data-adaptive spectral estimation algorithms and...

DATA-ADAPTIVE SPECTRAL ESTIMATION ALGORITHMS AND THEIR SENSINGAPPLICATIONS

By

ODE OJOWU JR.

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2013

c⃝ 2013 Ode Ojowu Jr.

2

I dedicate this to God, family and friends.

3

ACKNOWLEDGMENTS

This dissertation would not have been possible without the support of several

people. I would first of all like to thank my parents, my siblings and friends for the love

and moral support they have given me throughout the years.

I would also like to thank my advisor, Prof. Jian Li for taking me in as a student,

and taking the time and patience to guide me throughout this important phase of my

academic career; I will forever be grateful.

This dissertation also would not have been possible without the help of some of

my close colleagues, lab mates and friends at the Spectral Analysis Lab, which include:

William Rowe, Dr. Johan Karlsson, Dr. Duc Vu, Chris Gianelli, Kexin Zhao, Dr. Luzhou

Xu, Dr. Hao He, Dr. Jun Ling, Lim Deoksu, Qilin Zhang and Dr. Ming Xue. The daily

discussions and advice helped with my work tremendously.

I would finally like to thank my committee members, Prof. Henry Zmuda, Prof.

Jenshan Lin and Prof. Hugh Fan for their guidance and support, and also for taking the

time to be on my committee. I appreciate the sacrifice sincerely.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 REVIEW OF SPECTRAL ESTIMATION AND TECHNIQUES . . . . . . . . . . 13

1.1 Introduction: Spectral Estimation Problem . . . . . . . . . . . . . . . . . . 131.1.1 Energy Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 161.1.2 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 171.1.3 Power Spectral Density Estimation . . . . . . . . . . . . . . . . . . 17

1.2 Periodogram: Non-parametric Method . . . . . . . . . . . . . . . . . . . . 181.2.1 Resolution: Periodogram . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Filter-bank Interpretation: Periodogram . . . . . . . . . . . . . . . . 22

1.3 Data-adaptive Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.1 CAPON: Non-parametric . . . . . . . . . . . . . . . . . . . . . . . . 231.3.2 Amplitude and Phase Estimation (APES): Non-parametric . . . . . 251.3.3 Iterative Adaptive Approach (IAA): Non-parametric . . . . . . . . . 261.3.4 SLIM and SPICE Algorithms: Non-parametric . . . . . . . . . . . . 281.3.5 RELAX: Parametric . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 DATA-ADAPTIVE TECHNIQUES FOR NETWORK FREQUENCY EXTRACTIONFROM DIGITAL RECORDINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Network Frequency Characteristics and Database . . . . . . . . . . . . . 342.4 Extraction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Frequency Domain Analysis (STFT) . . . . . . . . . . . . . . . . . 362.4.2 IAA and TRIAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3 Frequency Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4.4 Matching the Extracted ENF to Database . . . . . . . . . . . . . . 44

2.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5.1 Data1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5.2 Data2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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3 DATA-ADAPTIVE RELAX FOR RFI SUPPRESSION FOR THE SYNCHRONOUSIMPULSE RECONSTRUCTION (SIRE) RADAR . . . . . . . . . . . . . . . . . 53

3.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 SIRE Equivalent Sampling Scheme . . . . . . . . . . . . . . . . . . . . . 563.4 Existing RFI Suppression Methods . . . . . . . . . . . . . . . . . . . . . . 603.5 Proposed RFI Suppression Method: RELAX and Averaging . . . . . . . . 63

3.5.1 Modelling of RFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5.2 RELAX Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5.3 Multi-snapshot RELAX Algorithm . . . . . . . . . . . . . . . . . . . 69

3.6 Autoregressive (AR) Modelling . . . . . . . . . . . . . . . . . . . . . . . . 723.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.7.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.7.2 Sniff Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 76

3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 DATA-ADAPTIVE, SPARSE SUPER-RESOLUTION IMAGING FOR THE SIREFLGPR RADAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3 Data Model: SIRE Impulse Based FLGPR . . . . . . . . . . . . . . . . . . 854.4 Back-projection/Delay-and-sum (DAS) Based Methods . . . . . . . . . . . 87

4.4.1 Back-projection/DAS . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4.2 Sparse: CLEAN Method . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Super-resolution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.1 Orthogonal Projection and Time Gating . . . . . . . . . . . . . . . 914.5.2 SLIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.5.3 SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.6 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . 994.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 CONCLUDING REMARKS AND FUTURE WORK . . . . . . . . . . . . . . . . 107

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6

LIST OF TABLES

Table page

1-1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2-1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2-2 Parameters for the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2-3 Correlation coefficients of Algorithms (Data1) . . . . . . . . . . . . . . . . . . . 47

2-4 Standard Deviation of error for Algorithms (Data1) . . . . . . . . . . . . . . . . 47

2-5 Correlation coefficients of Algorithms (Data2) . . . . . . . . . . . . . . . . . . . 48

2-6 Standard Deviation of error for Algorithms (Data2) . . . . . . . . . . . . . . . . 48

3-1 ARL Parameters for Synchronous Reconstruction Radar. . . . . . . . . . . . . 57

3-2 Suppression Algorithm: RELAX + Averaging . . . . . . . . . . . . . . . . . . . 68

3-3 Suppression Algorithm: M-RELAX + Averaging . . . . . . . . . . . . . . . . . . 71

3-4 RFI Suppression (dB): File 1 ( ~P = 1) . . . . . . . . . . . . . . . . . . . . . . . . 77

3-5 RFI Suppression (dB): File2 ( ~P = 1) . . . . . . . . . . . . . . . . . . . . . . . . 77

4-1 SLIM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4-2 CG SPICE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4-3 Subspace approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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LIST OF FIGURES

Figure page

1-1 Synthetic aperture radar (SAR) imaging . . . . . . . . . . . . . . . . . . . . . . 14

1-2 Spectrogram: Estimating the Electric Network Frequency (ENF) in audio anrecording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1-3 Bartlett window spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1-4 Spectra of two sinusoids with large frequency spacing . . . . . . . . . . . . . . 21

1-5 Spectra of two sinusoids with small frequency spacing . . . . . . . . . . . . . . 21

1-6 Spectrum: Comparison of adaptive methods to the periodogram . . . . . . . . 27

2-1 FDR Distribution in North America . . . . . . . . . . . . . . . . . . . . . . . . . 35

2-2 Segmentation of data for STFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2-3 Matching extracted ENF to database (Data1) . . . . . . . . . . . . . . . . . . . 46

2-4 Power Spectrum of one Frame (Data2): poor resolution of FFT . . . . . . . . . 49

2-5 Power Spectrum of one Frame (Data2): strong interference signal . . . . . . . 49

2-6 Extracted ENF via Frequency Tracking . . . . . . . . . . . . . . . . . . . . . . . 50

2-7 Matching extracted ENF to database (Data1) . . . . . . . . . . . . . . . . . . . 51

2-8 Absolute error of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3-1 Synchronous Impulse Reconstruction (SIRE) equivalent time sampling . . . . . 58

3-2 Spectrum of SIRE sampling after interleaving compared to the spectrum ofregular sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3-3 Spectrum SIRE sampling pattern: One fast time pulse . . . . . . . . . . . . . . 59

3-4 Spectrum SIRE sampling pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3-5 RFI Suppression (dB): Averaging method (M realizations) for simulated SIREsampled RFI signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3-6 RFI suppression (SIRE sampling) - using RELAX with P (real-valued) sinusoidsestimated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3-7 RFI suppression - RELAX algorithms with P (real-valued) sinusoids estimatedand suppressed from sniff data . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3-8 Echo retrieval (File1) - RELAX with P (real-valued) sinusoids compared toideal echo signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8

3-9 Echo retrieval (File1) - RELAX with P (real) sinusoids combined with M-RELAXwith ~P = 1 real sinusoid, compared to ideal echo signal . . . . . . . . . . . . . 79

3-10 RFI suppression - AR modelling with order q compared to averaging for sniffdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3-11 Echo retrieval (File1) - AR modelling with order q, compared to ideal echo signal 80

4-1 Forward looking ground penetrating radar . . . . . . . . . . . . . . . . . . . . . 84

4-2 SIRE FLGPR: 2D aperture for SAR imaging . . . . . . . . . . . . . . . . . . . . 86

4-3 Time gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4-4 Subspace dimension (s) for high resolution imaging . . . . . . . . . . . . . . . 100

4-5 FLGPR SAR Imaging - detection of weak target . . . . . . . . . . . . . . . . . . 101

4-6 FLGPR Imaging - resolution improvement . . . . . . . . . . . . . . . . . . . . . 102

4-7 Orthogonal projection comparison . . . . . . . . . . . . . . . . . . . . . . . . . 103

4-8 Real data - SIRE FLGPR SAR Imaging . . . . . . . . . . . . . . . . . . . . . . 104

4-9 ROC comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

DATA-ADAPTIVE SPECTRAL ESTIMATION ALGORITHMS AND THEIR SENSINGAPPLICATIONS

By

Ode Ojowu Jr.

December 2013

Chair: Jian LiMajor: Electrical and Computer Engineering

Spectral analysis of signals, or the problem of spectral estimation revolves around

estimating the distribution of power over frequency of a random signal. It has useful

applications in various fields of study (including Speech analysis, Medicine, RADAR and

SONAR) due to the fact that the frequency content of an observed signal can provide

very useful information in these fields.

A well known method for estimating the spectral content of a signal is the Pe-

riodogram (developed by Arthur Schuster), which is a data-independent method of

estimation. This method is based on computing the Fourier transform of the signal which

can be computed efficiently using the Fast Fourier Transform (FFT) algorithm. This

method however, is limited by relatively poor resolution and high sidelobe problems,

which can lead to degradation in retrieval of the desired information present within the

signal.

Data-dependent (adaptive) techniques both non-parametric and parametric can

offer superior performance over data-independent methods like the periodogram

at a cost of increased computational complexity. These data-adaptive approaches

however, can lead to improved spectral resolution and lower sidelobes, which can

reveal more information about the signal under study. These advantages have led to

increased interest in data-adaptive approaches to the problem of spectral estimation.

This dissertation revolves around analyzing and applying robust adaptive techniques

10

to real-world problems in a unique, effective and efficient way to achieve superior

performance over their data-independent counterparts.

The introduction chapter briefly reviews the problem of spectral estimation as well

as some of the methods for spectral estimation. We start this dissertation in Chapter

2 with the basic problem of frequency estimation (harmonic retrieval). In this chapter,

adaptive techniques are used in the problem of harmonic retrieval in the presence of

strong interference. The focus is on the problem of digital audio forensics, where the

goal is to extract the embedded network frequency from a digital recording and compare

it to a known database for digital audio verification. In the presence of significant

interference, extracting the network frequency using the standard method (Periodogram)

is ineffective and proves to be challenging due to poor resolution and high sidelobe

problems. We therefore use a robust adaptive algorithm (Iterative Adaptive Approach

- IAA) to improve the spectral resolution and suppress sidelobes hence effectively

separating the network frequency from interference. A frequency tracking method based

on dynamic programming is used in addition to this data-adaptive method to extract

the Network frequency accurately and hence provide more reliability for the verification

process compared to the current standard, which is based on the data-independent

Fourier transform.

Chapters 3 and 4 are the focus of this dissertation. In these chapters, the remote

sensing tool known as the Synchronous Impulse Reconstruction (SIRE) Ultra-wideband

radar (currently being built by the Army Research Lab (ARL) for landmine detection)

is analyzed and studied. In Chapter 3, we once again apply an adaptive technique

for harmonic retrieval. The goal here is to effectively suppress Radio Frequency

Interference (RFI) picked up by this UWB radar which samples its returned signals

using an equivalent sampling scheme. This equivalent sampling scheme makes RFI

suppression difficult due to its irregular and under-sampled data (aliasing). The current

method for RFI suppression for this UWB radar is simply averaging multiple realizations

11

of the measured data. In this chapter, we model the aliased RFI signals as a sum of

sinusoids and estimate the aliased frequencies and amplitudes accurately using a

robust algorithm - RELAX. A direct implementation of this algorithm is computationally

intensive, therefore, an efficient method for implementation is presented in this chapter,

which takes advantage of this equivalent sampling and improves computation. As RFI

suppression is the goal, the estimates are used to reconstruct the aliased RFI samples

accurately and are then suppressed from the data without altering the desired radar

signals.

In Chapter 4, we focus on radar imaging for landmine detection for this SIRE

UWB radar. The standard method currently used for this radar is the data-independent

backprojection or delay-and-sum (DAS) approach. This method suffers from high

sidelobe problems and poor resolution. A recursive sidelobe minimization (RSM)

algorithm was recently proposed by the army research laboratory for effective sidelobe

reduction. This data-independent approach however, has the same resolution limitation

as the backprojection algorithm. As imaging resolution is important for separating

desired targets (mines) from clutter, this chapter, focuses on sparse super-resolution

imaging techniques for imaging. A new technique for imaging based on applying

data-adaptive approaches post significant data reduction as well as interference

reduction via an orthogonal projection is proposed in this chapter. This approach is

able to achieve an improvement in imaging resolution by a factor of approximately 2,

based on simulated experiments. Chapter 5 provides the concluding remarks and

possible future work.

The contents of Chapter 2 are published in IEEE transactions on information

forensics and security Volume 7, no. 4. The contents of Chapter 3 are published in the

International Journal of Remote Sensing and Applications (IJRSA) vol 3 Issue 1. The

contents of Chapter 5 are to be submitted for publication.

12

CHAPTER 1REVIEW OF SPECTRAL ESTIMATION AND TECHNIQUES

1.1 Introduction: Spectral Estimation Problem

Most phenomena or signals that occur in nature or in practice are typically random

in nature, and are best modelled as random signals. Examples of such random signals

include but are not limited to speech/audio signals and thermal noise generated by

electronic devices. Due to the random fluctuation of these signals, they are best

characterized in terms of statistical averages. The autocorrelation function of a random

process is a statistical average used for characterizing these random signals in the time

domain. The power spectral density (spectrum) provides the frequency content of such

signals.

Spectral analysis of signals or the spectral estimation problem, involves estimating

the frequency content of a random signal. This is done by estimating the power

distribution over frequency from a stationary sequence of finite time samples, which

is known as the power spectrum of the signal [1–5].

Schuster in the late 19th century pioneered the most well-known spectral estimation

techniques called the Periodogram. This harmonic analysis approach allows for

detecting and measuring ”hidden periodicities” [6] in the observed data. Spectral

estimation can also be performed on non-stationary data, by dividing the data into

segments in time (each assumed to be stationary) [7],[8],[9]. A time-varying power

spectrum (image) can be displayed to provide information about the signal (also known

as the spectrogram [10]).

Power spectral estimation has applications in many fields [1–3, 11, 12]. Speech

signals which are periodic in nature are analyzed using the spectrogram. This frequency

domain analysis provides useful information that can lead to speech recognition and

generation. In the sensing fields of RADAR and SONAR, the spectral content of

received signals may provide information about the targets of interest [11, 13] in a

13

given scene of interest (see Fig. 1-1). Also the power spectrum of signals may provide

information about radio frequency interference in such a signal and hence lead to

effective suppression of the interference. In the field of MEDICINE, the power spectrum

of electroencephalogram (EEG) signals can be used to evaluate the different sleep

cycles in humans [14, 15]. These can are used to investigate and study narcoleptic

(disease characterized by the inability to properly regulate sleep-wake cycles) patients

[15]. More recently in audio analysis, the spectrogram of the audio signal can indicate

the presence of the electric network frequency (see Fig. 1-2), which can be used for

digital audio authentication [16].

A

SAR Image from Phase History data

−35

−30

−25

−20

−15

−10

−5

0

B

Figure 1-1. Synthetic aperture radar (SAR) imaging: (A) Photograph of object at 45o (B)SAR image formed using Spectral estimation (FFT)

There are two broad approaches to spectral estimation. The first approach is

called the non-parametric method and the other is called the parametric method. The

non-parametric methods assumes no prior information about the data, where as the

parametric methods assumes a specific model of the data, which then results in a

problem of parameter estimation. The parametric methods are more accurate than the

classical non-parametric techniques, when the assumed model is accurate. However,

they perform poorly when there are inaccuracies in the data-model.

14

time (secs)

Spectrogram of pre−filtered audio signal (ENF Harmonic =180 Hz)

0 500 1000 1500 2000 2500 3000 3500

179.5

179.6

179.7

179.8

179.9

180

180.1

180.2

180.3

180.4

180.5 −50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Figure 1-2. Spectrogram: Estimating the Electric Network Frequency (ENF) in audio anrecording for forensic analysis (see chapter 1 for more details)

In this chapter, the problem of power spectral density estimation of signals is briefly

described. Commonly used techniques for spectral estimation within these two broad

methods (non-parametric and parametric methods) for estimating the spectrum of a

signal, will be briefly discussed. The limitations of these methods in practice will also be

briefly discussed.

Some data-dependent (adaptive) algorithms (Capon, APES, IAA, SLIM and

SPICE which are non-parametric and RELAX which is paramteric) will be mentioned

along with their benefits [1, 2, 17] over the classical (data-independent) approaches in

practical scenarios. The core of this dissertation is effective and efficient application of

data-adaptive techniques to solving real-world problems.

Before delving into the problem of spectral estimation of random signals, let us

consider the case of spectral estimation of finite length deterministic signals. This

analysis is fairly straightforward as deterministic signals are predictable over time [18].

The results will then be extended to the case of random signals.

15

1.1.1 Energy Spectral Density

Consider a signal x[n] (discrete) with finite energy, that is,

E =

∞∑n=−∞

|x[n]|2< ∞ (1–1)

then its discrete time fourier transform (DTFT) exists and is given by:

X(ω) =∞∑

n=−∞

x[n]e−jωn (1–2)

where ω is the angular frequency variable measured in radians per sample. From

Parseval’s theorem equation (1–1) can be written as:

E =

∞∑n=−∞

|x[n]|2= 1

2π

∫ π

−π

|X(ω)|2 (1–3)

From the equation above the energy spectral density of x[n] which is the distribution of

the energy of the signal of frequency is therefore defined as:

Sxx(ω) = |X(ω)|2 (1–4)

Note that the energy spectral density Sxx(ω) can be written as the Fourier transform

of the autocorrelation sequence rxx(k) of the signal x[n]:

Sxx(ω) =∞∑

n=−∞

rxx(k)e−jωkdω (1–5)

where

rxx(k) =

∞∑n=−∞

x∗[n]x[n− k] (1–6)

The analysis above, is specifically for signals with finite energy (deterministic

signals). However, signals typically encountered in applications are characterized as

stochastic processes and do not have finite energy and hence do not posses a Fourier

transform. These random signals however, posses an average power and can be

described by their power spectral density.

