Reshetov LA Narrowband signal detection techniques in shallow ocean

Digital Signal Processing 23 (2013) 1645–1661

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Digital Signal Processing

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Narrowband signal detection techniques in shallow ocean by acousticvector sensor array

V.N. Hari a,∗, G.V. Anand b, A.B. Premkumar c

a CeMNet Annex, Block N4, B2b-05, School of Computer Engineering, Nanyang Technological University, Singapore 639798, Singaporeb Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, Indiac School of Computer Engineering, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e i n f o a b s t r a c t

Article history:Available online 28 June 2013

Keywords:Acoustic vector sensorGeneralized likelihood ratio testShallow oceanSubspace detection

This paper presents the formulation and performance analysis of four techniques for detection of anarrowband acoustic source in a shallow range-independent ocean using an acoustic vector sensor (AVS)array. The array signal vector is not known due to the unknown location of the source. Hence all detectorsare based on a generalized likelihood ratio test (GLRT) which involves estimation of the array signalvector. One non-parametric and three parametric (model-based) signal estimators are presented. It isshown that there is a strong correlation between the detector performance and the mean-square signalestimation error. Theoretical expressions for probability of false alarm and probability of detection arederived for all the detectors, and the theoretical predictions are compared with simulation results. It isshown that the detection performance of an AVS array with a certain number of sensors is equal to orslightly better than that of a conventional acoustic pressure sensor array with thrice as many sensors.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Acoustic vector sensors (AVS) are different from traditionalacoustic pressure sensors (APS) or hydrophones in that they mea-sure the components of particle velocity in addition to the acousticpressure [1]. AVS technology with co-located pressure and velocitysensors has grown rapidly in recent years due to the establishedsuperiority of AVS over APS in signal processing applications. Thissuperiority stems from the fact that particle velocity measure-ments provide additional information about the acoustic field, viz.,the direction of the impinging acoustic waves. While an APS arraycan extract directional information by measuring the propagationdelay between sensors, a single AVS can obtain this informationdirectly from the particle velocity measurements. These measure-ments also enable unambiguous bearing estimation of underwatersources using shorter and/or sparser AVS arrays, in contrast to APSarrays which suffer from several ambiguities [1–4].

Velocity sensors per se are not new. They have been availableand used for quite some time. However, the importance of simulta-

Abbreviations: APS, Acoustic pressure sensor; ASFD, Approximate signal formdetector; AVS, Acoustic vector sensor; ED, Energy detector; GLRT, Generalized like-lihood ratio test; HLA, Horizontal linear array; MFD, Matched filter detector; NMSE,Normalized mean-square signal estimation error; PDF, Probability density function;SD, Subspace detector; TSD, Truncated subspace detector; VLA, Vertical linear array.

* Corresponding author.E-mail addresses: [email protected] (V.N. Hari), [email protected]

(G.V. Anand), [email protected] (A.B. Premkumar).

1051-2004/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.dsp.2013.06.010

neous measurement of acoustic pressure and particle velocity wasfirst demonstrated experimentally by D’Spain et al. [4] in 1991,sparking interest in the development of acoustic vector sensors.Since then, several experimental setups such as the DIFAR array[5] and the recently conducted Makai experiment [6] have demon-strated the effectiveness of AVS. The impressive performance ofvector sensors has been demonstrated in numerous signal process-ing applications such as source direction-finding and localization[3,7–26], tracking [27–29], communication [30–32], geo-acousticinversion [33] and source detection [34,35]. The compactness andimproved performance of AVS arrays make them ideal for use infields such as underwater acoustic surveillance for port security[36] and underwater imaging [37].

The initial theoretical framework for signal processing appli-cations using an AVS array was developed by Nehorai and Paldi[1] who presented the AVS array measurement model for planewaves propagating in a homogeneous medium. Several theoreti-cal treatments of the AVS followed, and numerous algorithms fordirection-finding and localization of sources employing AVS havebeen developed. These include modifications of existing direction-of-arrival estimation (DOA) algorithms such as the Capon algo-rithm [3], MUSIC [10,38], ROOT-MUSIC [11], ESPRIT [8,9,12,13,16,17,38], MLE [18,19], and SIM [20,22] as well as other [23–26] al-gorithms. In a series of papers, Hawkes and Nehorai dealt withtheoretical aspects of AVS signal processing, such as characteriza-tion of the AVS array performance in terms of error measures andthe Cramer–Rao bound (CRB) [2,39], and characterization of intra-sensor and inter-sensor correlation of noise in an AVS array [40].

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1646 V.N. Hari et al. / Digital Signal Processing 23 (2013) 1645–1661

The performance bounds on direction finding using AVS have alsobeen investigated by Tam and Wong [41]. Hawkes and Nehorai alsoillustrated how a distributed AVS array can function as a sensornetwork for localization [14]. In this setup, each AVS can functionas a separate node of the network. Zha and Qiu [16] illustratedhow the AVS could be employed for bearing estimation when theenvironment is contaminated by impulsive noise. Tichavsky et al.[13] demonstrated that a single AVS can be used for direction-finding of an acoustic source in far-field or near-field. Wu andWong [21] have presented the near-field array manifold of the AVS,thus opening the doors for near-field AVS array signal processingapplications. Later, they presented a method of near-field sourcelocalization using one triad of velocity sensors and one APS [23].The problem of tracking source azimuth using an AVS was studiedby Felisberto et al. [27]. Zhong et al. [28] explored the possibil-ity of two-dimensional DOA tracking of acoustic sources with AVSusing particle filtering. The feasibility of using AVS in underwateracoustic communications was demonstrated by Song et al. [30,31]and Abdi et al. [32]. The AVS may also be employed as an effectivemeasurement system for the estimation of geo-acoustic parame-ters [33].

Work on the development of algorithms for signal detection inshallow ocean using an AVS array is of recent origin [34,35]. In[34] and [35], the authors have proposed several detection strate-gies such as the energy detector (ED), subspace detector (SD),truncated subspace detector (TSD) and approximate signal formdetector (ASFD), for narrowband detection of a source in a range-independent ocean with Gaussian ambient noise. All detectors arebased on a generalized likelihood ratio test (GLRT) [42] since thearray signal vector is unknown due to the unknown location ofthe source. The ED is formulated assuming that the signal vec-tor is completely unknown. The SD, which is formulated using thematched subspace detection approach [43], exploits the fact thatthe signal vector belongs to a low-dimensional modal subspace.Subspace detection is a well-discussed problem in the literature[44–46], starting with the works of Scharf and Lytle [44] who ap-plied the theory of invariance in hypothesis testing to the problemof CFAR signal detection. The classic paper by Scharf and Fried-lander [43] on matched subspace detection generalized the classof problems for detecting a subspace signal and showed that theGLRT yields a maximal invariant, uniformly most powerful detec-tor. This work has also been extended to the problem of adap-tive subspace detection [47–49]. Subspace detection has also beenstudied for signals in spherically invariant random vector noise[50] and generalized Gaussian noise with interference [51].

In the context of underwater source detection, the SD providesa better performance than the ED. But the SD requires the knowl-edge of the modal wavenumbers of the underwater channel whilethe ED does not require any prior information. A modification ofthe SD, namely the TSD, seeks to optimize the detector by trun-cating the dimension of the modal subspace. The ED, SD and TSDmethods can be employed with either AVS or APS arrays; but AVSarrays provide a significantly better detection performance thanAPS arrays. The ASFD, which is applicable to AVS arrays only, ex-ploits the fact that a simple approximate relation exists betweenthe components of measurement at each AVS. Neither the ED northe ASFD assume any knowledge of the channel, but the ASFD pro-vides a better performance than that of the ED. An extension of theabove-mentioned detection strategies to the problem of detectionin non-Gaussian noise has been explored recently in [52].

This paper presents improved versions of methods developedin [35] for narrowband detection of an acoustic source using anAVS array in a range-independent ocean with Gaussian environ-mental noise. We undertake a detailed theoretical investigation ofthese methods, that is missing in [35]. The performance analysisof the detectors is based on several performance measures and

simulation studies. We also discuss and illustrate the performanceimprovement obtained by these detection methods by exploitingthe advantages of the AVS array in comparison to the APS array.The paper is organized as follows. The data models for a horizontallinear array (HLA) and a vertical linear array (VLA) of AVS are pre-sented in Section 2. A detailed formulation of several alternativedetection strategies is presented in Section 3. The issue of opti-mization of the TSD is also discussed in this section. In Section 4,the theoretical performance analysis of the detectors is presentedfor the case of finite data as well as the case of asymptotically largedata records, and the relative merits and demerits of the differentdetectors are discussed. Theoretical and simulation results are pre-sented and compared in Section 5. A summary and conclusions arepresented in Section 6.

