Aliasing-Free Wideband Beamforming Using Sparse Signal Representation

Abstract

Sparse signal representation (SSR) is considered to be an appealing alternative to classical beamforming for direction-of-arrival (DOA) estimation. For wideband signals, the SSR-based approach constructs steering matrices, referred to as dictionaries in this paper, corresponding to different frequency components of the target signal. However, the SSR-based approach is subject to ambiguity resulting from not only spatial aliasing, just like in classical beamforming, but also from the over-completeness of the dictionary, which is typical to SSR. We show that the ambiguity caused by the over-completeness of the dictionary can be alleviated by using multiple measurement vectors. In addition, by considering the uniform linear array (ULA) structure, we argue that if the target signal contains at least two frequencies, whose absolute difference phrased in wavelengths is larger than twice the array spacing, the spatial aliasing corresponding to these frequencies will be completely distinct. These properties enable us to adapt the existing ℓ₁ algorithms to extract the target DOAs without ambiguity.

Full text

Page 1

3464IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 7, JULY 2011
Aliasing-Free Wideband Beamforming Using
Sparse Signal Representation
Zijian Tang, Gerrit Blacquière, and Geert Leus
Abstract—Sparse signal representation (SSR) is considered to be
an appealing alternative to classical beamforming for direction-of-ar-
rival (DOA) estimation. For wideband signals, the SSR-based approach
constructs steering matrices, referred to as dictionaries in this paper,
corresponding to different frequency components of the target signal.
However, the SSR-based approach is subject to ambiguity resulting from
not only spatial aliasing, just like in classical beamforming, but also from
the over-completeness of the dictionary, which is typical to SSR. We show
that the ambiguity caused by the over-completeness of the dictionary
can be alleviated by using multiple measurement vectors. In addition, by
considering the uniform linear array (ULA) structure, we argue that if the
target signal contains at least two frequencies, whose absolute difference
phrased in wavelengths is larger than twice the array spacing, the spatial
aliasing corresponding to these frequencies will be completely distinct.
These properties enable us to adapt the existing
the target DOAs without ambiguity.
algorithms to extract
Index Terms—Direction-of-arrival estimation, multiple-dictionary, or-
thogonal matching pursuit, sparse signal representation, spatial aliasing,
uniform linear array, wideband beamforming.
I. INTRODUCTION
DOA estimation by means of beamforming can be subject to
ambiguities, for instance due to spatial aliasing. For a uniform linear
array (ULA), spatial aliasing occurs when the spacing of the ULA
is not small enough (the spacing is larger than half the apparent
wavelength). On the other hand, a larger spacing, leading to a larger
aperture, is often desired to the benefit of angle resolution. Although
the aperture can also be enlarged by embedding more channels, this
is not always feasible due to hardware and processing costs as well as
restrictions in deployment and storage.
In practical sonar signal processing, many underwater objects emit
signals with a very wide band or at a very high frequency. For in-
stance,thetoothedwhales(odontocetes)canemitclickssweepingfrom
0 to 20 kHz. Another example, given at the end of this paper, is a
diverequippedwithanopen-circuitbreathingset(scuba).Theexhaling
sound of the diver is so wideband that it can almost be viewed as
white[1].Wideband“aliasing-free”beamforminghasbeenreportedin,
e.g., [2]–[5]. These works require statistical knowledge to reconstruct
Manuscript received September 01, 2010; revised December 02, 2010 and
March 25, 2011; accepted March 25, 2011. Date of publication April 07, 2011;
date of current version June 15, 2011. The associate editor coordinating the re-
view of this manuscript and approving it for publication was Prof. Jean-Yves
Tourneret. This work was supported by the TNO project 032.31642 and in part
by NWO-STW under the VICI program (project 10382).
Z. Tang is with the TNO Defence, Security and Safety, 2509 JG, The Hague,
the Netherlands, and also with the Delft University of Technology—Fac.
EEMCS, 2628 CD Delft, The Netherlands (e-mail: zijian.tang@tno.nl;