16

1.1.2 Power Spectral Density

Consider a stationary stochastic process y[n], where E{y[n]} = 0 for all n. The

autocovariance function (same as autocorrelation function for stationary stochastic

process with mean zero) of y[n] is given by

ryy(k) = E{y∗[n]y[n− k]} (1–7)

where E{·} is the statistical average over all realizations. The power spectral density

(PSD) of y[n] is defined as (Wiener-Khintchine theorem [1] ):

ϕyy(ω) =∞∑

n=−∞

ryy(k)e−jωk (1–8)

This simply the fourier transform of the autocorrelation function. Note that the inverse

transform of this PSD gives ryy(k) as shown below

1

2π

∫ π

−π

ϕyy(ω)ejωkdω =

∞∑s=−∞

ryy(s)

[1

2π

∫ π

−π

ejω(k−s)dω

]=

∞∑s=−∞

ryy(s)δks = ryy(k)

were δ denotes the Kronecker delta function. Note that, the average power of the

stochastic process y[n] is given by the zero lag of the autocorrelation function ryy(0):

E{|y[n]|2} = ryy(0) =1

2π

∫ π

−π

ϕyy(ω)dω (1–9)

This equation (1–9) leads to the motivation for defining the power spectral density in

(1–8). The PSD can also be defined as:

ϕyy(ω) = limN→∞

E

1

N

∣∣∣∣∣N∑

n=1

y[n]e−jωn

∣∣∣∣∣2 (1–10)

which is equivalent to the definition in (1–8) under the assumption that the autocovaraince

sequence (ACS) ryy(k) decays quickly.

1.1.3 Power Spectral Density Estimation

Obtaining the true power spectral density (PSD) ϕyy(ω) of a random process is

impossible from a finite set of measurements. This is due to the fact that one will need

17

to compute an an infinite number of values from a finite set of data, which is an ill-posed

problem [1, 2].

The problem of spectral estimation, then becomes getting an estimate ϕyy(ω) of

the true PSD ϕyy(ω) of a random process from a finite sequence of observations of

the signal. If the signal is statistically stationary, the longer the observed sequence the

more accurate the estimate. However if the signal is statistically non-stationary, then one

cannot select and arbitrarily long data length for estimation. This is a major limitation on

the quality of the PSD estimate.

Recall that the PSD describes how the power of a signal is distributed in frequency.

This can then be interpreted physically as filtering the random signal through a

narrowband filter around a specific frequency of interest (ωo). This process is then

repeated for all the frequencies of interest (−π ≤ ωo ≤ π). Fourier based methods

(computed efficiently using the Fast Fourier Transform (FFT)) of spectral estimation are

based on this technique [1] and are discussed next.

1.2 Periodogram: Non-parametric Method

As mentioned in the section above, the non-parametric methods of spectral

estimation provide an estimate of the power spectral assuming no prior information

of the data model. The periodogram which was introduced by Schuster in 1898 to detect

”hidden periodicities” in a signal, is a classical non-parametric method which is widely

used for spectral estimation. This fourier based method, along with its modified versions

are based directly on the definition in (1–10). The periodogram of a set of N samples of

random process {y[n]}Nn=1 is given as (the subscript yy in ϕyy(ω) has been dropped for

notational simplicity):

ϕp(ω) =1

N

∣∣∣∣∣N∑

n=1

y[n]e−jωn

∣∣∣∣∣2

(1–11)

Note that (1–11) is essentially thesame as the (1–10) with the expectation and limit

operation removed. This ommission is due to the fact that only N samples of the signal

18

are available. The periodogram can be computed using the discrete fourier transform

of the available samples (which can be efficiently computed using the fast fourier

transform (FFT). This yields samples of the PSD estimate at frequencies ωk = 2πk/N for

k = 0, 1, . . . , N − 1).

Note that equation (1–11) can be written as:

ϕp(ω) =

N−1∑k=−N+1

r[k]e−jωk (1–12)

where

r[k] =1

N

N−1∑k=−N+1

y[n]y∗[n− k] (1–13)

corresponds to the biased estimates of the ACS sequence. This is referred to as the

correlogram. The unbiased estimate of the ACS can also be used to compute the

correlogram.

One major limitation of the periodogram is limited spectral resolution, which is

discussed next.

1.2.1 Resolution: Periodogram

One key concept in spectral estimation is spectral resolution, which is the ability

to resolve or seperate closely spaced frequency components within a signal. The

resolution of the periodogram is one major drawback of this data-independent method of

spectral estimation. Note that the expected value of the periodogram can be written as:

E{ϕp(ω)} =

N−1∑k=−N+1

E{r[k]}e−jωk =

N−1∑k=−N+1

w[k]r[k]e−jωk (1–14)

where (based on (1–13))

w[k] =

1− |k|N

for n = ±1,±2, . . . ,±N

0 otherwise(1–15)

19

is the Bartlett window and r[k] is the true PSD. Equation (1–14) is the Fourier transform

of the product of two time sequences, which correspond to the convolution of their

individual Fourier transforms as given in (1–16).

E{ϕp(ω)} =1

2π

π∫−π

ϕ(β)W (ω − β) (1–16)

where W (ω) is the Fourier transform of the Bartlett window.

W (ω) =1

N

[sin(ωN/2)

sin(ω/2)

]2(1–17)

Figure 3 below shows W (ω) for N = 10 and N = 20. The 3dB (half-power) main lobe

−3 −2 −1 0 1 2 3−40

−35

−30

−25

−20

−15

−10

−5

0

ω (radians/sample)

dB

N = 10N = 30

Figure 1-3. Bartlett window spectrum: resolution limitation periodogram (window length= N )

width is approximately equal to 4π/2N = 2π/N radians per sample (1/N cycles per

sample). The spectral estimate of periodogram ϕ(ω) will not be able to resolve peaks in

the true PSD ϕ(ω) that have less than 1/N cycles per sample separation. Increasing the

number of observed samples will improve the spectral resolution (not be confused with

zero-padding).

The estimated spectrum can be computed using the DFT (and efficiently using

the FFT as mentioned earlier). Increasing the number of available samples by

zero-padding (adding zeros to the end of the signal) can provide more detail in the

20

spectrum computed using the FFT. This results in the interpolation of spectrum, however

it does not change the spectral resolution as shown in Figure 1-4 and 1-5. Figures 1-4

and 1-5 show the spectrum of sinusoids (N = 20 samples) with frequency spacing

�ω = 2π × (0.06) and �ω = 2π × (0.02) respectively. Each figure shows different

zero-padding factors. The periodogram suffers from relatively poor resolution and high

0 5 10 15 20 25 300

0.5

1

0 20 40 60 80 100 1200

0.5

1

Zeropad (128 samples)


Figure 1-4. Spectra of two sinusoids with frequency spacing �ω = 2π × (0.06)

0 5 10 15 20 25 300

0.5

1

0 20 40 60 80 100 1200

0.5

1



Figure 1-5. Spectra of two sinusoids with frequency spacing �ω = 2π × (0.02)

sidelobe problems as seen in Figure 1-3. These reasons have led to investigation into

data-adaptive methods of spectral estimation that can provide improved resolution and

sidelobe suppression capabilities. In the next subsection some of these data-adaptive

21

algorithms are discussed. Prior to this discussion we will review the periodogram in

different light which leads to one of the well known data-adaptive algorithms known as

the CAPON algorithm [19].

1.2.2 Filter-bank Interpretation: Periodogram

Recall that the PSD is the power distribution over frequency of the signal, which

as mentioned earlier can be interpreted as passing the signal through a bank of

narrowband filters (at different frequencies) and computing the output power (which

is then divided by the bandwidth of the filter). In this light, the periodogram estimator

ϕp(ω) at a given frequency ω can be written as:

ϕp(ω) =1

N

∣∣∣∣∣N∑

n=1

y[n]ejω(N−n)

∣∣∣∣∣2

= N

∣∣∣∣∣N−1∑n=0

h∗ω[k]y[N − k]

∣∣∣∣∣2

= N |z(N)|2 (1–18)

where

z(N) =

∞∑n=0

hω[k]y[N − k] (1–19)

and

h∗ω[k] =

ejωk for k = 0, 1, . . . , N − 1

0 otherwise(1–20)

Note that the periodogram can be interpreted as filtering the signal y = {y[k]}Nk=0

through a narrowband pass filter hω = {hω[k]}Nk=0 and selecting just a single output

z(N) of the filtering process hHω y for power calculation at the specified frequency (

{·}∗ and {·}H correspond to the conjugate (scalar) and conjugate transpose (vector)

operation). This fact leaves the periodogram with a large variance irregardless of the

data length (N ). The output power divided by the bandwidth (PSD) is then calulcated as

E|z[n]|2/� = |z[N ]|2/�, where � = 1/N cycles per sample is the filter’s bandwidth.

Modified versions of the periodogram such as the Bartlett and Welch which

segment (non-overlapping and overlapping respectively) the stationary sequence

22

in question and average the periodograms of the segments can be used to reduce

the variance [2]. In terms of the filter-interpretation, these methods can be seen as

computing the power with more than one sample (number of segments). However, the

periodogram is computed using a reduced length of the data, hence there is a trade-off

between statistical variance and resolution.

Some data-adaptive non-parametric methods have addressed the limitations

of the periodogram by designing a data-adaptive filter, to provide more accurate

PSD estimates with better resolution. In the next section methods like the CAPON,

APES (Amplitude and Phase Estimation), IAA (Iterative Adaptive Approach) which

are data data-adaptive non-parametric approaches are discussed. The data-adaptive

parametric approach known as RELAX (strictly for sinusoidal parameter estimation) is

also discussed.

1.3 Data-adaptive Approaches

In this section, we discuss some well known non-parametric data-adaptive

approaches (CAPON, APES) as well as recent non-parametric spectral estimators

(IAA, SLIM SPICE). These algorithms improve upon the periodogram spectral estimator

in terms of resolution and sidelobe reduction. A parametric approach specifically for

estimating parameters of line spectra (sinusoids) known as RELAX is also mentioned

and discussed in detail later on in Chapter 3.

1.3.1 CAPON: Non-parametric

From the last sub-section, the periodogram output at a specific frequency ω can be

interpreted as using a data-independent filter (bandpass filter) with an impulse response

{hω[k] = e−jωk}N−1k=0 corresponding simply to the Fourier Transform vector.

Unlike the data-independent filter used in the periodogram, the CAPON method

[19–21] (also known as the minimum variance method) designs a data-dependent

(adaptive) bandpass filter hω = {hω[k]}l−1k=0 to achieve some specific desired properties

23

(The CAPON method uses overlapping segments of length (l × 1) of the data to improve

statistical variance). These properties include:

1. Design a bank of filters hω that pass the frequency component (or sinusoid withfrequency) ω undistorted.

2. Filter should also effectively suppress (or minimize) all out-of-bound (any otherfrequencies) power within the signal.

This process can be expressed as follows. Let the output of the filter at any instant

n = [0, 1, . . . , N − 1] be given by:

z[n] =l−1∑k=0

h∗ω[k]y[n− k] = hHω yn (1–21)

where yn = [y[n], y[n − 1], . . . , y[n − l + 1]]T . The total output power of the filter is

then given as E{|z[n]|2= hHω Rhω. Where R = E{ynyHn } is the covariance matrix of the

data vector. The CAPON filter is designed to meet the properties in the aforementioned

steps by minimizing the total output power of the filter subject to the constraint that the

frequency ω is filtered without distortion given by the optimization equation (1–22).

minhω

hHω Rhω subject to hHω a(ω) = 1 (1–22)

where a(ω) = {e−jω}ln=0 is the sinusoid component with frequency ω to be passed

undistorted. The resulting filter is given by:

hω =R−1a(ω)

aH(ω)R−1a(ω)(CAPON filter) (1–23)

The PSD estimate can then be calculated as filter output power E{|z[n]|2 divided by the

bandwidth � ≈ 1/(l).

ϕCAPON(ω) =E{|z[n]|2

�=

l

aH(ω)R−1a(ω)(CAPON spectral estimate) (1–24)

24

The sample covariance matrix R based on the M = N − l + 1 overlapping segments

(each of length l) of the data is used to estimate the covariance matrix and is given by:

R =1

M

M−1∑n=0

ynyHn (1–25)

A very similar algorithm to the CAPON algorithm known as the Amplitude and

Phase Estimation algorithm (APES) is described next.

1.3.2 Amplitude and Phase Estimation (APES): Non-parametric

Note that in the description of the CAPON algorithm, the filter design was based

on passing a single frequency, while suppressing all other out-of-bound frequencies.

CAPON achieves the suppression by minimizing the total output power. APES algorithm

[22],[23],[24] uses the same idea but suppressing out-of-bound frequencies is achieved

by designing a filter such that the filtered sequence is as close as possible to the

a sinusoidal signal at the given frequency ω in the least squares (LS) sense. The

optimization equation is given by :

minα(ω),hω

M−1∑n=0

|hHω yn − α(ω)ejωn|2 subject to hHω a(ω) = 1 (1–26)

The cost function in (1–25) can be re-written as:

1

M

M−1∑n=0

|hHω yn − α(ω)ejωn|2= |hHω Rhω − α∗(ω)hHω ~yω − α(ω)~yHω hω + α(ω)|2

= |α(ω)− hHω ~yω|2+hHω Rhω − |hHω ~yω|2(1–27)

Note that the second and third terms in (1–27) do not depend on α(ω) and therefore the

minimization of this cost function with respect to α(ω) is given by α(ω) = hHω ~yω where

~yω = (1/M)∑M−1

n=0 yne−jωn . The optimization problem for designing the filter hω is given

as:

minhω

hHω Qωhω subject to hHω a(ω) = 1 (1–28)

25

where Qω = R− ~yω~yHω The APES filter is given by:

hω =Q−1

ω a(ω)

aH(ω)Q−1ω a(ω)

(APES filter) (1–29)

The amplitude spectrum of APES algorithm is given by:

α(ω) =a(ω)HQ−1

ω ~y(ω)

aH(ω)Q−1ω a(ω)

(APES amplitude spectrum) (1–30)

The APES and CAPON algorithms have been shown to provide higher resolution

compared to the classical non-parametric methods. The CAPON algorithm minimizes

the total output power subject to a constraint which tends to provide spectral estimates

that are biased downward due to the noise gain of the filter [23]. The APES algorithm

minimizes a least square function requiring the filter output to be as close as possible to

the a sinusoid. This provides more accurate spectral estimates.

However, in the cases where the data is not stationary for a long period of time (only

few snapshots are available), the APES and CAPON methods yield undesirable results.

The Iterative Adaptive Approach (IAA) algorithm improves on these algorithms by being

able to give good spectral estimates for a few snapshots (even a single snapshot), while

providing high spectral resolution, making it very suitable for practical applications. This

algorithm is discussed briefly in the next subsection and also in Chapter 1 where it is

used.

1.3.3 Iterative Adaptive Approach (IAA): Non-parametric

The IAA algorithm [25],[26],[27],[28] for spectral estimation is derived by minimizing

a weighted least squares cost function (described in Chapter 1). The spectral estimate

for the IAA algorithm for a single snapshot y is given below:

α(ω) =a(ω)HQ−1

ω y

aH(ω)Q−1ω a(ω)

(IAA amplitude spectrum) (1–31)

This estimate looks similar to the APES estimate, with the main differences being

that the IAA algorithm is iterative and also the computation of the covariance matrix of

26

the noise is given as Qωl= R −

∑Ki=0,i=l pia(ωi)a(ωi)

H , where R = APAH and P is a

diagonal matrix with elements corresponding to {pi}Ki=0 = {|α(ωi)|2}Ki=0 (powers at each

individual frequencies).

Note that unlike the CAPON and APES algorithms where the covariance matrices

are based on the data samples and computed once. The covariance matrix of the

IAA algorithm is dependent on the spectral estimate and hence refined iteratively, with

the initial estimates of the spectral powers computed using the periodogram. This

refinement allows the IAA algorithm to produce accurate spectral estimates with a single

snapshot and hence makes it useful for practical applications (where the available data

for estimation is usually limited to a single snapshot). Fig. 1-6 shows the spectrum of

−3 −2 −1 0 1 2 30

0.2

0.4

ω (rad/sample)

spec

trum

Periodogram (single snapshot)

−3 −2 −1 0 1 2 30

0.2

0.4

ω (rad/sample)

spec

trum

Adaptive Algorithm

PeriodogramSinusoid 1Sinusoid 2Sinusoid 3

IAA

Figure 1-6. Spectrum: Comparison of adaptive methods to the periodogram

three sinusoids in white noise (SNR = 30dB) with frequencies ω1 = 0.63 rad/samp,

ω2 = 1.26 rad/samp and ω3 = 1.33 rad/samp. The CAPON and IAA estimates are poor,

due to ill-conditioning of the matrices. However with a single snapshot the IAA spectra is

capable of picking out the sinusoids.

A comparison of the periodogram to the IAA algorithm in this figure shows how this

adaptive technique improves over the periodgoram in terms of spectral resolution and

sidelobe suppression.

27

1.3.4 SLIM and SPICE Algorithms: Non-parametric

The Sparse Learning via Iterative Minimization (SLIM) [29] and the Sparse Iterative

Covariance-based Estimation (SPICE)[30] algorithms are two super-resolution

algorithms capable of providing high resolution estimates even in the presence of a

single snapshot and coherent sources similar to the IAA algorithm. They both estimate

the covariance matrix R iteratively similar to the IAA algorithm and are hence also useful

for practical applications. The SLIM approach is a maximum a posteriori approach

(MAP) based on the hierrachial model. The goal is to use a sparse prior to promote

sparsity in the estimates which is useful for certain applications. The SPICE algorithm

on the other hand minimizes a covariance cost function that yields sparse estimates.

These two algorithms empirically yield less accurate that the IAA approach. However

they provide sparse and higher resolution estimates compared to the IAA algorithm and

can be useful in certain applications. They are described in more detail in Chapter 4.

The algorithms mentioned above are all non-parametric methods that do not

assume a specific model for the data. Next we briefly mention parametric methods,

which assume a specific data model for PSD estimation. A robust algorithm (which is

later discussed in more detail in Chapter 3) RELAX [31]; which is specific for estimating

parameters of line-spectra (sinusoids) is discussed next.

1.3.5 RELAX: Parametric

Parametric methods unlike the non-parametric methods assume a specific model

for the observed data. These methods essentially estimate the PSD, by assuming

the data takes on a specific model and then estimates the parameters of the model.

Auto-regressive (AR) methods such as Yule, Prony, Forward-Backward Prony methods

are used for estimating parameters for continuous spectra and Eigen-analysis methods

(MUSIC, ESPRIT) are used for estimating parameters of line spectra (sinusoids). The

AR methods model the data as the output of a linear system driven by white noise

and proceed to estimate the parameters of that system. One major limitation of these

28

parametric methods is that they a subject to errors due to poor model specifications.

The Eigen-analysis methods (for line spectra) estimate frequency components of

sinusoids buried in noise by an eigen-decompostion of the autocorrelation matrix. These

methods tend to perform poorly in practical applications due to data model inaccuracies.

The RELAX algorithm is an algorithm that is robust, and that estimates parameters

of sinusoids in an iterative manner. It estimates the parameters of the sinusoid

accurately even with modelling errors and colored noise [31]. This algorithm is described

in more detail in Chapter 3 where Radio Frequency Interference (RFI) is modeled as

sum of sinusoids. The RELAX algorithm is used there for identifying and suppressing

the RFI signals.

1.4 Conclusions

In this section, a brief discussion on the problem of spectral estimation is presented.

The periodogram which is a data-independent algorithm for spectral estimation and

also widely applied in practical applications is briefly discussed. This algorithm is then

re-interpreted as a filtering process with a data-independent filter. This re-interpretation

has led to some data-dependent (adaptive) filters which provide improved spectral

estimates.

We discussed some well-known data-adaptive (non-parametric) algorithms

(CAPON,APES) and the advantages provided by these data-adaptive approaches

over the classical non-parametric methods. However these algorithms perform poorly

in the case when only one snapshot of data is available. More recent data-adaptive

(non-parametric) algorithms, which are robust and perform well even in the single

snapshot case were briefly mentioned and will be discussed in more detail in later

chapters. The improved robustness of these algorithms allows for useful applications

in a practical setting, while providing better spectral properties (high resolution, lower

sidelobes) over the commonly used periodogram.

