DETECTION OF NARROW-BAND SONAR SIGNALS ON A RIEMANNIAN MANIFOLD

Transcription

1 DETECTION OF NARROW-BAND SONAR SIGNALS ON A RIEMANNIAN MANIFOLD

2 DETECTION OF NARROW-BAND SONAR SIGNALS ON A RIEMANNIAN MANIFOLD BY JIAPING LIANG, B.Eng a thesis submitted to the department of electrical & computer engineering and the school of graduate studies of mcmaster university in partial fulfilment of the requirements for the degree of Master of Applied Science c Copyright by Jiaping Liang, June 2015 All Rights Reserved

3 Master of Applied Science (2015) (Electrical & Computer Engineering) McMaster University Hamilton, Ontario, Canada TITLE: DETECTION OF NARROW-BAND SONAR SIGNALS ON A RIEMANNIAN MANIFOLD AUTHOR: Jiaping Liang M.Eng, B.Eng (Electrical Engineering) McMaster University, Hamilton, Canada SUPERVISOR: Dr. Kon Max Wong ii

4 Abstract We consider the problem of narrow-band signal detection in a passive sonar environment. The collected signals are passed to a fast Fourier Transform (FFT) delay-sum beamformer. In classical signal detection, the output of the FFT spectrum analyser in each frequency bin is the signal power spectrum which is used as the signal feature for detection. The observed signal power is compared to a locally estimated mean noise power and a log likelihood ratio test (LLRT) can then be established. In this thesis, we propose the use of the power spectral density (PSD) matrix of the spectrum analyser output as the feature for detection due to the additional cross-correlation information contained in such matrices. However, PSD matrices are structurally constrained and therefore form a manifold in the signal space. Thus, to find the distance between two matrices, the measurement must be carried out using Riemannian distance (RD) along the tangent of the manifold, instead of using the common Euclidean distance (ED). In this thesis, we develop methods for measuring the Fréchet mean of noise PSD matrices using the RD and weighted RD. Further, we develop an optimum weighting matrix for use in signal detection by RD so as to further enhance the detection performance. These concepts and properties are then used to develop a decision rule for the detection of narrow-band sonar signals using PSD matrices. The results yielded by the new detection method are very encouraging. iii

5 Acknowledgements Firstly, I like to express my deep gratitude to my supervisor Dr. Kon Max Wong. Without his tremendous help and coaching, my academic accomplishment would not have been possible. I will always remember and appreciate all his efforts and months of working overtime to help me succeed in my research. Dr. Wong exemplifies a great researcher and a dedicated professor who puts scientific research and students achievements as his first priorities. I am moved by his devotion and passion to research and education and his ceaseless hard work. Even right after a major eye surgery, he was back to his office two days later to meet with his students discussing research progress. He took up personal sacrifices to help me preparing my conference paper and my thesis, working till late at night, over the weekends, and even on the air plane. I will carry his encouragement and great inspiration with me and work hard through my future career. Making him proud of me is the least I can do to repay Dr. Wong. Secondly, I would express my gratitude to Dr. Jian-Kang Zhang for his insightful discussions. I would also like to thank Dr. Jun Chen who has provided me with valuable suggestions. I am deeply indebted to Dr. Kon Max Wong and Dr. Dongmei Zhao for the great lectures in my first year of graduate studies. Thirdly, I would like to thank all my group members in ITB A203 for their kindness support. The friendship we built has made my academic life a joyful and unforgettable iv

6 memory. I would also like to thank our graduate administrative assistant, Ms. Cheryl Gies. She has been very helpful and her hospitality has made my study here very enjoyable. Last but not least, I want to send a special Thank you! to my family for their unconditional love and support in every decision I made throughout these years. v

7 Contents Abstract iii Acknowledgements iv 1 Introduction Passive Sonar Signal Detection Contents and Contributions of the Thesis Notations and Abbreviation Conventional Sonar Signal Detection Signal and Noise Model Probability Density Function of Noise and Signal Plus Noise Probability Density Function of Noise Probability Density Function of Narrow-Band Gaussian Signals Signal Detection LRT Noise Estimate Using SWMA Simulation Results Noise generation vi

8 2.4.2 Signal generation Detector performance Numerical experiments Evaluation of theoretical performance Comparison of performance PSD Manifold The Power Spectral Density Matrix Riemannian Distance The Riemannian Distances d R The Riemannian Distance d R Weighted Riemannian Distance The Weighted Riemannian Distance d WR The Weighted Riemannian Distance d WR Fréchet Mean Fréchet Mean According to d R Fréchet Mean According to d R Weighted Fréchet Mean According to d WR Weighted Fréchet Mean According to d WR Optimum Weighting Optimum Weighting Matrix for d WR Signal Detection on Manifold LRT Evaluation of FM for RD and Weighted RD vii

9 4.1.2 Establishment of Detection Threshold Simulation Results Comparison of ROC results Variations of M Comparison on variations of window size L Conclusion Review of the Thesis Comment on the Results Ideas for Future Research Signal Detection Other applications viii

10 List of Figures 1.1 Block diagram of a passive sonar signal detection system Geometry of a linear towed array Noise PDF Signal PDF Probability of false alarm, probability of detection, and power threshold Theoretical PDF of SWMA estimated noise (no signal in window) Theoretical PDF of SWMA estimated noise with signals in window Beamformer power spectrum (noise only) Histogram (noise only) Beamformer power spectrum (signals only) Beamformer power spectrum (signals + noise) Signal histogram Receiver operating curves (SNR=5dB, Q=0.005) Noise and signal distributions at SNR=5dB for d R1, d R2 and d WR Detection regions on manifold ROC using PSD and power spectrum at 3db ROC using PSD and power spectrum at 3db zoomed ROC using PSD and power spectrum at 5db ix