z.tang@tudelft.nl).
G. Blacquière is with the TNO Defence, Security and Safety, 2509 JG, The
Hague,theNetherlands,andalsowiththeDelftUniversityofTechnology—Fac.
Civil Engineering and Geosciences, 2628 CN Delft, The Netherlands (e-mail:
gerrit.blacquiere@tno.nl; g.blacquiere@tudelft.nl).
G. Leus is with the Delft University of Technology—Fac. EEMCS, 2628 CD
Delft, The Netherlands (e-mail: g.j.t.leus@tudelft.nl).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2011.2140108
the signal/noise subspace, which is (for both signal and noise) not al-
ways available or reliable due to the volatile underwater acoustic envi-
ronment. For this reason, the delay-sum beamformer [6] is sometimes
preferredinpracticalsonarapplications,althoughitismoresusceptible
to spatial aliasing and renders a poor resolution. Most notably, [2]–[5]
are not free of spatial aliasing in a strict sense. First, it is often as-
sumed that the array manifolds of the target DOAs are independent for
all the frequencies, a situation that lacks theoretical guarantees. Even
if this assumption is true, the aliasing-induced ambiguity is not com-
pletely resolved: at most, it is suppressed to be indistinguishable from
the background noise.
Recently, the sparse signal representation problem, a topic bearing
a close affinity with compressive sensing (CS) [7], has attracted enor-
mousattention.TheideaofutilizingSSR(andCS)innarrowbandDOA
estimation has been reported in various contexts. In [8], CS is applied
in the time domain to reduce the ADC sampling rate for each channel
of the array. A more common approach is to use SSR, as an alterna-
tive to conventional beamforming, to improve the angle resolution [9],
[10],ortoreducehardwarecomplexity[11].Likethedelay-sum beam-
former, the SSR-based methods do not require signal statistics. On the
other side, they rely on the assumption that ambiguity is not present.
NotethattheambiguityinherenttoSSR-basedmethodscomesnotonly
fromspatialaliasing,butisalsoduetotheover-completenessofthedic-
tionary. The latter is related to the robustness of an SSR system, which
is often characterized by the restricted isometry property (RIP) [7].
This paper gives a rigorous ambiguity analysis for DOA estimation
in the context of wideband signals. First, we show that the SSR system
will be more robust to nonunique solutions if multiple measurement
vectors (MMV) are utilized [12]–[14]. Next, we exploit the property
that spatial aliasing is frequency-dependent and show that if there exist
at least two frequencies, whose absolute difference phrased in wave-
lengths is larger than twice the array spacing, the spatial aliasing re-
lated to these two frequencies will take a completely different form.
This implies that by judiciously choosing the frequencies and con-
structing the corresponding dictionaries, the resulting signal models
will share a common sparse structure, which means that the positions
of the nonzero coefficients correspond only to the true target DOAs, al-
though they may have different values across the dictionaries. This fact
resembles the joint sparsity model-2 (JSM-2) considered in [15], for
which we can utilize a similar solver to jointly reconstruct the sparse
signal for multiple dictionaries.
The remainder of the paper is organized as follows. Section II de-
scribes the wideband array signal model. We briefly review classical
beamforming in Section III and propose an SSR-based DOA estimator
in Section IV. Some numerical examples are provided in Section V.
Section VI concludes the paper.
Notation: We use upper (lower) bold face letters to denote ma-
trices (column vectors). ????, ????, and ????represent conjugate, trans-
pose and complex conjugate transpose (Hermitian), respectively. Cal-
ligraphic letters are used to denote sets. ? is reserved for the imaginary
unit???.Therankofamatrix?isexpressedbyrank???.???defines
the ?? norm. ???? stands for the ?th entry of the vector ?. Depending
on context, we use ? ? ? to represent either the cardinality of a set or the
absolute value of a scalar.
II. DATA MODEL
We consider a uniform linear array (ULA) comprised of ? chan-
nels indexed by ? ? ??????? ? ?, which are equally spaced on a
line with spacing ?. It receives signals radiated from ? point sources.
The sources are supposed to be located in the far field such that we can
assume a plane wave arrival. For wideband processing, the signal at
1053-587X/$26.00 © 2011 IEEE

