position of these non-zero elements will determine the symbol. This modulation,

as described above makes the most of UWB by transmitting several pulses, carrying

28

the information in the order that the pulses are placed within the frame. Obviously,

the more non-zero elements, the more information can be transmitted. However at

a given point the signal is not sparse anymore and compressed sensing is no longer

applicable. In the simulations the number of non-zero elements will be rose up to 128

in order to see how the density affects the performance of the different algorithms.

The output of the Pre-coding Filter will be a PN sequence of 128ns. Taking in

account the delay spread the length of the received signal will be over 256ns so a guard

period will be added at the end of each symbol. The total length of the symbol will be

of 512ns. Hence the link will be about 1.9 MSymbols/s. As depicted in the Table 3.1,

the number of pulses (k) used to encode will define the final bit-Rate. Nevertheless,

the more non-zero elements will require raising the Sampling-Rate, something against

one of the goals of UWB which is to reduce complexity in the receiver. This trade-off

will be solved depending on the requirements of each case bearing always in mind

that the signal has to have sparsity properties.

The measuring matrix (Phi) will be the combination of the PN filter and the

channel. Both the Pre-coding filter and the channel are modeled as Finite Impulse

Response (FIR) filters. Earlier simulations[13] showed that if the chip rate of the PN

sequence is equal to the bandwidth of the UWB pulse, the signal can be recovered

using the proposed algorithms. Thus the necessary chip rate will be 2 GHz. The

band conversion to the 4-6 GHz and 6-8 GHz will be done after the pre-coding filter.

29

Non-Zero Elements Bits Encoded Bit-Rate(Mbps)

1 8 15.6252 14 27.3444 27 52.7348 48 93.75016 83 162.10932 135 263.67164 203 396.484128 251 490.234

Table 3.1: Bit-Rate due to the number of pulses

Number of Measurements Sampling Rate (Msps) % of Nyquist Sampling Rate

8 15.625 0.3916 31.25 0.7832 62.5 1.5648 93.75 2.3464 125 3.1396 187.5 4.69128 250 6.25192 375 9.38

Table 3.2: Sampling Rate at the Receiver

Finally, the A/D Converter at the receiver does not need to down-convert but

sample directly. This is the real achieving of Compressed Sensing, how with a very low

sampling rate compared with the bandwidth, is possible to recover the transmitted

signal. The Table 3.2 shows the impressive sampling rate that the model can achieve.

Considering the Bandwidth of the signal of 2 GHz, these rates are at most a 10% of

the Nyquist(4 Gsps).

30

3.4 Summary

The Chapter describes the Compressed Sensing theory and gives mathematical

support. It is key to underline the importance of the Matrix and its properties in

order to reach a proper recovering after compression. At the end, a model to apply

CS sensing to UWB communication is suggested. The simulations as stated above,

will be based in this model.

CHAPTER 4

THE UWB CHANNEL

To verify the feasibility of Compressed Sensing the method will be tested for

different frequency bands and different environments. The environments chosen for

the research are completely different from each other. Its performance through this

different settings will show the reliability for changing conditions and will help to

understand better the communication model proposed.

4.1 Channel Measurement

The data from the channel has been obtained by a Vector Network Analyzer

(VNA), shown in Figure 4.1. The device performs channel frequency measurements.

These measurements include both frequency bands (4-6 and 6-8 GHz) and have 1

MHz frequency step with averages of 256 for the 4-5 GHz Band and 511 for the 6-8

GHz. Frequency measurements will be transformed into time domain thanks to the

Fast Fourier Transform (FFT) in order to implement the Matrix Φ.

The antennas employed are Azimuth Omni-directional and have linear phase

across frequency. The frequency range of these antennas, since they were designed for

UWB purposes, is from 3.1 to 10 GHz. In the simulations, these antennas are placed

1.5 meters high in different scenarios. They are connected to the VNA through a

31

32

Figure 4.1: Vector Network Analyzer (VNA)

Figure 4.2: 3.1-10 GHz UWB Antenna

coaxial cable which losses will be subtracted to the final channel measurement.

33

Figure 4.3: Hallway Environment

4.1.1 Hallway

As one of real possible indoor environments to use UWB a hallway has been

chosen for the simulations. This scenario will provide a realistic approach to a daily

context. It will allow to demonstrate the positive contribution that multipath has in

UWB. One Antenna will be placed at the end of the Hallway, right before a wall that

will act as a multipath reflector. The other one will be placed on the other end, 20

meters apart. As seen in the Figure 4.3 the ceiling is not flat so it will not generate

proper reflections. However, as all the doors are closed, the walls and the floor will

make a perfect cavity for the rays to propagate.

34

4.1.2 Metalic Box

This environment perfectly represents the inside of a Submarine or a Ship,

were all the walls are metallic. The principal property of the Metallic Box fixed to

ground, is that reflects everything. Thus the multipath will be extremely rich. Unlike

most of wireless technologies, especially narrow-band, the multipath is a problem in

these kind of situations. However in UWB it is used to collect larger quantity of

energy. The box is a square of 2.44 meters(8 feet) side as shown in the Figure 4.4.

Inside the box, the antennas will be placed perpendicular to one of the walls, 0.3

meters (1 foot) apart from the wall. Thus the distance between the antennas will be

2 meters (6 feet).

4.1.3 Inter-Vehicle

Since UWB has been proposed as a candidate for new upcoming Inter-vehicle

communication, the feasibility of a car-to-car transmission will be studied. The chal-

lenge of this scenario is that it is outdoors and is hard to collect most of the energy for

the receiver. Nevertheless the antennas have been placed inside the car, using it as a

receiving cavity, making the most of the multipath by collecting as much energy as

possible. One of the antennas will be placed inside the car in the backseat, close to the

car roof while the other will be situated outside in the roof of the other car. The cars

35

Figure 4.4: Metallic Box Environment

will be up to 8 meters away from each other and will simulate a urban environment.

The Doppler Effect will not be considered as stationary channel is assumed.

4.2 Channel Sounding Results

In this section, all the data obtained from Channel Measurements will be

presented and analyzed. The data obtained from VNA measurements of the Channel

in each one of the environments and its correspondent impulse respond.

The frequency measurements made on the Hallway, show a relatively stability

in terms of frequency. The several selective fading does not affect to the average signal

level, which is around -20 dB as shown in the Figure 4.6. For the 6 to 8 GHz interval,

36

Figure 4.5: Inter-Car Environment

the channel behavior follows the same pattern, very stable along the frequency and

a slightly lower level.

The Impulse Response (Figure 4.7), obtained from the Frequency Data, shows

that the multipath contribution is declining along the time with a slightly upturn at

the end probably due to the reflections in the background wall. The first component

is the greater one, this corroborates the fact that it is a line-of-sight communication

since it will be the fastest path. The other contributions along the time are the

multipaths, that will be collected from the receiver. While in narrowband this would

lead to a synchronization problem or to a distortion of the signal, in UWB is used to

collect more energy.