11 4.6 ROC using PSD and power spectrum at 5db zoomed Receiver operating curves using 8000 samples with different M ROC for data divided into 4 and 8 segments ROC for data divided into 4 and 8 equal length segments ROC using variation of window size in d R ROC using variation of window size in d W R x

12 Chapter 1 Introduction 1.1 Passive Sonar Signal Detection Sonar systems are widely applicable in many fields, such as military surveillance, navigation, fishery, search and rescue, etc. Sonar systems can be divided into two categories - active and passive. An active sonar emits acoustic signals and collects echoes bounced back from targets, whereas a passive sonar system [3] collects information by listening to the acoustic signals emanated from the underwater targets. In this thesis, we concentrate on the signal detection in a passive sonar system. Fig. 1.1 shows a passive sonar system consisting of an array of hydrophones which are used to collect signals to perform the functions of detection, localization, parameter estimation, classification, etc., of the targets. A common arrangement of sensors in a passive sonar system is in the form of a towed line of uniformly spaced hydrophones. Acoustic signals from different directions to the normal of the array are received at each sensor. The received signal is usually passed onto a Fourier analyzer to determine its frequency contents, followed by a frequency-domain beamformer for determining 1

13 its directional features. The beam output may undergo further processing before signal detection and other processing is performed [4]. Figure 1.1: Block diagram of a passive sonar signal detection system The signals of a target vessel originate from the rotation of the propulsion system, vibration of the propeller, as well as from the rotation and vibration of auxiliary machinery in the vessel. The signals are usually considered as random narrow-band Gaussian [4] because they are sets of harmonics propagated under multi-path environment. On the other hand, acoustic noise is generated by wind, waves, currents, ocean creatures, and even distant shipping. In our consideration, the noise will be modeled as an ergodic zero mean Gaussian process with a flat power spectrum over the total system bandwidth [3]. In this thesis, the problem of detecting such sonar signals in noise is examined. The detection process tests each frequency bin of a beam of the beamformer output and determines the presence of a signal. We first review the classical method of detection, which uses the power of the frequency bin as the feature, and examine its efficiency. We then consider the use of power spectral density (PSD) matrices 2

14 of the beamformer output as the feature for detection. Since the PSD matrices are structurally constrained and are not free elements in the signal space, they describe a manifold. Therefore, distances between these matrices must be carried out along the tangent of the manifold, i.e., the Riemannian distance (RD) should be used. This is akin to the measurements of the distance between two cities on earth. The Euclidean distance (ED) is neither informative nor accurate. In addition, the mean of these matrices should be evaluated in terms of RD. This reasoning leads to the derivation the Fréchet mean of PSD matrices. Furthermore, to separate signal from noise more distinctly, the Riemannian distance can be optimally weighted. This leads to a weighted Riemannian distance for better detection. Using these new concepts and new derivations, and following similar strategy as the Neyman-Pearson test, a decision rule is established. Simulations applying this novel detection method show very attractive results. The objective of this thesis is to introduce and examine the use of the PSD matrix as the feature for narrow-band signal detection. Our aim is to compare the performance of this approach with that of the classical approach which uses the signal power as detection feature, and examine the use of RD in contrast to the use of ED in signal detection. It is not our goal to compare different detection methods under different signal/noise models, nor signal detection under different methods of noise estimation. Therefore, the choice of the signal detection model is kept to the simple consideration of narrow-band Gaussian signal in wide-band Gaussian noise, and the noise estimation is also kept to the simple moving average method. More complicated signal/noise models and more sophisticated methods of noise estimation can be included in further investigations. 3

15 1.2 Contents and Contributions of the Thesis There are 5 chapters in this thesis. After the introductory chapter, chapter 2 briefly describes the passive sonar signal processing system and establishes signal and noise models. It also reviews the classical signal detection method. Chapter 3 proposes the use of PSD matrix as the signal feature for detection. The concept of PSD matrix manifold is introduced and the RD is established as the measurements of the distances on the manifold. Several of the important parameters concerning the PSD manifold are developed, including the Fréchet mean for RD d R1 and d R2, and for weighted RD d W R2, as well as the optimum weighting matrix facilitating detection. Chapter 4 applies the concepts and parameters established in chapter 3 to set up the hypothesis testing rule for narrow-band sonar signal detection. Simulations are carried out to verify the performance using PSD matrix as signal feature for detection. Chapter 5 concludes the thesis and suggests topics for future research. The contributions of the thesis can be listed as follows: 1. Develop algorithm for finding the Fréchet mean for RD d R1 2. Develop closed-form expression for finding the Fréchet mean for RD d R2 3. Develop the weighted Fréchet mean for RD d W R2 4. Obtain the optimum weighting matrix for d W R2 for the purpose of detection 5. Establish the hypothesis testing rule for detection on the Riemannian Manifold 6. Perform computer simulations to compare detection performance using different features and different distance measures. 4

16 1.3 Notations and Abbreviation Mathematical Notations The imaginary unit is denoted by j = 1. Vectors and matrices are represented by lower and upper cases of bold letters respectively, e.g., a and A. tr( ) and ( ) H denote respectively the trace and the Hermitian conjugate of a square matrix., denotes the inner product of vectors or matrices. Vector spaces and subspaces, as well as manifolds are denoted by upper case script letters, in particular, a Euclidean space is denoted by H, a manifold by M, the tangent space to a point in H is denoted by T H ( ), and the tangent space to a point in M is denoted by T M ( ), etc. The time domain signal is represented by lower case and the Fourier transform is represented by the corresponding upper case. Abbreviations ED DFT FFT FM LLRT LRT PSD RD ROC SNR SWMA Euclidean Distance Discrete Fourier Transform Fast Fourier Transform Fréchet Mean Log Likelihood Ratio Test Likelihood Ratio Test Power Spectrum Density Riemannian Distance Receiver Operating Characteristics Signal-to-noise ratio Split Window Moving Average 5