Page 2

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 7, JULY 20113465
each channel after time-sampling is partitioned into ? (possibly over-
lapping) segments, where for each segment, ? frequency subbands
are computed by e.g., a filter bank or the discrete Fourier transform
(DFT) [16]. Let us use ??????? to denote the ?th subband (frequency)
coefficient computed for the ?th segment of the signal that is radiated
from the ?th target with ? ? ??????? ? ?, ? ? ??????? ? ? and
? ? ?????????; similarly, we use ???????to denote the ?th subband
(frequency) coefficient for the ?th segment of the signal received at the
?th channel. If the bandwidth of each subband is much smaller than its
central frequency ??, the narrowband assumption applies, which en-
ables the following equality [17]:
???????? ?
???
???
?????? ??? ?
????????
(1)
where ? stands for the propagation velocity of the signal and ?? for
theDOAofthe?thtarget.1Intheaboveequation,wehaveusedthetime
of arrival at the first channel ?? ? ?? as a reference time and neglected
the additive noise term just for the sake of notational ease. The aim of
this paper is to estimate the target DOAs ??? ???????
If we stack the results from all the channels in a vector, ???? ??
?, it can be concisely expressed as
?.
?????????????????????
?????
???
???
???? ???????
(2)
with
?????? ??????
? ????
??????????
????? ????
??
(3)
also known as the array response vector. Throughout the remainder of
the paper, the following assumption will be adopted.
Assumption 1: The array response vectors corresponding to dif-
ferent targets are mutually independent.
III. CLASSICAL BEAMFORMING
Asan exampleofclassicalbeamforming,weconsiderthedelay-sum
beamformer in this paper [17]. The angle domain is divided into a grid
of ? points ? ?? ????????????? and scanned for each possible
angle ??. This is achieved by weighting the signal received by the
?th channel with a corresponding phase shift ??????
results from all the channels are then summed, which is equivalent
to ??
image is desired, which can be made by repeating the above procedure
for all the subbands; the outputs are then combined and transformed
back into the time domain by means of e.g., an inverse Fourier trans-
form. In the end, the signal for the ?th segment at the ?th angle in the
range-bearing image ?????? can be computed as [6]
? ????
. The
???????. In many sonar applications, a range(time)-bearing(angle)
?????? ??
?
?
???
???
??
???????????? ?
?
?
(4)
?????? can be interpreted as the power of the output of a spatial-tem-
poral filter steered to the direction of ??. The DOAs are estimated by
seeking those ?? whose corresponding values in
the largest.
The delay-sum beamformer is subject to spatial aliasing. In the case
of a ULA, for a certain frequency ?? and angle ??, if the channel
???
????????? are
1Notice that we use ? here to index the ?th target DOA. In the next section,
it will become clear that ? stands for the position of ?
set that will be defined later on.
within a larger angle
spacing ? is larger than half the apparent wavelength
possible to find at least another angle ?? such that
?
??
????
, it is
???
??????? ???
?????? ? ?
(5)
where ? is an arbitrary integer. As a result, ????? ????, which gives
rise to multiple peaks in the range-bearing image ??????. The am-
biguity due to spatial aliasing aggravates with a higher frequency or
larger spacing. The latter is, however, often desired in favor of a higher
angle resolution.
IV. DOA ESTIMATION VIA SSR
A. Problem Formulation and Existing Methods
Like in classical beamforming, we divide the whole angle search
range into a fine grid ? ? ?????????????, with ? being a pre-
defined parameter. Each ?? corresponds to a certain array response
vector ????defined in (3), which is dependent on the center frequency
??. Corresponding to ??, we construct an ? ? ? steering matrix
?? ?? ?????????????????. In the context of SSR, we will refer to
??in the sequel as a dictionary.
For the SSR-based approach, we adopt the following assumption.
Assumption 2: The DOAs of the targets are subsumed by the search
grid,2i.e.,
??? ???????
? ? ??
(6)
With Assumption 2, we are allowed to rewrite (2) in a more general
form as
????? ??????
(7)
where ???? stands for an ? ? ? vector, which contains all zeros ex-
cept for the positions whose indexes are contained in the set ? ??
?????????????. Obviously, ???????? ? ????????. With in total ?
snapshots available, we can extend (7) to
??? ????
(8)
where ?? is an ? ? ? matrix ?? ?? ????????????????? and ??
is an ? ? ? matrix similarly defined as ??. If we assume that the
target DOAs during the span of ? snapshots remain unchanged, ??
will obviously have at most ? rows containing nonzero elements.
Generally speaking, ??is an over-complete dictionary, which ham-
pers a straightforward estimation of ??. In practice, it is reasonable to
assume that the number of targets is very small compared to the size
of the dictionary and therefore ??admits a sparse representation. To
exploit this property, let us introduce the notation ???? to represent
the operation that collects the indexes of all the nonzero rows of a ma-
trix ?. Hence, ????? ? ? and ??????? ? ?. Armed with such a
notation, we can formulate a sparse problem out of (8) as
???
? ?
????????? subject to ??? ??????
(9)
Becausethecolumnsof??shareacommonsparsity,(9)fallsunderthe
framework of the multiple measurement vectors (MMV) approaches
considered in [12]–[14] and the group least absolute shrinkage and se-
lection operator (LASSO) proposed in [19], where various algorithms
are given to solve (9).
2If we deviate from this assumption, the effect of any discrepancy can be
captured by an additional observation noise term. An alternative is to model the
discrepancy explicitly as a disturbance on ? [18].