29

A robust parametric algorithm known as the RELAX for sinusoidal parameter

estimation is also mentioned briefly (discussed in more detail in Chapter 3). This

algorithm is capable of accurate sinusoidal parameter estimation even in the presence of

colored noise making it suitable for practical applications.

In this dissertation we focus on solving specific real world problems by analyzing

these data-adaptive techniques and coming up with effective and efficient ways to apply

them to give superior performance to the standard data-independent approaches.

1.5 Notations

Notation: Throughout this dissertation, Boldface upper-case and lower-case letters

are used to denote matrices and vectors, respectively. See Table 1-1 for more details on

notation.

Table 1-1. Notations

x a vector

X a matrix

diag(x) a diagonal matrix with elements of x on the diagonal

(·)H conjugate transpose of a matrix or vector

(·)T transpose of a matrix or vector

(·)(n) n th iteration of a scalar, vector or matrix in algorithm

||·||2 ℓ2 norm

x estimate of scalar x

, definition

30

CHAPTER 2DATA-ADAPTIVE TECHNIQUES FOR NETWORK FREQUENCY EXTRACTION FROM

DIGITAL RECORDINGS

2.1 Chapter Summary

A novel forensic tool used for assessing the authenticity of digital audio recordings

is known as the Electric Network Frequency (ENF) criterion. It involves extracting

the embedded power line (utility) frequency from said recordings, and matching it to

a known database to verify the time the recording was made, and its authenticity. In

this chapter, a non-parametric, adaptive, and high resolution technique known as the

Time-Recursive Iterative Adaptive Approach (TRIAA), is presented as a tool for the

extraction of the ENF from digital audio recordings. A comparison is made between this

data dependent (adaptive) filter, and the conventional Short-time Fourier Transform

(STFT). Results show that the adaptive algorithm improves the ENF estimation

accuracy in the presence of interference from other signals. To further enhance the

ENF estimation accuracy, a frequency tracking method based on dynamic programming

will be proposed. The algorithm uses the knowledge that the ENF is varying slowly with

time to estimate with high accuracy the frequency present in the recording.

2.2 Introduction

The use of digital recorders has become more prevalent in the world today due to

the advancement in digital technology and the significant progress made in the field of

digital signal processing (DSP). Prior to the increased use of digital recorders, forensic

audio analysis relied on different techniques of audio authentication. For instance, the

magnetic signatures that are left by the erase, record or play heads on the magnetic

tape of analog recorders, can be used to verify the authenticity of such recordings.

When it comes to digital recordings, alterations can be made very easily without

leaving behind such imprints, because digital recorders produce a recording by

converting sound variations to a series of numbers, making authentication of these

recordings a lot more difficult [32]. The importance of being able to verify the authenticity

31

of a recording can be seen in litigation cases [16] [33], where digital recordings are

brought forward as evidence in a trial. Therefore, more reliable methods of verifying the

authenticity of digital recordings need to be researched.

The Electric Network Frequency (ENF) Criterion was proposed by Grigoras [16], [34]

to address the issue of digital audio authentication. The ENF criterion is based on

extracting the utility frequency or ENF from a digital audio recording, and matching

the extracted frequency estimate to a reference database in order to determine the

authenticity, and also time of the digital recording. This process is possible because,

in some cases, digital recorders (even some battery powered recorders [35]), can

pick up the audible sound that is generated by the oscillation of a power grid’s

alternating current at this frequency. The frequency of oscillation is approximately

60 Hz in the United States, whereas in Europe it oscillates at approximately 50 Hz. The

corresponding harmonics of this frequency might also be present in the digital recording.

The ENF criterion is based on two assumptions. Firstly, the ENF for interconnected

networks is the same at all points within the network. Secondly, the frequency varies

randomly within a given interconnection, and hence, are not repeatable over a long

period of time. [33]

There are three known methods of extracting the ENF over time from a digital

recording [16], [34]. They are termed:

• time/frequency domain analysis - This method is based on computing thespectrogram of the signal and visually comparing it to the database.

• frequency domain analysis - This method is based on selecting the frequencylocation corresponding to the maximum amplitude of the power spectrum ofsegments (frames) of the data after applying a band-pass filter.

• time domain analysis - This method is based on measuring the zero crossingsof the signal in the time domain after a bandpass filter has been applied to therecording.

Recently in [36], a quadratic interpolation scheme was applied to the frequency

domain analysis method to estimate the spectral peak locations (frequencies) more

32

accurately. This reduces the estimation error resulting from the use of a fixed grid size in

the spectral estimation process.

Besides the time-domain analysis, the above methods estimate the ENF based on

computing the Fast Fourier Transform (FFT) of overlapping segments (frames) of the

data known as the Short-Time Fourier Transform (STFT) which is limited by the trade-off

between time resolution and frequency resolution [2]. Parametric methods such as the

Frequency Selective ESPRIT, which give superior resolution compared to the FFT, can

also be used successfully to extract the ENF from one frame to another. However, in the

presence of significant interference within a given frame, the parametric methods yield

poor frequency estimates because of their sensitivity to an assumed data model.

This chapter focuses on two methods of extraction. The first, builds upon the

frequency domain analysis with quadratic interpolation. However, in place of the FFT,

the spectrum is estimated for each segment of the data using a non-parametric and

high resolution adaptive algorithm known as the Iterative Adaptive Approach (IAA) [25].

In the presence of interfering signals with frequencies within the range of values the

ENF can take on, IAA yields more accurate estimates of the ENF compared to the FFT

as a result of the improved spectral resolution and interference suppression capability.

The second method involves applying a frequency tracking algorithm based on discrete

dynamic programming [37], which takes into account the slowly varying nature of the

ENF over time. This tracking algorithm is necessary because, in some frames of the

data, the maximum spectral peak might correspond to an interference signal rather

than the network frequency signal even within the acceptable ENF limits. The ENF is

then estimated inaccurately, which can result in a false diagnosis that the recording in

question has been edited.

It is worthwhile to point out that, in order for the proposed methods to work, the ENF

must be embedded in the recording, which is not always the case especially in some

battery operated recorders [35]. This is certainly a drawback of using the ENF criterion

33

for digital authentication. However, if the ENF is embedded in a digital recording, more

reliable methods of extraction need to be sought.

Table 2-1. Abbreviations

APES Amplitude and Phase Estimation

ENF Electric Network Frequency

ESPRIT Estimation of Signal Parameters by Rotational Invariance

FDR Frequency Disturbance Recorder

FIAA Fast Iterative Adaptive Approach

IAA Iterative Adaptive Approach

QN-IAA Quasi-Newton Iterative Adaptive Approach

STFT Short-time Fourier Transform

TRIAA Time-Recursive Iterative Adaptive Approach

Extraction can also be carried out using the harmonics of the ENF signal for the

frequency estimation process. In some cases, the harmonics may give better estimates

because of a higher signal-to-interference-and-noise ratio compared to the fundamental

frequency.

The remaining sections of this chapter are organized as follows. In Section 2.3, the

network characteristics and the network frequency database are described. In Section

2.4, the IAA and TRIAA algorithms are described along with the frequency tracking

algorithm for ENF extraction. In Section 2.5, the experimental results based on a set

of digital audio recordings are presented. Finally, Section 2.6 contains the conclusions

drawn from the results.

Abbreviations: The abbreviations are presented for easy reference in Table 2-1.

2.3 Network Frequency Characteristics and Database

The frequency at which alternating current is distributed to various customers from

power stations, corresponds to the utility frequency or ENF. For European and most

Asian countries the value of this frequency is 50 Hz, while the value is 60 Hz in North

America, and several countries in South America. Japan uses both frequencies (50 and

34

Figure 2-1. FDR Distribution in North America

60 Hz) for electricity distribution. This frequency is determined by the speed of rotation

of the turbines used to drive the generators at the various power plants [38]. Naturally,

the rotation speed is not constant and varies within a certain limit (approximately

±0.05 Hz) depending on the amount of load connected to the network, and amount

of power generated at a given time. Experiments carried out in some European

countries [16], [39], have shown that this frequency variation is random and unique

within specific geographic locations. This uniqueness in frequency variation within a

region, coupled with the fact that network frequency is not repeatable over a long period

of time is what makes the aforementioned ENF criterion possible.

A database of the network frequency is needed in order to match the extracted ENF

from a recording for verification. In [16], such a database is created by connecting the

sound card of a computer to a transformer which is then connected directly to an AC

power outlet. The database currently being built in North America involves deploying

several sensors termed frequency disturbance recorders (FDRs), which perform

accurate ENF measurements, up to about ±0.0005 Hz. The measured data collected

by the FDRs is transmitted over the internet to servers, where it can be analyzed

and stored in a system termed the Information Management System (IMS) [40]. This

collection forms the Frequency Monitoring Network (FNET).

35

There are two major interconnections in North America and three minor interconnections.

These regions have unsynchronized networks (frequency and phase) and are therefore

connected via High Voltage Direct Current Lines (HVDC) [41]. The Eastern and Western

Interconnections form the major interconnections, while the Quebec, Texas and Alaska

Interconnections form the minor. The Alaska Interconnection is isolated, in the sense

that it is not connected to any of the other interconnections. It is therefore generally

not considered to be part of the North American grid. Fig. 2-1 shows the distribution

of the FDRs in Western, Eastern, Quebec and Texas Interconnections. Frequency

measurements collected by the FDRs in these interconnections show that the frequency

pattern is different at a given time from one interconnection to another. However, the

frequency pattern is unique at different locations within each interconnection [42]. The

FNET system, therefore, provides a viable ENF database.

2.4 Extraction Algorithms

2.4.1 Frequency Domain Analysis (STFT)

Due to the fact that the ENF varies with time, the extraction process involves

analysing a non-stationary data sequence. STFT is a common method for time-frequency

analysis of signals. This analysis assumes the signal of interest is stationary within short

time windows (frames); the FFT of the signal is then computed for each frame. The

frequency domain analysis [16] method of extraction is based on this idea.

The process involves re-sampling the audio signal to a lower sampling rate, to

reduce the computational complexity of the analysis. A bandpass filter with a narrow

bandwidth is applied to the signal with center frequency 50/60 Hz as a preprocessing

step. The rest of the analysis is described as follows. Let,

z = [z0, z1 . . . zN−1]T (2–1)

denote the re-sampled and filtered discrete-time signal. This signal is then split into R

overlapping frames as shown in Fig. 2-2, with each frame having length M and a shift

36

from frame to frame of length T . Using the frequency domain analysis method, the ENF

of the rth frame is estimated by finding the frequency that maximizes the spectrum of

each frame which is computed using the FFT based periodogram.

In order to get a more accurate estimate of the frequency, quadratic interpolation is

used [36], [43]. This interpolation scheme, involves fitting a quadratic model of the form

log ϕ(ω) = m(ω − ωkmax − �)2 + c (2–2)

around the frequency point that maximizes the power spectrum:

ωkmax = argmaxωk

ϕr(ωk) (2–3)

where ωk = 2πk/K, k = 0, 1, . . . , K − 1 corresponds to the frequency grid point of a

frequency grid with size K, and ϕr(ωk) is power spectrum of the rth frame.

The value of ω that maximizes the model (2–2) is taken as the estimated peak

of the spectrum. This value is determined by fitting the model to the highest sample

of the power spectrum and the two adjacent points with corresponding frequencies

(ωkmax−1, ωkmax , ωkmax+1). This value of ω that maximizes the model is:

ω = ωkmax + � (2–4)

where

� =1

2

β−1 − β1β−1 − 2β0 + β1

(ωkmax+1 − ωkmax) (2–5)

βℓ , logϕr(ωkmax+ℓ), ℓ = −1, 0, 1. (2–6)

The corresponding frequency estimate of the rth frame in Hz is given by:

f(r) = 2π (ωkmax + �)Fs (2–7)

where Fs is the sampling frequency (in Hz) of the signal.

37

Figure 2-2. Segmentation of data for STFT

The use of STFT will result in a trade-off between frequency resolution and time

resolution. For a given frame length, this trade-off can be optimized by applying a

rectangular window to each frame, which will provide the best spectral resolution at a

cost of higher side lobes compared to other spectral windows.

In order to get improved spectral resolution over FFT, one has to resort to using

parametric methods or data-dependent (adaptive) non-parametric methods for spectral

estimation. Parametric methods, on the one hand, are not robust against data model

errors. On the other hand, non-parametric adaptive methods are more robust, since they

do not assume a specific parametric data model. Well-known adaptive methods include

the Capon algorithm and the Amplitude and Phase Estimation (APES) algorithm. These

algorithms also provide higher resolution and lower sidelobes than the periodogram.

However, these methods are inadequate because they require multiple realizations

(snapshots) of the random signal, which is not the case with the current data, as only

one snapshot is available for frequency estimation. Spatial smoothing (segmenting and

spectral averaging of the data) can be used to improve the spectral estimates of the

Capon and APES algorithms in the one-snapshot case; but the cost of doing this will be

a degradation in the spectral resolution, which is not desirable. The wavelet transform

is also a common tool for time-frequency analysis. Contrary to the STFT, which uses a

38

fixed window size, the wavelet transform uses short windows at high frequencies and

longer windows at low frequencies. The wavelet transform is therefore not suitable for

our problem because we are interested only in a small range of frequencies.

IAA is a non-parametric data-dependent algorithm based on Weighted Least

Squares (WLS), originally presented in [25] for Direction of Arrival (DOA) estimation

in array processing. The IAA algorithm is capable of yielding high resolution and low

sidelobes even in the case of a single snapshot [25], hence making it suitable for

estimating the ENF in the presence of interferences.

2.4.2 IAA and TRIAA

The ENF can be extracted with high accuracy in the presence of interference using

the IAA algorithm for a given frame. The proposed ENF extraction process follows

(2–2)-(2–7), with the FFT spectral estimate ϕr replaced by the IAA spectral estimate.

The IAA and TRIAA [44] used for spectral estimation of non-stationary data will be

discussed in this section.

The spectral estimation problem can be set-up as follows. Let y = [y0, y1 . . . yM−1]T

denote a uniformly sampled stationary data sequence and A = [a(ω0), a(ω1) . . . a(ωK−1)],

where a(ωk) = [1, ejωk , . . . , e(M−1)jωk ]T corresponds to a steering (frequency) vector, and

ωk = 2πk/K, k = 0, 1, . . . , K − 1, corresponds to a frequency grid point of a frequency

grid with size K. Also let α = [α(ω0), α(ω1), . . . , α(ωK−1)]T , with α(ωk) denoting the

complex spectral estimates of y at ωk. The following data model can be formulated:

y = Aα (2–8)

where the noise contributions of y are taken into account implicitly [25].

39

The IAA algorithm solves for the spectral estimates α by minimizing the following

quadratic cost function in (2–9) using weighted least squares (WLS):

||y − a(ωk)α(ωk)||2Q−1(ωk) (2–9)

where ||x||2Q−1(ωk) , xHQ−1(ωk)x,

Q(ωk) = R− pka(ωk)aH(ωk) (2–10)

R = APAH (2–11)

and P , diag[p0, p1, . . . pK−1], with pk for k = 0, . . . , K − 1, denoting the power estimate at

each frequency grid point, given by |α(ωk)|2. R1 is the covariance matrix of the data and

Q(ωk) is the covariance matrix of the interference and noise, where interference refers

to all the signals at frequency grid points other than the current grid point of interest ωk.

Minimizing the cost function in (2–9) with respect to the α(ωk) for k = 0, . . . , K − 1 gives

the following solution:

α(ωk) =aH(ωk)Q

−1(ωk)y

aH(ωk)Q−1(ωk)a(ωk), k = 0, 1, . . . , K − 1 (2–12)

The solution in (2–12) can be re-written as

α(ωk) =aH(ωk)R

−1y

aH(ωk)R−1a(ωk), k = 0, 1, . . . , K − 1 (2–13)

using the Woodbury matrix identity2 and (2–10). This prevents the computation of

the interference covariance matrix Q−1(ωk) for each frequency grid point. Note that

the computation of R−1 requires the knowledge of α(ωk) and vice versa. Hence this

algorithm is solved in an iterative manner, with the estimate of α initialized using the

1 R = APAH + σ2I for ill-conditioned matrices [45]

2 matrix inversion lemma

40

FFT. This iterative algorithm takes about 10 to 15 iterations to converge based on

experimental and numerical results.

Note also that without accounting for the interference from other frequency grid

points (without weighting), minimizing the cost function in (2–9) for K = M gives the

Discrete Fourier Transform (DFT) of the signal:

α(ωk) =aH(ωk)y

M, k = 0, 1, . . . ,M − 1. (2–14)

The IAA algorithm described above is used for spectral estimation of stationary data.

Analogous to the STFT, the spectral content of a non-stationary data sequence, such as

(1), can be estimated using the TRIAA [44]. The signal is split into overlapping frames

similar to Fig. 2-2 and the IAA spectral estimate is computed for each frame. However,

to reduce the computational complexity, each subsequent frame after the first frame

is initialized with the spectral estimate of the previous frame instead of the FFT based

periodogram as described in the IAA algorithm. The resulting algorithm yields better

spectral resolution and lower side lobes than the STFT.

There is still a significant increase in the computational complexity when using the

TRIAA algorithm compared to using STFT for spectral estimation. This computational

complexity is reduced slightly by reducing the number of iterations in subsequent frames

for the TRIAA. This is because convergence of the estimated spectrum will occur in

fewer iterations given the current frame is initialized by the spectral estimate of the

previous frame. When the dataset is significantly large, the use of this algorithm is

still impractical. The bottle-neck of the TRIAA algorithm is in the computation of the

denominator in (2–13) for each frame.

In [46], [47] the Toeplitz structure of the covariance matrix R is exploited and the

computation of R−1 is performed using the Gohberg-Semencul (GS) factorization of

this matrix [2]. Moreover, the denominator is obtained via evaluating a polynomial.

This reduces the computational complexity of the denominator in (2–13) (which is the

41

bottleneck of the IAA algorithm) from O(M2K) to O(M2) floating point operations

(flops) [46] for a given frame, without a loss in performance. The algorithm is termed the

Fast IAA (FIAA), which is a significant improvement but still computationally expensive

for large datasets. The computational complexity of IAA and FIAA are O(M2K) and

O(M2 + K logK), respectively, where M is the data length and K is the grid size, with

K >> M .

An approximate algorithm to the IAA algorithm with significantly faster computational

time is described in [48] and referred to as the Quasi-Newton IAA (QN-IAA). The

QN-IAA algorithm estimates the covariance matrix as if it were from a low-order (L)

autoregressive (AR) process, where L << M with M being the data (frame) length.

The inversion of the lower-order covariance matrix Q ∈ CL×L is carried out in place of

R ∈ CM×M , yielding an approximate solution to the IAA spectral estimate (2–13) with

significant reduction in the computational complexity and just a slight degradation in the

resolution. The computational complexity of this algorithm is O(L2 +K logK).

The FIAA or QN-IAA can be used in a time-recursive manner for non-stationary

data as is the case with the ENF signal. This algorithm reduces the trade-off between

frequency resolution and time-resolution for a given frame length compared to the FFT

based periodogram during the ENF extraction process. The extraction process is the

same as the frequency domain analysis (2–2)-(2–7) with ϕr replaced by either of the

aforementioned algorithms.