17 Chapter 2 Conventional Sonar Signal Detection Assume that we use the passive sonar towed array of Fig. 2.1 to receive a plane wave signal s(t (k 0 r)/c) which propagates in a known direction with respect to the array represented by unit vector k 0 at a velocity c. The signal is received at a position r by a hydrophone in a background of spatially white noise [4]. The signals are then passed on to an FFT which evaluates the spectral contents in a set of frequency bins using Fourier analysis. The FFT of the received signals at the different hydrophones constitute the output of a beamformer. Based on the spectral contents in each frequency bin, the presence of signal in a fixed direction is detected by comparing to a threshold which is to be set. In general, a target signal is embedded in ambient acoustic noise which will affect the threshold setting. Thus an estimate of the background noise power must be obtained. 6

18 Figure 2.1: Geometry of a linear towed array 2.1 Signal and Noise Model Suppose the sonar array has P hydrophones. At the pth hydrophone sensor, the output discrete-time signal at nt is denoted by x p (nt ) where T is the sampling period. Suppose we collect a finite record of this signal, n = 0,, N. We also assume that the ambient noise is spatially white, thus, at any instant of time, the noise is uncorrelated from sensor to sensor. However, the signal has the same value along each wavefront and hence the signal received by the sensors are correlated. A reasonable means of processing the array data for detection is thus to combine the outputs of the sensors so that the signal is enhanced relative to the noise. This is done by appropriately delaying and then summing the outputs of the sensors. Hence, if d is the separation between sensors and θ is the angle of arrival of the signal, then the summed output can be written as: b(nt, θ) = P x p (nt pd sin θ/c) (2.1) p=1 7

19 where we have used the fact that k 0 r = pd sin θ in Fig The quantity b(nt, θ) in Eq. (2.1) is normally referred to as a beam in the direction θ. To achieve more stable power spectrum estimation, the output signal at each sensor is divided into M segments: x pm (nt ), m = 1,, M. The corresponding beam is denoted as b m (nt, θ). The duration of the segment T s determines the width of the frequency bin after the signal is processed by FFT. Therefore, for a fixed record length, as the number of segments increases, T s decreases, the stability of the power spectrum estimate increases, but the frequency resolution decreases. Taking the DFT of b m (nt, θ), we have: B m (kω, θ) = P X pm (kω)e jp(kω/c)d sin θ (2.2) p=1 where B m (kω, θ) and X pm (kω) are the DFT of b m (nt, θ) and x pm (nt ) respectively. To decide whether a frequency bin contains a signal plus noise or noise only, we examine measurement of the power spectrum at that particular bin. The power spectrum of a bin can be obtained by averaging over M segments of the magnitudesquare of the beam B m (kω), i.e., Z x (kω, θ) = 1 M M B m (kω, θ) Bm(kΩ, θ) = 1 M m=1 M B m (kω, θ) 2 (2.3) m=1 Since we utilize the power spectrum, Z x (kω, θ), as the signal feature for detection, we have to analyze and develop its probability density function. To do that, we let β k β(kω, θ) = [B 1 (kω, θ) B 2 (kω, θ) B M (kω, θ)] T (2.4) 8

20 Then, we can write Eq. (2.5) as Z x (kω, θ) = 1 M [βh k β k ] = 1 M β k 2 (2.5) The notation of in Eq. (2.5) is the Euclidean norm of a vector. The distance between two vectors β i and β k induced by this norm is known as the Euclidean distance (ED) [9, 10] and is given by d E (β i, β k ) = (β i β k ) (2.6) Now, in general, each of the vectors β is a random vector. Therefore, we can form its covariance matrix such that K = E[ββ H ] = VΛV H = (VΛ 1/2 )(Λ 1/2 V H ) (2.7) where V and Λ are respectively the eigenvalue and eigenvector matrices of K. We note that K is Hermitian and positive-definite, i.e., K mn = K nm, det K > 0 and also Toeplitz: K mn = K m n Furthermore, we notice that if a unitary matrix U is attached to each of the factors in 9

21 Eq. (2.7), the same expression of covariance matrix is satisfied. Thus, we can write, K = E[ββ H ] = E[VΛ 1/2 uu H Λ 1/2 V H ] = VΛ 1/2 E[uu H ]Λ 1/2 V H = VΛ 1/2 IΛ 1/2 V H = VΛV H (2.8) where u = [u 1,, u M ] T is a random unit vector, with each element being a complex Gaussian variable of zero mean and unity power. Hence, β can be written as the transformation of uncorrelated Gaussian variates u: β = VΛ 1/2 u (2.9) Replacing β k in Eq. (2.5) by (2.9), we have: Z x (kω, θ) = 1 M (V kλ 1/2 k u k) H (V k Λ 1/2 k u k) = 1 M uh k Λ k u k = 1 M M λ km u km 2 (2.10) m=1 where λ km is the mth element on the diagonal of the diagonal matrix Λ k and is a positive random quantity, and u km is the mth element of u k and is a complex Gaussian variable of unity power. Therefore, E[Z x (kω, θ)] = 1 M M λ km E[ u km 2 ] = 1 M m=1 M λ km (2.11) m=1 10

22 Comparing Eq. (2.11) with the expectation of Eq. (2.3), we see that E[ B m (kω, θ) 2 ] = λ km (2.12) Since u km is a complex Gaussian variable, therefore u km 2 is of a Chi-Square distribution with n = 2 degree of freedom. [6] Let χ km = u km 2. The pdf of χ km is given by f χkm (χ km ) = χ km n/2 1 Γ(n/2) exp( χ km) = exp( χ km ) (2.13) If we write the right-side of Eq. (2.10) such that z km = 1 M λ km u km 2 = 1 M λ kmχ km (2.14) then, the pdf of z km can be derived as f zkm (z km ) = f χkm (χ km ) dχ km dz km ( = f χkm χ km = M ) M z km λ km = M [ exp M ] z km λ km λ km λ km (2.15) The characteristic function for each z m is Φ km (ζ) = 0 f zkm (z km ) exp(jζz km )dz km = [ 1 jζλ ] 1 km (2.16) M Since Z x is a sum of independent variables z km, its characteristic function becomes 11