Page 3

3466IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 7, JULY 2011
B. Aliasing Suppression
Notice that unlike classical beamforming where the DOAs are
sought by steering a beamformer to different potential angles, the
SSR-based method given in (9) takes an intermediate step to recover
the value of ??first and then estimates the DOAs by locating the rows
of ?? that contain dominant entries. To facilitate a robust recovery
of ??, the restricted isometry property (RIP) [7] should be satisfied
for ??, or in other words, all subsets of ? columns taken from ??
need to be nearly orthogonal. However, the RIP does not always hold,
which gives rise to nonunique solutions to (9) and thus ambiguity in
DOA estimation.
In this paper, we differentiate between two cases of ambiguity. First,
if the array spacing ? is larger than half the apparent wavelength,it will
be possible to find at least two angles ??and ??, for which (5) holds.
Equivalently, the corresponding columns in ??will be identical. This
effectiswidelyknownasspatialaliasing,whichclassicalbeamforming
is also subject to. The second cause of ambiguity is inherent to the
’fat’ matrix ??, where some of its columns, even though not identical,
are linearly dependent. We will label this case as algebraic aliasing in
this paper to differentiate it from spatial aliasing. Note that algebraic
aliasing will not pose any problem in classical beamforming since the
latter examines the columns of ?? individually. Although both spa-
tial aliasing and algebraic aliasing are related to the linear dependency
among the columns of ??, they demand a separate counter-strategy.
Let us first introduce the following proposition that facilitates the
elimination of algebraic aliasing.
Proposition 1: Under Assumption 1, if the number of targets ? and
channels ? satisfy
? ? ?? ? rank????
(10)
then the SSR-based method in (9) will not suffer from algebraic
aliasing.
Proof: Let us first rule out the effect of spatial aliasing by coining
anewmatrix???,whichiscomprisedofallthedistinctcolumnsof??.
Resulting from Assumption 1 which states that the array response vec-
torscorrespondingtodifferenttargetsarelinearlyindependent,wehave
?? ???????, where???is comprised of the corresponding entries of
??and ?????? ? ?.Obviously, algebraic aliasing will not exist ifwe
can find a unique solution???? satisfying ?? ????????. This is only
possible according to [14, Theorem 2.4] if the Kruskal-rank3of???is
larger than ??? rank????. At the same time, we understand that???
is a Vandemonde matrix, whose Kruskal-rank will therefore be equal
to its rank which is ? [21]. With this, we conclude the proof.
Remark1: Notethatalthough???canbeunambiguouslyrecovered,
the true DOAs are still unknown. This is because the way to generate
???is not unique and hence there exist multiple mappings between???
and ??.
Remark 2: Proposition 1 underwrites the significance of using mul-
tiple measurement vectors. If there is only one snapshot available, i.e.,
? ? ?, we have rank???? ? ? and hence we can discriminate at
most?
?, (10) suggests that the number of targets that can be handled is
upper-bounded by the number of channels ?. It is interesting to note
that the same constraint is imposed on classical beamforming but for a
different reason.
Sofar,wehaveconcentratedonthedatamodelforasinglefrequency
??. Actually for all the frequencies contained in the target signal, we
can follow the same procedures leading to (8), giving rise to different
?targets. On the other side, because rank???? ? rank???? ?
3The Kruskal-rank of a matrix ? is defined as the largest integer ? for which
every set of ? columns of ? is linearly independent.[20]
dictionaries ?? ?? ??if ?? ?? ??. We will next show that using mul-
tipledictionariesenablesustoeliminatespatialaliasing.Tothisend,let
us introduce the symbol ??to denote the support of all possible DOA
solutions for the ?th dictionary:
???? ????????
??????????
????????
(11)
with?????
Spatial aliasing is frequency-dependent, which means that for dif-
ferent center frequencies, the resulting ambiguitywill not (completely)
overlap. Therefore, we can imagine that if we solve (9) for several
center frequencies: ???????????and combine the solutions in a judi-
ciousway,theambiguitydue tospatialaliasingwillatleastbe reduced.
In fact, we can invoke the following theorem to eliminate ambiguity
completely.
Theorem 1: With Proposition 1 met, if there exist at least two dic-
tionaries, whose corresponding frequencies, say ??and ??, satisfy
?
being one of the solutions of (9).
? ? ???? ??? ?
?
???
(12)
then the intersection of the solution support related to different dictio-
naries will contain exclusively the target DOAs, i.e.,
?
??? ? ?
(13)
Proof: WithProposition1satisfied,wecanexcludetheambiguity
due to algebraic aliasing and need to focus only on spatial aliasing. Let
us proceed with a counter-example. Suppose ?? is one of the target
angles and ?? ?? ??is spatial aliasing contained in both dictionaries
corresponding to ?? and ??, which implies that ??????? belongs to
both ??and ??. In accordance with (5), we then have
???
??????? ???
???
?????? ????
??????? ???
??????????
?????? ????
???
??????????
??????????
?????? ??????
???
where??,??,and??areallintegers.Noticethatbecauseoftheport/star-
board ambiguity inherent to the ULA, the search space is limited to
???? ? 90?and hence, ??and ??cannot be zero. Utilizing the trigono-
metric identities, the above equations can be rewritten as
?????
??????? ??
?????
?
?????? ??
?
???
(14)
??????? ??
??????? ??
?
?????? ??
?
???
(15)
?????? ????
?
?????? ??
?
????
(16)
In light of (12), it is only possible for (16) to hold if the integer ?? ?
?????? ?. On the other hand, (14) and (15) suggest that if ??and ??
are not zero, ??cannot be equal to ??. Therefore, the angle ?? cannot
be contained simultaneously in ??and ??, which concludes the proof.
Remark 3: In classical beamforming, in order to avoid spatial
aliasing, it is required that all the frequencies ?? should satisfy
?? ?
constraint since, instead of avoiding spatial aliasing, it only aims at
avoiding the overlap of spatial aliasing corresponding to different
frequencies.
Remark 4: It is interesting to note that many classical wideband
beamforming works, e.g., [4] and [5], consider the results to be readily
“aliasing-free”if(13)issatisfied.However,[4]doesnotgivearigorous
?
??. In this sense, Theorem 1 can be viewed as a relaxed