However, even if a good algorithm is used for frequency estimation based on (2–7),

specific frames might be corrupted by interference signals with frequency components

within the ENF limits. This could lead to errors in frequency estimation, if the frequency

location corresponding to the maximum value of the estimated spectra belongs to an

interference signal. A robust method of tracking the ENF that exploits the slowly varying

nature of this frequency is needed. The next section describes the proposed frequency

tracking algorithm.

42

2.4.3 Frequency Tracking

A method of estimating the ENF by tracking it from one frame to another is

formulated here from a mathematical point of view. The proposed method uses discrete

dynamic programming [37] to find a minimum cost path. A cost function as shown in

this section is selected which takes into account the slowly varying nature of the actual

network frequency. This cost function penalizes significant jumps in frequency from

frame to frame and the corresponding path is used to estimate the ENF.

This algorithm involves finding the peak locations from the spectrum of each frame

and assigning costs based on the difference between a peak location in one frame and

a peak location in another frame. The magnitude of the assigned cost is related to the

difference in the frequency from one frame to another. The minimum cost path from the

first frame to the last frame is computed to estimate the ENF.

To estimate the number of relevant peaks (sinusoids) in a given frame, a model

order selection tool known as the Bayesian Information Criterion (BIC) is used. The BIC

for complex sinusoids in noise is given by (refer to [2] [49] for a full derivation):

BIC(nr) = M ln

(||y −

2nr∑k=1

a(ωk)α(ωk)||2)+ 5(2nr)lnM. (2–15)

The number of peaks (real sinusoids) nr, is estimated as the minimizing argument of the

above BIC criterion. The first term in (2–15) is a Least-Squares data fitting term, which

decreases as the number of estimated peaks nr increases, where as, the second term

is a penalty term that prevents ’over-fitting’ of the data model. Once the nr largest peaks

and corresponding locations are determined in each frame, the frequency tracking

problem is formulated and solved as follows.

Assume that for a given frame r, a set of estimated peak locations (frequencies)

is denoted by �r = {Pr1, Pr2, . . . Prnr}. We would like to find a path {fr}Rr=1, such that

fr ∈ �r and where the difference fr − fr−1 is as small as possible for r = 1, 2, ..., R.

This set corresponds to the estimated ENF over all frames and can be obtained as the

43

minimizing argument in the following optimization problem:

J = minfr∈�r

r=1,...,R

R∑r=2

(fr − fr−1)2. (2–16)

Calculating this cost using an exhaustive search is impractical. However, using dynamic

programming [37] the path that minimizes this cost can be computed recursively and

efficiently by minimizing the cost from a given frame j < R, to the last frame, denoted by

J(j, fj).

J(j, fj) = minfr∈�r

r=j+1,...,R

R∑r=j+1

(fr − fr−1)2, fj ∈ �j. (2–17)

This optimal cost satisfies the recursive equation,

J(j, fj) = minfj+1∈�j+1

{(fj+1 − fj)2 + J(j + 1, fj+1)}, fj ∈ �j (2–18)

which can be calculated for j = R − 1, R − 2, . . . , 1, with the initialization, J(R, fR) =

0, fN ∈ �N . Note that

J = minf1∈�1

J(1, f1), fN ∈ �N (2–19)

is the cost from the first frame to the last frame R and the set {fr}Rr=1 that minimizes this

cost function corresponds to the extracted ENF signal as mentioned above. Dynamic

programming has a computational complexity of O(R�2max), where R corresponds to the

total number of frames and �max is the number of spectral peaks in the frame with the

maximum number of peaks.

2.4.4 Matching the Extracted ENF to Database

Once the ENF signal has been extracted, a method of matching the estimated

signal to the database signal is required. The goal is to find the location/time within

the database that is similar in pattern to the extracted ENF. In [36] a method based on

minimizing the squared error between the ENF and database is used for automated

44

matching. A method of correlation matching proposed in [50] for short digital recordings

(10-15 minutes) is used in place of this MSE method. The process of correlation

matching is described as follows. Assume that f = [f1, f2, . . . , fR] is the extracted ENF

signal and d = [d1, d2, . . . , dL] corresponds to the database signal with L > R. The

matching process requires finding lmax such that:

lmax = argmaxl

c(l), l = 1, 2, . . . , L−R (2–20)

where c(l) is the correlation coefficient between f and the vector [dl, dl+1, . . . , dl+R−1].

An important point to make is that, the maximum correlation coefficient c(lmax) is

used here only for matching the estimated ENF to the database and comparing the

accuracy (reliability) of the different algorithms presented. Once a match has been

made, determining locations of edits to a recording should be based on the differences

between the ENF estimate and the database.

Table 2-2. Parameters for the Experiment

PARAMETERS Data1 Data2

T (Time Shift) 1s 1s

M (Length of Frame) 20s 33s

R (Number of Frames) 1800 1800

2.5 Experimental Results

The algorithms presented in the previous section are applied to two different

digital audio datasets referred to as Data1 and Data2. The two datasets are recorded

simultaneously and therefore, should contain the same ENF pattern over time. The

first data set (Data1) is acquired by connecting an electric outlet via a voltage divider

directly to the internal sound card of a desktop computer, resulting in an ENF signal

with a rather high signal-to-interference-and-noise ratio. On the other hand, the second

dataset (Data2) is an actual speech recording played from a speaker and picked up by

the internal microphone of a laptop computer.

45

Each of these recordings are originally sampled at 44.1 kHz at a bit rate of 16 bits

per sample. Each dataset is re-sampled to 441 Hz, hence keeping only the fundamental

frequency (1st harmonic) and the two higher harmonics of the ENF. A bandpass

filter with a narrow bandwidth around the network frequency is applied to the data to

eliminate as much interference as possible without distorting the ENF signal. Based on

Fig. 2-2 each data set is split using the values shown in Table 2-2. This set-up results in

an ENF estimate every second for a total of 30 minutes for each dataset.

An increase in the frame length improves the signal-to-noise ratio of the signal [36]

and the spectral resolution at the cost of lower time resolution. Therefore, a larger frame

length is used for Data2 which has a weak ENF signal compared to Data1 which has a

strong ENF signal.

Figure 2-3. Matching extracted ENF to database (Data1 - scaled to 60 Hz)

Fig. 2-3 shows the extracted ENF signal (shifted by 0.05 Hz for illustration

purposes) from Data1, matched with the truth obtained from the FDRs, when the

data set has not been altered in any form (using STFT and (2–7)). Fig. 2-7 shows the

extracted ENF using the STFT based method and our proposed method (also shifted

for comparison purposes). Tables 2-3 and 2-5 give the maximum correlation coefficient

c(lmax) of the various methods for Data1 and Data2, respectively, also when the signals

have not been altered. The maximum correlation coefficient values are used to compare

46

the accuracy of the algorithms and hence determine which is more reliable for ENF

estimation. We have also included similar MSE (actually standard deviation) analysis

in Tables 2-4 and 2-6 for the datasets, where the MSE is computed by averaging the

squared difference between the True ENF and the estimated ENF. It is important to point

out that the estimated ENF can sometimes have a constant offset [39], [50]. Therefore,

the correlation is the preferred method for accuracy measure. The datasets used for

this experiment do not have such an offset. They have also been made available at

http://www.sal.ufl.edu/download.html.

Table 2-3. Correlation coefficients of Algorithms (Data1)

Algorithm

Harmonic STFT STFT(Track) TRIAA TRIAA(Track) F-ESPRIT

60 Hz 0.9912 0.9917 0.9895 0.9900 0.9800

120 Hz 0.9911 0.9949 0.9902 0.9946 0.9470

180 Hz 0.9968 0.9968 0.9961 0.9961 0.9962

Table 2-4. Standard Deviation of error for Algorithms (Data1)

Algorithm


60 Hz 2.772e−3 2.650e−3 3.032e−3 2.919e−3 5.364e−3

120 Hz 2.774e−3 2.145e−3 2.822e−3 2.198e−3 6.570e−3

180 Hz 1.900e−3 1.851e−3 1.999e−3 1.999e−3 2.830e−3

2.5.1 Data1 Analysis

Fig. 2-3 shows the extracted harmonic (180 Hz) of the ENF signal scaled to 60

Hz and matched (using the location corresponding to the maximum correlation (2–20))

to the actual database frequency obtained from the FDRs. For each of the algorithms

used, the third harmonic gave the most accurate results for this dataset as shown in

Table 2-3. This is because for a fixed grid size, the estimation error when using the

third harmonic is reduced by a factor of three compared to the fundamental frequency.

Harmonics with frequencies higher than 180 Hz can be used for the estimation process

47

at a cost of increased computational complexity due to the increased sampling rate. Also

from Table 2-3, It can be seen that each of the STFT and TRIAA algorithms, produce

accurate estimates of the ENF using (2–7) because of the rather strong ENF signal. The

signal at the second harmonic is weak relative to the first and third harmonics, and in

a few frames the estimate was inaccurate. However, the frequency tracking algorithm

mitigated these inaccuracies successfully by tracking the correct spectral peaks.

The parametric method, frequency selective (F-ESPRIT) [2],[51] also yields

accurate estimates of the ENF for Data1 when the signal model assumes there is

only one sinusoid per frame. However, this method and other parametric methods are

not appropriate for ENF estimation in the presence of interference, because they are

sensitive to model assumptions.

For this dataset, the STFT yields slightly better results, compared to the adaptive

method (TRIAA). This can be explained by the fact that the periodogram is optimal for

estimating spectral lines (sinusoids) in the presence of white noise when they are well

resolved [2]. However, when there are interfering signals present, the poor resolution

of the periodogram will yield inaccurate estimates as is the case with Data2, a typical

digital recording.

Table 2-5. Correlation coefficients of Algorithms (Data2)

Algorithm


120 Hz 0.9125 0.9857 0.9305 0.9907 0.8446

Table 2-6. Standard Deviation of error for Algorithms (Data2)

Algorithm


120 Hz 7.948e−3 3.369e−3 7.225e−3 2.914e−3 1.086e−2

48

119.8 119.85 119.9 119.95 120 120.05 120.1 120.15 120.20

0.5

1

1.5

2

2.5x 10

−7

frequency (Hz)sq

uare

d m

agni

tude

FFTIAA

Freq. estimate IAA (119.965 Hz)Freq. estimate FFT (119.970 Hz)

True frequency (119.963 Hz)

Figure 2-4. Power Spectrum of one Frame (Data2): poor resolution of FFT

2.5.2 Data2 Analysis

For Data2, the second harmonic (120 Hz) is used to estimate the ENF, because

the first and third harmonics are too weak to be used for estimation. Table 2-5 shows

the maximum correlation coefficient values for the STFT and TRIAA using (2–7), the

frequency tracking algorithm using the spectral peaks of the FFT and IAA and the

parametric method (F-ESPRIT) with one assumed sinusoid. The ENF estimation

accuracy is improved using the adaptive method (IAA) because of improved spectral

resolution for several frames. Fig. 2-4 shows a comparison of the spectrum of one frame

of the Data2, where the poor frequency resolution of the FFT results in a relatively poor

estimate of the network frequency compared to the IAA algorithm.

119.8 119.85 119.9 119.95 120 120.05 120.1 120.15 120.20

0.5

1

1.5

2

2.5

3

3.5

4x 10

−7

frequency (Hz)

squa

red

mag

nitu

de

FFTIAA

True frequency (119.952 Hz)

Figure 2-5. Power Spectrum of one Frame (Data2): strong interference signal

49

Figure 2-6. Extracted ENF via Frequency Tracking (Data2 - scaled to 60 Hz)

Fig. 2-7 shows this extracted ENF harmonic using the STFT and (2–7) matched

with the database. From this figure, there are several frames where the ENF is

estimated inaccurately, due to the fact that the frequency corresponding to the maximum

spectral peak for those frames do not correspond to the ENF. This can occur if there

is another signal present with frequency within the limits of the acceptable range of

the ENF as illustrated in Fig. 2-5. This figure shows that for both spectral estimation

techniques used (IAA, FFT) the ENF harmonic estimate using (2–7) will be 120 Hz,

whereas the true frequency is approximately 119.95 Hz.

This problem can be rectified using our dynamic programming based frequency

tracking algorithm presented above.

Fig. 2-6 shows the spectral peak locations computed using the TRIAA and the

corresponding ENF estimate using dynamic programming. The estimate of the network

frequency using this tracking algorithm is then matched to the database in Fig. 2-7,

which provides a better match when compared to using (2–7), which can also be seen in

this figure, Fig. 2-8 (absolute error) and also from Table 2-5.

A few important points to make are that the frequency tracking algorithm uses the

peak locations for each frame estimated either by the adaptive algorithm (IAA) or the

FFT. The results show that the estimated ENF is more accurate when the peak locations

50

Figure 2-7. Matching extracted ENF to Database (Data2 - scaled to 60 Hz)

Figure 2-8. Absolute error of Algorithms: STFT and TRIAA (Track)

of IAA are used. This is as a result of the inaccurate estimates in some frames caused

by the poor resolution of using FFT. Also, all the numbers presented can be improved

upon slightly by using the entire dataset (44.1kHz) for analysis. For example, the STFT

maximum correlation of 0.9125 will be improved to 0.9158 without re-sampling, which

may not be worth the increased computational complexity.

2.6 Conclusions

When it comes to digital audio verification, the reliability of the method used for

authentication cannot be overemphasized. This chapter demonstrates a reliable method

of extracting the network frequency from a digital recording when the ENF cannot be

51

extracted from some of the frames using the FFT based periodogram either because

of poor spectral resolution or a stronger interference signal within said frame. These

problems were solved by using an iterative adaptive method (IAA), which provides better

spectral resolution than the FFT based approach. Also a frequency tracking method

based on dynamic programming was used for accurate extraction of the ENF even in the

presence of a strong interference signals within ENF limits.

From the results presented, the FFT gives slightly better estimates of the network

frequency when the signal-to-interference-plus-ratio is very high as is the case with the

first dataset. However, in most digital recordings, there will be significant interferences

from the recorded speech signals and other surrounding sounds that could lead to poor

estimation performance using the FFT due to its poor resolution and high side lobe

problems. As the results have shown, the adaptive techniques and frequency tracking

method should be adopted for ENF estimation, especially in challenging environments.

52

CHAPTER 3DATA-ADAPTIVE RELAX FOR RFI SUPPRESSION FOR THE SYNCHRONOUS

IMPULSE RECONSTRUCTION (SIRE) RADAR

3.1 Chapter Summary

This next two chapters focus on remote sensing applications, specifically on the

Synchronous impulse reconstruction radar (SIRE) UWB radar built by the Army research

laboratory for landmine detection. This chapter focuses on suppression of Radio

Frequency Interference (RFI) for ultra-wideband (UWB) radar signals, sampled using

this synchronous impulse reconstruction (SIRE) time equivalent sampling scheme. This

equivalent sampling scheme is based on the Army Research Lab’s (ARL) efforts to

build an ultra-wideband (UWB) radar in forward looking mode that samples returned

radar signals using low rate and inexpensive analog-to-digital (A/D) converters. The

cost effectiveness of this SIRE UWB radar makes it plausible for actual ground missions

for detecting buried explosive devices. However, the equivalent time sampling scheme

complicates RFI suppression as the RFI samples are aliased and irregularly sampled

in real time. In this chapter, the data-dependent RELAX and multi-snapshot RELAX

algorithms are presented as an intermediate step to the previously proposed averaging

scheme by the Army Research Laboratory, in order to enhance RFI suppression for

this sampling scheme. A direct application of the RELAX algorithm is computationally

intensive so an efficient method for generating the spectrum of this equivalently sampled

data is proposed in this chapter that provides a factor of 10 improvement in computation.

The proposed suppression technique involves modelling the narrowband RFI signals as

a sum of sinusoids and applying the aforementioned algorithms. The RELAX algorithm

improves the RFI suppression performance without altering the target signatures

compared to AR modelling. The multi-snapshot RELAX algorithm which provides a more

accurate sinusoidal model than the RELAX algorithm, improves on the RELAX algorithm

in terms of suppression. However, the target signatures are suppressed as the number

of sinusoids increases. The analysis of the algorithms is performed using sniff (passive)

53

data collected using the SIRE radar in addition to simulated wide-band echo signals

(point-target signatures).

3.2 Introduction

Ultra-wideband (UWB) radar is a commonly used tool for various remote sensing

applications. Such applications include but are not limited to the use of low frequency,

high bandwidth pulses for detecting improvised explosive devices (IEDs) and land mine

targets. The effective detection of land mines and other IEDs could lead to increased

safety for various ground related missions [52].

The use of low frequencies in UWB radar is necessary for foliage or ground

penetration, whereas the use of wideband pulses are necessary for good resolution

(ability to detect targets from clutter) [53]. However, because of these requirements, the

data (target returns) collected by the UWB radar will be corrupted by signals in the radio

frequency spectrum (specifically the UHF/VHF bands). These signals include FM Radio,

TV broadcasts and other narrowband and wideband communication signals. The ability

to effectively detect targets is reduced by the presence of these radio frequency signals.

General methods of suppressing RFI and their limitations are discussed in detail in [53]

for conventional UWB radar, which is the case when the returned signals are sampled

regularly at or above the Nyquist rate.

Due to the large bandwidth of the returned radar signals, conventional sampling

will require high rate analog-to-digital (A/D) converters to digitize the returned signals.

These high speed A/D converters are expensive to build and makes practical applications

improbable. In other to improve on the cost of UWB radars, the Army Research

Laboratory (ARL) is currently working on an equivalent time sampling UWB radar in

forward looking mode, referred to as the Synchronous Impulse Reconstruction (SIRE)

radar [54]. This radar uses low rate (inexpensive and commercially available) A/D

converters to sample the returned signals (approximately 3GHz bandwidth), which

makes the radar more feasible for adoption in practice. This equivalent time sampling

54

scheme takes advantage of the fact that the scene is not changing with time, hence

aliasing of the returned target signals can be prevented. However, this is not the case

with the radio frequency signals which are changing with time. Aliasing and the irregular

sampling caused by the time-equivalent scheme becomes an issue when it comes to the

subject of RFI suppression as discussed in the next two sections.

This chapter focuses on suppression of RFI signals for this equivalent time-sampling

scheme. The goal is to model the narrowband interference as a sum of sinusoids in real

time and estimate and subtract the sinusoids before averaging to achieve further

suppression.

A cyclic optimization algorithm known as RELAX [31] is proposed for estimating the

parameters of the sinusoids (in an iterative manner). This algorithm is an asymptotic

maximum likelihood approach [55] and is computationally and conceptually simple. It

has been applied to problems like non-contact vital sign detection for more accurate

estimates of respiratory rates and heart rates [56]. It has also been shown to estimate

the parameters of sinusoids accurately even in the presence of colored noise [55]. The

multi-snapshot RELAX [57] algorithm, which is an extension of the RELAX algorithm, will

be used to provide a more accurate sinusoidal model for the SIRE sampling scheme.

The RELAX algorithm or multi-snapshot RELAX are implemented as an intermediate

step to the already proposed averaging method [58] to achieve further suppression.

In Section 3.3, the time equivalent SIRE sampling scheme is described. Section

3.4 briefly describes the limitations of some conventional methods to the SIRE sampled

data for RFI suppression. The averaging method proposed in [58] for RFI suppression

of SIRE sampled data is also discussed in this section, along with its performance.