37

Figure 4.6: Hallway Channel in Frequency Domain

Figure 4.7: Hallway Channel Impulse Response

38

Figure 4.8: Metallic Box Channel in Frequency Domain

In order to simulate a marine environment, a box which walls are covered by

metallic material is used as a background. For this channel, due to the reflection

in the walls the level of the signal in for each frequency will be more irregular than

other environments. As portrait in Figure 4.8 The level of the signal decreases along

the frequency but is scarcely noticeable. And as there is much more multipath the

selective fading is greater than in the Hallway due to destructive interferences.

The Time Response (Figure 4.9), confirms the large number of multipath gen-

erated by the reflections on the walls. The energy decreases progressively due to

propagation, because very few is absorbed by the surroundings. This environment,

unlike for narrowband will not entail any challenge but it may help recovery. As the

matrix that represent the channel will have more non-zero elements, will make the re-

39

Figure 4.9: Metallic Box Channel Impulse Response

sulting measuring matrix having better isometry properties therefore easing recovery.

However this recovery is not guaranteed.

For Inter-Vehicle Environment, different measurements were made choosing

the worst one in order to prove the feasibility of the Technology. As expected, it

was the farthermost and without line-of-sight one. In these scenarios, the receiving

antenna was placed inside the car to use this as a cavity to collect as much energy

as possible for a proper transmission. This is shown in Figure 4.10, where the level

of the signal is remarkably lower than the other settings. The main reason is the

fact that there is no direct path and the antennas are placed further than the other

simulations. Also the slope is significantly steeper than the others scenarios, so the

pulse is sensitive with the frequency although is safe to say it is within an acceptable

40

Figure 4.10: Inter-Car Channel in Frequency Domain

range.

On the other side, the Impulse Response does not have the rich-multipath that

other environments do. This is due to the open-air surroundings in which there are

not many reflections. In fact the only reflections are the ones collected by the inside

car structure. The sudden drop in the signal along the time displayed in Figure 4.11

is due to this lack of reflections. It can become an issue for further distances and will

definitely limit the application of this technology to the Inter-Car applications.

As seen in the channels, almost all the energy of the pulse is transmitted within

the first 40 ns. Knowing the delay spread, and the length of the pre-coding filter, is

safe to say that the signal will be spread out within the 256ns frame as expected.

41

Figure 4.11: Inter-Car Channel Impulse Response

4.3 Summary

Throughout the Chapter, the different environments are presented in terms of

frequency and Time. Their features are discussed stressing the ones that may affect

the most to CS implementation such as multipath and the delay spread.

CHAPTER 5

CS ALGORITHMS COMPARISON

Wireless Communications technology is moving forward dramatically in the

last few years. However, sometimes the advances in hardware cannot fulfill the even

higher demanding levels in all kinds of fields. Then more efficient software is needed

to reach the expectations as well as improve the performance of the avant-garde

hardware. Besides, while high technology on hardware will always imply more ex-

pensive devices and manufacturing process, software will try to simplify some of the

requirements, reducing this way the prices.

In our case the algorithms used for recovering in the Compressed Sensing will

address de tradeoff between the high sampling rate which demands a very sophisti-

cated A/D Converter and the great computational time inherent to the process of

reconstruction which requires a complex receiver. Neither of the options is desirable

but in some cases any of them could be less restrictive.

This study will consider two main approaches to the reconstruction problem,

once based on Linear Programming (BP) and another using Greedy Methods(OMP

and StOMP).

42

43

5.1 Basis Pursuit (BP)

Basis Pursuit itself is not exactly an algorithm but a principle that stands for

finding signal representations in over-complete dictionaries by complex optimization[15].

By dictionary a collection of parameterized waveforms is meant so it can be Frequency

dictionary using sinusoidal waveforms, Time dictionary based on wavelets or even a

simply collection of waveforms that are zero except in one point. They will determine

the complexity and the feasibility of the method. Applied to Compressed Sending,

this dictionary will be given by the sensing matrix so it will contain information of the

channel and the pre-coding matrix. Thus BP addresses perfectly the reconstruction

problem on Compressed Sensing in which the sensing matrix (Φ) is an over-complete

dictionary and the signal representation will be the recovered signal (x) from the

original:

y = Φx

Broadly, the principle is to find a representation of the signal minimizing the

coefficients, making them sparse. To reach this the sparsest solution is necessary to

minimize in L1norm due to its singular property, with L2norm, the solution would

never preserve the sparsity. This can be explained by the heuristics[39], the solution

for the L2norm is the contact point where the smallest Euclidean ball and the sub-

space y meet, as it is shown in the Figure 5.1. In contrast, for L1norm minimization,

44

this ball becomes an octahedron so that the solution is the meeting point between the

subspace and any of the vertex of it. Any of the vertices of the octahedron will be a

sparse solution, hence the method guarantees the solution being the optimal, will be

the sparse. All of this is due to the fact that for most large underdetermined systems

of linear equations the minimal L1norm solution is also the sparsest solution.

Figure 5.1: Heuristic approximation to the minimization problem

5.1.1 Algorithm Definition

Defining the problem as an optimization in L1norm

min ‖x‖1 subject to Φx = y

Becomes a convex, non-quadratic optimization problem so that a translation

into Linear Programming problem is made:

min cTX subject to AX = bX ≥ 0

45

where X ∈ <m, X := (u; v), c := (1; 1), A := (Φ,−Φ), b := y in order to find

the recovered signal as:

x := u− v

During the process, the nonzero coefficients are associated with m columns of

the matrix A, building a basis of <m. The solution will be given by this basis after

optimizing through an iterative process. This process involves swapping columns of

the basis in order to find the combination that minimizes the solution X. In other

words, finding the solution to the Linear Programming is equivalent to a process of

Basis Pursuit.

Of all the Linear Programming algorithm, the most interesting ones for BP

resolution are the Simplex and Interior due to their characteristics:

The first one, BP-Simplex, starts from a linear independent columns of A

for which the product with y is feasible to iteratively improving the basis swapping

one term in the basis for one term is not. On each iteration the swap used will

be the one that best improves the objective function. Studies[16][17] have shown

how to select terms to guarantee convergence. The method achieves improvement

in each swap except at the optimal solution and the speed will be given by the

number of constraints, bounds on variables are implicitly handed and provide little

computational cost[40].

46

On the other hand, Although BP-Interior works swapping columns, not al-

ways choose the optimal swapping. Considering all the feasible points as a convex

Polyhedron, Simplex would be travelling around the border while Interior would find

the solution travelling inside reaching the border in the last iteration. This method

requires processing more information at every iteration so that In some cases some

intermediate iterations may not be feasible and find the feasibility eventually[15].

The algorithm used for the simulations will be Interior, is called PDCO[41],

and is a Primal-Dual interior method for Convex Objectives. It is the one that

only requires matrix-vector products with A and AT [42] instead of implicit functions,

something incompatible with Compressed Sensing for UWB purposes. Unfortunately

it often requires many of these matrix-vector products to converge representing higher

processing time, the main drawback of Basis Pursuit. This dead end situation is due

to the impossibility to find a implicit matrix to emulate the channel and the pre-

coding matrix even in a stationary environment.