2. Data model

We consider the problem of using an N-element AVS array forthe detection of a narrowband point source with a center fre-quency f , using T snapshots of data. The ocean is modeled as arange-independent channel, having a water column of density ρwand sound speed cw over sediment of density ρb and sound speedcb . The source is located at azimuth φ, depth zs and range r withrespect to the reference element of the array. The sensor array isassumed to be in the far-field region with respect to the source,i.e. 2π( f /cw)r � 1. We consider the uniform HLA and uniformVLA configurations for detection. In the case of an HLA, the sen-sor closest to the source is considered to be the reference sensorwhereas in the case of a VLA the topmost sensor is considered tobe the reference. The elements of the array are separated by a uni-form distance of d meters. The source azimuth angle is measuredwith respect to the endfire direction of the HLA and an arbitrarydirection for the VLA.

Each AVS in the array has three outputs, viz., the acoustic pres-sure and two orthogonal horizontal components of particle veloc-ity. The vertical component of particle velocity is not consideredsince the inclusion of this additional measurement is found to in-crease the complexity of the detection algorithm without yieldingany significant additional improvement in performance. We shallscale the velocity components by the factor

√2ρw cw to maintain

dimensional uniformity of the measured quantities and also to en-sure that all the noise components have equal variance [20]. Letx(t), s(t) and w(t) denote, respectively, the tth snapshot of the3N × 1 data vector, signal vector and noise vector at the N-sensorAVS array. The signal vector s(t) can be represented as

s(t) = [s1(t), . . . , s3N(t)

]T,

s3n−2(t) = pn(t), s3n−1(t) = √2ρw cw vxn(t),

s3n(t) = √2ρw cw v yn(t), n = 1, . . . , N, (1)

where pn(t) denotes the complex amplitude of acoustic pressureat the nth sensor, and (vxn(t), v yn(t)) denote the correspondingcomplex amplitudes of the horizontal (x, y) components of particlevelocity. According to the normal mode theory of sound propaga-tion, the acoustic field in the channel can be expressed as the sumof a discrete spectrum of propagating normal modes and a con-tinuous spectrum of evanescent normal modes [53]. For a sourcein the far-field region, the contribution of the continuous spectrumcan be neglected and the expression for the acoustic pressure pn(t)can be written as

s3n−2(t) = pn(t) =M∑

m=1

pmn(t), (2)

where pmn(t) is the contribution of the mth mode to the acousticpressure at the nth sensor, given by

V.N. Hari et al. / Digital Signal Processing 23 (2013) 1645–1661 1647

pmn(t) = B(t)Ωmn(φ)ψm(zn)ψm(zs)exp(ikmr − ζmr)/√

kmr, (3)

Ωmn(φ) ={

exp(i(n − 1)kmd cos(φ)) for HLA,

1 for VLA,(4)

where km and ζm are the modal wavenumber (real part of theeigenvalue) and attenuation coefficient (imaginary part of theeigenvalue) of the mth normal mode at frequency f , ψm(z) is thenormalized eigenfunction of the mth normal mode, zn is the depthof the nth sensor and B(t) is a slowly varying complex quantitywhose root-mean-square magnitude is proportional to the strengthof the source. For a given channel, the number of propagatingmodes M is, in general, an increasing function of the signal fre-quency.

The relation between acoustic pressure p and particle veloc-ity v at a point r = (x, y, z) at time t is governed by the law ofconservation of momentum

ρwdv

dt+ ∇p = 0. (5)

In most applications of practical interest, the source is located inthe far-field, satisfying the condition kmr � 1 for all m. Using (2),(4) and (5), and invoking the far-field condition, and defining k =2π f /cw , we can write

s3n−1(t) = √2ρw cw vxn(t) = √

2 cos(φ)

M∑m=1

(km/k)pmn(t), (6)

s3n(t) = √2ρw cw v yn(t) = √

2 sin(φ)

M∑m=1

(km/k)pmn(t). (7)

Hence the tth snapshot of the signal vector can be expressed as

s(t) = A(φ)b(t), (8)

where A(φ) is a 3N × M modal steering matrix and b(t) is themode amplitude vector. In the case of an HLA (denoted by thesubscript H), b(t) and A(φ) are defined as

bH (t) = B(t)[ψ1(za)ψ1(zs)exp(ik1r − ζ1r)/

√k1r · · ·

ψM(za)ψM(zs)exp(ikMr − ζMr)/√

kMr]T

, (9)

AH (φ) = [a1,H (φ) · · · aM,H (φ)

], (10)

where za is the array depth and {am,H (φ), m = 1, . . . , M} are themodal steering vectors for the HLA, defined as

am,H (φ) = dm(φ) ⊗ gm(φ), (11)

dm(φ)

= [1 exp

(ikmd cos(φ)

) · · · exp(i(N − 1)kmd cos(φ)

)]T,

(12)

gm(φ) = [1

√2 cos(φ)(km/k)

√2 sin(φ)(km/k)

]T, (13)

where the symbol ⊗ denotes the Kronecker product. In the case ofa VLA (denoted by the subscript V ), b(t) and A(φ) are defined as

bV (t) = B(t)[ψ1(zs)exp(ik1r − ζ1r)/

√k1r · · ·

ψM(zs)exp(ikMr − ζMr)/√

kMr]T

, (14)

AV (φ) = [a1,V (φ) · · · aM,V (φ)

], (15)

and the modal steering vectors for a VLA are defined as

am,V (φ) = [ψm(z1)gT

m(φ) · · · ψm(zN)gTm(φ)

]T. (16)

The signal is assumed to be contaminated by additive tempo-rally white Gaussian noise. Let the noise vector at the nth sensor

of the AVS array be denoted by [w3n−2(t) w3n−1(t) w3n(t)]T . For agiven time t , the three components of noise at an AVS are uncor-related and identically distributed [20]. But the noise is spatiallycorrelated and it has a depth-dependent variance. Let the correla-tion matrix of the noise vector w(t) be denoted by

Rw = E[

w(t)w H (t)] = σ 2R0, (17)

where σ 2 is the variance of each component of noise at the ref-erence depth z1 and R0 is a 3N × 3N matrix. The spatial cor-relation decays rapidly in the horizontal direction, and it can beignored when the horizontal separation between sensors is λw/2(half-wavelength) or larger. But the vertical spatial correlation be-tween similar components of noise remains strong even at largedistances. Hence, for an HLA at depth z1, the matrix R0 is given by

R0,H = I3N , (18)

where I3N is the 3N × 3N identity matrix, and for a VLA whoseelements are located at depths z1, . . . , zN , the matrix R0 is givenby [54]

R0,V =[ r(z1, z1) · · · r(z1, zN)

· · · · · · · · ·r(zN , z1) · · · r(zN , zN)

]⊗ I3, (19)

where

r(zk, zn) =∑M

m=1 sin(γmzk) sin(γmzn)∑Mm=1 sin2(γmz1)

, (20)

where γm =√

(k2 − k2m). The vertical spatial correlation function

r(z1, z1 + Δ) is shown in Fig. 1 for a shallow iso-velocity oceanwhose parameters are listed in Section 5. Fig. 1(a) shows a plot ofr(z1, z1 + Δ) versus Δ for z1 = 10 m and frequency f = 350 Hz.Fig. 1(b) shows the variation of the correlation with frequency forz1 = 10 m and Δ = λw , where λw = cw/ f is the wavelength inwater. It is seen from these figures that the correlation is small forΔ � λw , and exhibits small irregular fluctuations with frequency.Hence, for a VLA, it is desirable to choose inter-sensor spacing d ∼=λw to minimize the degradation of detector performance due tointer-sensor noise correlation. In the following analysis, we shallassume that the matrix R0 is known for both HLA and VLA. Thevariance σ 2 is generally not known.

3. Formulation of detectors

In this section, we shall present several alternative methods fordetecting a narrowband source. The detection problem can be castin the form of the following binary hypothesis testing problem:

H0: x(t) = w(t),

H1: x(t) = s(t) + w(t), t = 1, . . . , T , (21)

where H j denotes the hypothesis that j sources are present, and

x(t) = R−1/20 x(t), s(t) = R−1/2

0 s(t),

w(t) = R−1/20 w(t), (22)

are the pre-whitened versions of the data vector, signal vector, andnoise vector respectively. The joint likelihood functions of T snap-shots of the data vector under hypotheses H0 and H1 are givenby

f(

X∣∣σ 2; H0

) = 1

(πσ 2)3NTexp

[− 1

σ 2

T∑t=1

xH(t)x(t)

], (23)


Fig. 1. (a) Vertical spatial correlation function r(z1, z1 + Δ) at f = 350 Hz. (b) Variation of r(z1, z1 + λw ) with f .

f(

X∣∣σ 2,S; H1

)= 1

(πσ 2)3NTexp

[− 1

σ 2

T∑t=1

(x(t) − s(t)

)H(x(t) − s(t)

)], (24)

where X = [xT(1), . . . , xT

(T )]T , S = [sT(1), . . . , sT

(T )]T . The loga-rithm of the likelihood ratio (LR) is given by

log L(X) = 1

σ 2

T∑t=1

[2 Re

(xH

(t)s(t)) − sH

(t)s(t)]. (25)

The primary hurdle faced in the detection of a signal in theocean is that the signal vectors {s(t), t = 1, . . . , T } at the sensorarray are unknown due to lack of knowledge of the source locationand other uncertainties in source, receiver or environment param-eters. Hence we perform a generalized likelihood ratio test afterreplacing each vector s(t) in (24) by its maximum likelihood (ML)estimate s(t). The ML estimate may use any prior information thatmay be available. The different detectors presented in this paperuse different types of prior information. The noise variance σ 2 isalso not known in general. Sometimes, it may be possible to obtaina prior estimate of σ 2 from noise-only data. If a reliable a prioriestimate of σ 2 is not available, it has to be treated as an addi-tional unknown parameter. Hence we shall consider the detectionproblem for two different cases, viz. (a) the noise variance σ 2 isknown, and (b) σ 2 is unknown.