In Section 3.5, the RELAX algorithm, along with a fast computation of the spectrum

of irregularly sampled SIRE data for this algorithm is presented; the multi-snapshot

RELAX algorithm is also described in this section. The results are presented in Section

3.7, starting with simulations that show how the RELAX algorithm suppresses aliased

55

sinusoids for simulated data. The sniff (passive data collected using the SIRE UWB

radar) is then used to test the effectiveness of the proposed algorithms, which are

compared to AR modelling of the interference based on this sampling scheme (see

Appendix). Finally, the conclusions of this chapter are presented in Section 3.8.

3.3 SIRE Equivalent Sampling Scheme

In this section, the Synchronous Impulse Reconstruction (SIRE) equivalent time

sampling technique as detailed in [54] is briefly described. This time equivalent sampling

scheme poses some challenges on identifying and hence suppressing RFI sources, due

to the fact that the RFI sources are changing with time as will be discussed.

The SIRE sampling scheme involves sampling the returned radar signals from a

scene at a significantly lower sampling rate fs, (with corresponding sampling period

�s), than the Nyquist rate, which leads to aliased samples. N aliased samples are

collected per pulse repetition interval (PRI) or fast time, and for each subsequent PRI,

N more samples are collected with the range profile shifted by �e (in time). After K

pulse repetition intervals (PRIs) or slow time, a total of K × N aliased samples are

collected. These samples are interleaved as shown in Fig. 3-1, which gives an effective

sampling rate of fe = 1/�e that is equal to, or greater than the Nyquist rate. Because the

scene of interest in not changing with time, the returned samples from a given range bin

theoretically should also remain unchanged in time. Therefore, the interleaved samples

are theoretically effectively sampled above the Nyquist rate and should be unaliased.

The measurements from each range profile are typically repeated M times and

added coherently to improve the signal-to-noise ratio (SNR). Fig. 3-1 shows the special

case of M = 1. Table 1 summarizes the parameters used by ARL in the SIRE radar

pertaining to RFI suppression [54].

The RFI signals, which are collected in addition to the desired target returns, on the

other hand, are changing with time. Therefore, when the collected data are interleaved,

they do not represent the true time samples of the RFI signals.

56

Table 3-1. ARL Parameters for Synchronous Reconstruction Radar.

Radar A/D sampling rate fs = 40 MHz

Radar A/D sampling period �s = 25 ns

Pulse repetition frequency PRF = 1 MHz

Pulse repetition interval PRI = 1 µs

Number of range profiles (per slow time) N = 7

Interleaving factor K = 193

Total number of range profiles K ×N = 1351

Effective sampling period �e = 129.53ps

Effective sampling rate fe = 7.72 GHz

For instance, consider a complex sinusoid sampled at fe (see Tab. 3-1), with time

samples h[n] = ejωon. The periodogram estimate of the spectrum of h[n] is given by

ϕ(ω) = (1/L)|H(ω)|2, where

H(ω) =L∑

i=1

ej(ωo−ω)n (3–1)

is the discrete-time Fourier transform (DTFT) of h[n] and L = K ×N is the total number

of samples. If this complex sinusoid is sampled using the SIRE technique (M = 1), the

time samples of the interleaved signal will be given by:

~h[l] =

h[l(T + 1)] for l = 0, 1, . . . , K − 1

h[(l −K)(T + 1) +K] for l = K, . . . , 2K − 1

· · ·

h[(l − 6K)(T + 1) + 6K] for l = 6K, . . . , 7K − 1

where T = (fe/PRF) (the other variables are described in Tab. 3-1). The corresponding

periodogram is given by ~ϕ(ω) = (1/L)| ~H(ω)|2, where ~H(ω) is the DTFT of ~h[l] and is

57

given by:

~H(ω)=K−1∑l=0

ejωol(T+1)e−jωl +

K−1∑l=K

ejωo((l−K)(T+1)+K)e−jωl

· · ·+7K−1∑l=6K

ejωo((l−6K)(T+1)+6K)e−jωl (3–2)

Therefore,

~H(ω) =

(6∑

s=0

ej(ωo−ω)sK

)(K−1∑r=0

ej(ωo(T+1)−ω)r

)(3–3)

Fig. 3-2 shows the periodogram spectral estimate of the regularly sampled sinusoid

ϕ(ω) and the interleaved SIRE sampled signal ~ϕ(ω). The spectrum of the complex

sinusoid is not only distorted, but it peaks at a different frequency. Note that ~H(ω) can

be re-written as:

~H(ω) =

(6∑

s=0

ej(ωo−ω)sK

)(K−1∑r=0

ej(ωo−ω)rejωoTr

)(3–4)

Figure 3-1. Synchronous Impulse Reconstruction (SIRE) equivalent time sampling.

Therefore, if ωo = 2πm/T , where m ∈ Z, then ~H(ω) reduces to H(ω). This condition

implies that the frequency (in Hz) of the complex sinusoid f = feωo/2π = m × PRI,

is an integer multiple of the pulse repetition frequency. Unless this condition is true,

interleaving will lead to distortion of the complex sinusoid.

58

0 0.1 0.2 0.3 0.4 0.5

−70

−60

−50

−40

−30

−20

−10

0

Normalized frequency (cycles/sample)

dB

SIRE (interleaved)Regularly sampled

Figure 3-2. Spectrum of SIRE sampled complex sinusoid sampled after interleavingcompared to the spectrum of sampled regularly above the Nyquist rate.

0 0.2 0.4 0.6 0.8 1

x 10−6

0

0.5

1

time (s)

Am

plitu

de

0 20 40 60 80 100 1200

0.5

1

frequency (MHz)

Mag

nitu

de

Figure 3-3. Spectrum SIRE sampling pattern: One fast time pulse (N = 7 samples).

A single complex sinusoid, sampled regularly below the Nyquist rate (fs), should

consist of a single peak at an ambiguous frequency in the frequency domain (in a

bandwidth of fs). However, due to the irregular sampling pattern of the SIRE sampling

technique, a single sinusoid will be seen as multiple peaks within this bandwidth.

Fig. 3-3 shows the SIRE sampling pattern (in real time) and its corresponding

spectrum for a single fast time pulse (N samples). As expected, this will result in a

sinc like function every 40 MHz (fs) in the frequency domain. However repeating this

sampling pattern K times will correspond to sampling in the frequency domain as

seen in Fig. 3-4. Therefore, the spectrum of a single sinusoid sampled using the SIRE

59

sampling scheme will correspond to convolving the spectrum of this sampling scheme

with that of a sinusoid resulting in multiple peaks.

In the next section, we discuss some of existing algorithms for RFI suppression as

well as the limitations posed by this sampling scheme, based on the analysis above.

0 0.5 1 1.5 2 2.5

x 10−6

0

0.5

1

time (s)

Am

plitu

de

0 20 40 60 80 100 1200

0.5

1

frequency (MHz)

Mag

nitu

de

Figure 3-4. Spectrum SIRE sampling pattern (N ×K = 1351 samples).

3.4 Existing RFI Suppression Methods

One popular technique for RFI suppression based on conventional sampling

involves the use of notch filters. This method involves estimating the spectrum of the

corrupted signal and removing the spikes in this spectrum using a notch filter. This

method works well for narrowband interference sources. However, it will introduce

sidelobes in the time-domain [59–61]. Filtering techniques in general, suffer from filter

transients and reduced data length. The notch filtering problem is even more severe

because of the ambiguity in frequency for the SIRE sampling scheme based on the

analysis in the previous section and Fig. 3-2 for the interleaved signals. Also, if the

analysis of the corrupted signal is performed in real-time (before interleaving), one

interference source will appear to have multiple peaks in the spectrum due to irregular

sampling as seen in Fig. 3-4, which makes this method not applicable.

Modelling the RFI using AR models can also be used for suppression (see

Appendix). The irregular sampling of the SIRE data makes this endeavour challenging.

60

However, the SIRE sampled data is sampled regularly in fast time and slow time, and

this can be exploited for AR modelling. There are only N = 7 samples sampled regularly

in fast time, whereas there are K = 193 regularly sampled samples in slow time. These

slow time samples can be used for AR modelling with N = 7 snapshots allowing for

more freedom in the choice of the AR model order (see Appendix for AR modelling of

SIRE sampled data).

Modelling the narrowband RFI as a sum of sinusoids, and estimating their

parameters has been shown to be effective for suppressing RFI with little signal

distortion [59],[62]. This method involves estimating the amplitude, frequency and

phase of each interfering sinusoid and subtracting the resulting sinusoid from the

corrupted data. The effectiveness depends on how accurate these parameters are

estimated, and is reduced if the sinusoidal model for the RFI signals starts to breakdown

[62]. This occurs when the duration of data is greater than the modulation time (inverse

of modulation bandwidth) of the RFI signals. For instance, a 3 kHz narrowband voice

channel will have a modulation time of approximately 0.3 ms, whereas wideband TV

signals with bandwidth of several kHz will have a much smaller modulation time [62]. If

the duration of the processed data is greater than this modulation time, the estimated

parameters will change during the acquisition time, leading to less effective suppression.

These methods are also computationally expensive when many interference sources are

estimated.

When the RFI signals are sampled using the SIRE equivalent sampling scheme,

estimation of these parameters becomes even more challenging due to the irregular

sampling pattern and aliasing introduced, even if the model is accurate.

Another technique for RFI suppression is using passive data to adaptively suppress

RFI from active radar data by projecting the measured active data to a signal subspace

created by the passive data. This method assumes orthogonality between the desired

target signatures and the RFI, and has been shown to be effective for RFI suppression

61

in [63] for conventionally sampled data. However, as noted in [54], this method is

inadequate for suppression of SIRE sampled data, due to the irregular sampling pattern

and aliased samples of the RFI. These challenges have prompted the need for new RFI

suppression techniques for the SIRE sampling scheme.

The averaging method proposed in [58] and also detailed therein, has been shown

to suppress wideband and narrowband interferers. The method is based on repeating

the measurements from the same range profile M times and averaging the repeated

measurements. The averaged samples are then interleaved and used for generating

SAR images. Fig. 3-2 shows the amount of suppression as a function of the number of

repeated measurements based on simulated RFI sources. A similar plot can be seen

in [58]. An important point to note is that this method of suppression does not take

into account any properties of the RFI signal, which is the motivation for improving the

performance.

0 200 400 600 800 1000−35

−30

−25

−20

−15

−10

Number of Pulses Averaged (M)

Sup

pres

sion

(dB

)

Figure 3-5. RFI Suppression (dB): Averaging method (M realizations) for simulatedSIRE sampled RFI signals.

In this chapter, the averaging method is improved by analyzing the data in ’real-time’

(before interleaving). The aliased samples of the data in ’real-time’ are modelled as a

sum of sinusoids, in other to achieve further suppression with little signal distortion. The

parameters of the sinusoids are estimated and the resulting sinusoids are subtracted

62

from the data using the RELAX algorithms [31],[57] (to provide accurate estimates of the

parameters) before averaging.

Based on the analysis in the previous section, a single sinusoid appears as multiple

peaks due to the irregular SIRE sampling pattern in ’real-time’. However, in theory,

estimating the parameters of a single sinusoid from the maximum peak location of the

spectrum, and subtracting this from the data, will correspond to its removal from the

spectrum. This will eliminate all the multiple aliased peaks (Fig. 3-4). This analysis will

be shown on simulated sinusoids in the results section.

The RELAX algorithm and its multi-snapshot counterpart are described in the next

section and the steps for RFI suppression are also presented.

3.5 Proposed RFI Suppression Method: RELAX and Averaging

3.5.1 Modelling of RFI

The proposed suppression method, entails modelling RFI signals of length L

collected in real time (before interleaving) as a sum of P complex-valued aliased

sinusoids as described in Eq. (3–5):

z =

P∑p=1

αpa(fp) (3–5)

where αp and fp are the complex amplitude and frequency of the pth sinusoid and

a(fp) =[1 ej2πfp · · · ej2π(L−1)fp

]TThe received measurement signal can be written as y = z+ s+ n, where z, s, and

n, are the RFI signal, desired target returns, and receiver noise, respectively. The target

returns have a wide bandwidth relative to the RFI signals and can be modelled as white

noise [53] ,[64]. RFI suppression, then, becomes a case of estimating the parameters

of multiple sinusoids in the presence of white noise. The non-linear least squares

(NLS) approach (an asymptotic Maximum Likelihood approach [55],[2]) estimates these

63

parameters by minimizing the following non-linear least squares cost function in Eq.

(3–6),

{αp, fp

}P

p=1= argmin

{αp,fp}P

p=1

∣∣∣∣∣∣∣∣∣∣y −

P∑p=1

αpa(fp)

∣∣∣∣∣∣∣∣∣∣2

(3–6)

where P is the number of sinusoids, which can be estimated using a model-order

selection tool like the Bayesian Information Criterion (BIC) [65]. This method can

approach the Cramer-Rao bound in performance, but it involves a multi-dimensional

search and hence involves complex computations for the case of multiple sinusoids.

It can also be sensitive to initializations [2],[66]. The RELAX algorithm can be used

for solving the problem in an iterative manner reducing the computational complexity

significantly [31]. This conceptually and computationally simple algorithm was shown to

estimate sinusoidal parameters accurately and robustly even in the presence of colored

noise [55]. The parameters are estimated for the above non-linear least squares fitting

problem in an iterative manner as described below.

3.5.2 RELAX Algorithm

The RELAX algorithm estimates the parameters as follows: Let

yp , y −P∑

i=1,p=i

αia(fi) (3–7)

The frequency and complex amplitude estimates of the pth sinusoid are, respectively,

estimated by:

fp = argmaxfp

|aH(fp)yp|2 (3–8)

and

αp =aH(fp)yp

L︸︷︷︸DTFT of yp

|fp=fp(3–9)

64

The RELAX algorithm steps are given by:

• Step 1: Assume P = 1. Estimate f1 and α1 from y.

• Step 2: Assume P = 2. Compute y2 based on estimates from the previousstep and estimate f2 and α2. Compute y1 and re-estimate f1 and α1. Re-iterateprevious steps until practical convergence.

• Step 3: Assume P = 3. Compute y3 and estimate f3 and α3. Re-compute y1 andre-estimate f1 and α1 from f2, α2, f3 ,α3. Re-iterate until convergence or a fixednumber of iterations.

• Remaining Steps: Continue until P = P , which is an estimated or desired number.

Note that, the frequencies and complex amplitudes in (3–8) and (3–9), respectively,

are estimated using the DTFT of the signals yp. This can be efficiently computed using

the FFT and zero-padding for conventionally (regularly) sampled data.

Based on Fig. 3-3 and the analysis leading to Eq. (3–4), as previously discussed,

the interleaving process of a SIRE sampled sinusoid leads to a distortion of that signal

except for a specific case, being that the frequency of the sinusoid is an integer multiple

of the PRI. The analysis of the RFI using the RELAX algorithm will, therefore, be

performed on the data in real-time (before the interleaving process). As will be shown

in the results section, the estimated complex amplitudes and frequencies (although

possibly ambiguous), can be used to accurately reconstruct the aliased RFI samples

and yield effective RFI suppression using the RELAX algorithm.

The RELAX algorithm requires the computation of the spectrum of the received

samples. For irregularly sampled SIRE data, this spectrum can be computed using an

FFT after re-sampling the data (interpolating with zeros). Re-sampling this data to give a

regularly sampled data with effective sampling frequency of fe, will lead to a significantly

long data sequence with most of the samples being zero. For instance, one realization

(M = 1) of a SIRE sampled data, sampled at fs = 40 MHz , contains N = 7 aliased

samples per PRI. After re-sampling to an effective rate of fe = 7.72 GHz, each PRI will

consist of T = fe × PRI = 7720 samples. Hence a total of T × K = 7720 × 193 ≈ 1.5

65

million samples per realization. Therefore, applying a direct FFT (with zero-padding)

to this re-sampled data to estimate the frequencies and complex amplitudes becomes

computationally intensive with a computational complexity of O(TK logTK).

Note that similar to Fig. 3-4, a single sinusoid sampled using the SIRE sampling

technique and re-sampled as discussed above to give an effective sampling rate

of fe = 7.72 GHz will repeat itself approximately every 40MHz (A/D rate) in the

frequency domain, due to aliasing. In order to reduce the computational complexity

of this re-sampling scheme, the regular sampling of the data in both fast and slow time

can be exploited and the spectrum can be computed only on a 40 MHz bandwidth to

save on computations. The analysis is performed as follows:

The spectral estimate for SIRE sampled data in real time (before interleaving) based

on parameters in Tab. 3-1 is given by:

X(f) =∑n

∑m

xm,ne−j2πf(m�m+n�n) (3–10)

where

n = 0, 1, 2, · · · , N − 1 (N = 7)

m = 0, 1, 2, · · · , K − 1 (K = 193)

�n = �s = 25 ns, (ADC sampling rate),

�m = PRI +�e.

A direct computation of the spectrum in Eq. (3–10) is obviously computationally

intensive, especially for a fine grid size in frequency. Assuming

f =k1�m

+ k2�f (3–11)

66

where

k1 = 0, 1, 2, · · · , K1 − 1 K1 = T = 7720

k2 = 0, 1, 2, · · · , K2 − 1 K2 = 1/(�m�f)

�f = fixed grid size (in Hz)

Eq. (3–11) is the frequency grid (in Hz) on which the spectrum in Eq. (3–10) will be

computed. Note that the choice of K2 determines the grid spacing �f and the choice

of the k1 values determines the portion of the bandwidth in which the spectrum is to be

estimated. For instance, k1 = 0, 1 · · ·T = 7720 computes the spectrum over the entire

7.72 GHz (effective sampling rate) bandwidth.

For the frequency grid specified in (3–11), the spectrum in (3–10) can be re-written

as follows:

X(f) =∑n

∑m

xm,ne−j2π(

k1m+k2�)(m�m+n�n) (3–12)

which simplifies to:

X(k1, k2) =∑n

e−j2π(k1�m

+k2�f )n�n∑m

xm,ne−j2π

k2K2

m

X(k1, k2) =∑n

e−j2π(k1�m

+k2�f )n�nXn(k2) (3–13)

From Eq. (3–13), the spectrum is computed by summing up multiple FFTs. Also

because the signal is aliased, the spectrum needs only to be computed over a small

portion (40 MHz - A/D sampling rate) of the entire bandwidth. The computational

complexity of this algorithm is O(NK2logK2 + K1K2N). Note that the bottle neck of

this algorithm is in the second term. When the spectrum is computed over the entire

frequency grid, K1 = T = 7720, the computational complexity is on the same order as

re-sampling and applying an FFT. However, when the spectrum is computed over a 40

67

MHz bandwidth (K1 = 40), this algorithm drastically improves on the computation. For

example, for a frequency grid with spacing (�f ) of approximately 2 kHz, the spectrum in

Fig. 3-4 was computed in 0.26 secs when the SIRE sampled data was re-sampled and

the FFT was applied directly using the MATLAB software. However, the spectrum based

on Eq. (3–13) was computed in 0.035 secs on a 40 MHz grid.

Table 3-2. Suppression Algorithm: RELAX + Averaging

Step 1: RELAX (P sinusoids estimated).

- Compute the DTFT of the measured data y from (3–13) and estimate f1

and α1, using (3–8), (3–9) and (3–13).

- Compute y2 using (3–7) and its DTFT using (3–13). Estimate f2 and α2.

Re-estimate f1 and α1 from y1 and iterate. Continue for yp, fp and

αp (3 ≤ p ≤ P ) (Section IV.B).

Step 2: Reconstruct aliased RFI samples using {fi}Pi=1 and {αi}Pi=1. Subtract

from each realization (y).