Page 4

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 7, JULY 20113467
formulation of the related choice of frequencies, but simply asserts that
thecasewhere(13)isnotsatisfied“isveryunlikely”.Inamorespecific
manner,[5]claimsthatallwidebandsignalswillnot sufferfromspatial
aliasing, but the wideband signals considered in [5] have a continuous
frequencyspectrumandthusitistrivialtofindtwodiscretefrequencies,
which are very close to each other to comply with (12). Compared to
[5], Theorem 1 does not only give a more rigorous formulation of the
frequency choices, but it is also of practical significance. For instance,
wecandeviseanactivesystem,whichemitsasensingsignalcontaining
just as few as two frequency components. Further, in many real-life
situations, not all the frequencies are usable for DOA estimation. We
will give such an example in the next section.
Remark 5: The optimum DOA estimates should be attained by
solving (9) using all the frequencies present. In the simulations,
however, we will give a real example showing that this is often infea-
sible. A judicious choice of frequencies can not only prevent spatial
aliasing, but also enhance the performance in a noisy environment. An
analogous problem is noted in [22], where the choice of frequencies
is shown to be significant for Doppler frequency estimation. How to
apply a similarapproach to DOA estimation remains to be future work.
C. Aliasing-Free SSR Recovery
The analysis in the previous section suggests that we can formulate
a multiple-dictionary (MD) joint optimization problem as
???? ?
????????? for ? ? ??????? ? ?
??? ?????? and
?????? ? ?????? for ? ?? ?
subject to
(17)
whose solution will be free of any ambiguity under Theorem 1.
Note that the MD problem ??? ????for ? ? ??????? ?? has
been explored in various works. In its strictest form, the collaborative
spectrum sensing problem handled in [23] requires that ?? ? ??for
? ?? ?.Thismodelactuallyreducestoasingle-dictionaryproblemifwe
stack ??vertically as ? ?? ???
where ? is defined similarly as ? and ? ? ?? ? ??? ? ????. The
multichannel-image problem considered in [24] imposes a less strict
constraint on ?? by requiring
some sparsity-inducing function which should be differentiable. This
property facilitates a dual descent method, which can be viewed as a
variant of basis pursuit (BP) [25]. In its loosest form, the problem de-
scribedin(17)isgeneralizedin[15]asjointsparsitymodel–2(JSM-2),
where different signals share the same sparsity, but are measured by
distinct dictionaries.
In [15], orthogonal matching pursuit (OMP) is utilized to solve the
MDproblem.Althoughthealgorithmisbasedonasinglemeasurement
vector and defined in the real domain, its extension to the MMV case
and complex domain, as considered in this paper, is straightforward.
OMP seeks the solution iteratively. A judicious choice of the stop-
ping criterion can be critical to the performance as shown in [13] and
should be tailored to different applications. The following considera-
tionsaretakeninthechoiceofthestoppingcriterioninthispaper.First,
the problem tackled in this paper is essentially a detection problem: we
are in principle not interested in the signal coefficients themselves, but
rather in the positions of the nonzero coefficients. Second, in a typ-
ical underwater environment, the signal-to-noise ratio (SNR) is usu-
ally very low. For instance, for diver detection in a harbour, we would
prefer using high-frequency acoustic signalsto avoid interference from