For case (a), the logarithm of the generalized likelihood ratio(GLR) is obtained from (25) on replacing s(t) by s(t). Hence thetest statistic for this case can be written as

γCase(a)(X) =T∑

t=1

[2 Re

(xH

(t)s(t)) − sH

(t)s(t)], (26)

where the signal-vector estimates s(t) are different for differentdetectors, and Re(·) denotes the real part of (·).

For case (b), the following estimates of σ 2 under H0 and H1are readily obtained by maximizing the likelihood functions in (23)and (24):

σ 20 = 1

3NT

T∑t=1

xH(t)x(t),

σ 21 = 1

3NT

T∑[x(t) − s(t)

]H[x(t) − s(t)

]. (27)

t=1

On substituting (27) into (23) and (24) respectively, we get thefollowing expression for GLR for case (b):

LG,Case(b)(X) =[σ 2

0

σ 21

]3NT

, (28)

and further simplification yields the following expression for thetest statistic:

γCase(b)(X) =∑T

t=1 xH(t)x(t)∑T

t=1[(x(t) − s(t))H (x(t) − s(t))] . (29)

We shall now introduce the different detectors and discuss theirperformance in detail.

3.1. Matched filter detector (MFD)

It is well known that, if {s(t), t = 1, . . . , T } and σ 2 are known,the optimal detector in the Neyman–Pearson sense is the replica-correlator which may be implemented as a matched filter detector(MFD) [42]. Even though this case is of little practical interest, weshall consider it solely for the purpose of comparison since theperformance of the MFD provides an upper bound on the perfor-mance of other realizable detectors. In this case, the log-likelihoodratio in (25) leads to the following likelihood ratio test (LRT): De-cide H1 if γMFD(X) > η, where η is the detection threshold, and

γMFD(X) =T∑

t=1

Re(xH

(t)s(t))

(30)

is the test statistic of the MFD.

3.2. Energy detector (ED)

If {s(t), t = 1, . . . , T } are totally unknown, the ML estimate ofs(t) is given by

sED(t) = x(t) for all t. (31)

If the noise variance σ 2 is known, the test statistic is given by

γED(X) =T∑

t=1

xH(t)x(t). (32)

This is the well-known energy detector (ED) [42], and it is pre-sented here only for the sake of comparison.


We note for later reference that the normalized mean-squaresignal estimation error (NMSE), defined as

εED�=

∑Tt=1 E[(sED(t) − s(t))H (sED(t) − s(t))]∑T

t=1 E[sH(t)s(t)] ,

is given by

εED = 3NT

λ, (33)

λ = Es

σ 2, Es =

T∑t=1

E[sH

(t)s(t)], (34)

where Es is the total energy of the modified signal vector s(t)summed over all the snapshots, and λ is the signal energy to noisepower ratio (ENR).

The GLRT does not provide a meaningful solution if the signalis completely unknown and the noise variance is also unknown.

3.3. Subspace detector (SD)

The ED described in Section 3.2 assumes no prior informationabout the structure of the signal vector. It is possible to achievebetter performance by using any prior information that may beavailable. One such detector is the subspace detector (SD) [34,35].This detector is based on the fact that the 3N-dimensional sig-nal vector belongs to an M-dimensional modal subspace. We shallpresent here a simpler formulation of the subspace detector andits theoretical performance analysis.

We know that the pre-whitened signal vector can be expressedas s(t) = R−1/2

0 s(t) = R−1/20 A(φ)b(t), where A(φ) = [a1(φ) · · ·

aM(φ)] is the 3N × M modal steering matrix and {am(φ), m =1, . . . , M} are modal steering vectors defined in (10) and (11) or(15) and (16) respectively. Here and in the sequel, we omit thesubscripts H and V for the sake of brevity. The columns of A(φ)

are linearly independent if 3N � M . Assuming that 3N � M the3N-dimensional signal vector s(t) belongs to the M-dimensionalmodal subspace V (φ) spanned by the columns of R−1/2

0 A(φ), i.e.

V (φ) = span{

R−1/20 a1(φ) · · · R−1/2

0 aM(φ)}

= span{

u1(φ) · · · uM(φ)}, (35)

where {u1(φ) . . . uM(φ)} is the orthonormal basis of V (φ) ob-tained through a Gram–Schmidt transformation process. We notethat s(t) depends on φ though this dependence is generally sup-pressed for the sake of brevity. The signal vector s(t) may thereforebe expressed as

s(t;φ) = U(φ)β(t), U(φ) = [u1(φ) · · · uM(φ)

]. (36)

The M-dimensional vector β(t) and the bearing φ are unknown.For a given φ, the subspace V (φ) is known if the modal wavenum-bers {km, m = 1, . . . M} are known in the case of an HLA, andif the modal wavenumbers {km, m = 1, . . . , M} as well as themode functions {ψm(z), m = 1, . . . , M} are known in the case ofa VLA. Assuming that this information is available, the problemof estimating the 3N-dimensional signal vector s(t) is reduced tothe problem of estimating the M-dimensional vector β(t) and thebearing φ. The conditional ML estimator of β(t) is given by

β(t|φ) = [UH (φ)U(φ)

]−1UH (φ)x(t) = UH (φ)x(t). (37)

Hence, for a given φ, the estimate of s(t;φ) can be written as

sSD(t|φ) = U(φ)β(t|φ) = D(φ)x(t), (38)

where

D(φ) = U(φ)UH (φ) (39)

is the projection matrix onto the subspace V (φ). On replacing s(t)by sSD(t|φ) in the likelihood function (24) and maximizing with re-spect to the unknown parameter φ, we get the following estimateof φ:

φSD = arg max

[T∑

t=1

xH(t)D(φ)x(t)

]. (40)

From (38), the unconditional estimate of s(t) can be written as

sSD(t) = D(φSD)x(t). (41)

On substituting (41) into (26) and (29), we obtain the following ex-pressions for the test statistic for case (a) (σ 2 known) and case (b)(σ 2 unknown)

γSD,Case(a)(X) =T∑

t=1

xH(t)D(φSD)x(t), (42)

γSD,Case(b)(X) =∑T

t=1 xH(t)D(φSD)x(t)∑T

t=1 xH(t)(I3N−D(φSD))x(t)

. (43)

3.4. Truncated subspace detector (TSD)

The SD method presented in Section 3.3 can be employed onlyif the columns of A(φ) are linearly independent, i.e. if M � 3N .Since the number of modes M increases as the frequency is in-creased [53], the applicability of SD is limited by an upper cut-off frequency fc that increases with the number of sensors N; alonger array (larger N) is required for detection of signals of higherfrequency. Moreover, it is observed that, for a given array length N ,the performance of SD suffers degradation as the signal frequencyf is increased even if f < fc . This progressive degradation can beexplained by considering the normalized mean-square signal esti-mation error (NMSE) εSD

εSD�=

∑Tt=1 E[(sSD(t) − s(t))H (sSD(t) − s(t))]∑T

t=1 E[sH(t)s(t)] .

It can be readily shown that

εSD = MTσ 2 + E[∑Tt=1 sH

(t;φ)(I3N − D(φSD))s(t;φ)]Es

,

where Es is the total signal energy, φSD is defined in (40). Thequantity sH

(t;φ)(I3N − D(φSD))s(t;φ) is equal to zero if the esti-mate φSD is equal to the true value φ. We may therefore assumethat E[∑T

t=1 sH(t;φ)(I3N − D(φSD))s(t;φ)] is small compared to

MTσ 2 to arrive at the following result

εSD = MT

λ, (44)

where λ is the ENR defined in (34). The NMSE increases linearlywith increasing M and hence it increases with increasing fre-quency.