5.1.2 Characteristics

The unique properties of the Basis Pursuit, have made it one of the references

of solving Compressed Sensing problems. Many studies and researches have come

up with a large number of alternative algorithms trying to overcome BP with faster

methods but none of them reach the accuracy of BP with such a few number of

47

samples.

BP is founded on a solid theoretical basis, so if a signal is strictly sparse in

certain transform domain it can be exactly recovered. The number of samples needed

and the processing time will be given by the sparsity of the signal and the properties

of the matrix as stated above. This relationship and the way they are related will be

studied throughout the simulations below in the thesis.

Because is based in global optimization it can stably super-resolve in ways

other methods cannot[15]. The way the iterations are made, enable to use signifi-

cantly less measurements than any other method. Although it implies more complex-

ity in the receiver, the sampling rate is reduced noticeably. In contrast with other

algorithms BP is based on a linear programming approach to the sparse representa-

tion problem, where instead of minimizing the number of nonzero coefficients in the

approximation, minimization of the sum of the absolute values of the coefficients.

Since the channel cannot be implemented implicitly, the BP-algorithm that

better suits the problem is the Interior and it can be easily obtained from the software

package SparseLab[41]. It is slower than other algorithms but there is no formula to

construct the combination of the channel matrix and the pre-coding matrix.

BP is stable in presence of noise, in fact there is a variation of BP that still

finds the optimal answer in presence of certain level of noise. It is called BP De-

noising (BPDN) and is based in the same principle but relaxing the constraint of the

48

original problem[43]:

min ‖x‖1 subject to ‖Φx− y‖2 ≤ σ

Where σ is an estimation of the noise level in the data. Although in this thesis

the noise will not be considered it is important to point out that BP solves this kind

of problems with high level of reliability.

Unfortunately life is not all that beautiful and there are several drawbacks that

are needed to be taken in account. Since BP is a relatively computationally expensive

algorithm and the number of iterations is unbounded, it is hard to set a limit or make

safe approximations about the processing time. Along the simulations will observe

a tendency but there can always be cases in which, due to the characteristics of

the channel, or the combination between the pre-coding matrix and the signal, the

computational time suddenly increases.

The properties of the Matrix will also affect the performance, and although

the method is stable the matrix still have to meet the Isometry conditions. Good

restricted Isometry constants are required to reach an acceptable performance. The

isometry conditions are held with overwhelming probability if the matrix presents

entries that are independent and identically distributed (iid). This is reached for

dense matrices that represent high multipath channels. However, the more dense the

matrix, the worse performance in terms of timing.

49

5.2 Orthogonal Matrix Pursuit (OMP)

Due to the density of the Sampling Matrix a new alternative to Linear Pro-

gramming was needed in terms of simplicity and speed. Dense matrices entail great

computational burden that some real-time transmission cannot permit. A main al-

ternative is Orthogonal Matching Pursuit (OMP) due to its speed and its ease of

implementation. It is an alternative approach that is not based on optimization; it

does not seek for any optimization goal but identify which components of the sam-

pling matrix are related with the non-zero elements to build the sparse signal. OMP

still maintain the property of recovering a k − sparse signal when the number of

measurements m is nearly proportional to k [18]. However, the sparsity level of the

signal is needed in advance for the method to resolve the problem.

OMP method starts from an empty model and builds up a smaller matrix with

all the columns that are related with the non-zero elements, picking one column at

each iteration. OMP is considered a greedy algorithm because selects the columns in

a greedy fashion. At each iteration chooses the column that is more correlated to the

sampled signal y to build this way the matrix with the chosen atoms as is shown in

the Figure 5.2.

After k iterations, the algorithm should have chosen all the columns related

to the non-zero elements of the original sparse signal.

50

Figure 5.2: Atoms related to non-zero elements (OMP)

5.2.1 Algorithm Definition

Defining r as the residual, γt as the number of the column with greater corre-

lation level with the received signal y at iteration t and Γt as the vector of the index

of the selected columns from which the estimation of the signal (x) will be obtained,

the matrix with the selected columns(Λ) will be obtained after k iterations.

In the first iteration, the data is initialized:

r0 = y , Γ = � and t = 1

The first column will be selected according to the correlation with the received

signal:

γt = arg maxi=1,...,n |〈rt−1, φi〉|

If there is several columns with the highest correlation level, the method will

choose one deterministically[18]. Then the column is added to the set of index Γ and

to the matrix Λ:

51

Γt = Γt−1

⋃γt , Λt = [Λt−1 φγt ]

Then it is just solving a least squares problem in order to get the signal esti-

mation. This signal estimation will be an all non-zero elements that together with

the index of the matrix will give the final solution after the last iteration:

xt = arg minx ‖y − Λt · x‖2

To solve this projects orthogonally y onto all selected atoms:

x = Λ†ty

where Λ†t is the pseudo-inverse calculated based on QR or Cholesky factor-

ization. The coefficient vector is the orthogonal projection of the signal onto the

dictionary elements selected up to this iteration. This property gives the method the

name and ensures the algorithm selects a new element in each iteration. If it is not

the last iteration, the residual is updated:

rt = y − Λtx

Then the correlation with the remaining columns is made again, repeating

the same steps until the last iteration that corresponds with the number of non-zero

elements of the original signal.

As stated before, the signal estimation will be the elements of x placed in the

positions of Γ.

52

5.2.2 Characteristics

As an alternative for BP, Orthogonal Matching Pursuit is much faster, both

theoretically and experimentally. According to several studies [44] [45] [19] OMP is

observed to perform faster and is easier to implement than L1-minimization. OMP,

iteratively selects the vectors from the sampling matrix that contain most of the

energy of the measurement vector y. The selection at each iteration is made based on

inner products between the columns of the matrix and a residual. The residual reflects

the component of y that is orthogonal to the previously selected columns[46][47]. It

takes k iterations, where each iteration amounts to a multiplication by a mxn matrix

Φ and includes solving a least squares problem in dimensions at most mxk, yielding

a strongly polynomial running time.

Besides the simplicity of each iteration, the iterations are limited, it has been

proven is that OMP selects a correct term at each iteration, and terminates with the

correct solution after just k iterations. In fact studies [17] pointed out that the k-step

solution property is not a necessary condition for OMP to succeed in the recovery of

the sparsest solution although it is sufficient. The fact that the iterations are bounded

makes OMP even simpler making the complexity of OMP significantly smaller than

that of LP methods, especially when the signal sparsity level K is small[48].

53

It has been demonstrated OMP is simpler and faster, but its success in Com-

pressed Sensing against other fast algorithms finding the sparsest solution is its or-

thogonality. Thanks to orthogonal projection used, the residual rn is always orthog-

onal to all previously selected elements and then these elements are not selected

repeatedly[49]. Methods such Matrix Pursuit (MP) can converge to a solution that

explains the data but it is not guaranteed that it is a sparse solution. Instead OMP

additionally orthogonalizes the residual against all previously selected measurement

vectors. Despite this step increases the complexity of the algorithm, it improves its

performance and provides better reconstruction guarantees compared to plain old

MP. Experiments[50] shown OMP as the algorithm with superior performance from

the family of matching pursuits.