the busy traffic. However, high frequencies are subject to a much larger
attenuation through the water, resulting in an extremely low SNR [26].
Third, the number of targets ina realisticsituation is quite small. These
considerations suggest that a criterion that halts OMP after a fixed
????????
?????such that ? ? ??
??????? ? ?. Here, ????? denotes
number, say ?, of iterations would suit best. In practice, we predefine
? as an upper-bound on the possible number of targets.
A criticism to OMP is that unlike those convex relaxation methods
such as BP [27], OMP does not provide the same guarantees on perfor-
mance, which is phrased as upper-bounds on ?????????. To formu-
late our MD problem in terms of BP, we have
???
? ? ?????? ?
?
????????????? ? ?
???
???
???? ???????
(18)
where ? is a parameter controlling the trade-off between the sparsity of
??andtheerror?????????;????definessome(differentiable)func-
tion that enforces the joint sparsity constraint ?????? ? ?????? for
? ?? ?. An elegant example of ???? is given in [28], which exploits the
property that when the joint sparsity constraint is satisfied, the number
of nonzero rows of the bigger matrix ??????????????? will be mini-
mized. Accordingly, the following function is coined:
?
?????????????
?
???
???
?? ???
?????????????????
(19)
where ?? denotes an all-zero vector except for its ?th position which
is one and
????? ? ??
?
(20)
with ? denoting some small number. Note that (19) serves as an
approximation of the number of nonzero rows of the bigger matrix
??????????????? when ? ? ??.
In the numerical examples we show later on, we will stick to OMP
foritsmuch fasterimplementation,whichisessentialtomanypractical
applications. In practice, the performance degradation of OMP is often
limited.Notethatan improvedversionofOMP,referred toascompres-
sive sampling matching pursuit (CoSaMP), has been proposed in [27],
which provides a good compromise between complexity and perfor-
mance with theoretic performance guarantees. Its extension to the MD
problem remains to be future work.
V. NUMERICAL EXAMPLES
We will examine the performance of the proposed method using an
OMP algorithm, which halts at ? ? ? iterations. The numerical exam-
ples are based on both synthetic data and real data. For both cases, we
consider a ULA comprised of ? ? ?? hydrophones, with a spacing of
? ? 0.06 m.Thespeedofthesignalwaveisassumedtobe? ? ?????
(underwater acoustic signal).
1) Test Case 1. Synthetic Data: In this test case, we will generate
the received signals as a superposition of several harmonics corrupted
by additive white Gaussian noise. The signal received by the ?th hy-
drophone at the ?th time-instance admits an expression as
?
???
? ??
???
???
?????? ???
??? ?
???
?? ??????
(21)
where?? ? 96 kHz standsforthesamplingfrequency;?? represents
the phase corresponding to the ?th target, which is chosen as a uniform
random variable in the range ??????; ????stands for the additive noise
having a normal distribution ?????? and ?? is a constant satisfying
??? ?
Finally, ??stands for the ?th central frequency.
We assume that the target signal is present in ? ? ??? snapshots.
With the search grid defined as ? ? ??90???89.75??????90??, each
dictionary defined in (8) has a dimension of 16?720.
Inthefirstnumericalexample,letusassumethattherearetwotargets
whose DOAs are ?35??39?? and utilize two frequencies ?? ? 25 kHz
?
?. Throughout this test case, we will assume ??? ? 3 dB.