In order to extend the applicability of the SD to shorter ar-rays/higher frequencies and also to arrest the degradation associ-ated with increasing frequency, we propose a detector called trun-cated subspace detector (TSD) which uses a truncated model of thesignal vector obtained by projecting s(t, φ) onto a truncated modalsubspace defined as

V ′(φ) = span{

R−1/20 a1(φ) · · · R−1/2

0 aM ′(φ)}

= span{

u1(φ) · · · uM ′(φ)}, M ′ < M. (45)


The set of spanning vectors of V ′(φ) is a subset of the set of span-ning vectors of the full modal subspace V (φ) defined in (35). Thetruncated signal vector s′

(t, φ) can be written as

s′(t, φ) = R−1/2

0 A′(φ)b′(t) = U′(φ)β ′(t), (46)

where

A′(φ) = [a1(φ) · · · aM ′(φ)

],

U′(φ) = [u1(φ) · · · uM ′(φ)

]. (47)

The conditional ML estimator of the M ′-dimensional vector β ′(t)is given by

β′(t|φ) = U′(φ)H x(t). (48)

Hence, for a given φ, the estimate of s(t, φ) can be written as

sTSD(t|φ) = U′(φ)β′(t|φ) = D′(φ)x(t), (49)

where

D′(φ) = U′(φ)U′(φ)H (50)

is the projection matrix onto the truncated modal subspace V ′(φ).Expressions for the estimate of φ and the unconditional estimateof s(t) can be obtained using a procedure analogous to that for SDdescribed in Section 3.3. Thus we get

φTSD = arg max

[T∑

t=1

xH(t)D′(φ)x(t)

], (51)

sTSD(t) = D′(φTSD)x(t). (52)

It can be readily shown that the NMSE of the signal estimator de-fined in (52) is given by

εTSD(M ′, φ

) �=∑T

t=1 E[(sTSD(t) − s(t))H (sTSD(t) − s(t))]∑Tt=1 E[sH

(t)s(t)]= ε

(1)TSD

(M ′) + ε

(2)TSD

(M ′, φ

), (53)

where

ε(1)TSD

(M ′) = M ′T

λ, (54)

ε(2)TSD

(M ′, φ

) = E[∑Tt=1 sH

(t;φ)(I3N − D′(φTSD))s(t;φ)]Es

∼= 1 − E[∑Tt=1 sH

(t)D′(φ)s(t)]Es

= 1 − λ′

λ, (55)

where

λ′ = Es′

σ 2, Es′ =

T∑t=1

E[s′

(t)H s′(t)

]. (56)

In (56), Es′ is the total energy of the truncated signal vectorsover all snapshots and λ′ is the ENR for the truncated signal. TheNMSE εTSD(M ′, φ) has two components ε

(1)TSD(M ′) and ε

(2)TSD(M ′, φ).

The first component ε(1)TSD(M ′) is the error due to noise, and it is

analogous to the MSE εSD defined in (44). Eq. (54) indicates thatε

(1)TSD(M ′) decreases linearly as the number of retained modes M ′

is reduced. The second component ε(2)TSD(M ′, φ) is the error due to

truncation of the normal mode expansion of the signal vector.It is seen from (55) that ε

(2)TSD(M ′, φ) = 0 when M ′ = M . As M ′

is reduced, ε(2)TSD(M ′, φ) increases, but this increase is very slow.

Illustrative plots of the total MSE and its components versus thenumber of retained modes M ′ are shown in Fig. 2 for a 6-sensor

HLA and three different values of SNR viz. 0, 10 and 30 dB. Sim-ilar plots for a 6-sensor VLA are shown in Fig. 3 for −10, 0 and10 dB. The signal frequency is 350 Hz and the number of modesis M = 15. The channel parameters, array parameters, and sourceposition for these figures are the same as those listed in Section 5.It is seen from Figs. 2 and 3 that, as M ′ is reduced from M to 1,the total MSE εTSD(M ′, φ) reduces and reaches a minimum at anoptimal value of M ′ which is quite small (especially in the case ofthe HLA), and may even be equal to 1. Let the optimal value ofM ′ be denoted by M ′

opt . We note that εTSD(M ′opt, φ) is significantly

smaller than εSD . We can therefore expect the performance of TSDto be significantly better than that of SD. The use of a truncatedsignal model in TSD also has the additional advantages of (1) re-ducing the need for channel information to modal wavenumbers ofthe first M ′ modes only, and (2) reducing the computational com-plexity.

The value of M ′opt depends primarily on the signal-to-noise ratio

(SNR), the degree of correlation among the modal steering vectors{a1(φ) · · · aM(φ)}, and the total number of modes M , and to alesser extent on the channel parameters, and the location of thesource. In general, the value of M ′

opt for the HLA is smaller thanthat for the VLA since the modal steering vectors of the HLA aremore highly correlated than those of the VLA. It is seen from (54)that the rate of reduction of ε

(1)TSD(M ′) with decreasing M ′ becomes

higher if λ is reduced. Consequently, the value of M ′opt decreases

as λ is reduced. Figs. 2 and 3 illustrate the dependence of M ′opt on

the more commonly used measure of SNR defined in (104). It isseen that M ′

opt has a larger value at higher SNR, and also that thevalues of M ′

opt for an HLA are smaller than those for a VLA.

The error ε(2)TSD(M ′, φ) due to truncation of the modal expansion

is small because (1) the modal vectors {a1(φ) · · · aM(φ)} in theexpansion of s(t, φ) are highly correlated, and (2) amplitudes ofthe discarded higher order modes are generally quite small due tofaster attenuation of the higher order modes. Let PM′am(φ) denotethe projection of am(φ) on the truncated modal subspace V ′(φ)

for m > M ′ . The L2 norm of the projection error vector is given by

E M ′,m(φ) = ∥∥am(φ) −PM ′am(φ)∥∥2

. (57)

It can be readily shown that E M′,m(φ) increases as |φ − π/2| isincreased. Hence, E M′,m(φ) is maximum at φ = 0. The proximitybetween V (φ) and V ′(φ) is illustrated in Fig. 4, which shows plotsof E M′,m(0) versus m at frequency of 350 Hz for a 6-sensor HLAand four different values of M ′ , viz. M ′ = 1,2,3,4. The channel pa-rameters, signal frequency, and source range-depth have the samevalues as in Section 5. It is evident that the difference betweenthe full modal subspace V (φ) and the truncated modal subspaceV ′(φ) is negligible for all m and for all M ′ > 2. It follows that thesignal vector may be modeled using a drastically truncated modalsubspace without causing a significant modeling error. This con-clusion can be tested by considering the modeling error due toapproximation of the signal vector s(t, φ) by its truncated versions′

(t, φ) defined in (46). The L2 norm of the modeling error vectoris given by

K M ′(φ) = ∥∥s(φ) − s′(φ)

∥∥2. (58)

In (58), the dependence of the signal vector on t is suppressed.Fig. 5 shows the plots of K M′ (0) versus M ′ for a 6-sensor HLA andalso for a 6-sensor VLA for 3 different values of SNR. All the otherparameters have the same values as in Fig. 4. It is seen from Fig. 5that the signal modeling error due to modal subspace truncation isnegligible for M ′ � 3 for the HLA and M ′ � 7 for the VLA. Conse-quently, the optimal value of M ′ for minimizing the mean-squaresignal estimation error εTSD(M ′, φ) is very small for an HLA andsomewhat larger for a VLA.


Fig. 2. Plots of NMSE εTSD and its components (ε(1)TSD(M ′) and ε

(2)TSD(M ′, φ)) vs. M ′ for HLA. (a) SNR 0 dB, (b) SNR 10 dB, (c) SNR 30 dB.

Fig. 3. Plots of NMSE εTSD and its components (ε(1)TSD(M ′) and ε

(2)TSD(M ′, φ)) vs. M ′ for VLA. (a) SNR −10 dB, (b) SNR 0 dB, (c) SNR 10 dB.

Expressions for the test statistics for TSD can be obtained usinga procedure similar to that for SD described in Section 3.3. Thuswe get

γTSD,Case(a)(X) =T∑

t=1

xH(t)D′(φTSD)x(t), (59)

γTSD,Case(b)(X) =∑T

t=1 xH(t)D′(φTSD)x(t)∑T

t=1 xH(t)(I3N−D′(φTSD))x(t)

. (60)

3.5. Approximate signal form detector (ASFD)

The SD and TSD seek to achieve better performance than that ofthe ED by exploiting the knowledge of the modal wavenumbers. Ifthis information is not available, it is still possible to achieve bet-ter detection than that of the ED by using the knowledge of thestructure of the vector gm(φ) (see (13)) associated with each AVS.

Since the modal wavenumbers are subject to fairly tight bounds,viz. 1 > k1/k > · · · > kM/k > cw/cb , where cw = sound speed inwater, cb = sound speed in the ocean bottom, and k = 2π f /cw ,we may use the approximation km/k ≈ 1 for all m and hence ap-proximate gm(φ) as

gm(φ) ≈ g(φ) = [1

√2 cos(φ)

√2 sin(φ)

]T. (61)

This approximation considerably simplifies the detection problemand leads to a much simpler detection algorithm which we referto as the approximate signal form detector (ASFD) [35]. The signalvector s(t, φ) = R−1/2

0 s(t, φ) can now be approximated as

s′′(t, φ) = R−1/2

0 H(φ)p(t), (62)

where

H(φ) = IN ⊗ g(φ) = [h1(φ) · · · hN(φ)

](63)


Fig. 4. EM ′,m(0) versus m for different values of M ′ for HLA. f = 350 Hz, M = 15.