As a trade-off, OMP lacks of stability where other algorithms do not. As

demonstrated in some studies[35][21] if OMP selects a wrong element in some step

it might never recover the right signal. Like other greedy algorithms OMP cannot

provide uniform recovery guarantees as other methods like convex relaxation does.

Being a heuristic mode, there is not solid theoretical foundation about the reliability

of the methods but empirical experiences have shown it works in most of the cases.

Perhaps building up an approximation one step at a time by making locally optimal

choices at each step makes it more vulnerable to fail in certain scenarios than l1

minimization which uses a global optimization.

54

Another drawbacks are the requirements of OMP needs to solve the problem.

To implement the algorithm, the level of sparsity (k) is needed and it is used as the

upper bound in the iterations. Also requires that the correlation between all pairs

of columns of the matrix is at most 1/2k to operate successfully representing a more

restrictive constraint than the Restricted Isometry Property [48].

Comparing again with MP, calculating the pseudo-inverse of the sub-matrix

through QR or Cholesky factorization will require computationally more demanding

than Matching Pursuit but ensuring that the algorithm selects a new element in each

iteration and that the error is minimal for the currently selected set of elements[22].

These factorizations require additional storage, not very significative when for small-

sized problems but when it comes to large problems, the storage requirements can

became an issue and sometimes Λ cannot be stored. New studies [49] try to develop

fast approximate OMP algorithms that require less storage.

Finally, the vulnerability against noise is similar to algorithms such BP when

the level noise is not too high as shown in some studies[21][35]. For higher level

of noise, the performance of OMP gets worse due to the instability of the method

although it has not been demonstrated theoretically. As stated before this study will

not go deeper in the matter since the simulations are noiseless.

OMP has been the reference of many other models that tried to make speed and

simplicity their principal feature, achieving highly efficient computations comparable

55

to existing CS algorithms.

5.3 Stagewise Orthogonal Matrix Pursuit (StOMP)

So far two main trucks has been portrayed to solve the recovery of sparse

solutions problem with CS, the fast and simple option (OMP), and the accurate and

less demanding (BP). However in the case where of large scale problems both methods

become extremely slower, sometimes unacceptable. Addressing these cases, StOMP

was implemented. The nomenclature Stage-wise OMP is due to the fact that the

algorithm is able to select most of the columns related with the non-zero elements of

the vector in one iteration or step.

StOMP as an extension of OMP, is a fast greedy algorithm but its singular

characteristics make it one of the state-of-the-art fast CS algorithms[51]. Its main

difference with original OMP is the way to select the columns for the matrix that

contains the non-zero atoms. Instead of selecting one column at a time, fixed a

threshold, all the columns whose correlation value is over the threshold will be selected

as matched columns. Hence with just few iterations all the columns can be selected.

Nevertheless the proper performance of the algorithm will rely in the correct choice

of the threshold. As shown below it can become an issue.

Due to the demanding requirements of the method, it is not that efficient for

high sparsity (few non-zero components), which is a contradiction because it should be

56

easy to set a threshold able to differentiate the columns corresponding with the non-

zero elements. The values of the correlation in that case should be clearly different,

but in most of the cases the columns of the matrix are not taken from the Uniform

Spherical Ensemble (USE)[20] they are not Independent Identically Distributed (iid)

as the method requires.

5.3.1 Algorithm Definition

As in OMP, a matrix with the selected columns from non-zero elements will be

build up to solve a system of equations. Using the same notation: r will represent the

residual, γt the number of the column selected at iteration t and Γt as the vector of

the index of the selected columns and Λ the actual matrix with the selected columns.

Besides, with StOMP a vector of residual correlations will be saved (Ct).

In the first iteration, the data is initialized:

r0 = y , Γ = � and t = 1

The correlation vector is obtained by applying a matched filtering:

Ct = ΦT rt−1

In this case, instead of selecting one column, all the columns whose correlation

is above the threshold will be selected:

57

Γ = j : |Ct(j)|

Then the set of selected columns is added to the the matrix Λ:

Λt = [Λt−1 φj] : j ∈ Γ

Again, the problem is reduced to solving a least squares problem:

xt = arg minx ‖y − Λt · x‖2

where the elements of x are the values of the non-zero components of the

solution.

Unlike OMP the xt approximation will be reached by:

x =(ΛTt Λt

)ΛTt y

At this point, the residual is updated to find the vector of correlations again:

rt = y − Λtx

If the none of the components of Ct = ΦT rt−1 is above the threshold, the

algorithm considers the process done. Otherwise the procedure will continue with

the newly selected columns.

As in OMP, the signal estimation will be the elements of x placed in the

positions of Γ.

58

5.3.2 Threshold

The main advantage of StOMP over the rest of methods is the capability of

solving the problem within just few iterations due to the use of the threshold. The

threshold allows to select several atoms at each iteration reducing the complexity of

the algorithm and hence the computational time. However the choice of the threshold

is not an easy task. A wrong election of the threshold can lead to a poor performance.

Two criteria have been proposed to set the threshold. After analyzing both of

them, this study will provide empirical values for a good performance in the concrete

cases that are treated. The two ways of fixing the value of the threshold try to limit

the number wrong selected columns also called False Alarms. Once fixed this limit,

each step will try to maximize the number of detections to find the solution as fast

as possible. The limits are controlled by setting either the False Alarm Rate(FAR)

or the False Discovery Rate(FDR):

5.3.2.1 False Alarm Rate (FAR). This rate measures the number of

False Alarms over the total possible False Alarms that can occur:

FAR = FalseAlarmsm−k

To find the threshold from this rate the inverse of the normal Cumulative

Distribution Function (CDF) of 1−FAR/2 will be calculated. This is possible thanks

to the Gaussian behavior of the vector of correlations. It is considered a non-zero

59

entries combined with large number of ’Gaussian Noise’[20]. Furthermore it has been

proved[20] that if the matrix belongs to Uniform Spherical Ensemble(USE)(iid points

on the unit sphere Sn−1), the residual Z = x̃ − x0 has a histogram that is nearly

Gaussian.

Obviously, with this criterion is necessary to know in advance the sparsity of

the signal in order to set it properly. Holding the threshold with a tight FAR can

lead to miss any essential atom so the recovery will be unreachable. On the other

hand, a very loose may affect in the timing performance of the method and some

accuracy. The method still can reach the solution even selecting more columns than

necessaries. The values of the solution that correspond with zeros will have very low

-close to zero-, but not zero so some accuracy will be lost.

5.3.2.2 False Discovery Rate (FDR). It also measures the number of

False Alarms but over the discoveries already made:

FDR = FalseAlarmsk′

where k′ is the number of discoveries that not necessarily has to be the same

as non-zero elements.