Step 3: Average residue from each realization and interleave.

This spectrum is used in the RELAX algorithm (3–8) and (3–9), to estimate the

parameters of the sinusoids present. The RELAX algorithm is applied here to one

realization (M = 1) of SIRE sampled data (which correspond to a data with an

acquisition time of 0.193 ms based on Tab. 3-1). Therefore a narrowband interference

source with a modulation bandwidth of 5 kHz or less can be accurately approximated as

a single tone, whereas multiple sinusoids are needed to model an interference source

with wider bandwidth. The sinusoidal model begins to break down for very wideband

interferers. In the next subsection we propose the multi-snapshot RELAX algorithm

for SIRE sampled data that provides a more accurate sinusoidal model for the RFI

signals by using fewer samples (smaller modulation time), for suppression. The overall

proposed RFI suppression algorithm can be summarized in the following steps as shown

in Tab. 3-2 for RELAX.

68

3.5.3 Multi-snapshot RELAX Algorithm

The multi-snapshot RELAX algorithm [57] uses N = 7 samples (150 ns acquisition

time) for RFI suppression. Interference sources with modulation bandwidth of 6.7 MHz

or less can be accurately modelled as sinusoids, which includes wideband interferers

like TV broadcasts etc.

The multi-snapshot RELAX algorithm (M-RELAX for short) proposed for angle and

waveform estimation in [57] is a modification of the originally proposed RELAX algorithm

[31]. The algorithm estimates the angle of arrival (using multiple snapshots of the data)

and the corresponding waveform for each snapshot.

This algorithm is proposed here for RFI suppression of SIRE sampled data to

provide a more accurate sinusoidal model for the RFI signals. Here, each set of N =

7 fast time samples is treated as a snapshot. The data is split into K = 193 total

snapshots based on the parameters in Tab. 3-1. The frequency of a single tone is

estimated by averaging the periodogram of each snapshot and finding the frequency

that maximizes the average. The complex amplitudes of each snapshot is estimated by

finding the complex value of the spectrum of each snapshot at the estimated frequency.

Note that for a single complex sinusoid, K = 193 complex amplitudes are estimated

from each snapshot, whereas only one frequency is estimated. The parameters are

estimated as given in Eq. (3–14) (modification of NLS for the multi-snapshot case [57]):

{αp, fp

}P

p=1= argmin

{αpfp}Pp=1

K∑m=1

∣∣∣∣∣∣∣∣∣∣x(m)−

P∑p=1

αp(k)a(fp)

∣∣∣∣∣∣∣∣∣∣2

(3–14)

where αp = [αp(1), αp(2), . . . , αp(K)] contains the estimated complex amplitudes

of the pth sinusoid for each of the K snapshots, fp is the estimated frequency of the

pth sinusoid for all snapshots and x(m) is the mth snapshot. These parameters are

estimated as follows. Let

xp(m) , x(m)−P∑

i=1,p=i

αi(m)a(fi) (3–15)

69

The estimates described above are given by:

fp = argmaxfp

K∑m=1

|aH(fp)xp(m)|2 (3–16)

and

αp(m) =aH(fp)xp(m)

L︸︷︷︸DTFT of xp(m)

|fp=fp, m = 1, 2, . . . , K (3–17)

The multi-snapshot RELAX algorithm steps are as follows:

• Step 1: Assume P = 1. Estimate f1 and α1(m) from x(m), for m = 1, 2, . . . , K.

• Step 2: Assume P = 2. Compute x2(m) based on the estimates from the previousstep and estimate f2 and α2(m), for m = 1, 2, . . . , K. Compute x1(m) andre-estimate f1 and α1(m), for m = 1, 2, . . . , K. Re-iterate previous steps untilpractical convergence.

• Step 3: Assume P = 3. Compute x3(m) using {αp, fp}2p=1 and estimate f3 andα3(m), for m = 1, 2, . . . , K. Re-compute x1(m) and re-estimate f1 and α1(m) from{αp, fp}3p=2, for m = 1, 2, . . . , K. Then re-compute x2(m) and re-estimate f2 andα2(m) from {αp, fp}p=1,3, for m = 1, 2, . . . , K. Re-iterate until convergence.

• Remaining Steps: Continue until P = P , which is an estimated or desired number.

For the SIRE sampled data, the spectrum (DTFT) of each snapshot can be

described as follows. Let x(m) = {dm(n)}N−1=7n=0 correspond to the mth snapshot,

where dm(n) denotes the nth sample of the mth fast time pulse (m = 1, 2, . . . K). Note

that each snapshot is regularly sampled (at the A/D rate). The spectral estimate can

therefore be computed using an FFT multiplied by a corresponding phase shift (over a

40 MHz bandwidth). The spectrum of each snapshot is given by:

Xm(f) =6∑

n=0

dm(n)e−j2πf(n�n+m�m) (3–18)

where (f ∈ (0, fs)) is the frequency (in Hz), �n = 1/fs and �m are the sampling period

and the time difference from one snapshot to the next, respectively. Equation (3–18)

70

above can be simplified as follows:

Xm(f) = e−j2πf(m�m)

6∑n=0

dm(n)e−j2π f

fsn (3–19)

which simplifies to:

Xm(r) = e−j2π rR(m�m

�n)

6∑n=0

dm(n)e−j2π r

Rn (3–20)

for a discrete frequency grid r = 0, 1 . . . R − 1. It is important to note that parameter

identifiability (maximum number of sinusoids that can be uniquely identified) [67–69],

becomes an issue with this approach. Given N = 7 real valued samples, only up

to P = 2 sinusoids (amplitude, frequency, and phase), can be uniquely identified.

Estimating more than P = 2 sinusoids will significantly distort the target signatures.

The overall proposed RFI suppression algorithm can be summarized in the steps as

shown in Tab. 3-3 for multi-snapshot RELAX.

We also consider Auto-regressive(AR) modelling of the RFI data for suppression,

however due to desired signal distortion, this approach is not effective. AR for RFI

suppression for the SIRE radar is described in the next section.

Table 3-3. Suppression Algorithm: M-RELAX + Averaging

Step 1: M-RELAX (P sinusoids estimated)

- Compute the DTFT of the mth snapshot x(m) of the measured data y

from (3–20) and estimate f1 and α1(m), using (3–16), (3–17).

- Compute x2(m) using (3–15) for each snapshot and its DTFT using

(3–20). Estimate f2 and α2(m). Re-estimate f1 and α1(m) and iterate.

Continue for yp, fp and αp (3 ≤ p ≤ P ). (Section IV.C).

Step 2: Reconstruct aliased RFI samples using {fi}Pi=1 and {αi(m)}Pi=1 for each

snapshot. Subtract from each realization (y).

Step 3: Average residue from each realization and interleave.

71

3.6 Autoregressive (AR) Modelling

Auto-regressive (AR) models, which is commonly used for modelling narrowband

(”peaky”) signals, can be used for estimating and suppressing RFI signals. The

measured signal (RFI, desired target returns, and thermal noise) is modelled as an

AR process [62]. The AR modelling (linear prediction modelling) equation is written as:

y[tn] = −q∑

i=1

a[i]y[tn − i] + u[tn] (3–21)

where, y[tn] is the measured data sequence, u[tn] corresponds to the white noise term

at a time instant tn and q corresponds to the AR order, which is determined by the

number of spectral peaks and their widths. The assumption is that the first term on the

right hand side of Eq. (3–21) corresponds to the RFI signal. The suppression process

therefore involves estimating {a[i]}qi=1 and using the coefficients to suppress the RFI

signals. Note that Eq. (3–21) can be re-written as:

y[tn] = H(z)u[tn] (3–22)

where H(z) = 1/A(z) = 1/(1+ a[1]z−1+ . . .+ a[q]z−q), with z−1 being the delay operator.

The RFI suppression process involves passing the measured data through the inverse

filter 1/H(z) = A(z) (from the estimated AR coefficients {a[i]}qi=1).

The well-known methods for solving for the AR coefficients in (3–21) include the

Yule-Walker (YW) method, Prony method and the modified Prony method [2]. The YW

and Prony methods give similar results for large data samples. However, for smaller data

records the Prony method tends to give gives more accurate AR estimates [2].

If both sides of the forward linear prediction equation (3–21) are multiplied by

y[tn − tm], and the expectation is taken, the well-known Yule-Walker equations are

obtained.

72

r(1)

...

r(n)

= −

r[0] · · · r[−q + 1]

......

r[q] · · · r[0]

a[1]

...

a[q]

(3–23)

which can be re-written as r = −Ra. Where r and R are the covariance vecotr and

matrix of the data. The AR coefficients (a) are estimated by solving Eq. (3–23). The

Yule-Walker method estimates the coefficients by replacing r with the standard biased

autocorrelation sequence (ACS) estimator [2]. The Prony method solves the forward

linear prediction equation (3–21) using least squares (LS). The problem reduces

to (3–23), with the covariance sequence estimated by the standard unbiased ACS

estimator [2].

The Modified covariance (Prony) method (which improves on the Prony method)

combines the forward linear prediction in (3–21) and the backward linear equation given

below in (3–24) to solve for the AR coefficients using least squares:

y[tn] = −q∑

i=1

ab[i]y[tn + i] + ub[tn] (3–24)

where ab[i] = a[i].

This Modified covariance method is applied to the SIRE sampled data which is

sampled regularly in fast-time and slow-time. N = 7 sets of the slow-time regular

samples (K = 193 samples per set) are used for AR modelling. The Modified covariance

73

equations can be written in matrix form as follows (for each set of slow time samples):

y(q)

...

y(K − 1)

y(0)

...

y(K − q − 1)

= −

y[q − 1] · · · y[0]

......

y[K − 2] · · · y[K − q − 1]

y[1] · · · y[q]

......

y[K − q] · · · y[K − 1]

a[1]

a[2]

...

a[q]

(3–25)

where q is the AR order. The equation can be re-written as

yn = −Yna, for n = 1, 2, . . . N with N = 7 (3–26)

The least-squares solution of this overdetermined linear system of equations is given by:

a = −(YTnYn)

−1YTn yn for n = 1, 2, . . . N with N = 7 (3–27)

where (YTnYn)

−1 estimates the covaraince matix and YTnyn estimates the ACS in (3–23).

A more accurate estimate of the covariance matrix in (3–27) is derived by averaging the

N = 7 snapshots.

3.7 Experimental Results

3.7.1 Simulations

In this section, a signal consisting of three sinusoids in white noise (SNR = 10dB)

is simulated and sampled using the SIRE equivalent scheme based on the parameters

in Tab. 3-1, with no repeated measurements (M = 1). The sinusoids have frequencies

f1 = 111.111 MHz, f2 = 300 MHz and f3 = 650.255 MHz, all with amplitudes of 1.

Note that the samples obtained will also correspond to a signal containing sinusoids with

frequencies fa1 = f1 mod fs, fa2 = f2 mod fs and fa3 = f3 mod fs, as well as a signal

containing sinusoids fa1 + kfs, fa2 + kfs and fa3 + kfs (where k ∈ Z and fs is the A/D

rate). This ambiguity in frequency is caused by aliasing due to the low A/D rate of the

radar. The RELAX algorithm can be used to accurately estimate the complex amplitudes

74

of these sinusoids as well as the ambiguous frequencies. This is achieved using the

spectrum in (3–13) estimated only on a 40 MHz bandwidth to save on computations.

The estimated parameters are then used to reconstruct the aliased samples, in order to

suppress the sinusoids through subtraction.

1200 1250 1300 1350

−2

0

2

Original Signal

Samples

Am

plitu

de

0 10 20 30 40

0

0.2

0.4

Spectrum

Frequency (MHz)

Mag

nitu

de

A

1200 1250 1300 1350

−2

0

2

Residue

Samples

Am

plitu

de

0 10 20 30 40

0

0.2

0.4

Spectrum

Frequency (MHz)

Mag

nitu

deB

1200 1250 1300 1350

−2

0

2

Residue

Samples

Am

plitu

de

0 10 20 30 40

0

0.2

0.4

Spectrum

Frequency (MHz)

Mag

nitu

de

C

1200 1250 1300 1350

−2

0

2

Residue

Samples

Am

plitu

de

0 10 20 30 40

0

0.2

0.4

Spectrum

Frequency (MHz)

Mag

nitu

de

D

Figure 3-6. RFI suppression (SIRE sampling) - Signal and spectrum of simulated datacontaining 3 real-valued sinusoids in white noise after suppression usingRELAX with P (real-valued) sinusoids estimated (A) Original data, (B) P = 1,(C) P = 2, (D) P = 3.

Fig. 3-6, shows the original signal, its spectrum and the progression of suppression

as the number of estimated aliased parameters increases. From Fig. 3-6, we observe

that the spectrum of the three sinusoids contains multiple peaks, due to the irregular

sampling as described previously. By estimating the ambiguous frequency and complex

amplitudes of each of the sinusoids (on only a 40 MHz bandwidth), multiple aliased

peaks are suppressed. The purpose of the above simulations is to show the ability of the

75

RELAX algorithm to estimate the ambiguous frequency and complex amplitudes of the

sinusoids on a small bandwidth correctly, to effectively suppress the sinusoids (including

the multiple aliased peaks).

In the next subsection the sniff dataset collected using ARL’s SIRE UWB radar is

analyzed and the RFI is suppressed using both the RELAX and multi-snapshot RELAX

algorithms. Comparison with AR modelling of the RFI is also provided.

3.7.2 Sniff Experimental Data

The sniff data to be analyzed, was collected by ARL using the SIRE UWB radar

in passive mode based on the parameters in Tab. 3-1. Each set of data consists of

L = K × N = 1351 samples. In this subsection this RFI data will be analyzed using

the proposed algorithms. Two sets of RFI data with different energy levels are analyzed.

For simplicity they will be referred to as File1 and File2. Each set of the data, consists of

M = 88 realizations. The amount of suppression achieved are presented in Table3-4.

A wideband echo signal which represents a return from a single point target is

simulated. This signal is added to each realizations (M = 88) of the sniff data, in a way

that the echo signal adds up coherently. The goal is to show how much distortion is

introduced to the desired signals after the application of the RFI suppression algorithms.

Fig. 3-7 shows the amount of suppression achieved when the RELAX algorithm

with P real-valued sinusoids are suppressed for each realization and the residues are

averaged (File1). These results are compared to straightforward averaging, also in this

figure. A similar analysis is performed for the multi-snapshot RELAX algorithm and the

amount of suppression can be seen in Table 3-4 and 3-5. The average power of the

signals {s(i)}Li=1, are computed using (3–28):

10 log10

(L∑

i=1

|s(i)|2)/L (3–28)

From Table 3-4 and 3-5, it is clear that the amount of suppression increases as the

number of real-valued sinusoids increases for the RELAX algorithm. This improvement

76

Table 3-4. RFI Suppression (dB): File 1 ( ~P = 1)

Avg. RELAX M-RELAX RELAX/M-RELAX ( ~P ) AR (q)

20.89 23.46 (P = 1) 24.19 (P = 1) 27.04 (P = 1) 23.21 (2)

26.61 (P = 4) *PI (P = 4) 29.11 (P = 4)

27.79 (P = 7) *PI (P = 7) 30.38 (P = 7) 24.52 (20)

28.06 (P = 10) *PI (P = 10) 31.27 (P = 10)

Table 3-5. RFI Suppression (dB): File2 ( ~P = 1)

Avg. RELAX M-RELAX RELAX/M-RELAX( ~P ) AR (q)

18.49 20.02 (P = 1) 20.22 (P = 1) 22.40 (P = 1) 20.03 (2)

21.55 (P = 4) *PI (P = 4) 23.47 (P = 4)

22.30 (P = 7) *PI (P = 7) 23.65 (P = 7) 20.37 (20)

22.61 (P = 10) *PI (P = 10) 25.97 (P = 10)

*PI - Parameter identifiability not met.

comes at a cost of increased computational complexity. However, the target signatures

are left basically unaltered as can be seen in Fig. 3-8.

The multi-snapshot RELAX algorithm shows a significant amount of suppression

of the data as the number of sinusoids increases. Due to the issue of parameter

identifiability discussed in the previous section, estimating more than P = 2 real-valued

sinusoids (in theory), using only N = 7 real-valued samples per-snapshot will effectively

suppress all the samples to zero. This leads to the suppression of the target energy, as

can also be seen in Fig. 3-8.

The multi-snapshot RELAX algorithm can be seen to improve on the suppression

with little target distortion for P = 1 based on the real RFI data collected using the SIRE

radar. This algorithm is combined with the RELAX algorithm to effectively suppress both

wideband and narrowband interferers. This improvement is seen in Table 3-4 and 3-5

and Fig. 3-9 shows the reconstructed echo after suppression.

A similar analysis is performed for AR modelling. The AR modelling improves on the

suppression compared to averaging as can be seen in Fig. 3-10. However, this inverse

77

0 200 400 600 800 1000 1200 1400−30

−20

−10

0

10

20

30

Samples

Am

plitu

de

AveragingRELAX (P = 1) and averaging

A

0 200 400 600 800 1000 1200 1400−30

−20

−10

0

10

20

30

Samples

Am

plitu

de

AveragingRELAX (P = 10) and averaging

B

Figure 3-7. RFI suppression - RELAX algorithms with P (real-valued) sinusoidsestimated and suppressed from sniff data (File1) compared to averaging. (A)P = 1, and (B) P = 10.

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de

Echo signalRELAXM−RELAX

A

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de


B

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de


C

Figure 3-8. Echo retrieval (File1) - RELAX with P (real-valued) sinusoids compared toideal echo signal. (A) P = 1, (B) P = 2, and (C) P = 10.

78

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de

Echo signalRELAX/M−RELAX

A

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de


B

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de


C

Figure 3-9. Echo retrieval (File1) - RELAX with P (real) sinusoids combined withM-RELAX with ~P = 1 real sinusoid, compared to ideal echo signal. (A)P = 1, (B) P = 2, and (C) P = 10.

filtering technique leaves the desired signal distorted. This distortion is increased as

the model order increases (due to filtering transients) as seen in Fig. 3-9. Hence the

combined RELAX and multi-snapshot RELAX outperforms the AR approach in terms of

both RFI suppression and desired target echo preservation.

3.8 Conclusions

In this chapter, we have proposed a method for RFI suppression for the SIRE

UWB radar, which is a cost efficient system of sampling returned radar signals used for

detecting land mines and IEDs developed by ARL. The low sampling rate and irregular

sampling pattern of this radar poses a challenge for Radio Frequency Interference

(RFI) suppression as the measured RFI signals will be severely aliased. In this chapter,

we have discussed the challenges of RFI suppression for this radar and proposed

79

0 200 400 600 800 1000 1200 1400−30

−20

−10

0

10

20

30

Samples

Am

plitu

de

AveragingAR−2

A

0 200 400 600 800 1000 1200 1400−30

−20

−10

0

10

20

30

Samples

Am

plitu

de

AveragingAR−20

B

Figure 3-10. RFI suppression - AR modelling with order q compared to averaging forsniff data (File1). (A) q = 2, and (B) q = 20.

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de

EchoAR−2

A

650 660 670 680 690 700−40

−20

0

20

40

60

Samples

Am

plitu

de

EchoAR−20

B

Figure 3-11. Echo retrieval (File1) - AR modelling with order q, compared to ideal echosignal. (A) q = 2, and (B) q = 20.

using the RELAX algorithm and its multi-snapshot counterpart as an intermediate

step to the already proposed averaging scheme for RFI mitigation, for the SIRE UWB

radar. The results show that the RELAX algorithm can suppress RFI further than just

averaging without altering desired target echo signals. The RELAX algorithms are easy

to implement since they just involve FFTs. They have been shown to outperform AR

modelling of the RFI singals.