Fig. 5. KM ′ (0) versus M ′ for (a) HLA and (b) VLA, f = 350 Hz, M = 15.

is a 3N × N matrix with linearly independent columns, and p(t) =[p1(t) · · · pN(t)]T where pn(t) denotes the acoustic pressure atthe nth sensor. The approximate signal vector s′′

(t, φ) belongs tothe N-dimensional subspace V ′′(φ) defined as

V ′′(φ) = span{

R−1/20 h1(φ) · · · R−1/2

0 hN(φ)}

= span{

u′′1(φ) · · · u′′

N(φ)}, (64)

where {u′′1(φ) · · · u′′

N (φ)} is the orthonormal basis of V ′′(φ). Wecan therefore rewrite (62) as

s′′(t, φ) = U′′(φ)β ′′(t),where U′′(φ) = [

u′′1(φ) · · · u′′

N(φ)]. (65)

The signal vector approximation defined in (62), (63), and (65) isqualitatively different from that defined in (46). The approximationin (46) involves truncation of the normal mode expansion, whereas(62) and (63) involve an approximation of the relation among thecomponents of the signal vector measured by an AVS. In (46), theunknown vector β ′(t) is M ′-dimensional, whereas the unknownvector β ′′(t) in (65) is N-dimensional. The conditional ML estimateof β ′′(t) and the corresponding estimate of s(t, φ) are given by

β′′(t|φ) = U′′(φ)H x(t), (66)

sASFD(t|φ) = D′′(φ)x(t), (67)

D′′(φ) = U′′(φ)U′′(φ)H . (68)

Expressions for the estimate of φ and the unconditional estimateof s(t) can be obtained using a procedure analogous to that for SDand TSD. Thus we get

φASFD = arg max

[T∑

t=1

xH(t)D′′(φ)x(t)

], (69)

sASFD(t) = D′′(φASFD)x(t). (70)

The NMSE of the signal estimator defined in (70) is given by

εASFD(φ)�=

∑Tt=1 E[(sASFD(t) − s(t))H (sASFD(t) − s(t))]∑T

t=1 E[sH(t)s(t)]

= ε(1)ASFD + ε

(2)ASFD(φ), (71)

where

ε(1)ASFD = NT

λ, (72)

ε(2)ASFD(φ) = E[∑T

t=1 sH(t;φ)(I3N − D′′(φASFD))s(t;φ)]

Es

1 − E[∑Tt=1 sH

(t)D′′(φ)s(t)]Es

= 1 − λ′′

λ, (73)

λ′′ = Es′′

σ 2, Es′′ =

T∑t=1

E[s′′

(t)H s′′(t)

]. (74)

In (74), Es′′ is the total energy of the approximate signal vec-tors s′′

(t) over T snapshots and λ′′ is the ENR of the approximatesignal. The NMSE εASFD(φ) also has two components ε

(1)ASFD and

ε(2)ASFD(φ). The first component ε

(1)ASFD is the error due to noise, and

it is analogous to the NMSE component ε(1)TSD defined in (54). The

second component ε(2)ASFD(φ) arises from the signal modeling error

due to the approximation (61). We have ε(1)ASFD = 1

3 εED while the

other component ε(2)ASFD(φ) is quite small in comparison to ε

(1)ASFD(φ)

because the approximation in (61) introduces only a small error. Itfollows that εASFD 1

3 εED . Hence the ASFD is expected to performbetter than the ED. This prediction is confirmed by the asymptoticanalysis in Section 4.6 and the simulation results presented in Sec-tion 5.

Expressions for the test statistics of ASFD, which are analogousto the corresponding expressions for SD and TSD, are given by


γASFD,Case(a)(X) =T∑

t=1

xH(t)D′′(φASFD)x(t), (75)

γASFD,Case(b)(X) =∑T

t=1 xH(t)D′′(φASFD)x(t)∑T

t=1 xH(t)(I3N − D′′(φASFD))x(t)

. (76)

4. Performance analysis

4.1. Matched filter detector (MFD)

For the MFD, the probability of detection P D and probability offalse alarm PFA are given by [42]

PFA = Q

(η/√

Esσ 2

2

), P D = Q

(Q −1(PFA − √

2λ )), (77)

where Es is the energy of the pre-whitened signal summed overall snapshots and λ = Es/σ

2 is the signal energy to noise powerratio (ENR) defined in (34), Q (·) denotes the right-tail probabilityfunction of the standard normal distribution and Q −1(·) denotesthe inverse of this function.

4.2. Energy detector (ED)

The test statistic γED of the ED, defined in (32), is the total en-ergy of the pre-whitened data vectors. It is the sum of squares of6NT independent Gaussian random variables with variance σ 2/2under both hypotheses. Under H0, the means of these randomvariables are zero; and under H1, the sum of their means is equalto Es = ∑T

t=1 E[sH(t)s(t)]. It follows that, under hypothesis H0,

we have ( 2σ 2 γED) ∼ χ2

6N T , i.e., the random variable 2σ 2 γED has chi-

squared distribution with 6NT degrees of freedom, and under hy-pothesis H1, we have ( 2

σ 2 γED) ∼ χ ′ 26N T (2λ), which is a noncentral

chi-squared distribution with 6NT degrees of freedom and non-centrality parameter 2λ. Thus we have

PFA,ED = Q χ26NT

(2η

σ 2

), (78)

P D,ED = Q χ ′ 26NT (2λ)

(2η

σ 2

), (79)

where the right-hand sides of (78) and (79) denote the right-tailprobability functions of the respective distributions.

4.3. Subspace detector (SD)

Derivation of expressions for P D and PFA requires determina-tion of the probability density functions (PDFs) of the test statisticunder both hypotheses. The test statistic is defined in (42) for case(a), noise variance known, and in (43) for case (b), noise vari-ance unknown. It is difficult to determine the PDFs since the teststatistics are highly nonlinear functions of the data. But we candetermine approximate expressions for the PDFs by assuming thatthe estimate φSD is equal to the true value φ. Validity of this ap-proximation is discussed in Section 5. We note that xH

(t)D(φ)x(t)is a quadratic form involving a complex circular Gaussian randomvector x(t), D(φ) is an idempotent matrix of rank M , and

sH(t)D(φSD)s(t) ∼= sH

(t)D(φ)s(t) = sH(t)s(t) = Es. (80)

We can apply Graybill’s theorem [55] to find the probabilitydistribution of the quadratic form xH

(t)D(φ)x(t). With this, itcan be readily shown that ( 2

σ 2 γSD,Case(a)) ∼ χ22MT under H0, and

( 2σ 2 γSD,Case(a)) ∼ χ ′ 2

2MT (2λ) under H1. Therefore, the expressionsfor PFA and P D under case (a) can be written as

PFA,SD,Case(a) = Q χ22MT

(2η

σ 2

), (81)

P D,SD,Case(a) = Q χ ′ 22MT (2λ)

(2η

σ 2

). (82)

In case (b), the matrices D(φSD) and I3N − D(φSD) in the numer-ator and denominator of (43) are idempotent matrices of rank Mand 3N − M respectively. Hence, on applying Graybill’s theorem[55,56], we find that the numerator of (43), scaled by the factor(2/σ 2), has chi-squared distribution with 2MT degrees of freedomunder hypothesis H0 and noncentral chi-squared distribution with2MT degrees of freedom and noncentrality parameter 2λ underH1. The denominator of (43), scaled by the factor (2/σ 2), has chi-squared distribution with (6N − 2M)T degrees of freedom underboth hypotheses since sH

(t)(I3N − D(φSD))s(t) = 0 in view of (80).The numerator and denominator of (43) are statistically indepen-dent since they represent the squares of the norms of projectionsof x(t) onto mutually orthogonal subspaces. It follows that underhypothesis H0 the test statistic defined in (43) has F distributiondenoted by F2MT ,(6N−2M)T , and under hypothesis H1 it has non-central F distribution with noncentrality parameter 2λ, denotedby F ′

2MT ,(6N−2M)T (2λ). Hence the expressions for PFA and P D canbe written as

PFA,SD,Case(b) = Q F2MT ,(6N−2M)T (η), (83)

P D,SD,Case(b) = Q F ′2MT ,(6N−2M)T (2λ)(η), (84)

where Q F (.) and Q F ′ (.) denote the right-tail probability functionsof the respective distributions.

4.4. Truncated subspace detector (TSD)

As in the case of SD, we can once again determine the approx-imate distributions of the test statistics, defined in (59) and (60),under the assumption that φTSD = φ. Before proceeding further, wenote some important differences between SD and TSD. It can beshown that

sH(t)D′(φ)s(t) = s′ H

(t)s′(t), (85)

where s′(t) = U′β ′(t) is the truncated signal vector defined in (46).