FDR also assumes that the correlations vector have Gaussian behavior. How-

ever it does not consider the sparsity so, although it is a bit more complex than the

FAR method, FDR can be used when the sparsity is unknown a priori.

60

To apply the criteria to obtain the threshold the Complementary Error Func-

tion values of the correlation vector are calculated and sorted. These values are

compared with a curve defined by the FDR value. the values under this curve will

define the threshold.

The value of the threshold in both cases will have to be different depending

on the percentage of non-zero elements of the signal. Even if the exact number is

unknown, an approximated idea of the non-zero elements will be necessary in order

to set the threshold accurately.

StOMP, in theory is the fastest method to recover the signal but the threshold

setting makes it unstable and dependent. Only by finding a pattern or a reliable

criterion to define it, will make it efficient and competitive with the other suggested

algorithms. Along this study different thresholds will be analyzed in order to find

any trend that allows a stable and reliable setting for proper reconstruction.

5.4 Simulation in Matlab

The Simulations in Matlab will show the processing that would take part in the

receiver. The received signal will be processed assuming that the channel and the pre-

coding information are known by the receiver. Then the three methods commented

61

above will be applied to recover the original signal and obtain results related to the

Error Rate, Processing Time and Number of Missed Detections.

5.4.1 Measuring Matrix

To obtain the measuring matrix the Pre-coding Filter and the Channel will be

combined. The Pre-coding Filter as well as the Channel will be considered as FIR

filters so they will be implemented with Toepliz matrices. So will be the resultant

matrix Φ. However, since the receiver samples with lower rate than the transmitter

does, the number of rows will be less than the columns. This downsampling makes the

matrix a Quasi-Toepliz Matrix in which its rows are selected rows from the original

Toepliz. In the new matrix, each row will be the row above shifted several positions.

The number of positions the rows will be shifted will be given by the downsampling

factor. For Example, if the 256 elements signal that is intended to transmit, is

sampled with 64 measurements, the downsampling factor will be 4, so the rows will

be shifted 4 positions from the row before. As shown in the Figure 5.3.

The procedure will repeat transmission at the same environment changing the

order of the non-zero elements for a given sparsity. This will offer a widest perspective

of recovering, so that the results just will take in account the sparsity and not the

particular combination of the signal and the measuring matrix. The methods will be

tested in order to set a minimum number of measurements that are needed to recover

62

Figure 5.3: Quasi-Toeplitz Matrix

a signal for a given sparsity. This number of measurements will be compared with

former approximations that have been made in earlier studies.

The sparsity of the signals will vary from one single spike to 128, being this,

the half of the total elements. It is with 128 non-zero elements when the maximum

degrees of freedom is reached. Nonetheless, to reduce the hardware complexity, one of

the goals for the future, is necessary to reduce the sampling rate as much as possible.

Thus, lowering the number of measurements will be the principal premise.

63

5.5 Simulation Results

In this section, simulation data will be obtained. Firstly, the accuracy on

the recovery will be shown as the Error Rate. Later on, the limit of reducing the

sampling rate will be defined thanks again to the data from simulations. And finally,

the Computational Time will be compared in order to analyze the complexity of the

algorithms.

5.5.1 Accuracy on the Recovery

For this section, two types of accuracy are defined on the recovery, the error of

the recovered signal as a whole and the concrete error of each one of the components

of the signal.

The Error Rate that will measure the global error is defined as:

ErrorRate =∑256

i=1 (x−x̂)2∑256i=1 x

2

It will indicate the total accuracy so that, setting a threshold is possible to

define whether is properly recovered or not. In this case, if the Error Rate of a

recovering is < 0.001 the signal will be considered as well recovered.

Starting with Basis Pursuit, in the Figure 5.4 corresponding with the Hallway

channel simulations, is possible to observe the differences in accuracy. They are

mainly related to the number of measurements. As the level of sparsity grows, the

64

Figure 5.4: Error Rate of BP Recovery for different levels of sparsity

necessary number of measurements to maintain the Error Rate below the desired level

will also grow, but a different rate. The different traces represent different levels of

sparsity, being each one twice the previous. However, the number of measurements

required are less than that, sometimes even less than 40% more.

Comparing the behavior in an upper frequency band (Figures 5.45.5), very

little difference is observed. If any, the curves representing lower levels of sparsity

(higher number of non-zero elements)are slightly left shifted, which means that for

the higher range of frequencies, the BP performs better in this particular scenario.

For the other environments, as portrait in the Figures 5.65.7 the method per-

forms equally well, showing an outstanding stability toward changes in the channel.

It will be very useful for Inter-Car applications where the environment is changing

constantly.

65

Figure 5.5: Error Rate of BP Recovery (6-8 GHz)

Figure 5.6: Error Rate BP Recovery Metallic Box (4-6 GHz)

66

Figure 5.7: Error Rate BP Recovery Inter-Car (4-6 GHz)

For Orthogonal Matching Pursuit, significantly more measurements will be

needed for a successfully recovery. The Figure 5.8 indicates that the signal cannot

be recovered with less than 40 measurements what means about 80 Msps. Also for a

number of non-zero elements above 16, the method cannot recover the signal. These

have no theoretical demonstration, but empirically, is safe to say that 2 out of 3

times the algorithm fails to recover for this sparsity levels. Thus the curves for the

few successfully recoveries are not displayed in the Graphic.

This time, changes in the frequency range do affect the performance making

it remarkably worse. The several vertical traces for the same sparsity indicate the

instability and the lack of consistency. Thus, to ensure a recovery is necessary to

sample with a slightly higher sampling rate in order to ensure the recovering in the

case of a transmission in an upper frequency range.

67

Figure 5.8: Error Rate OMP Recovery (4-6 GHz)

Figure 5.9: Error Rate OMP Recovery (6-8 GHz)

68

Figure 5.10: Error Rate OMP Inter-Car(4-6 GHz)

Figure 5.11: Error Rate OMP Metallic Box(4-6 GHz)

When it comes to a Metallic surrounding environment, the OMP reaches the

recovery with less than 40 measurements. For Inter-Car environment, the perfor-

mance is similar to the Hallway, guaranteeing the recovery with 40 measurements for

one non-zero element signal. However the visibly difference with BP is still present

showing inconsistency in the recovery. This, as stated before can lead to an even

higher sampling rate.

If OMP has problems solving dense signals, the StOMP is the opposite case.

According to Figure 5.12, the algorithm is unable to recover the sparsest signal with

less than 100 samples. Nevertheless, when it reaches a top (128 samples in the FAR

69

Figure 5.12: Error Rate StOMP Hallway (4-6 GHz)

case) it can keep recovering until the half of the elements are non-zero. It can be

very useful for large sized problems and can complement the OMP deficiency. In the

graphic the two threshold techniques are compared. Setting the optimal threshold for

each case, a better performance is observed in the case of False Alarm Rate (FAR)

case. It strictly follows the observation above that once it reaches 128 samples, it

does not need more to successfully recover the signal. Apparently a matrix twice wide

as high has favorable conditions for StOMP to recovering.