23 the product of Φ km, i.e., Φ k (ζ) = = M m=1 M m=1 [ 1 jζλ ] 1 km M [ 1 jζe[ B ] m(kω, θ) 2 1 ] (2.17) M 2.2 Probability Density Function of Noise and Signal Plus Noise Probability Density Function of Noise For noise only signal model, its power is normalized to unity. Therefore the term E[ B m (kω, θ) 2 ] is replaced by 1, hence the product of the M terms is simplified to: Φ Zx (ζ) = The PDF in this case is therefore given by: f Zν (z) = 1 2π = [ 1 jζ ] M (2.18) M [ 1 jζ ] M e jζz dζ M M M (M 1)! zm 1 e Mz for z 0 (2.19) The plot of the noise PDF is shown in Fig

24 Figure 2.2: Noise PDF Probability Density Function of Narrow-Band Gaussian Signals When there is signal present, λ m is the average power of the signal and noise in the kω frequency. λ m = E[ B m(signal) + B m(noise) 2 ] (2.20) Because the noise is normalized to unity, λ m can be rewritten as λ m = P ower signal + P ower noise P ower noise = 1 + ρ (2.21) where ρ is the normalized signal-to-noise ratio. Therefore, Eq. (2.17) can be reduced to: Φ Zx (ζ) [ 1 jζ ] M (1 + ρ) (2.22) M 13

25 and the corresponding pdf is given by: f Zx (z) = M/(1 + ρ)m (M 1)! z M 1 e Mz/(1+ρ) for z 0 (2.23) The plot of the PDF of the signal having a signal-to-noise ratio of ρ = 10 is shown in Fig Figure 2.3: Signal PDF 2.3 Signal Detection LRT Our goal is to be able to detect a signal embedded in a background of acoustic noise. The decision made between the two possible hypotheses is a binary detection problem. We denote H 0 as the scenario when there is only noise present, and H 1 as the scenario when there is signal plus noise present. Denoting by f(z H 0 ) and f(z H 1 ) respectively as the pdf when there is only noise, and the PDF of signal plus 14

26 noise, then, the probability of false alarm is given by P F A = Z T f(z H 0 )dz (2.24) and the probability of detection is P D = Z T f(z H 1 )dz (2.25) Here, the threshold Z T is usually set to be larger than the mean noise power Z ν, such that Z T = (1 + r) Z ν (2.26) where r is a positive constant power ratio. When a priori probability and costs of detection error are unknown, we use the Neyman-Pearson criterion [7] so that the probability of false alarm is fixed at an acceptable value and the probability of detection is maximized. In our particular case of sonar detection, we have a one-dimensional problem such that if the probability of false alarm is fixed, the probability of detection is automatically fixed. Fig. 2.4 shows the relationship between the probability density functions f(z H 0 ) and f(z H 1 ) as well as the appropriate threshold z T leading to the respective values of P F A and P D. Thus, the z-axis can now be divided into two regions where H 0 and H 1 would respectively be the accepted hypotheses. For a maximum value of false-alarm constraint, we set a fixed threshold Z T. If the true mean of the noise power Z ν is known, to make a decision between the two hypotheses H 0 and H 1, the power in the frequency bin of interest, Z x (kω), is compared to the threshold (1+r 0 ) Z ν. The log likelihood ratio test (LLRT) can then be written 15

27 Figure 2.4: Probability of false alarm, probability of detection, and power threshold. as: Z x (kω) Z ν H 1 H 0 (1 + r 0 ) (2.27) To obtain the nominal constant power ration r 0, we use the expression of f Zν (z) in Eq. (2.19) to evaluate the probability of allowable false alarm so that P F A = (1+r 0 ) M M (M 1)! zm 1 e Mz dz In practice, the true noise mean Z ν is unknown. An estimate Ẑν of the mean noise power has to be made. Ẑ ν is a random variable which depends on the data and the estimator applied. The expected values of P F A and P D can be expressed in terms of Ẑ ν : and E[P F A ] = E[P D ] = 0 0 (1+r)Ẑν (1+r)Ẑν f(z H 0 )dz fẑν (Ẑν)dẐν (2.28) f(z H 1 )dz fẑν (Ẑν)dẐν (2.29) 16

28 where fẑν (Ẑν) is the pdf of Ẑν. For a given value of E[P F A ] together with a knowledge of fẑν (Ẑν), the value of r can be obtained from Eq. (2.28). Now, the decision rule becomes: Z x (kω) Ẑ ν H 1 H 0 (1 + r) (2.30) Noise Estimate Using SWMA There are several methods of estimating noise power [5] and each of these estimators can be applied along the frequency axis of a beam of the data Z x, and the corresponding noise power is obtained for each frequency bin. In this thesis, we concentrate on application of the moving average filter. The moving average filter is a simple linear processor. It utilizes the principle of averaging, whereby the noise power estimate in a particular frequency bin is obtained by averaging the samples in the neighboring frequency bins of the same beam. This method is unbiased and efficient as long as the neighboring bins contain noise samples only. If signals are present in the neighboring frequency bins, the estimation is biased. For a certain angle of arrival θ, the average noise power is thus given by L Ẑ x (k) = a l Z x (k + l) (2.31) l= L where the parameters of Z x in Eq. (2.11) has been simplified by omitting the dependence on Ω and θ. k =..., 1, 0, 1,... where Ẑx(k) and Z x (k) are the output and input sequences to the filter. 2L + 1, L being an integer, is called the window size, and a l are constants chosen such that a L = a L+1 = = a L 1 = a L = 1/(2L + 1). 17