It follows that sH(t)D′(φ)s(t) �= sH

(t)s(t), whereas sH(t)D(φ)s(t) =

sH(t)s(t). Also, rank(D′(φ)) = M ′ , whereas rank(D(φ)) = M . Hence,

we have ( 2σ 2 γTSD,Case(a)) ∼ χ2

2M′T under H0, and ( 2σ 2 γTSD,Case(a)) ∼

χ ′ 22M′T (2λ′) under H1, and

PFA,TSD,Case(a) = Q χ22M′T

(2η

σ 2

), (86)

P D,TSD,Case(a) = Q χ ′ 22M′T

(2λ′)

(2η

σ 2

), (87)

where λ′ is the ENR for the truncated signal defined in (56). Forcase (b), the numerator of (60), scaled by the factor (2/σ 2), haschi-squared distribution with 2M ′T degrees of freedom under hy-pothesis H0, and noncentral chi-squared distribution with 2M ′Tdegrees of freedom and noncentrality parameter 2λ′ under H1. Thedenominator of (60), scaled by the factor (2/σ 2), has chi-squareddistribution with (6N − 2M ′)T degrees of freedom under hypoth-esis H0 and noncentral chi-squared distribution with (6N − 2M ′)Tdegrees of freedom and noncentrality parameter 2(λ − λ′) un-der H1. The numerator and denominator are statistically indepen-dent. It follows that under hypothesis H0 the test statistic definedin (60) has F distribution denoted by F2M′T ,(6N−2M′)T , and underhypothesis H1 it has a doubly noncentral F distribution denotedby F ′′

2M′T ,(6N−2M′)T (2λ′,2λ − 2λ′). Hence the expressions for PFA

and P D can be written as


PFA,TSD,Case(b) = Q F2M′T ,(6N−2M′)T(η), (88)

P D,TSD,Case(b) = Q F ′′2M′T ,(6N−2M′)T

(2λ′,2λ−2λ′)(η), (89)

where Q F (·) and Q F ′′ (·) denote the right-tail probability functionsof the respective distributions.

4.5. Approximate signal form detector (ASFD)

The test statistics of ASFD for cases (a) and (b), given by (75)and (76), are analogous to the corresponding test statistics for theTSD, with D′(φ) replaced by D′′(φ). Since D′′(φ) is an idempotentmatrix of rank N , the expressions for PFA and P D are similar tothose for the TSD. Thus we have

PFA,ASFD,Case(a) = Q χ22NT

(2η

σ 2

), (90)

P D,ASFD,Case(a) = Q χ ′ 22NT (2λ′′)

(2η

σ 2

), (91)

PFA,ASFD,Case(b) = Q F2NT ,4NT (η), (92)

P D,ASFD,Case(b) = Q F ′′2NT ,4NT (2λ′′,2λ−2λ′′)(η), (93)

where λ′′ is defined in (74).

4.6. Asymptotic performance analysis

The performance analysis in the five preceding subsections isbased on 3NT data samples provided by an AVS array of N sensorsover T snapshots. It is of interest to also consider an asymptotic(3NT → ∞) performance analysis since such an analysis facili-tates an easy comparison of the performance of different detectorsand provides a better insight into their relative merits. For case(a) and a given φ, the test statistics of the detectors, defined in(32), (42), (59), and (75) are asymptotically (3NT → ∞) Gaussian.Hence, if the noise variance is known, the asymptotic expressionfor the receiver operating characteristic (ROC) of each detector canbe written as [42]

P D,Case(a) ∼ Q

(ν0

ν1Q −1(PFA) − μ1 − μ0

ν1

)

= Q

(Q −1(PFA) − δ

τ

), (94)

where ∼ denotes asymptotic equality in the above expression, and

τ = ν1

ν0, δ = μ1 − μ0

ν0, μ j = E[γ ; H j],

ν2j = var(γ ; H j), j = 0,1. (95)

In (94), μ j and ν j are respectively the mean and standard devi-ation of γ under hypothesis H j , τ is the ratio of standard devia-tions, and δ is the parameter known as the deflection coefficient. Itis seen from (94) that, for a given PFA , the probability of detectionP D increases when δ and/or τ increases. Hence the relative mag-nitudes of δ and the relative magnitudes of τ provide measures forcomparing the performance of the different detectors under con-sideration.

For the case of known noise variance, each detector has a teststatistic of the form defined in (26) which is reproduced below:

γCase(a)(X) =T∑

t=1

[2 Re

(xH

(t)s(t)) − sH

(t)s(t)].

Using the expressions for s(t) for different detectors defined in(31), (41), (52) and (70), and noting that x(t) = w(t) under H0 andx(t) = s(t) + w(t) under H1, we can obtain the following results

μ0,ED = 3NTσ 2, μ1,ED = Es + 3NTσ 2,

ν20,ED = 3NTσ 4, ν2

1,ED = 2σ 2 Es + 3NTσ 4, (96)

τED =√

1 + 2λ

3NT, δED = λ√

3NT, (97)

μ0,SD = MTσ 2, μ1,SD = Es + MTσ 2,

ν20,SD = MTσ 4, ν2

1,SD = 2σ 2 Es + MTσ 4, (98)

τSD =√

1 + 2λ

MT, δSD = λ√

MT, (99)

μ0,TSD = M ′Tσ 2, μ1,TSD = Es′ + M ′Tσ 2,

ν20,TSD = M ′Tσ 4, ν2

1,TSD = 2σ 2 Es′ + M ′Tσ 4, (100)

τTSD =√

1 + 2λ′M ′T

, δTSD = λ′√

M ′T, (101)

μ0,ASFD = NTσ 2, μ1,ASFD = Es′′ + NTσ 2,

ν20,ASFD = NTσ 4, ν2

1,ASFD = 2σ 2 Es′′ + NTσ 4, (102)

τASFD =√

1 + 2λ′′NT

, δASFD = λ′′√

NT. (103)

We can draw some useful conclusions from Eqs. (96)–(103).We recall that 3N > M > M ′ . It follows from (97) and (99) thatδSD > δED and τSD > τED . It can be readily verified that the signalenergy-to-noise power ratios (ENRs) λ (for the signal vector s) andλ′′ (for the approximate signal vector s′′) are very close to eachother. It can also be verified that the value of λ′ (ENR of the trun-cated signal vector s′) decreases very slowly as M ′ is reduced, andthe rate of reduction of λ′ becomes large only at very small val-ues of M ′ . Therefore, it follows from (101) that the values of δTSDand τTSD keep increasing as M ′ is reduced until an optimal valueof M ′ is reached, as shown in Fig. 6. (We recall that a differentcriterion of optimality, viz., minimization of the normalized mean-square signal estimation error εTSD , was considered in Section 3.The two criteria may lead to slightly different optimal values ofM ′ .) It is seen from Fig. 6 that δTSD > δSD and τTSD > τSD as longas the truncation of the signal vector in TSD is not too severe. Wemay therefore make the following predictions: (1) TSD performsbetter than SD, which in turn performs better than ED, and (2) theperformance of TSD keeps improving as M ′ is reduced from M to1, till it reaches its optimal value. In the case of ASFD, it is obviousfrom (97) and (103) that δASFD > δED and τASFD > τED , and thereforewe can predict that ASFD performs better than ED. The perfor-mance of ASFD relative to that of the SD and TSD depends on therelative magnitudes of N , M and M ′ . It is seen from (99) and (103)that δASFD > δSD and τASFD > τSD if M > N . Consequently ASFD per-forms better than SD if M > N , and worse than SD if M < N . Itis seen from (101) and (103) that δTSD > δASFD and τTSD > τASFDif M ′ < N . For HLA, optimal value of M ′ is very close to 1 andtherefore optimal M ′ is almost always less than N . Hence, for HLA,TSD with optimal M ′ performs better than ASFD almost always.For VLA, it turns out that optimal M ′ ∼= N and consequently, foroptimal M ′ , δTSD ∼= δASFD and τTSD ∼= τASFD . Therefore, for VLA, theperformance of ASFD is very close to that of TSD with optimal M ′ .

In order to illustrate the conclusions and predictions of thepreceding paragraph, we shall consider an example with N = 6,T = 20, f = 350 Hz, M = 15, and SNR = −10 dB. SNR (in dB) isdefined as

SNR = 10 log10

(1

N

N∑n=1

|pn|2σ 2

n

), (104)

where pn and σ 2n are respectively the signal component of acous-

tic pressure and the variance of noise at the nth sensor. Only the


Fig. 6. Variation of (a) standard deviation ratio τ and (b) deflection coefficient δ of TSD with number of retained modes M ′ for different values of SNR, for HLA (solid lines)and VLA (dashed lines), for f = 350 Hz, array length N = 6, source bearing φ = 20◦ .

Table 1Values of deflection coefficient δ, standard deviation ratio τ , NMSE ε and probabil-ity of detection P D for different detectors, at SNR = −10 dB, T = 20 snapshots, fora 6-sensor AVS array.