The changes in frequency just affect the StOMP when False Discovery Rate

is used to set the threshold. The graphic from the Figure 5.13 shows the poor per-

formance of the FDR case. Yet for the FAR case, the Error does not change very

70

Figure 5.13: Error Rate StOMP Hallway (6-8 GHz)

much with respect to the 4 to 6 GHz frequency range. Is observable the fact that for

signals with 8 or more non-zero elements, the method reaches a top in the number

of samples needed to recover. With this number fixed it is obvious that the more

non-zero elements will enable to transmit more information with the same sampling

rate. This is very useful in the case of constraints in sampling rate and processing

time are equally demanding.

As OMP, StOMP is very sensible to changes in the channel. As expected,

the Metallic Box channel will make the measuring matrix denser and together with

the random contribution ease the recovering. Both techniques seem to be similarly

effective in the Metallic box as portrait in Figure 5.15, opposite to Inter-Car envi-

71

Figure 5.14: Error Rate StOMP Inter-Car(4-6 GHz)

Figure 5.15: Error Rate StOMP MetallicBox (4-6 GHz)

ronment where the algorithm shows instability using the FDR threshold technique.

Although the method looks instable if the FAR technique is applied, great stability

can be reached, even for most challenging environments as Inter-Car (Figure 5.14).

The property of half of the length sampling is held for the FAR case.

5.5.2 Number of Measurements

Analyzing the error in every component of the reconstructed signal, a new

concept is defined: the number of missed detections in the recovery. This concept

is used to define if a given number of measurements really guarantees the recovery.

72

If for a 5000 simulations, this ratio is 0, a reliable recovery is reached with the

given parameters. Following this criteria, the results of the minimum number of

measurements needed for recovery in every situation are presented in the Figures

5.16-5.21.

In the graphics, the fact that OMP appears just for the sparsest signals (k ≤

n/8) is that in the other cases, although it works, is not completely reliable. In

some cases (1 out of 200 simulations), a fatal error occurs. The error can be due to

the high number of columns belonging to non-zero elements and the method fails to

differentiate between them once the algorithm has reached and advanced point.

However, OMP has presented great features for high sparsity (k ≤ n/8) in

every environment, being hardly affected by changes in frequency. If any, it requires

slightly more samples to reconstruct in the case of Inter-Car environment.

The curious case of Stage-wise OMP shows that, for FAR threshold technique

achieves a superb performance for signals with large number of non-zero element.

The main attractive of the method is its independency with the sparsity. This in-

dependency makes it equally effective transmitting different amounts of information.

The top of information is reached when half of the elements are non-zero (k ∼= n/2).

The main drawback of this method is its efficiency and inconsistency dealing with

few non-zero elements.

73

Figure 5.16: Error Rate OMP Inter-Car(4-6 GHz)

Figure 5.17: Error Rate OMP Metallic Box(4-6 GHz)

Figure 5.18: Error Rate OMP Inter-Car(4-6 GHz)

Figure 5.19: Error Rate OMP Metallic Box(4-6 GHz)

Figure 5.20: Error Rate OMP Inter-Car(4-6 GHz)

Figure 5.21: Error Rate OMP Metallic Box(4-6 GHz)

74

Figure 5.22: Sampling Rate (High Sparsity)

Figure 5.23: Sampling Rate (Low Sparsity)

Above all, the results show that BP algorithm overcomes the rest in terms of

samples needed, stability and reliability.

The Figure 5.22, indicate the average of Minimum Sampling Rate that every

algorithm can reach depending on the sparsity. This graphic gathers information of

75

Non-Zero Elements (k) Number of Measurements (m) Constant C = m√k logn

1 10.1 4.152 14.5 4.254 21.3 4.428 29.6 4.3516 42.8 4.4432 61.8 4.5364 88.6 4.6128 117.5 4.29

Table 5.1: Sampling BP Constant

all the performances. It gives a general idea of the behavior of each of the algorithms.

In the first place, remark the sparsity-independent performance of StOMP.

Highly inefficient for high sparsity (Figure 5.22) but very interesting when the Data-

Rate goes up (Figure 5.23).

In the case of OMP and BP, both cases the sampling-rate happen to depend

on the sparsity and logarithmically on the length of the signal. The number of

measurements for BP, as shown in the graphic follows the pattern:

mBP ≥ CBP√k log n

where CBP ≈ 4.3 is a constant. This constant values for the given results, sets its

approximate value in 4.3:

Therefore, unlike some studies point out a direct dependency of this constant

on the sparsity, the study reveals that empirically, it can be considered as square-root

dependent.

76

Finally, for the OMP case also can see a root dependent behavior, but in this

case the best approximation is reached by:

mOMP ≥ COMP4√k log n

where COMP would have a value around 18.9.

mStOMP :6= f (k)

5.5.3 Processing Time

As the computational time is highly related to the size of the measuring matrix,

the more samples, the more time will be needed to reconstruct the signal. However the

BP curve is not really that pronounced as in the number of measurements. Seems like

the computational burden relies on the density of the procedure, which is the number

of non-zero elements of the signal. The complexity does not change that much along

the frequency neither the channel as a sign of its robustness.

Unlike BP, it does get affected by the channel, showing better timing on the

Metallic Box surrounding and unexpected bad performance on the Hallway case (Fig-

ure 5.25). This vulnerability to changing conditions can be very dangerous in changing

environments such Mobil communications and Inter-Vehicle.

For StOMP, as shown and commented above, the threshold technique that

provides better results is the FAR against the FDR. As BP does, StOMP also presents

77

Figure 5.24: Processing Time of BP

Figure 5.25: Processing Time of OMP

78

Figure 5.26: Processing Time of Stagewise OMP

Figure 5.27: Processing Time (High Sparsity)

completely independence to changes in channel and frequency (Figure 5.26).

Comparing the three algorithms in terms of Processing Time and Complexity,

the one that optimizes the computing time for high sparsity is OMP. Apparently is the

79

Figure 5.28: Processing Time (Low Sparsity)

simplest, and it works on its favor for the less information demanding transmissions.

From k ≥ n/32 on, the simplest recovery method becomes StOMP. Both two methods

are far simpler and faster than BP. The dependency on the sparsity of both BP and

OMP, leaves StOMP as the best performing algorithm in terms of less complexity.

As shown in the Figure 5.28 the more information, the more efficient it becomes.

5.6 Summary

Compressed Sensing offers different ways to apply the Compressed Sensing

to communications. The different algorithms provide advantages and disadvantages

depending on the characteristics of the transmission. As seen in the results, there

is a clearly trade-off when it comes to choose an algorithm to achieve a recovering.

80

While BP recovers outstandingly well and with relatively low sampling rate, it has

very demanding computational burden. The Orthogonal Matching methods, being

far less stable and needing higher sampling rate are noticeable simpler and can be

used when sampling-rate is not the main priority.

