Iterative Broadband Source Localization Coleman DeLude Rakshith Sharma Santhosh Karnik Christopher Hood Mark Davenport and Justin Romberg

2025-05-03 0 0 1.58MB 27 页 10玖币

侵权投诉

Iterative Broadband Source Localization

Coleman DeLude, Rakshith Sharma, Santhosh Karnik, Christopher Hood,

Mark Davenport, and Justin Romberg

October 20, 2022

Abstract

In this paper we consider the problem of localizing a set of broadband sources from a ﬁnite

window of measurements. In the case of narrowband sources this can be reduced to the problem

of spectral line estimation, where our goal is simply to estimate the active frequencies from a

weighted mixture of pure sinusoids. There exists a plethora of modern and classical methods that

eﬀectively solve this problem. However, for a wide variety of applications the underlying sources

are not narrowband and can have an appreciable amount of bandwidth. In this work, we extend

classical greedy algorithms for sparse recovery (e.g., orthogonal matching pursuit) to localize

broadband sources. We leverage models for samples of broadband signals based on a union of

Slepian subspaces, which are more aptly suited for dealing with spectral leakage and dynamic

range disparities. We show that by using these models, our adapted algorithms can successfully

localize broadband sources under a variety of adverse operating scenarios. Furthermore, we

show that our algorithms outperform complementary methods that use more standard Fourier

models. We also show that we can perform estimation from compressed measurements with

little loss in ﬁdelity as long as the number of measurements are on the order of the signal’s

implicit degrees of freedom. We conclude with an in-depth application of these ideas to the

problem of localization in multi-sensor arrays.

1 Introduction

At its core this paper revisits and extends the classical problem of source localization. Speciﬁcally,

we assume that we observe the superposition of Lsources x1(t), . . . , xL(t) corrupted by noise η(t):

y(t) =

`=1

x`(t) + η(t).(1)

We then sample y(t) at Npoints {tn}N

n=1 and denote the vector of these samples by y. Broadly

speaking, source localization can be accomplished by identifying certain key parameters of the

component signals from the observations y. Diﬀerent applications will entail varying choices in

the sampling domain and the speciﬁc structure imposed on the x`(t). Perhaps the most commonly

studied variation is when when x`(t) = c`ej2πf`tfor c`∈C, f`∈Rand we seek to estimate

{(c`, f`)}L

`=1 from y, in which case this problem is called spectral line estimation (SLE). SLE arises

in a variety of source localization problems. For example, in direction-of-arrival estimation (DOA),

yconsists of a “snapshot” from a multi-sensor array and we wish to estimate the angle at which

each x`(t) is impinging on an array.1Classically, DOA is most often performed under a narrowband

1In this instance each x`(t) manifests as a plane-wave moving through space.

arXiv:2210.11669v1 [eess.SP] 21 Oct 2022

assumption which makes it eﬀectively equivalent to SLE. In fact, by leveraging a narrowband signal

model a wide variety of important problems in communications, seismology, and radar can be cast

as a SLE problem [1].

The narrowband signal model coupled with existing works on SLE form the foundation upon

which a more general class of source localization algorithms can be built. For this reason Section 2

oﬀers a thorough overview of both recent and classical approaches to this problem. However, our

goal in this paper is to explicitly diverge from this narrowband assumption and develop robust

source localization algorithms that can accommodate broadband sources. With even a relatively

modest increase in the bandwidth of the sources, the narrowband assumption fails to hold, even

approximately. As will be discussed in further detail in Section 3this leads to a breakdown in

the performance of existing localization algorithms. This forms the motivation for us to explore

models and methods that more aptly suit broadband signals and can operate under a wider range

of adverse conditions.