29 In order to improve on the estimation of the noise power, the split window moving average (SWMA) filter is often employed in place of the MA filter. This method assumes that the frequency bins in the neighborhood of the bin of interest contain noise only, taking the average of 2L random samples, L-samples on either side of kω. In other words, the coefficients of the SWMA filter of window size 2L + 1 are given by a L = a L+1 = = a 1 = a 1 = = a L 1 = a L = 1/2L a 0 = 0 (2.32) The SWMA improves the estimation by not taking the signal in the frequency bin kω into account for averaging. We denote the sample of estimated noise power using SWMA in the kω bin by Ẑν. The pdf of Ẑν is of a chi-square distribution of 4LM degrees of freedom: fẑν (Ẑν, 0) = (2LM)2LM (2LM 1)!ẑ(2LM 1) ν e 2LMẑν (2.33) The theoretical pdf is plotted in Fig Numerical integration shows that the area under the PDF curve in Fig. 2.5 is unity. It can be observed that the variance of the SWMA filtered noise PDF curve in Fig. 2.5 is considerably smaller than the noise PDF (Fig. 2.2) without filtering. When, within the 2L samples in the window of the SWMA filter, there are n (signal + noise) samples, while the remaining 2L n are noise only samples, the characteristic function of the resulting average Ẑν will be a product of the noise and signal + noise characteristic functions. Using the expressions of the characteristic 18

30 Figure 2.5: Theoretical PDF of SWMA estimated noise (no signal in window) functions in Eqs. (2.18) and (2.22), the characteristic function of Ẑν is given by: n ΦẐν (ζ, n) = [Φ Zν (ζ)] 2L n Φ Zxi (ζ) = ( 1 jζ 2LM i=1 ) M(2L n) n i=1 [ 1 jζ ] M 2LM (1 + ρ i) (2.34) The PDF fẑν of the estimated power Ẑν can be obtained from ΦẐν (ζ) by using the transform relationship: fẑν (Ẑν, n) = 1 ΦẐν (ζ, n)e jζẑν dζ (2.35) 2π The theoretical PDF for signal plus noise are plotted in Fig.2.6 in which the curves represent the PDF of having one, two and three signals, each with ρ = 13, falling within the SWMA window (2L = 6). It can be observed that as the number of signals 19

31 N=1 N=2 N= Figure 2.6: Theoretical PDF of SWMA estimated noise with signals in window falling into the SWMA window increases, the bias of the noise estimate increases. Numerical integration shows that the area under the PDF curves in Fig.2.6 is also unity. False Alarm Analysis The expression of the expected value of probability of false alarm, E[P F A (n)], for an estimated noise level Ẑν has been given in Eq. (2.28). However, in trying to estimate the noise level using a SWMA filter, signal samples may be included in the window which may bias the estimation, changing the pdf of the estimated noise to fẑν (Ẑν, n) as given in Eq. (2.35). Let Q denote the probability of finding a signal in a sample of the beam. Then the probability of noise only is 1 Q. Let P n be the probability of finding n signals in the 2L samples other than that at kω of the window. Thus the overall probability of false alarm P F A given that the kω frequency sample contains noise only can be expressed as: 20

32 P F A = = 2L n=0 2L n=0 P n E[P F A (n)] 2L n Q n (1 Q) 2L n 0 (1+r)ẑ v f(z H 0 )dzfẑv (ẑ v, n)dẑ v (2.36) Detection Analysis When a signal exists in a particular frequency bin, what is concerned is whether this signal can be detected. For the SWMA estimator, since one of the n signals falling within the filter window occurs at kω and has no effect on the outcome of the averaging (a 0 = 0), there are only n 1 signals in the rest of the 2L samples of the filter window that will be taken into the average. Therefore Eq. (2.29) should be rewritten as: E[P D (n)] SW MA = 0 (1+r)Ẑν f(z H 1 )dz fẑν (Ẑν, n 1)dẐν (2.37) The probability of n 1 interfering signals distributed among the other 2L samples of the window is given by P n 1 = 2L n Q n 1 (1 Q) 2L n+1 (2.38) Hence the overall probability of the detection is: 21

33 P D = 2L+1 n=1 Substituting Eqs. (2.37) and (2.38) into (2.39), we have: P n 1 E[P D (n)] (2.39) P D = 0 (1+r)Ẑν f(z H 1 )dz f(ẑν H 1 )dẑν (2.40) where f(ẑν H 1 ) is the overall pdf of the estimated noise power given that there is a signal at kω. For the SWMA estimator, f(ẑν H 1 ) is: f(ẑν H 1 ) = 2L n=0 2L n 1 Q n 1 (1 Q) 2L n+1 fẑν (Ẑν, n 1) (2.41) 2.4 Simulation Results Here, we verify by numerical experiments the detection performance of the sonar system equipped with an FFT beamformer using a SWMA filter for data normalization. In our simulations, 4 segments, each containing 1000 samples of white Gaussian noise and narrow band signals, are generated. The probability of having a signal present in a frequency bin is taken to be in each beam Noise generation We first examine when there is noise only in the frequency bins of the beam. We generate a sequence of Gaussian random samples of zero mean and 0.01 variance. This sequence is passed through an FFT and a beamformer as mentioned in Section 22

34 2.1. The power spectrum of the beamformer is then obtained as shown in Eq. (2.10) and is shown in Fig The normalized histogram (100 power, making mean =1) of this noise power spectrum, generated using 1,000,000 noise samples, is also plotted and is shown in Fig In comparison to the theoretical PDF in Fig. 2.2 which is re-shown in the curve in Fig. 2.8, we can see that the two agrees very well. Figure 2.7: Beamformer power spectrum (noise only) Figure 2.8: Histogram (noise only) 23