CHAPTER 6

CONCLUSION

6.1 CS applied to UWB

According to the theoretical analysis and the empirical results obtained, it has

been proved that Compressed Sensing is a perfect suitable candidate to support UWB

wireless transmissions. Even for the worst conditions and hostile environments, it has

presented as a more than interesting option to address the demanding requirements

of UWB. Compressed Sensing has the potential to substantially lower the sampling-

rate, while selects the key information. This way the receiver is simplified and the

costs of the analog-digital conversion are reduced.

The model is perfectly adjusted to the UWB communications by reformulating

the problem to be a linear-time invariant system, using filters to represent the codi-

fication and the channel. The resultant Toeplitz matrix, as demonstrated in earlier

studies [52] can be perfectly used as the measuring matrix. Throughout this thesis it

has also been proved that if the matrix satisfies the Restricted Isometry Propierties

(RIP), using CS to successfully recover the signal with any of the given algorithms.

This thesis has also presented a deep study on the number of samples nec-

essary, proposing empirical-based models to approximate the minimum number of

81

82

measurements. It also suggests values for the constants that can be useful to orien-

tate the reconstruction based on the sparsity and the length of the signal. However

the methods have slight variations depending on upon the channel and it relation

with the encoding matrix that need to be taken into account.

6.2 Reconstruction Algorithms

The other main point of this thesis is the comparison between the different

methods that CS offers to implement the transmission. The two paths for the recov-

ering are based either on Linear Programming (BP) or on Linear Algebra (OMP &

StOMP).

As pointed above, the demanding sampling rate that UWB requires, calls for

less number of samples to simplify the conversion from analog to digital, the principal

bottleneck of UWB. The lowest sampling rate can only be achieved by BP. Thanks

to the Linear Programming, the information can be recovered with ostensible less

measurements than other methods. However, it requires a complexity that makes

the procedure extremely complex and computationally slow compared with Basis

Matching. The algorithms based on OMP, due to the Linear Algebra operations in

which are based, are faster and easier to implement.

Along the thesis it has been demonstrated that for different environments

BP had a stable performance unlike OMP that suffers a noticeable worsening when

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the signal level gets very low. This can happen due to a lack of line-of-sight, lack

of contributions due to reflections or because the receiver is placed that far that

the propagation losses make the signal level drop dramatically. Thus for a harsh

environment BP should be chosen to ensure a good and stable performance.

In the case the requirements are of timing and processing simplicity, OMP will

be the option for high sparsity (k ≤ n/8) and StOMP for low sparsity, of course using

the FAR threshold technique. Nevertheless, they will require extra complexity on the

hardware part of the receiver that will mean a substantially increase of the cost.

As the advances in technology are greater in the field of software than the ones

improving the hardware, in the long run it may be interesting for future studies to

focus on developing the BP algorithm rather than the ones based on Linear Alge-

bra. The processing time can always be improved much more easily than the A/D

converters.

As seen along the study, they offer different advantages, but also some draw-

backs that need to be addressed in future studies. One of the drawbacks is the

instability of OMP for high sparsity due most likely to the measuring matrix, so

new encoding methods can be developed to adapt better to given channels. Even

new algorithms that define the encoding once they read the channel to optimize the

transmission.

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6.3 Future Work

The study has proved that the information is successfully recovered by Com-

pressed Sensing algorithms using dedicated software. However for fast and efficient

performance, it should be implemented in Hardware Description Languages (HDL).

The following goal of Compressed Sensing application to UWB then, should be achiev-

ing the implementation in low level languages.

Since both OMP and BP have de-noising version that are proved that work,

a similar study should be done in order to verify if they preserve the properties and

behavior.

The great complexity of BP can be optimized since it was not designed specif-

ically for communication. The algorithm can be modify since in communication the

receiver has more information than what BP uses.

Also find theoretical explanations of the non-information dependency of StOMP,

which can mean a great breakthrough on the Compressed Techniques.

Knowing that the channel affects in the results of the recovering, sometimes

in a great deal, the properties of the matrix should be analyzed and studied.

Finally, a study should be perform addressing the Channel sounding when this

is not sparse, and then Compressed Techniques cannot be applied.

BIBLIOGRAPHY

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86

[1] R. C. Qiu, R. Scholtz, and X. Shen, “Ultra-Wideband Wireless Communications–A New Horizon,” IEEE Trans. Veh. Technol., Editorial on Special Issue onUWB, vol. 54, September 2005.

[2] X. Shen, M. Guizani, R. C. Qiu, and T. Le-Ngoc, UWB Wireless Communica-tions and Networks. Wiley, 2006.

[3] J. Reed, “An Introduction to Ultra Wideband Communication Systems,” Pren-tice Hall Communications Engineering and Emerging Technologies Series, 2005.

[4] M. Ghavami, L. B. Michael, and R. Kohno, UWB Signals and Systems in Com-munications Engineering. Wiley, 2004.

[5] F. Nekoogar, UWB Communications Fundamentals and Applications. PatienceHall Communications Engineering and Emerging Technologies Series, 2006.

[6] K. Siwiak and D. McKeown, UWB Radio Techonology. Wiley, 2004.

[7] M. Di Benedetto and G. Giancola, UWB Radio Fundamentals. Patience HallCommunications Engineering and Emerging Technologies Series, 2004.

[8] E. Yeung and M. J., “Modelling of Ultra-wideband (UWB) Radio System,”Department of Electronic & Electrical Engineering, 2002.

[9] D. Porcino and W. Hirt, “Ultra-Wideband Radio Technology: Potential andChallenges Ahead,” IEEE Commun. Mag., vol. 41, pp. 66–74, July 2003.

[10] D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inform. Theory, vol. 52,pp. 1289–306, April 2006.

[11] E. Candes, “Compressive Sampling,” in Proceedings of the InternationalCongress of Mathematicians, vol. 3, pp. 1433–1452, 2006.

[12] J. Haupt and R. Nowak, “Compressive Sampling for Signal Detection,” Acous-tics, Speech and Signal Processing, 2007. ICASSP 2007. IEEE InternationalConference on, vol. 3, 2007.

[13] Z. Q. R. Zhang, P. Hu, “A Compressed Sensing Based Ultra-Wideband Commu-nication System,” 2008.

[14] E. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inform.Theory, 2005.

87

[15] D. L. Donoho, S. Chen, and M. Saunders, “Atomic Decomposition by BasisPursuit,” SIAM Journal on Scientific Computing, vol. 20, pp. 33–61, 1998.

[16] E. Candes and T. Tao, “Decoding by Linear Programing,” IEEE Transactionson Information Theory, vol. 51, no. 12, 2005.

[17] D. Donoho and Y. Tsaig, “Fast Solution of l 1-norm Minimization Problemswhen The Solution may be Sparse,” tech. rep., Stanford University Departmentof Statistics, 2006.

[18] J. Tropp and A. Gilbert, “Signal Recovery From Random Measurements Via Or-thogonal Matching Pursuit,” IEEE Transactions on Information Theory, vol. 53,no. 12, pp. 4655–4666, 2007.