Detectors Deflectioncoefficient δ

Standard deviationratio τ

NMSE ε Probability ofdetection (P D )

TSD (HLA) 7.698 2.108 0.561 0.9974ASFD (HLA) 3.196 1.258 3.366 0.5383SD (HLA) 2.022 1.111 8.57 0.1565ED (HLA) 1.846 1.093 10.28 0.1178TSD (VLA) 2.486 1.224 3.401 0.2963ASFD (VLA) 2.644 1.218 3.811 0.3453SD (VLA) 1.675 1.093 10.34 0.0886ED (VLA) 1.529 1.078 12.41 0.0668

pressure components of signal and noise are considered in thedefinition of SNR as per normal convention [20], to facilitate afair comparison between the performance of AVS and APS (acous-tic pressure sensor) arrays. Values of the channel parameters andsource coordinates used in this example are the same as thoselisted in Section 5. For TSD, we have chosen the optimal valuesof M ′ , which turn out to be M ′ = 1 for the HLA and M ′ = 5 forthe VLA as shown in Fig. 6. For this example, we have the fol-lowing values of ENRs: λ = 35.03, λ′ = 34.44 (for M ′ = 1), andλ′′ = 35.02 for the HLA, and λ = 29.02, λ′ = 24.86 (for M ′ = 5),and λ′′ = 28.96 for the VLA. Values of δ, τ , the normalized mean-square signal estimation error ε, and the asymptotic probability ofdetection P D (for false alarm probability PFA = 0.001) for differ-ent detectors are tabulated in Table 1. The values of δ, τ , and P Dconfirm the predictions of the preceding paragraph. We also notethat a reduction in the value of ε is almost always accompaniedby an increase in the value of P D , as expected. For VLA, we haveδTSD = 2.486 < δASFD = 2.644 and τTSD = 1.224 > τASFD = 1.218in the present example. On substituting these values of δ andτ in (94), we see that P D of ASFD is larger than P D of TSDif {Q −1(PFA) − 2.486}/1.224 > {Q −1(PFA) − 2.644}/1.218, i.e. ifPFA > Q (34.72). Since Q (34.72) � 1, it follows that P D of ASFDis larger than P D of TSD for almost all values of PFA . It is how-

ever emphasized that, under different conditions, P D of TSD–VLAwith optimal M ′ may be higher than that of ASFD–VLA, but thedifference in performance is always small.

Additional results based on the same example are provided inFigs. 6 and 7. Fig. 6 shows the variation of δTSD and τTSD with M ′for different values of SNR. For an HLA, the values of δTSD and τTSDare maximized at M ′ = 1, whereas for a VLA, they are maximizedat M ′ = 5. It is also seen from Fig. 6 that the maximum values ofδTSD and τTSD for the VLA are consistently smaller than those forthe HLA at all values of SNR. Hence, for the TSD, the performanceof an HLA may be expected to be consistently better than that ofa VLA. The variation of δ and τ with SNR for different detectorsis shown in Fig. 7. In this figure, plots for TSD are shown for theoptimal values of M ′ , viz. M ′

opt = 1 for the HLA and M ′opt = 5 for

the VLA. Fig. 7 confirms once again that, if N < M < 3N and M ′ isoptimal, we have δTSD > δASFD > δSD > δED and τTSD > τASFD > τSD >

τED for HLA, and δASFD ∼= δTSD > δSD > δED and τASFD ∼= τTSD > τSD >

τED for VLA.It is known that the number of modes M increases as the fre-

quency is increased. It follows from (99) that both δSD and τSDreduce with increasing frequency. We may therefore expect theperformance of the SD to degrade as the frequency is increasedtill M reaches the threshold 3N . For all other detectors, the perfor-mance is expected to be independent of frequency. All predictionsin the preceding paragraphs will be verified through simulation re-sults presented in Section 5.

If noise variance is not known (case (b)), the test statistics de-fined in (43), (60) and (76) converge asymptotically to ratios ofindependent Gaussian random variables with non-zero means. Theasymptotic PDFs of the test statistics have complicated expressionswhich do not provide any useful insights. Therefore, we shall notpursue the asymptotic analysis of case (b).

5. Simulation results

This section presents a detailed study of the performance ofall the detectors. The ocean is modeled as a Pekeris channel [53]


Fig. 7. Variation of (a) standard deviation ratio τ and (b) deflection coefficient δ with SNR for different detectors, f = 350 Hz, array length N = 6, source bearing φ = 20◦ .

Fig. 8. Comparison of non-asymptotic and asymptotic theoretical results for the case of known noise variance. P D vs. SNR at PFA = 0.001. (a) HLA, (b) VLA.

comprising a homogeneous water layer of constant depth overa fluid half-space. The following values of channel, source and ar-ray parameters have been assumed unless otherwise stated: oceandepth h = 70 m, sound speed in water cw = 1500 m/s, soundspeed in ocean bottom cb = 1700 m/s, density of water ρw =1000 kg/m3, density of ocean bottom ρb = 1500 kg/m3, attenua-tion in ocean bottom ζ = 0.2 dB/λb , where λb = cb/ f is the wave-length in the ocean bottom, number of sensors N = 6 for both HLAand VLA, array depth z1 = 40 m for HLA and z1 = 15 m for top-most sensor in the VLA, inter-sensor spacing d = λw/2 = 15/7 m,where λw = cw/ f is the wavelength in water, source at ranger = 5000 m, depth zs = 40 m, bearing φ = 20◦ with respect to theendfire direction of the HLA, and frequency f = 350 Hz. At thisfrequency, the number of normal modes in the channel is M = 15.It is assumed that the signal does not vary from one snapshot toanother. Detection is done using T = 20 snapshots of data, and theprobability of false alarm is fixed at PFA = 0.001.

Before presenting the results, we recall that the ED and theASFD do not require any prior information about the channel. TheTSD requires the knowledge of wavenumbers of M ′ lowest or-der modes and the SD requires the knowledge of all the modalwavenumbers if an HLA is deployed. If a VLA is used, the SD andTSD require the knowledge of the modal eigenfunctions also, inaddition to the modal wavenumbers.

Fig. 8 shows the theoretical plots of P D versus SNR at PFA =0.001 for different detectors for the case of known noise vari-ance (case (a)). The non-asymptotic results ((77)–(79), (81)–(82),(86)–(87), (90)–(91)) are shown as dashed lines and the asymptoticresults ((94) in conjunction with (96)–(103)) are shown as solidlines. The TSD results have been obtained using values of M ′ thatminimize the NMSE εTSD . Performance of the unrealizable MFD isalso shown to indicate the upper bound on P D for any realizabledetector. It is seen that the asymptotic and non-asymptotic resultsexhibit the same trend, even though there are some quantitative


Fig. 9. Comparison of theoretical (non-asymptotic) and simulation results for the case of known noise variance. P D vs. SNR at PFA = 0.001. (a) HLA, (b) VLA.

differences between the two sets of results. Among the realizabledetectors, the ranking of HLA-based detectors in decreasing orderof performance is TSD, followed by ASFD, SD, and ED. For VLA-based detectors, the order of TSD and ASFD is reversed. Theseresults are consistent with the values of the deflection coefficientδ shown in Table 1 and Fig. 6, and are in conformity with the pre-dictions in Section 4.6. It is noteworthy that the ASFD, that doesnot require any channel information, compares favorably with theTSD that requires the knowledge of wavenumbers/eigenfunctionsof the lowest M ′ modes of the channel. The ASFD compensates forthe lack of channel information by exploiting the knowledge of anapproximate relationship among different components of the sig-nal measured by an AVS. The performance of the SD is poor eventhough it uses the modal wavenumber/eigenfunction informationof all the M modes. This is so because the number of modes Mis large and hence the NMSE of the SD is high, which leads toa degradation in performance. The performance of the ED is thepoorest because it does not use any prior information.

In Fig. 9, the theoretical (non-asymptotic) results (dashed lines)are compared with simulation results (solid lines). For, SD, TSD andASFD, two types of simulation results are considered, viz. simula-tions of realizable detectors that do not assume prior knowledge ofthe bearing φ (solid lines), and simulations of unrealizable detec-tors which assume that φ is known (dotted lines). The variance ofnoise is assumed to be known (case (a)). The theoretical results arebased on the expressions for PFA and P D given in Sections 4.1–4.5.The following observations can be made from Fig. 9. For MFD andED, the simulation results match the theoretical predictions veryclosely. In the case of SD, TSD and ASFD, there is very good agree-ment between theoretical results and the simulation results thatare based on the assumption that φ is known. But the P D of therealizable versions of these detectors are significantly lower thanthe theoretical predictions at low SNR. The difference between the-oretical predictions and actual performance can be explained asfollows. The theoretical results are based on the assumption thatthe bearing estimates φSD , φTSD and φASFD may be approximatedby the true value φ. This assumption has been made to simplifythe theoretical analysis. But the means and standard deviations ofthe bearing estimation errors keep increasing as SNR is reduced,as illustrated in Fig. 10 which shows the variation of the (a) biasand (b) root-mean-square error of the bearing estimates of the SD,

TSD and ASFD with variation in the SNR. Consequently, for SD, TSDand ASFD, the gap between theoretical predictions and actual per-formance also keeps increasing as SNR is reduced. Even though thetheoretical results tend to overestimate the actual performance, theformer provide a useful insight into the comparative performanceof different detectors.