In developing our algorithms, we make use of a specialized subspace model for broadband sig-

nals that is based on the Discrete Prolate Spheroidal Sequences (DPSSs) [2]. The ﬁnitely truncated

DPSSs (when appropriately modulated), also known as Slepian basis vectors, provide an optimal

subspace representation for broadband signals sampled over a ﬁnite interval and form the back-

bone of our algorithms [3]. Our algorithms borrow from existing “greedy” algorithms for sparse

approximation. In an iterative fashion we alternate between projecting the signal (or residual) onto

all possible Slepian subspaces and identifying the subspaces that capture the most energy to form

an estimate of each component signals spectral support. Slepian subspaces are also used at each

iteration to “null” the spectral components identiﬁed in previous iterations. The use of Slepian

subspace models allows our algorithms to be robust to a large dynamic range between the signal

powers when there are multiple active sources as well as operate at lower SNRs than traditional

methods.

To demonstrate the utility of our proposed algorithms, we provide a host of experimental

results. We ﬁrst review existing narrowband source localization methods and show that they fail

to generalize to the broadband case. Our proposed algorithms are then compared in a variety of

scenarios. We then consider the problem of spectral support estimation from compressed samples.

We show that when the number of samples is proportional to the inherent dimensionality (explained

in detail in Section 3.2) of the signal, our algorithms successfully identify the spectral support of

each source. We then consider a multi-sensor array receiving several broadband signals from various

DOAs. We apply our algorithms to this realistic scenario and demonstrate that they successfully

identify the parameters associated with each source.

We provide here a brief snapshot of the nature of the experimental results given throughout

the rest of the paper. In Figure 1(a), we demonstrate a speciﬁc instance of one of our proposed

algorithms for broadband DOA estimation and compare it to the classical narrowband version of the

same algorithm. While classical algorithms fail to identify all the sources due to the non-negligible

bandwidth, our algorithm is able to correctly identify them. In Figure 1(b), we demonstrate the

utility of the Slepian subspace model as compared to the traditional Fourier model for broadband

signals. Speciﬁcally, we consider two active broadband signals, but with a vast diﬀerence in signal

powers, with one signal being 105times stronger than the other. The traditional broadband Fourier

model, in which a signal is represented by orthogonal discrete Fourier transform (DFT) basis

vectors, fails to entirely capture the stronger source leading to repeated identiﬁcation of the same

source. Our Slepian signal model however is successful in identifying the full spectral support of

(a) (b)

Figure 1: (a) Comparison of standard spectral line estimation based OMP frequency estimates (red circles)

and broadband adapted OMP estimates (green circle). (b) For a two broadband signals with signiﬁcant

dynamic range disparities (top), we can null out sources using a Fourier dictionary projection (middle) or a

Slepian dictionary projection (bottom).

the stronger source, hence leading to the correct identiﬁcation of the other sources in subsequent

iterations.

2 Existing Approaches to Source Localization

2.1 Localization via spectral line estimation

We now brieﬂy return to the previously described case of spectral line estimation (SLE). In the

noiseless case, we observe the superposition of Lcomponent sinusoidal tones:

s(t) =

`=1

x`(t) =

`=1

c`ej2πf`t(2)

This is an idealized narrowband signal model. Here the Fourier transform of (2) is a superposition

of Dirac delta distributions b

S(f) = PL

`=1 c`δ(f−f`) and the component signals have eﬀectively

zero bandwidth.

If we had access to the entire temporal extent of the signal, then the SLE problem becomes triv-

ial. Simple peak picking in the frequency domain would suﬃce to determine the spectral support

while a least-squares problem could be used to determine the coeﬃcients. However, we observe (2)

through a ﬁnite set of Nsamples. We assume that these samples are taken uniformly over some

interval with sampling frequency fssatisfying the Nyquist rate criterion (determined by the com-

ponent with the largest frequency). A point of emphasis is that f`∈Rcan generally be arbitrary,

meaning that f`/fsis likely not an integer multiple of 1/N . These “oﬀ-grid” frequencies subject

the signal to spectral leakage during the sampling process wherein energy from the component

signals bleed into one another, biasing the spectral peaks and amplitudes. A visualization of this

phenomena is provided in Figure 2, where the presence of sidelobes about the sources is indicative

of spectral leakage. The bias incurred from spectral leakage makes the SLE problem non-trivial,

and we cannot determine the parameters of (2) perfectly through standard methods of spectral

estimation. This inconvenient fact has motivated an impressive body of work that spans more than

two centuries of research.