Page 5

3468IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 7, JULY 2011
Fig. 1. Comparison of classical beamforming with the proposed method.
Upper-leftsubplot: thefrequency-bearing
upper-right subplot: the time-bearing image after beamforming; lower-left
subplot: the integrated energy of the time-bearing image; lower-right subplot:
the result yielded by the proposed method.
imageafter beamforming;
and ?? ? 35 kHz to construct the dictionaries. In Fig. 1, we com-
pare the results of the proposed method with classical wideband beam-
forming. The frequency-bearing image after classical beamforming is
given in the upper-left subplot, where we can see that aliasing arises
at different angles for the two frequencies. The time-bearing image
is given in the upper-right subplot. The energy-bearing image in the
lower-left subplot is attained by integrating the signals along the time
axisofthisplot.TheresultsyieldedbytheproposedSSR-basedmethod
are given by the lower-right subplot. Compared to the previous sub-
plots, the spatial aliasing disappears. In addition, the two targets seem
to be easier to discriminate. Aside from the peaks at the target DOAs,
we can also observe three small peaks at other bearings, which corre-
spond tothefactthatwehaveassumedat mostfivetargetstobepresent
and used ? ? ? as the stopping criterion. Consequently, OMP will in
generalgenerate? nonzerocoefficientsforanoisysystem.Ontheother
side, the ambiguity due to overestimating the number of targets is not a
serious problem and can be quite straightforwardly removed by setting
a proper threshold. Note that thresholding is in practice always indis-
pensable for conventional beamformers.
Fig. 2 indicates the impact of the number of frequencies (dictio-
naries) utilized on the mean-squared error (MSE) between the esti-
mated and true DOAs. The MSE is calculated as
??? ? ?
???
???
???? ? ?? ??
(22)
where??? stands for the estimate of ?? . The above is evaluated for
? ? ? and utilizing 20 possible center frequencies ?? ? ?????kHz?
for ? ? ????????. Fig. 2 suggests that utilizing more dictionaries
increases the estimation precision. The MSE curve flattens off after
? ? ?? due to the fact that during each Monte Carlo run, a pair
of random DOAs is generated, which does not necessarily fall on the
search grid. When ? increases, this kind of modeling error becomes
more pronounced, preventing the performance from being further im-
proved.Wereferto[18]foramoredetailedimpactanalysisofthemod-
eling error on the SSR problem.
2) Test Case 2. Real Data: In this test case, we construct the test
datafromtherawdiverdatawhichwerecollectedduringanexperiment
conducted in June 2009, near Sea Bright, New Jersey, in the U.S. [1].
Fig. 2. MSE performance against the number of utilized frequencies.
Fig. 3. Spectrogram of the diver.
Thediver,atadepthof10m,isequippedwithanopen-circuitbreathing
set (scuba), and it can be seen in Fig. 3 that the exhaling sound of the
diver has a very large bandwidth.
The raw data is actually recorded at a distance of about 10 m from
the hydrophone. In the simulation, we plug the raw diver data into a
certain underwater propagation model [26], such that we can emulate
the effect of different distances and/or directions of the diver. In this
example, we generate the received signal as if there are two divers,
who are 150 m away from the hydrophone array at angles 52?and
60?, respectively. The results yielded by both classical beamforming
and the proposed method are presented in Fig. 4. In the upper-left sub-
plot where the frequency-bearing image is given, we can see that the
frequencieslowerthan10 kHz arecompletelyuselessforDOAestima-
tion: the diver signal is subdued by the ambient noise dominated by the
ship traffic in the harbour. In the midfrequency range (between 10 and
???? kHz), where the hydrophone array is not subject to aliasing, there
is a strong interference signal at a direction around ?40?, which pos-
sibly comes from a departing ship blowing the horn. The time-bearing
image given by the upper-right subplot in Fig. 4 is obtained based on
higher frequencies (above 25 kHz), where the spatial aliasing prevents
an unambiguous DOA estimate, as in the lower-left subplot. In con-
trast, the SSR-based method, as shown in the lower-right subplot, ren-
ders a correct estimate for the two closely-positioned divers. Note that