35 2.4.2 Signal generation We now examine the generation of random narrowband Gaussian signals. Each beam from a direction θ κ may have up to I signals. The ith signal occurring in the beam is generated according to the following model: s i (nt ) = a i cos(k i ΩnT + φ i ) + b i sin(k i ΩnT + φ i ) (2.42) Here, a i and b i are independent Gaussian random variables with zero mean and variance S i. We note that S i is the power of the ith signal. φ i is the phase of the signal which is modeled as a random variable evenly distributed between π and π. k i Ω is the frequency of the random signal and is fixed as long as the signal lasts in the beam. Fig. 2.9 shows the occurrence of 3 signals (without noise) in the output power of a beam respectively at frequencies 102, 187, 443 Hz, having average signal power S i = The power spectrum of the signal plus noise is shown in Fig The Figure 2.9: Beamformer power spectrum (signals only) histogram of the one signal in the kω bin (generated using 3,000 random Gaussian signals), again, as in the case of noise only, normalized by 100 power, is shown in 24

36 Figure 2.10: Beamformer power spectrum (signals + noise) Figure 2.11: Signal histogram Fig In comparison to the theoretical PDF in Fig. 2.3 which is re-shown in the curve in Fig. 2.8, the fitting is not as closed as in the case of the noise. There is a small shift of the normalized histogram to the left of the signal PDF Detector performance We now examine the theoretical and experimental performance of the detector using SWMA data normalization. 25

37 2.4.4 Numerical experiments We first verify the performance of the sonar detector using SWMA filter by numerical experiments: A range of threshold values from r [ 1, 2.5] are selected. For each threshold value, a detection process is performed on a beam of signal and noise, each having 1,000 frequency bins. The noise and signals in a beam is generated using the processes described in Sections and with the occurrence of a signal in a bin being kept at Q = The signal power, S, in each beam is calculated from the average of the power of all the generated signals, whereas the noise power, N, is calculated from the average of all the noise samples in the beam. The signal-to-noise ratio ρ = S/ N for each experiment is kept in the range of 8 to 10. A SWMA filter is slid through the whole beam. At each step, a normalized noise power is evaluated, which, when multiplied by (1 + r), forms the threshold value. The sample in the frequency bin being examined is then compared with the generated threshold value, and is determined to be a signal if it exceeds the threshold, or to be a noise sample otherwise. In our study here, we keep the window size of the SWMA filter (2L) to be 6. In order to cater for the starting 3 samples and the last 3 samples of the beam so that they too, can be examine and determined to be signal or noise, we add 3 noise samples each to the beginning and to the end of the beam so that in total, there are 1,006 samples with the first and last 3 being noise samples. Then the SWMA is now slid through these samples of the beam. The examination of each sample starts from the originally first sample (Sample 1) to the originally last sample (Sample 1,000). A detection is successfully made if the amplitude of a known signal in the beam is 26

38 larger than the threshold. The estimated probability of detection is then calculated as ˆP D = N d /N s (2.43) where N d is the counted number of known narrow-band signals being larger than the threshold and N s is the total number of known signals in the beam. A false alarm occurs if the amplitude of a noise sample (not one of the known signals) is larger than the threshold. The estimated probability of false alarm is calculated as ˆP F A = N fa /N n (2.44) where N fa is the counted number of noise samples larger than the threshold, and N n is the total number noise samples being examined, i.e., 1, 000 N s. The success rate of detection and the corresponding false alarm rate are recorded each time the SWMA filter is slid through a beam. This detection process is repeated 1,000 times for each value of r, and the detection and false alarm rates for the same value of r averaged over the number of trials. The signal-to-noise ratio, ρ for the complete set of experiments is evaluated by averaging all the signal-to-noise signals in each run. The plotting of P D against the corresponding values of P F A form the receiver operating curve (ROC) describing the performance of the receiver. An example of the ROC obtained by numerical experiments at SNR = 5dB and Q = is shown in red in Figs

39 1 0.9 Experimental ROC 0.8 Theoretical ROC Pd Pfa Figure 2.12: Receiver operating curves (SNR=5dB, Q=0.005) Evaluation of theoretical performance We have studied the theoretical performance of the sonar detector using SWMA filter in Section Eqs (2.36) and (2.40) provide us with the expressions for the average probabilities of false alarm and of the average detection. In the evaluation of these probabilities, we assume the average probability of finding a signal in a frequency bin to be Q = and the signal-to-noise ratio to be ρ as obtained from the numerical experiments. With these values of Q and ρ, the theoretical values of P F A and P D can be evaluated for the whole range of threshold (1 + r)ẑν using Eqs. (2.36) and (2.40). The theoretical ROC is shown in dotted curve in Fig Comparison of performance It can be seen that the experimental performance of the detector is quite close to its theoretical performance. The discrepancy is marginal for values of P F A > 0.3, being less than 2%. The discrepancy of P D in the lower range values of P F A is more substantial, being as much as 10 to 12%. This can be partly explained by the fact 28

40 that the probability model of the signal may not be completely accurate. It can be observed from Fig that the theoretical PDF curve of the signal power spectrum is shifted to the right of the histogram obtained by simulation. This, for the same probability of false alarm, will have higher probability of detection for the theoretical evaluation. 29

41 Chapter 3 PSD Manifold 3.1 The Power Spectral Density Matrix In chapter 2 we learn that for classical sonar signal detection, we need to establish the concepts of distance, mean and threshold to complete the binary hypothesis test. All these measures are based on the feature of power Z x of the sample at frequency kω. In this thesis, instead of using Z x, we propose to use the estimated power spectral density matrix (abbreviated to simply PSD in the ensuing sections) the signal feature for detection. Since PSD matrices possess additional correlation information between measured signals from different segments, we expect more accurate detection results. Parallel to conventional detection, we need to establish the concepts of distance between the features, the mean of the features, as well as the threshold distance. In addition, to better separate noise features from signal features, we can apply a weighting matrix to the distance measure so that certain parts of the PSD matrix can be enhanced or de-emphasized leading to the concept of weighted distances. The PSD matrix of the the received narrow-band sonar signal can be obtained by 30