The multi-snapshot RELAX uses a shorter time-duration (and fewer samples)

for suppression, which yields a more accurate wideband model of the RFI as sum of

sinusoids compared to the RELAX algorithm. However, this algorithm significantly

80

suppresses target signatures as the number of sinusoids increases and is limited to

estimating only one sinusoid. Combining this algorithm assuming just one sinusoid, with

the RELAX algorithm increases the suppression performance with little signal distortion.

81

CHAPTER 4DATA-ADAPTIVE, SPARSE SUPER-RESOLUTION IMAGING FOR THE SIRE FLGPR

RADAR

4.1 Chapter Summary

In the previous chapter, the problem of RFI suppression was presented for the

Forward Looking Ground Penetrating Radar (FLGPR) known as the SIRE radar built by

the Army Research Laboratory. FLGPR has multiple applications, one of which includes

its use for detecting landmines and other buried improvised explosive devices (IEDs). In

this chapter, we focus on data-adaptive high resolution imaging for this SIRE FLGPR.

The standard method for generating SAR images for this radar is the back-projection

algorithm, which is limited by poor resolution and high side-lobes problems. In this

chapter, we consider using the Sparse Iterative Covariance-based Estimation (SPICE)

and the Sparse Learning via Iterative Minimization (SLIM) algorithms for generating

sparse high-resolution images for FLGPR. The pre-processing involves an orthogonal

projection of the received measurements to a subspace related to the region of

interest for data and clutter reduction. These user-parameter free algorithms are

capable of providing sparse results as well as improved resolution synthetic aperture

radar (SAR) images. We also examine the well-known CLEAN approach based on

a signal model in the time domain for imaging. We show using simulated data that

the SPICE/SLIM algorithms provide higher resolution than CLEAN and standard

backprojection algorithm. Imaging using real data collected via the Synchronous

Impulse Reconstruction (SIRE) radar, a multiple-input multiple-output (MIMO) FLGPR

radar developed by the Army Research Laboratory (ARL)) is also used for analysis.

4.2 Introduction

The global problem of landmines and other buried improvised explosive devices

(IEDs) is affecting both military and civilians alike [70–74], and effective as well as

efficient methods for detecting these devices is very important in the world today.

Methods for detecting landmines include but are not limited to the use of metal

82

detectors, infrared sensing of the land surface [75], biological sensors such as animals

(dogs) [76] and more recently detecting fumes of the mines using lasers to ionize the air

[74]. Radar is an excellent tool for remote sensing applications [77]. Ground penetrating

radar (GPR) which transmits an electromagnetic wave into the ground and examines

the back-scattered returns to determine buried objects has become a useful tool for

effectively detecting landmines and IEDs [78–80].

By operating in forward looking mode, GPR can be applied to the problem of

landmine detection as it inspects the ground surface with a safe stand-off range as

can be seen in Fig 4-1. Impulse based forward looking ground penetrating radar

(FLGPR) typically transmits a mono-cycle pulse with typical operating frequency

range spanning the UHF, and L bands [54, 79]. The low frequency of GPR provides

the necessary ground penetrating properties of the radar and the large bandwidth

provides the necessary down-range resolution. The cross-range resolution on the other

hand is limited by the antenna beamwidth [81]. Increasing the antenna physical size

can improve cross-range resolution. However, this is limited by physical antenna size

constraints. Side-looking synthetic aperture radar techniques improve cross-range

resolution by synthesizing a virtual aperture much larger than the physical aperture

[11, 82–84]. However, in forward looking mode, the cross-range resolution is limited

by the physical radar size. A multi-input multi-output (MIMO) radar [85] can be used to

enhance this resolution [86][87]. For example, Fig. 4-1 shows the FLGPR for landmine

detection built by the Army Research Laboratory (ARL) known as the synchronous

impulse reconstruction (SIRE) radar [54]. This radar consists of 2 transmitters and 16

receivers and exploits waveform diversity [54, 86, 88] to enhance cross-range resolution

by alternatively transmitting between its two transmitters.

The well-known conventional method for imaging for this type of radar is the

standard back-projection method [89, 90]. This approach also known as the delay-and-sum

(DAS) approach, suffers from high sidelobe problems and is limited by poor resolution.

83

Figure 4-1. Forward looking ground penetrating radar [54]

High resolution imaging is important for separating closely spaced targets as well as

distinguishing targets from clutter and such imaging techniques should be investigated.

In this chapter, we focus on sparse high-resolution imaging for impulse based

FLGPR. A signal model in the time domain is established since the transmitted impulse

is well localized in time. Based on this model, the well-known CLEAN [91], [92] approach

is analyzed for imaging compared to the standard backprojection algorithm. This

technique eliminates side-lobes, but it provides no improvement in imaging resolution

over the standard backprojection algorithm.

Two recently proposed, user-parameter free and data-adaptive methods are

considered here for imaging. The Sparse Learning via Iterative Minimization (SLIM)

[29] and the SParse Iterative Covariance-based Estimation (SPICE) [93] methods are

capable of providing sparse, as well as high resolution imaging results. These methods

are applied to the data of significantly lower dimension for impulsed based FLGPR

to achieve sparse high resolution imaging results. The data-reduction is achieved via

a pre-processing technique, which involves an orthogonal projection of the received

data to the subspace spanned by the dominant singular vectors of a steering matrix

corresponding to the imaging ROI. An efficient decomposition of the steering matrix is

performed using an eigenvalue decomposition of a matrix of much reduced dimension.

84

A conjugate gradient SPICE (CG-SPICE) algorithm is also introduced in this paper

similar to the CG-SLIM algorithm [29] to speed up the computation of SPICE.

The improvement in imaging results over the standard backprojection method

are shown via simulated results and also real measured experimental data. For real

experimental data, the SIRE radar developed by ARL [54, 94] is used for analysis. This

radar was also presented as a MIMO radar in [88].

The remaining sections of this chapter are organized as follows. In Section 4.3, a

proposed data model is presented for forward looking GPR based on the ARL’s SIRE

radar. This MIMO radar is also briefly described therein. In Section 4.4, the standard

back-projection method for imaging is analyzed as well ARL’s Recursive Sidelobe

Minimization (RSM) algorithm which is based on the BP algorithm and iteratively and

effectively suppresses sidelobes [54, 95]. Based on the proposed model we also show

that the CLEAN approach can be used for sparse imaging for impulse based FLGPR.

In Section 4.5 we present the sparse and adaptive methods for improved resolution as

well as the pre-processing step of orthogonal projection for clutter and data reduction.

Section 4.6 contains the numerical results based on simulated and real data. Finally the

conclusions are drawn in Section 4.7.

4.3 Data Model: SIRE Impulse Based FLGPR

For impulse based FLGPR, we consider the SIRE radar which is designed by ARL

and mounted on an SUV for landmine detection [54]. The radar geometry as seen in

Fig. 4-1 consists of two transmitters and sixteen receivers. Each transmitter transmits

an impulse with a frequency range of 0.3-3.0 GHz, which determines the downrange

resolution. The cross range resolution is determined by the physical 2 m aperture of

this radar. This radar can be described as a practical example of a MIMO radar which

exploits waveform diversity by transmitting orthogonal waveforms from the two transmit

antennas located at the edges of the receive array [88]. These orthogonal waveforms

are achieved by alternatively transmitting narrow pulses (in ”ping-pong” mode [96]) from

85

Figure 4-2. SIRE FLGPR: 2D aperture for SAR imaging

each transmitter. This creates a virtual aperture which is effectively almost double the

physical 2 m aperture of the radar hence improving cross-range resolution [97].

The 2D-aperture of received measurements shown in Fig 4-2 is used for image

formation where there are k = 1 · · ·K receive measurements for a desired imaging area

consisting of i = 1 · · ·L targets (pixels). Let rk(t) denote the kth receive measurement.

This measurement can described by the following equation:

rk(t) =L+M∑i=1

αk,izis(t− τk,i) + nk(t) (4–1)

where αk,i = 1Rt(k,i)

1Rr(k,i)

is the propagation path-loss, zi is the reflectivity of the ith

target, τk,i = Rt(k,i)+Rr(k,i)c

is the trip time delay from transmitter to target to receiver,

where Rt(k, i), Rt(k, i) are the distances of transmitter to target and the target to

receiver, respectively. The speed of propagation is given by c and nk(t) is thermal noise

associated with kth measurement. Note that the model of the received measurement

in (4–1) takes into account the contribution of the M scatterers outside the ROI, i.e.,

86

desired imaging area which consists of L pixels. The model in (4–1) can be simplified

into the following linear equation:

(4–2)

y = Cz+ n

=

A B

βγ

+ n

where y = [r1(0), . . . , r1(T − 1), . . . , rK(0), . . . , rK(T − 1)]T , which is a vector of received

measurements stacked together; β = {βi}Li=1 are the pixel values in the desired imaging

grid to be estimated, γ = {γi}Mi=1 corresponds to the pixel values outside the ROI and n

is the noise vector. The matrix A of dimensions (TK)×L consists of delayed and scaled

versions of the transmitted signal, given by:

A =

α1,1s(τ1,1) · · · α1,Is(τ1,I)

... . . . ...

αK,1s(τK,1) · · · αK,Is(τK,I)

(4–3)

where the vector s(τk,i) = {s(t − τk,i)}T−1t=0 is the transmitted impulse delayed by τk,i.

This data model is used for FLGPR SAR imaging in this paper. In the next section

the backprojection algorithm as well as the CLEAN algorithm are described for SAR

imaging.

4.4 Back-projection/Delay-and-sum (DAS) Based Methods

The standard backprojection (BP) algorithm is a well-known and widely used

algorithm for FLGPR SAR imaging (also known as the delay-and-sum (DAS) algorithm).

This algorithm is limited in downrange resolution by the bandwidth of the transmitted

impulse and in cross-range resolution by the physical (or virtual) aperture of the radar.

One other limitation of this algorithm is that it produces images with high sidelobes.

A Recursive Sidelobe Minimization (RSM) algorithm based on the BP algorithm was

proposed in [54], [95] for effectively suppressing sidelobes. The CLEAN approach

[91] (which is also based on this BP/DAS algorithm) can also be used for eliminating

87

sidelobes [92] as well as accurately estimating weak targets by iteratively subtracting

the contributions of stronger targets from the received data based on the proposed data

model. In this section, we describe these algorithms as well as analyze the CLEAN

approach based on the proposed data model for impulse based FLGPR SAR imaging.

4.4.1 Back-projection/DAS

Based on Fig 4-2, the backprojection algorithm is described as follows: Consider

the ith pixel in this figure with location (xi, yi, zi) relative to a predefined reference point

or origin. For a specific transmit-receive pair, let (xr,k, yrk , zrk) and (xt,k, ytk , ztk) denote

the transmitter and receiver locations in this coordinate system for k = 1, . . . , K. The

delay due to the transmitted EM pulse from the transmitter corresponding to the kth

receive measurement to the ith pixel back to the corresponding receiver is given as:

(4–4)τk,i = τacq +

1

c(√(xt,k − xi)2 + (yt,k − yi)2 + (zt,k − zi)2

+√(xr,k − xi)2 + (yt,k − yi)2 + (zt,k − zi)2)

where τacq is the acquisition time delay associated with the radar system. The estimate

of the reflection coefficient at the ith pixel given by βi is given as:

βi =1

K

K∑k=1

wkrk(t− τk,i) (4–5)

This estimate is simply a summation of delayed receive measurements with the

propagation loss compensated by a weighting factor wk. This is referred to as coherent

processing of the received measurements which improves SNR by a factor K compared

to using a single received measurement.

The backprojection/DAS algorithm is limited in resolution and suffers from poor

sidelobes. The recursive sidelobe minimization (RSM) algorithm proposed by ARL

for sidelobe suppression effectively suppresses sidelobes by generating multiple DAS

images with apertures with randomly missing measurements and selecting the minimum

value across all images [54]. In the next subsection we analyze a CLEAN approach

88

based on the data model in (4–1) for sparse imaging. These DAS based algorithms

do not improve imaging resolution compared to the standard DAS algorithm. We then

present new approaches to imaging using the SLIM and SPICE algorithms for improved

resolution in imaging after preprocessing via an orthogonal projection of the data.

4.4.2 Sparse: CLEAN Method

The data-dependent CLEAN algorithm [91] (also known as matching pursuit) for

image formation based on the data model in (4–1) is briefly described and analyzed

here for imaging. This technique was introduced to produce ’CLEANer’ images

(where prior knowledge of the point spread function was required) [92]. The standard

DAS algorithm suffers high sidelobe problems. CLEAN can be used to eliminate

the side-lobes of strong returns so that weak targets can be revealed by eliminating

contributions of strong targets from the receive measurements.

The CLEAN algorithm can therefore be used iteratively to find the pixel location

of the strongest target and then subtract all the contributions of that target from the

data. The next strongest point is then computed more accurately based on the updated

measurements. The RELAX algorithm [31] can therefore be used to get even more

accurate estimates as well as improved imaging resolution. The RELAX approach

involves estimating the strongest target, subtracting the contributions of this target and

estimating the next strongest target. The initial pixel value of the strongest target is

then re-estimated based on the new estimate of the next strongest target. These two

estimates are then iterated back and forth to achieve more accurate estimates. The

process is then repeated for all the targets in the scene of interest.

Due to the exponential increase in computation as the number of targets increases,

this process proves to be too computationally intensive for practical purposes when

there are many scatterers in the ROI.

The much faster CLEAN approach is described in the following steps based on the

data model in (4–1).

89

• Step 1: Determine the brightest estimate P(io) and corresponding location io fromthe backprojection image (based on 4–5).

• Step 2: Subtract the contribution of brightest point from the received signals(update receive measurements).

– rk up = rk − λ(io)s(τk,i0) k = 1 · · ·K

– λ(io) =P(io)K

× 1αk,io

• Step 3: Generate a new image by filling in the ioth pixel with P(i0).

• Step 4: Use updated received signals to regenerate back-projection image.

• Iteration: Repeat previous steps with regenerated image until reaching apredefined threshold. η > P(io)

σ2.

– η threshold (typically chosen as 1).

– σ2 Noise variance.

This CLEAN approach, although effective for accurate and sparse imaging, is

limited in resolution, which is similar to the standard backprojection/DAS algorithm. We

therefore, investigate super-resolution methods for FLGPR SAR imaging.

4.5 Super-resolution Methods

In this section, a new approach for high-resolution imaging is presented for impulse

based FLGPR. The 2D aperture in Fig. 4-2 for imaging of the specified grid will result

in a data vector in (4–1) y ∈ RTK×1 that is large (on the order of 106), making practical

applications of adaptive techniques infeasible. Also the availability of a single data

vector makes well known high resolution data adaptive approaches, such as the

CAPON, APES, as well as subspace based methods [2], not directly applicable. Another

challenge is that the data vector will contain clutter reflections from regions outside the

imaging ROI, which need to be effectively suppressed prior to imaging.

We propose a data filtering and reduction approach via time gating and orthogonal

projection to reduce interference from scatterers outside the ROI. This approach

involves a singular value decomposition of the steering matrix and a projection of the

90

Figure 4-3. Time gating

data using the dominant singular vectors, which effectively reduces the data dimension

to practical levels. This projection also filters the data to effectively utilize received

energy from only the ROI. A computationally efficient method of this projection is

performed via an eigenvalue decomposition to obtain an updated data model.

Using the updated model, two recently proposed algorithms (SPICE and SLIM) are

used for imaging to produce sparse, accurate and high resolution imaging results, even

with a single data vector. The process is described in the next subsections.

4.5.1 Orthogonal Projection and Time Gating

For time gating, consider the kth receive measurement {rk(t)}T−1t=0 . Based on

(4–4), the delays of the pixels in the ROI imaging grid, τ k = [τk,1, τk,2, . . . , τk,I ], to

the corresponding transmit-receive pair for this measurement can be computed. The

minimum and maximum delays between this transmit-receive pair and the pixels in the

ROI imaging grid are given by τkmin = min(τ k) and τkmax = max(τ k), respectively. The kth

receive measurement can then be updated to {rk(t)}τkmaxt=τkmin

by discarding data outside

these computed delays.

Without loss of generality, this procedure is shown in Fig. 4-3 for a colocated

transmit-receive pair centered below (with a pre-specified standoff-range) the imaging

ROI grid. Interference from the regions below the minimum delay line and above the

maximum delay line are discarded. The data vector in (4–1) can then be updated to

91

yg = [r1(τ1min), . . . , r1(τ1max), . . . , rK(τKmin), . . . , rK(τKmax)]T , and the updated data model is:

yg = Agβ + Bgγ + ng. (4–6)

with Ag ∈ RQ×L and Bg ∈ RQ×M (Q < TK).

Note from Fig. 4-3, interference from scatterers outside the ROI still contribute

to the receive measurements. In this paper, an orthogonal projection of the the data

to relevant singular vectors of the data matrix is performed to reduce the effects of

the scatterers outside this grid as well as to reduce the data dimension significantly

(by a factor of 103 in practical applications), allowing for practical applications of high

resolution imaging techniques. This interference reduction via orthogonal projection

is performed in a computationally efficient manner via an eigen-decomposition and is

motivated and described as follows.

Consider the following penalized least squares optimization problem used for

square-root LASSO [98] (for notational simplicity, we eliminate the subscript g from

(4–6)).

argminβ,γ

||y − Aβ − Bγ||2+λ||β||1 (4–7)

Note here that the sparsity promoting ℓ1 constraint is placed on β (the desired

estimates), whereas no constraint is placed of γ due to the fact that even if γ is sparse,

the subsequent transformation eliminates this sparsity. Let Bγ = UB�BVBTγ , UB~γ

via a singular value decomposition of B. The optimization problem in (4–7) for ~γ

(which is no longer sparse) given β is unconstrained and the solution is given as

~γ = UBT (y − Aβ). Given ~γ, the constrained optimization problem in (4–7) is now given

as follows:

β = argminβ

||(I−UBUBT )(y − Aβ)||2+λ||β||1 (4–8)

Since B and hence UB is unknown, we assume that UB and UA, where UA are the

singular vectors of A are orthogonally compliment to each other. Then Eq. (4–8)

92

becomes:(4–9a)β = argmin

β||UAUA

T (y − Aβ)||2 + λ||β||1

(4–9b)β = argminβ

||UAT (y − Aβ)||2 + λ||β||1

Consider the following singular value decomposition of the steering matrix A ∈

RQ×L, with Q > L in practice1 :

(4–10)

A = U�VT

=

UA~UA

�A

0

[V] T

= UA�AVT

where the columns of [UA, ~UA] are the left singular vectors of A and the columns of

V are their right counterparts, and the singular values of A are on the diagonal of the

diagonal matrix �A.

Due to the find grid used for the ROI, some of the singular values of A in �A

are quite small. By discarding the small singular values of A, we approximate A as

A ≈ Us�sVsT . Then the optimization problem in Eq. (4–9b) becomes:

β = argminβ

||UsT (y − Aβ)||2+λ||β||1 (4–11)

Then using Us for orthogonal projection yields:

(4–12a)UsTy = Us

TAβ + ϵ,

(4–12b)~y = ~Aβ + ϵ

Via a series of simulations, we found that the number of columns in Us is not sensitive

to the image formation performance. We choose the dimension of Us by analyzing the

1 Note that since the radar illuminates a large area M > Q, but we do not consider Mas the contributions of the clutter (Bγ) are eliminated based on the decomposition

93

metric ||A−Us�sVsT ||F/||A||F . Note here that ~y ∈ Rs×1 and ~A ∈ Rs×1 where s << Q (a

factor of 103 reduction in practical applications).