It is also of interest to compare the performance of HLA-baseddetectors (Figs. 8(a) and 9(a)) and VLA-based detectors (Figs. 8(b)and 9(b)). It is seen that the performance of an HLA-based detec-tor is better than that of a similar VLA-based detector. One reasonfor this difference is that noise at the sensors of the HLA is i.i.d.,whereas noise at the sensors of the VLA is correlated and it hasspatially varying variance. The whitening transformation leads to areduction of the energy of the VLA signal vector and a consequentreduction in the performance of all VLA-based detectors. In thecase of the TSD, there is an additional reason for the better perfor-mance of HLA. The optimal value of M ′ for the HLA is lower thanthat for the VLA due to the higher degree of correlation among themodal steering vectors of the HLA. Therefore the NMSE εTSD for theVLA is higher than that for the HLA, and the higher NMSE trans-lates into a lower detector performance. Finally, it is seen fromFig. 10 that (1) for SD and TSD, the bearing estimation error of aVLA-based detector is larger than that of a similar HLA-based de-tector, and (2) for ASFD, the bearing estimation errors of HLA andVLA are almost the same. Therefore, for SD and TSD, the actualdifferences between the performance of HLA-based and VLA-baseddetectors (shown in Figs. 9(a) and 9(b)) are larger than the theo-retically predicted differences (shown in Figs. 8(a) and 8(b)).

Fig. 11 shows a comparison of theoretical (finite data) and sim-ulation performances of TSD, ASFD and SD for the case of unknownnoise variance σ 2 (case (b)). A comparison of Figs. 9 and 11 showsthat lack of knowledge of σ 2 causes degradation in performanceof all the detectors. But the degradation is less severe in the caseof TSD. It is well known that this degradation can be arrested byemploying secondary data vectors which are statistically identicalto the noise-only data vectors.

We shall now study the variation of detector performance withfrequency. Fig. 12 shows simulation results for the variation ofprobability of detection P D with frequency f , for PFA = 0.001 andSNR = −9 dB. All the other parameters have the values mentionedat the beginning of this section. When f = 30 Hz, the number


Fig. 10. Bearing estimation errors of SD, TSD and ASFD. (a) Bias vs. SNR. (b) Root-mean-square error vs. SNR.

Fig. 11. Comparison of theoretical (non-asymptotic) and simulation results for the case of unknown noise variance. P D vs. SNR at PFA = 0.001. (a) HLA, (b) VLA.

Fig. 12. Variation of P D (at PFA = 0.001) with frequency for HLA with 6 sensors. SNR = −9 dB.


Fig. 13. Comparison of simulated performance of AVS and APS HLAs. P D vs. SNR at PFA = 0.001. (a) ED and (b) TSD.

of normal modes M is equal to 1, and thus the SD and the TSDare equivalent. As f is increased, M increases and consequentlythe performance of the SD suffers a progressive degradation. Forf > 120 Hz, we have M > N = 6, and hence the performance ofthe SD dips below that of the ASFD. At f = 398 Hz, M = 3N = 18,and the performance of SD is the same as that of ED. At still higherfrequencies M is greater than 3N , and hence the SD cannot beused for detection. The performance of the other detectors doesnot vary with frequency. These results are in agreement with thepredictions at the end of Section 4.6.

In the context of source localization applications, the superior-ity of AVS arrays over acoustic pressure sensor (APS) arrays hasbeen well established [3,7–23]. It is therefore interesting to com-pare the detection performance of AVS and APS arrays. Such acomparison is shown in Fig. 13. It is seen from Fig. 13(a) that forthe ED, the performance of the 6-sensor AVS HLA is equal to thatof the 18-sensor APS HLA. This result can be explained by notingthat the performance of the ED depends only on the ENR λ, andthat the ENR at an N-sensor AVS array is equal to the ENR at a3N-sensor APS array. However, in the case of the TSD, the perfor-mance of the 6-sensor AVS HLA is slightly better than that of the18-sensor APS HLA. This difference is attributed to the superiorbearing estimation capability of the AVS array due to additionaldirectivity provided by the velocity measurements of the AVS [57].Better bearing estimation translates into a better estimation of thesignal vector and thus a better detection performance. These twinadvantages of an AVS array, viz. higher SNR and directivity, over anAPS array of the same size N and aperture (= (N − 1)d for a uni-form linear array) are recognized and documented in the literatureon AVS array signal processing [3,57]. The ability of an N-sensorAVS array to attain the same level of performance as a 3N-sensorAPS array is a result of great practical interest. In an array-basedmeasurement system, a considerable portion of the cost is relatedto the construction, deployment and location calibration of sensorpackages, and these costs depend on the number of sensor pack-ages deployed [3]. Therefore, a reduction from 3N sensor packagesin an APS array to N sensor packages in an AVS array will lead to asignificant cost reduction. Moreover, in a towed array system, the

drag on the towed array depends on the length of the array [58].The drag can be reduced considerably if an APS array is replacedby a shorter AVS array.

6. Conclusion

A detailed investigation of four methods, viz. ED, SD, TSD andASFD, for narrowband detection of an acoustic source in a range-independent shallow-ocean using an AVS array has been presentedin this paper. Expressions for probability of false alarm and proba-bility of detection were derived for the asymptotic case and thefinite-data case, and the theoretical predictions were comparedwith simulation results. The signal vector at the sensor array isnot known due to the unknown location of the source. Hence alldetectors employ a generalized likelihood ratio test (GLRT) whichinvolves a maximum likelihood estimation of the array signal vec-tor. The ED does not use any model for the array signal vector.The SD and TSD employ models based on the normal mode the-ory. The ASFD employs a model that exploits the relation betweenthe acoustic pressure and particle velocity at each sensor. Differ-ent models lead to different signal-vector estimation errors. Ex-pressions for the normalized mean-square signal estimation error(NMSE) were derived for each detector, and it was shown thatthere exists a strong negative correlation between the NMSE andthe detector performance.

If an HLA is employed, the signal model used by the SD re-quires the knowledge of all the modal wavenumbers, while theTSD requires the knowledge of wavenumbers of a small numberof the lowest order modes. If a VLA is employed, the knowledgeof the corresponding mode functions is also required. No chan-nel information is required by the ED and the ASFD. The ASFDexploits the knowledge of an approximate relationship among dif-ferent components of the signal vector to achieve a significantlybetter performance than the ED.

Theoretical derivations as well as simulation results indicatethat, in the case of an HLA, the best performance is achieved byTSD, followed by ASFD, SD, and ED in that order. These results areconsistent with the NMSEs associated with these detectors. In the


case of a VLA the performance of TSD and ASFD are very closeto one another, and either of them may perform better depend-ing on the values of various parameters. For all the detectors, theperformance of the HLA is significantly better than that of the VLAbecause (1) the HLA provides a better estimate of the bearing ofthe source, (2) noise at the VLA is spatially correlated whereas itcan be assumed to be uncorrelated in the case of an HLA, and (3)in the case of the TSD, the NMSE is lower when used with theHLA than with the VLA, due to a smaller value of M ′

opt . TSD, SD,and ED are detection strategies which can be used by both APS andAVS arrays. For a given detector and array size, the performance ofan AVS array is significantly better than that of the APS array. TheASFD is a detection strategy that is unique to an AVS array.

The analysis presented in this paper is based on the assump-tion that the environmental noise is Gaussian. It is known that theassumption of Gaussianity is not valid in all environments. There-fore, it would be of interest to enlarge the scope of the analysis toinclude non-Gaussian/impulsive noise.

Acknowledgments

The authors would like to thank the anonymous reviewers andMr. P.V. Nagesha for useful comments and suggestions. This workwas partly supported by a grant from National Institute of OceanTechnology, Chennai, India, under the Ocean Acoustics Programme.

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Hari Vishnu completed his B.Tech. in Electron-ics and Electrical Engineering from National Insti-tute of Technology, Calicut (NITC), India, in 2008. Hecompleted his Ph.D. in Computer Engineering fromNanyang Technological University (NTU), Singapore, in2013. He currently works at the Acoustic ResearchLaboratory, National University of Singapore. His re-search interests are in underwater acoustic modelling,acoustic vector sensor processing, signal processing in

impulsive noise including detection and localization and parameter esti-mation using Kalman filtering.

G.V. Anand received the Ph.D. degree in Engineer-ing from Indian Institute of Science (IISc), Bangalore,in 1971. He served on the faculty of the Departmentof Electrical Communication Engineering, IISc, from1969 to 2006. He was a Visiting Academic Staff Fellowat University College, London, in 1978–1979, VisitingScientist at Naval Physical and Oceanographic Labora-tory, Kochi, India, in 1996–1997, and Visiting Professorin University of Angers, France, in 2002, and Cankaya

University, Ankara, Turkey, in 2003–2004. He is currently Visiting Profes-sor at IISc and the MRD Chair Professor at PES Institute of Technology,Bangalore. He is a Fellow of Indian Academy of Sciences, Indian NationalAcademy of Engineering, Institution of Electronics and CommunicationEngineers, India, and Acoustical Society of India. His research interestsinclude statistical signal processing, ocean acoustics, and nonlinear dy-namics.

A.B. Premkumar received his B.Sc. and B.E. de-grees in India. He briefly worked in the R&D divisionof a large communication industry in Bangalore, India.He received his M.S. and Ph.D. degrees in digital sig-nal processing from the University of Idaho, Moscow,ID, USA. He has held various teaching positions since1992 in the U.S. and Singapore. Currently he is an As-sociate Professor in the School of Computer Engineer-ing, NTU, Singapore. His research interests are digital

filters and their applications in wireless communication, software definedradio, and underwater tracking and target localization.















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Reshetov LA Narrowband signal detection techniques in shallow ocean

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