42 forming the outer product of the beamformer output β of Eq. (2.4) such that P k P(kΩ, θ) = 1 M β(kω, θ) βh (kω, θ) = 1 M β k β H k (3.1) where, as in Eq. (2.4), we have shortened the notation of the vector β(kω, θ) to β k. Thus, at each frequency bin of the sonar beamformer, we can form an M M PSD matrix and use it as a feature for detection. 3.2 Riemannian Distance The binary hypothesis testing process of detection involves the separation of the signal feature into signal and noise. As such, the concept of distance measuring the similarity between like signals and dissimilariry between unlike signals is very important, the greater is the similarity, the closer is the distance, and the greater is the dissimilarity, the greater is the distance. As shown in Chapter 2, a powerful measure of distance is the ED which is induced by the Euclidean norm. ED is the most commonly used distance because it coincides with the usual concept of distance in a 3-dimensional space and represents many important physical quantities. To deal with PSD matrices, we may also look upon an M M complex matrix as a point in the M 2 complex signal space so that the same idea of distance [12] between two such matrices A = [a ij ] and B = [b ij ] can also be applied: M M d F (A, B)= a ij b ij 2 i=1 j=1 1/2 = tr[(a B)(A B) H ] (3.2) Eq. (3.2) is often called the Frobenius distance which is in fact induced by the inner product norm and can be considered as the ED between A and B since, if we form 31

43 the vectors veca and vecb using the vec-functions [11], it is easy to see that d 2 F(A, B)= (veca vecb),(veca vecb) = d 2 E(vecA, vecb) (3.3) The Frobenius distance is by far, the most commonly used distance measure between M M matrices. However, we notice that the PSD matrices are Hermitian, positive semi-definite, and are structurally constrained. Therefore they form a manifold, M, in the signal space. It is concluded that the distance between two PSD matrices must be carried out along the tangent to the manifold [2], and not by the usual ED of a straight line. This is analogous to the measurement between two cities on earth: the ED is neither informative nor accurate. To find the distance between two PSD matrices, we notice that the curve on the manifold linking two PSD matrices P m and P n having the minimum length is called a geodesic, and the length of the geodesic is called the Riemannian distance (RD) between the two points [1], i.e., d R (P m, P n ) min {l(p(θ))} P(θ):[θ m,θ n] M In general, the evaluation of the RD on a manifold is difficult, however, with particular mappings together with a certain Riemannian metric [1], two points P m and P n on a manifold can be linked by fibres (mapping routes) to two other image points P m and P n in an isometric Euclidean subspace. The RD between the two points on the manifold can then be obtained from the equivalent ED between P m and P n. Three different closed-form expressions of RD for the PSD matrix manifold have been developed [2]. In this thesis, we focus only on RD d R1 and d R2 since their performance 32

44 in signal detection on the PSD manifold can be improved using a weighting. For this reason, the use of d R3 is not discussed in this thesis because it is weighting invariant The Riemannian Distances d R1 Consider the mapping: P = P P H (3.4) We note that given P, P is not unique and can be written as P = P 1/2 U with U being any unitary matrix. Also, P is an M M complex matrix which may no longer be positive definite or Hermitian. Hence, P is in a subspace H of the Euclidean space H of all M M complex matrices such that H = { P : P P H = P M}, H H (3.5) We can thus view P as an image on M of the points P in H. Eq. (3.4) can be looked upon as a projection mapping such that for every PSD matrix P, there exists another matrix P H which, though not unique, can be viewed as a representation of P in the Euclidean space H. The manifold M and the Euclidean space H are often called the base space and the total space respectively. The linking between the two spaces can be conceived [1] such that each point P M is linked by the fibre above it with points in H through the mapping P = P P H. Any point along the fibre satisfies this mapping. At any P in H, the tangent space T H( P) is the vector space local to the neighbourhood of P and can be resolved into its horizontal and vertical subspaces U H( P) and V H( P) respectively. Using the mapping in Eq. (3.4) together with a chosen metric, Li 33

45 and Wong [2] showed that the tangent space T M (P) of the manifold and the subspace U H to which a curve between two points P m and P n in M is horizontally lifted are isometric, and exploited the isometry to obtain a closed-form expression for the RD between P m and P n on the manifold. The reasoning can be summarized as follows: Since H is a Euclidean space, and the ED between any two points P m and P n along the two fibres above P m and P n is a straight line, the length of which is given by P n P m, then, a geodesic (path of minimum length) between P m and P n in M, must equal the length of the shortest straight line joining P m and P n, i.e., d 2 R 1 (P m, P n )= min P m= P m PH m P n= P n PH n P m P [ n 2 = min tr {( P m P n )( P m P }] n ) H (3.6) P m= P m PH m P n= P n PH n For U m and U n being any two unitary matrices, P m = P 1/2 m U m and P n = P 1/2 n U n, satisfies the mapping P = P P H. Then, the minimization of Eq. (3.6) is satisfied if U m = Ũm and U n = Ũn where Pn 1/2 P 1/2 m = ŨnΣŨH m is the singular value decomposition with Σ being the singular value matrix, and Ũn and Ũm being the left and right singular vector matrices of P 1/2 n P 1/2 m respectively [12]. Hence, the RD d 2 R 1 can be expressed as: d 2 R 1 (P m, P n ) = Pm 1/2 Ũm P 1/2 2 n Ũ n = trp m + trp n 2tr [ (P 1/2 m P n P 1/2 m ) 1/2] 2 (3.7a) (3.7b) The Riemannian Distance d R2 We use the same mapping as in Eq. (3.4). However, instead of choosing U n and U m to be the left and right singular vector matrices of P 1/2 n P 1/2 m, we choose U m = U n = I, 34