The projection described in (4–12b) can be obtained in a computationally efficient

way via an eigenvalue decomposition in lieu of an SVD. Note that in (4–10) UA ∈

RQ×L, which makes the SVD decomposition in this equation computationally intensive

(O(Q2L)) in practical scenarios.

Consider the following eigenvalue decomposition2

ATA = V�VT (4–13)

where V ∈ RL×L and � = �2A. Then UA = AV�−1

A , where the diagonal matrix �−1A can be

written as:

�−1A =

�−1s 0

0 �−1n

(4–14)

with the diagonal of the sub-matrix �−1s consisting of the inverse of the dominant

singular values of A. Then Us = AVs�−1s , with Vs corresponding to the s dominant

eigenvectors of ATA.

The updated data vector in (4–12b) is given as:

~y = UsTy = (�−1

s )VsTATy (4–15)

whereas the updated steering matrix is given as:

(4–16)

~A = UTs A

= �−1s Vs

TATA

= �−1s Vs

TV�VT

= �sVsT

2 Complexity of eigenvalue decomposition - O(L3)

94

The updated data model in (4–12b), obtained using Us can be obtained using the

decomposition in (4–13). This decomposition can be performed offline as long as prior

knowledge of the imaging grid is known, which is typically the case.

A different approach for orthogonal projection that improves on computation is

described next. Consider the steering matrix G which corresponds to the imaging

ROI with a much coarser grid. The matrix G can then be generated by selecting

the appropriate columns of A. The matrix G is used to approximate Us. However,

unlike Us this matrix is not semi-unitary. The updated data vector is then given as

~y = (GTG)−1GTy and the updated steering matrix is given as ~A = (GTG)−1GTA. This

new projection approach, skips the computation ATA and its subsequent decomposition,

significantly improving computation at cost of less interference suppression..

Based on the updated model in (4–12b), we present below two recently proposed,

data-adaptive and iterative approaches for high resolution FLGPR SAR imaging. They

are the Sparse Learning via Iterative Minimization (SLIM) [29] and the SParse Iterative

Covariance-based Estimation (SPICE) [93], which is equivalent to square-root LASSO

with λ = 1 [99]. These two approaches are user-parameter free and are capable

of producing sparse and high resolution estimates when only a single data vector is

available for imaging; they are described next.

4.5.2 SLIM

The SLIM method [29] is a maximum-aposteriori (MAP) approach for sparse

signal recovery. The sparse recovery problem is can be solved by optimizing the ℓ1

optimization cost function in (4–17) based on the linear model in (4–12):

β = argminβ

||~y − ~Aβ||2+λ||β||1 (4–17)

The SLIM algorithm can be considered as an ℓq norm approach norm for 0 < q ≤ 1, that

considers the following hierarchial Bayesian model (with a sparsity promoting prior) for

95

Table 4-1. SLIM Algorithm

Initialization: Obtain initial estimate β(0) with the DAS algorithm and σ(0)

based on β(0) and (4–20)

SLIM (nth) Iteration: Repeat the following steps until convergence

Step 1: Compute: R(n+1) = ~Adiag(|β(n)|2−q)~AT + σ(n)I

= ~AP(n)~AT + σ(n)I.

Step 2: Compute: β(n) = P(n)~AT R−1

(n)~y.

Step 3: Update: σ(n) = 1L||~y − ~Aβ(n)||22.

estimation [29].(4–18a)~y|β, σ ∼ CN (~Aβ, σI)

(4–18b)f(β) ∝∏i

e−2

q(|βi|q−1)

(4–18c)f(σ) ∝ 1

SLIM estimates the desired sparse vector β, and the noise variance σ, iteratively by

minimizing the negative logarithm cost of the posterior density given by:

cq(β, σ) = L logσ +1

σ||~y − ~Aβ||22+

∑i

2

q(|βi|q−1) (4–19)

The choice of q = 1 simplifies this cost function to the well-known ℓ1 norm constraint

for sparse estimation [29]. Minimizing the cost function in (4–19) yields the following

estimates:(4–20a)β = PAT (~AP~AT + σI)−1~y

= P~ATR−1~y

(4–20b)σ =1

L||~y − ~Aβ||22

where P = diag(p) and p = |β|2−q. These estimates are obtained in an iterative manner

based on the steps in Tab. 4-1.

4.5.3 SPICE

The SPICE method for parameter estimation [30, 93] modifies the model in (4–12)

as follows:(4–21)~y = ~Aβ + ϵ

= Dx

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Table 4-2. CG SPICE Algorithm

Initialization: Obtain initial estimate xj(0) = dTj ~y/dTj dj, and pj(0) = |xj(0)|/ωj

j = 1, . . . , L+ s.

SPICE (nth) Iteration: Repeat the following steps until convergence

Step 1: Compute R(n) = Ddiag(p(n))DT = DP(n)DT .

Step 2: Compute (using CG): s(n) = R−1(n)~y.

CG Initialization: s(n)(0) = 0; r(0) = q(0) = ~y

CG Iterations (mth):

- α(m) = (rT(m)r(m))/(qT(m)q(m))

- s(n)(m+1) = s(n)(m) + α(m)q(m)

- r(m+1) = r(m) − α(m)R(n)q(m)

- q(m+1) = q(m) + (rT(m+1)r(m+1))/(rT(m)r(m))

Step 3: Update x(n+1) = P(n)DTR−1

(n)~y. j = 1, . . . , L+ s.

Step 4: Update pj(n+1) = xj(n+1)/ωj j = 1, . . . , L+ s.

where D = [~A, I] and x = [βT , ϵT ]T . This method is a covariance fitting approach to

parameter estimation that minimizes the following covariance fitting cost function:

||R1/2(~y~yT − R)||F (4–22)

where R = E{~y~yT} = DPDT is the covariance matrix of the data and the diagonal matrix

P is now given as:

P =

Ps 0

0 Pϵ

(4–23)

with Ps = diag(ps) and Pϵ = diag(pϵ) being the diagonal matrices containing the power

estimates of β and the noise and interference residue ϵ, respectively.

The criterion in (4–22) simplifies to the following minimization problem to estimate

both x and p = [psT ,pTϵ ]

T .

{x, p} = argminx,p

xTP−1x+

L+s∑j=1

ωjpj s.t. Dx = ~y (4–24)

where ωj = ||dj||/||~y|| and ||dj|| is the jth column of D. The estimates are given as:

97

(4–25a)x = PDTR−1~y

(4–25b)pj =|βj|ωj

, j = 1, . . . , L+ s

which are solved iteratively till convergence [30]; with x = {xj}L+sj=1 and p = {pj}L+s

j=1 .

To improve on the computationally efficiency of the SPICE algorithm for FLGPR

SAR imaging, a conjugate gradient based SPICE algorithm (CG-SPICE) is presented in

this paper. This approach is similar to the conjugate gradient SLIM algorithm described

in [29], [100]. The steps of this CG-SPICE algorithm are described in Tab 4-2.

Based on [101], the SPICE optimization problem can be re-written as:

argminx

||x||1=L+s∑j=1

|xj| s.t. Dx = ~y (4–26)

For real-valued data, let ~hj , max(xj, 0) and �hj , −min(xj, 0). Note that xj = ~hj − �hj

and |xj|= ~hj + �hj. The optimization problem in Eq. (4–26) can then be augmented to

[102]:

argmin^h

uTh s.t. �Dh = ~y and h ≥ 0 (4–27)

where h = [~h1, . . . ~hL+s, �h1 . . . �hL+s], �D = [D,−D] and u = [1, 1 . . . , 1]T . The linear

program in (4–27) can be solved efficiently to provide sparse estimates for FLGPR

SAR imaging. This approach is on the same order (computationally) as the cyclic

optimization approach in Tab 4-2 implemented using the conjugate gradient, and faster

(approximately 2 times) without CG based on numerical simulations.

Note that the estimates in SLIM and SPICE have the same form with the difference

lying in the estimation of the noise and interference residue. SPICE estimates the

reflection coefficients and the noise and interference residue simultaneously with the

noise and interference residue variance of each elements not necessarily being equal

unlike the case in SLIM. However, simulation results show similar performance of the

two algorithms with SPICE being less susceptible to the noise and interference residue.

This algorithms are robust and effective for generating sparse results.

98

Table 4-3. Subspace approximation

No. of singular vectors - s ||A−Us�sVsT ||F/||A||F

15 0.902

100 0.512

300 0.182

Unlike most well-known adaptive algorithms, SLIM can dynamically estimate the

user parameter (which in this case, corresponds to the noise power estimate) of the

original LASSO cost function (for sparse parameter estimation) [103] which is sensitive

to the choice of this parameter (λ). The SPICE criteria can also be reduced to the

criteria in (4–17) with λ = 1 [99] (a special case of the square-root lasso [98] which is

insensitive to the choice of the user-parameter λ).

These robust user parameter free algorithms are applied here to problem of

FLGPR SAR imaging. Analysis is performed on both simulated and real experimentally

measured SIRE FLGPR data for imaging, and the results are presented in the next

section.

4.6 Numerical and Experimental Results

In this section, we perform sparse high resolution imaging for FLGPR using

orthogonal projection (using Us) for clutter and data reduction. The SPICE and

SLIM algorithms are considered for high resolution imaging. We also analyze the

well-known CLEAN approach for imaging based on the proposed data model and show

the ability of this well-known algorithm to yield sparse and accurate results. The CLEAN

approach, however, is limited in imaging resolution and does not improve resolution

over the standard BP algorithm. The coarse grid approach using G is also analyzed for

projection.

For FLGPR SAR imaging analysis, we use the SIRE radar designed by ARL for

imaging based on the setup in Figs. 4-1 and 4-2. For simulations, we consider an

imaging area which has a range swath of 4 m and a cross-range swath of 5 m. A

99

0 200 400 600 800 10000

5

10

15

20

25

30

35

Eigenvalue number

Eig

enva

lues

EigenvaluesThreshold = 15Threshold = 100Threshold = 300

A

downrange (meters)

cros

s−ra

nge

(met

ers)

−2 −1 0 1 2

−2

−1

0

1

2

−40

−35

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−25

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−10

−5

0

B

downrange (meters)

cros

s−ra

nge

(met

ers)

−2 −1 0 1 2

−2

−1

0

1

2

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downrange (meters)

cros

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ers)

−2 −1 0 1 2

−2

−1

0

1

2

−40

−35

−30

−25

−20

−15

−10

−5

0

D

Figure 4-4. Subspace dimension (s) for high resolution imaging (A) Eigenvalues of ATA,(B) s = 15, (C) s = 100, and (D) s = 300

minimum standoff distance of 8 m is used for simulations, with a maximum standoff

distance of 14 m. Simulation was run with three targets placed at various [x, y, z]

locations in meters marked by the symbol ’X’. The imaging area consists of L = 10000

pixels.

An analysis of the subspace dimension, (i.e., the number of dominant singular

values of A) is performed first. Targets at locations [0,0,0], [-0.3, 1, 0], and [1.5, 1.5,

0] are simulated. Fig. 4-4 shows the SPICE algorithm applied with various thresholds.

From Tab. 4-3, thresholds as large 0.5 based on the criterion ||A−Us�sVsT ||F/||A||F

100

downrange (meters)

cros

s−ra

nge

(met

ers)

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−2

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0

1

2

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(met

ers)

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2

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cros

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ers)

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2

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downrange (meters)

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(met

ers)

−2 −1 0 1 2

−2

−1

0

1

2

−40

−30

−20

−10

0

E

Figure 4-5. FLGPR SAR Imaging - detection of weak target (A) Back-projection, (B)RSM, (C) CLEAN, (D) SLIM, and (E) SPICE

(normalized scale of 0 to 1) yield desirable results with all the targets detected as can be

seen in Fig. 4-4.

The next analysis involves detecting weak targets buried by the sidelobes of much

stronger targets. Three targets are again simulated at locations [0,0,0], [-0.1, 0.8, 0],

101

downrange (meters)

cros

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(met

ers)

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ers)

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E

Figure 4-6. FLGPR Imaging - resolution improvement (A) Back-projection, (B) RSM, (C)CLEAN, (D) SLIM, and (E) SPICE

and [1.5, 1.5, 0], with the strong targets 10 times stronger than the weak target as

shown in Fig. 4-5. From this figure we can see that the weak target is buried by the

sidelobes of the stronger target using the standard backprojection algorithm. The

CLEAN approach, which iteratively subtracts out the contributions of the strongest target

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2

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D

Figure 4-7. Orthogonal projection using (A) Us (High SNR), (B) G (High SNR), (C) Us

(Low SNR), and (D) G (Low SNR)

from the receive measurements, can effectively and accurately detect this weak target.

The SPICE and SLIM algorithms are also applied post orthogonal projection (with

threshold ||A−Us�sVsT ||F/||A||F= 0.2) and the weak target is revealed.

The CLEAN approach, although effective in providing sparse and accurate results,

is limited in resolution and has no improvement over the standard backprojection

algorithm. The high resolution imaging methods for FLGPR, provide improvement in

resolution over the backprojection-based algorithms (BP, RSM and CLEAN). Fig. 4-6

shows three targets at locations [0,0,0], [0, 0.75, 0], and [1.5, 1.5, 0]. Two of these

targets are ’closely’ spaced and are clearly resolved by SLIM and SPICE. The SPICE

and SLIM algorithms provide almost a factor of 2 improvement in imaging resolution.

103

downrange (meters)

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eter

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Figure 4-8. Real data - SIRE FLGPR SAR Imaging: (A) Back-projection, (B) RSM, and(C) SPICE

104

0 1 2 3 4 5 6 7 8 9 10 11 12 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Receiver Operating Characteristics (ROC)

Number of False Alarms

Pro

babi

lity

of D

etec

tion

(P

D)

BackprojectionRSMSPICE

Figure 4-9. Receiver Operating Curve (ROC) comparison: FLGPR SAR Imaging

This experiment is repeated using G for orthogonal projection. The results are

compared to using the semi-unitary matrix Us for projection and are shown in Fig. 4-9. A

gain in computation is achieved at a cost of less interference suppression.

The proposed approach is verified using real experimentally measured data.

Results based on real SIRE data provided by the Army research lab can be seen in

Fig. 4-8. In this figure a subimages 2m in range are continuously formed based on

overlapping 2D apertures to generate the entire image [54]. Based on this figure,

significant interference reduction can be seen by the high resolution SPICE compared to

the standard backprojection algorithm, with some of the targets of interest marked in red

oval circles.

A quantitative numerical analysis is performed to show the effectiveness of the

high-resolution SPICE approach for FLGPR SAR imaging. Several targets with varying

strengths are simulated and the receiver operating characteristics curve (ROC) is shown

in Fig. 4-9. This curve which shows the probability of detection versus the number

of false alarms is generated by using a simple threshold detector. The image under

analysis is segmented into regions, and the maximum pixel value in each region is

retained. The threshold is incremented in steps and for each threshold, the number

105

of alarms are recorded. The alarms that fall outside the regions with targets present

(based on prior knowledge) are considered false alarms. As shown in Fig. 4-9, when

there is full detection, the SPICE approach has less false alarms the the BP algorithms.

4.7 Conclusions

In this chapter, we have considered new approaches to imaging for forward looking

ground penetrating radar. The pre-processing involves a proposition of a data model

in the time domain, which takes into account the contributions of clutter outside the

imaging area. An orthogonal projection of the measured data to a subspace spanned by

the steering matrix corresponding to the imaging ROI is then used for clutter reduction

as well as significant data reduction, making it feasible for practical applications of high

resolution methods. The steering matrix decomposition is performed efficiently and

depends only on the prior knowledge of the desired imaging area and hence can be

performed offline. Two recently proposed, data-adaptive approaches, SPICE and SLIM

are used for FLGPR SAR imaging. They are user parameter free algorithms and have

the ability to provide sparse and high resolution images using a single data vector, unlike

other well-known high resolution methods. The results using simulated data show that

SLIM and SPICE provide improvement in resolution close to a factor of two compared

to the backprojection based algorithms including BP, RSM, CLEAN. A new conjugate

gradient based SPICE algorithm is also introduced in this paper for more efficient

computations of the estimates.

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CHAPTER 5CONCLUDING REMARKS AND FUTURE WORK

Due to limitations of data independent methods for spectral estimation, data-adaptive

methods are currently being investigated for improved performance. In this dissertation,

we focused on efficient and effective applications of data-adaptive methods to real world

sensing problems.

In Chapter 2, the basic problem of harmonic retrieval is investigated. The problem

pertains to digital audio forensics. The contribution we make to this problem involves

coming up with a more reliable and accurate way of estimating the network frequency

buried in an audio recording using data adaptive techniques. The proposed approach

involves spectral analysis using a robust high resolution algorithm and tracking the

network frequency via a dynamic programming approach. The approach yields

significant improvement in the estimation of the embedded network frequency when

this signal is weak compared to the audio recording (a major challenge for this problem).

Chapters 3 and 4 are the focus of this dissertation. In these chapters, the

Synchronous Impulse Reconstruction (SIRE) radar, which is a remote sensing tool for

landmine detection is analyzed and studied. In Chapter 3, we propose a new approach

of Radio Frequency Interference suppression for this radar. This new approach can

provide an improvement of close to 7 dB in RFI suppression without distorting the

desired target signatures. This approach is implemented in an efficient way by exploiting

the equivalent sampling technique of this radar.

Chapter 4 focuses on sparse high resolution imaging for this SIRE Forward Looking

Ground Penetrating radar (FLGPR). In this chapter, we establish a signal model in the

time domain since the transmitted impulse is well localized in time. This data model

takes into account the contributions of clutter outside the imaging region of interest

(ROI). We propose a pre-processing step of orthogonal projection to mitigate the effects

of clutter outside the ROI which is present in the collected data. Recently proposed

107

robust, sparse high resolution algorithms for imaging are then applied to provide

improved imaging resolution. We achieve close to a factor of 2 improvement in imaging

resolution compared to the standard methods currently used for SAR imaging for this

radar.

Our current and future work focuses on more efficient ways to implement the

pre-processing step of orthogonal projection. In lieu of a decomposition of the steering

matrix corresponding to imaging ROI and then projection of the data, we propose a

direct projection of the data to the steering matrix corresponding to the ROI with a much

coarser grid to improve computation. This matrix approximates the set of orthogonal

vectors that spans the subspace of the matrix corresponding to the ROI. Results show

significant improvement in computation at cost of less interference suppression.

108

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BIOGRAPHICAL SKETCH

Ode Ojowu Jr. was born in Zaria, Nigeria. He came to the United States in 2001

to pursue an academic career. He received a Bachelor of Arts in physics from Grinnell

College in 2005, as well as a Bachelor of Science and Master of Science in electrical

engineering from Washington University in 2007. He is currently with the Spectral

Analysis Lab (SAL) supervised by Prof. Jian Li at the University of Florida. He will

receive a Doctor of Philosophy in electrical engineering from the University of Florida in

the Fall of 2013.

His general research interest lies in the field of signals and systems with a focus on

data-adaptive spectral estimation techniques, array signal processing and radar signal

processing.

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data-adaptive spectral estimation algorithms and...

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