[19] D. Needell and R. Vershynin, “Uniform Uncertainty Principle and Sig-nal Recovery via Regularized Orthogonal Matching Pursuit.” Arxiv preprintarXiv:0707.4203, 2007.

[20] D. Donoho, Y. Tsaig, I. Drori, and J. Starck, “Sparse Solution of Under-determined Linear Equations by Stagewise Orthogonal Matching Pursuit,” inPreprint, 2006.

[21] S. Kunis and H. Rauhut, “Random Sampling of Sparse Trigonometric Polyno-mials II - Orthogonal Matching Pursuit versus Basis Pursuit,” June 2007.

[22] T. Blumensath and M. Davies, “Iterative Thresholding for Sparse Approxima-tions,” in To appear in Journal of Fourier Analysis and Applications, specialissue on sparsity, 2008.

[23] R. Saeed, S. Khatun, B. Ali, and M. K. Abduallah, “UWB Ad-hoc Networks:Review and Trends,” Journal of Computer Sciencie, March 2005.

[24] D. Porcino, “UWB Regulations & Coexistence,” IEEE WCN’04, 2003.

[25] I. Oppermann, M. Hmlinen, and J. Iinatti, UWB Theory and Applications. Wi-ley, 2004.

[26] R. C. Qiu, H. P. Liu, and X. Shen, “Ultra-Wideband for Multiple Access,” IEEECommun. Mag., vol. 43, pp. 80–87, February 2005.

[27] I. Lahaie, UWB Radar:SPIE Proceedings. 1992.

88

[28] X. Chu and R. Murch, “Performance Analysis of DS-MA Impulse Radio Commu-nications Incorporating Channel-Induced Pulse Overlap,” IEEE Trans. WirelessCommun., 2006.

[29] S. Zhao and H. Liu, “On the optimum Linear Receiver for Impulse Radio Systemin the Presence of Pulse Overlaping,” IEEE Commun. Letters, vol. 9, pp. 340–342, April 2005.

[30] R. H. Walden, “Analog-to-Digital Converter Survey and Analysis,” IEEE Jour-nal on Selected Areas in Communications, vol. VOL. 17, p. 539, April 1999.

[31] B. Le, T. W. Rondeau, J. H. Reed, and C. W. Bostian, “Analog-to-DigitalConverters,” IEEE Signal Processing, 2005.

[32] E. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Projec-tions: Universal Encoding Strategies?,” IEEE Transactions on Information The-ory, vol. 52, no. 12, pp. 5406–5425, 2006.

[33] H. Rauhut, K. Schnass, and P. Vandergheynst, “Compressed Sensing and Re-dundant Dictionaries,” IEEE Transactions on Information Theory, vol. 54, no. 5,pp. 2210–2219, 2008.

[34] M. Iwen, “A Deterministic Sub-Linear Time Sparse Fourier Algorithm via Non-Adaptive Compressed Sensing Methods,” in Proceedings of the nineteenth annualACM-SIAM symposium on Discrete algorithms, pp. 20–29, Society for Industrialand Applied Mathematics Philadelphia, PA, USA, 2008.

[35] J. Tropp and D. Needell, “CoSaMP: Iterative Signal Recovery from Incompleteand Inaccurate Samples.” Arxiv preprint arXiv:0803.2392, 2008.

[36] E. J. Candes and T. Tao, “Near-Optimal Signal Recovery From Random Pro-jections: Universal Encoding Strategies?,” IEEE Trans. Infor. Theory, vol. 52,no. 12, pp. 5406–5425, 2006.

[37] E. J. Candes, J. Romberg, and T. Tao, “Robust Uncertainty Principles: ExactSignal Reconstruction From Highly Incomplete Frequency Information,” IEEETrans. Inform. Theory, vol. 52, pp. 489–509, Feb. 2006.

[38] E. Candes and J. Romberg, “The Role of Sparsity and Incoherence for ExactlyReconstructing a Signal from Limited Measurements,” California Inst. Technol.,Pasadena, Tech. Rep., 2004.

89

[39] M. Rudelson and R. Vershynin, “Geometric Approach to Error Correcting andReconstruction of Signals,” 2005.

[40] S. Jokar and M. Pfetsch, “Exact and Approximate Sparse Solutions of Underde-termined Linear Equations,” in ZIB-Report 07-05, 2007.

[41] M. Saunders, PDCO, Matlab software for convex optimization.http://www.stanford.edu/group/SOL/software/pdco.html, 2005.

[42] E. Berg and M. Friedlander, “Probing the Pareto Frontier for Basis PursuitSolutions,” tech. rep., Department of Computer Science, University of BritishColumbia Technical Report TR-2008-01, 2008.

[43] E. van den Berg and M. Friedlander, “In Pursuit of a Root,” tech. rep., UBCComputer Science Technical Report TR-2007-16, 2007.

[44] M. Elad, “Optimized Projections for Compressed Sensing,” IEEE Transactionson Signal Processing, 2007.

[45] R. Ward, “Cross Validation in Compressed Sensing Via the Johnson Linden-strauss Lemma,” March 2008.

[46] D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham,K. F. Kelly, and R. G. Baraniuk, “A New Compressive Imaging Camera Archi-tecture using Optical-Domain Compression,” in SPIE Electronic Imaging, 2006.

[47] D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk,“Distributed Compressed Sensing,” November 2005.

[48] W. Dai and O. Milenkovic, “Subspace Pursuit for Compressive Sensing: Closingthe Gap Between Performance and Complexity.” Arxiv preprint arXiv:0803.0811,2008.

[49] T. Blumensath and M. Davies, “Gradient Pursuits,” IEEE Transactions on Sig-nal Processing, vol. 56, no. 6, pp. 2370–2382, 2007.

[50] P. Boufounos, M. Duarte, and R. Baraniuk, “Sparse Signal Reconstruction fromNoisy Compressive Measurements using Cross Validation,” in IEEE/SP 14thWorkshop on Statistical Signal Processing, pp. 299–303, 2007.

[51] S. Ji and L. Carin, “Bayesian Compressive Sensing and Projection Optimiza-tion,” Department of Electrical and Computer Engineering, Duke University,Durham, 2007.

90

[52] J. Haupt, W. U. Bajwa, G. Raz, and R. Nowak, “Toeplitz Compressed SensingMatrices with Applications to Sparse Channel Estimation,” IEEE Transactionson Information Theory, 2008.

VITA

Daniel Zahonero Inesta was born in Alicante, Spain, on November 28, 1984.

He attended elementary schools and graduated from the Sagrado Corazon Maristas

High School in June 2002. In August 2002 he entered Politecnic University of Valencia

and in June 2007 he completed the First Cycle of Telecommunication Engineering.

He entered in University of Tromso, Norway in August 2007, he completed the first

year of Master in Physics. In August 2008 he entered in Tennessee Technological

University. He is a candidate for the Master of Science Degree in Telecommunication

Engineering.

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