46 our mapping then becomes P = P 2 Now, we also choose a suitable Riemannian metric g P (A, B) = A, K (3.8) such that PK + KP + 2 PK P = B, where A, B T M (P). Writing P m = P 2 m and P n = P 2 n, the RD d R2 between P m and P n on M is given by [2]: d 2 R 2 (P m, P n ) = P m P [ n 2 = tr ( P m P n )( P m P ] n ) = trp m + trp n 2tr(P 1/2 m P 1/2 n ) (3.9) 3.3 Weighted Riemannian Distance Often in signal processing, due to prior information, it may be desirable to enhance or de-emphasize certain parts of the data feature [9]. It is no different in signal detection for which, to better distinguish between signal and noise, it is desirable to enhance or deemphasize certain part of the feature PSD matrices. This leads to a weighted RD. For the purpose of weighting, we may apply a positive definite Hermitian weighting matrix W to the PSD matrices such that we write W = ΩΩ H, where Ω is M K, K M. We let P mw = Ω H P m Ω and P nw = Ω H P n Ω be the weighted versions of P m and P n, respectively. We note that P mw and P nw are also positive definite Hermitian matrices, i.e., P mw, P nw M. Using the same method of lifting the weighted matrices along the fibres to the isometric Euclidean subspace U H, the weighted RD can be evaluated. 35

47 3.3.1 The Weighted Riemannian Distance d WR1 Let W = ΩΩ H, where Ω is M K, K M, and let P mw = Ω H P m Ω and P nw = Ω H P n Ω. It is easy to see that P mw, P nw M. Now, we can write P mw = P mw PH mw and P nw = P nw PH nw where P iw = Ω H P 1/2 i U i, U i being a unitary matrix, for i = m, n, and we have [2], d 2 WR1(P m, P n ) = min U m,u n P mw P nw 2 = min tr(wp m )+tr(wp n ) 2R [ tr ( U m U H n P 1/2 n ΩΩ H P 1/2 m U m,u n )] (3.10) Again, the last term in Eq. (3.10) is maximized if U m and U n are chosen to be the right and left singular vector matrices of P 1/2 n ΩΩ H P 1/2 m, giving the expression of the weighted Riemannian distance as [ d 2 WR1(P m, P n ) = trwp m + trwp n 2 tr ( ) P 1/2 m WP n WP 1/2 1/2 ] m (3.11) The Weighted Riemannian Distance d WR2 Again, we define W = ΩΩ H, where Ω is M K, K M, and let P mw = Ω H P m Ω and P nw = Ω H P n Ω. In the case of d R2, we have P m = P 1/2. Thus, for the weighted version, we define P mw = P 1/2 mw = (ΩH P m Ω) 1/2. Similarly, PnW = (Ω H P n Ω) 1/2. Then, PnW, n are in the Euclidean space isometric with the tangent space of the M. Following the same concept development as with that of d R2, we have, d 2 WR 2 (P m,p n ) = P mw P [ nw 2 = tr ( P mw P nw )( P mw P ] nw ) =tr(ω H P m Ω)+tr(Ω H P n Ω) 2tr[(Ω H P m Ω) 1/2 (Ω H P n Ω) 1/2 )](3.12) 36

48 3.4 Fréchet Mean As in the case of classical signal detection, the noise mean plays a very important role in the signal detection on PSD manifold. The mean of a group of random entities is defined as the centre from which the sum of the squared distances to all members is minimum. For a group of random matrices P n, this mean matrix is known as the Fréchet Mean (FM) defined as C F = arg min C N d 2 (P n, C) (3.13) n=1 In this thesis, d applies to the RD d R1, d R2 as well as to the weighted Riemannian distances d WR1 and d WR Fréchet Mean According to d R1 The problem of finding the Fréchet Mean according to d R1 can be stated as follows: Problem Statement: Given the Riemannian distance d R1 (P m, P n ) between two positive semi-definite matrices P m and P n as in Eq. (3.7), we seek C F, the Fréchet mean of {P n, n = 1..., N} such that C F = argmin C N d 2 R 1 (P n, C) (3.14) n=1 Solution: From Eq. (3.7), we note that the Riemannian distance d 2 R 1 (P m, P n ) between two points P m and P n on M is equal to the Euclidean distance between the two corresponding points P m and P n in the Euclidean horizontal subspace H. To find the solution for Eq. (3.14) we need the following lemma: 37

49 Lemma 1. Let { Dm } N m=1 be set of N points in the Euclidean space H. Then, N N m=1 n m D m D n 2 2 = N N D n C n=1 2 2 (3.15) where C = 1 N N n=1 D n. Proof : N D n X n=1 2 2 = = N D n C + C X n=1 2 2 N D n C 2 + 2( C X). 2 n=1 N ( D n C) + N C X n=1 2 2 From the definition of center of mass C, we have N n=1 ( D n C) = 0. Thus, N D n X 2 = 2 n=1 N D n C 2 + N C X 2 n=1 2 2 (3.16) Now suppose X varies over the set the N equations together, we have: { D1, D 2,..., D N } in Eq. (3.16), then adding up D m D 2 n m,n 2 = N = 2N N D m C 2 + N 2 m=1 N D m C m=1 2 2 N C D 2 n n=1 2 (3.17) However, we note that m,n D m D 2 n 2 = 2 m n m and substituting this into Eq. (3.17), the result of Eq. (3.15) follows. D m D 2 n 2 To find a solution for the problem in Eq. (3.14), we first find the corresponding 38