The SLE problem has roots dating back to Prony’s method [4]. Methods based on statistical

signal models leveraging eigendecompositions of the signal covariance matrix were later pioneered

in [5] and subsequently rediscovered in [6]. Initially considered to be less proﬁcient at SLE than

Prony’s method [7], this methodology would form the inspiration for the MUSIC algorithm [8].

Within a similar timeframe methods leveraging the rotational invariance of the signal subspace

to directly estimate the component frequencies were developed to form the ESPRIT algorithm [9].

Accompanying these statistical methods, deterministic Prony-like methods such as the matrix pencil

were later developed [10,11]. The generalized eigenvalue problem solved in the matrix pencil method

(with Prony’s methods enveloped as a special case) is more stable to noise than the standard root

ﬁnding process associated with Prony’s method [12]. The matrix pencil, MUSIC, and ESPRIT

algorithms form what we refer to as the “classical” SLE methods.

More recently, advances in sparse approximation has led to the creation of a more optimization

based perspective on SLE. Applying `1−`2optimization to source localization, speciﬁcally in

the context of arrays, was ﬁrst presented in [13]. The work in [14,15] showed that when the

components signals reside on on-grid frequencies and have positive weightings the signals can be

perfectly recovered via `1-minimization. Sparsity constraints were also shown to yield meaningful

performance improvements in more practical radar based applications in [16].

Works such as [17–19] adapt compressed sensing recovery algorithms such as OMP, CoSaMP,

subspace pursuit, and `1-minimization to SLE when the observations in (1) are viewed through a

dimensionality reducing sensing matrix. The approach in each is similar, the component signals

are assumed to admit a sparse representation in an overcomplete discrete Fourier dictionary. To

overcome the dictionary’s coherence the elements are split into coherent frames, and the sources

can be coarsely localized to a subset of these frames. This approximation is then reﬁned via a

local optimization step that each author handles diﬀerently. In a similar manner to compressed

sensing, the authors of [20] proposed a 2-D SLE technique posed in a structured matrix completion

framework. The process leverages the “matrix enhancement” method developed in [11] to yield

recovery guarantees in traditionally adverse scenarios. However, the paper is largely centered on

the matrix completion aspect of the problem and defers the SLE portion to the methods proposed

in [11,12].

A common theme amongst the above sparse approximation and compressed sensing SLE meth-

ods is the assumption that the signal is sparse in a ﬁnite dictionary. This precludes the scenario

where the underlying frequencies of the component signals lie oﬀ-grid. As previously discussed,

in the traditional SLE problem frequencies are permitted to lie on a continuum. The use of over-

complete dictionaries is meant to help circumvent this problem, but is ultimately avoiding the true

nature of the issue at hand.

TV/atomic norm minimization based methods2reconcile this issue, and are eﬀectively a gen-

eralization of `1-minimization to a continuous setting [22]. Utilizing TV norm minimization, the

authors of [23] were able to extend the work of [14,15] to operate on a continuum of frequencies

(assuming a positive weighting of the sinusoids). Subsequently, in [24] it was shown that, under a

mild separation constraint, exact recovery of the component signal parameters could be achieved

in the absence of noise via TV norm minimization. In [25], the sequel to [24], strong theoretical

guarantees on the accuracy of reconstruction from noisy measurements were established for the

same atomic norm denoising framework presented in [26]. The work of [27] decoupled the support

and amplitude estimation errors and showed that the support estimation in atomic norm denoising

2In the context of SLE, the total variation (TV) norm and atomic norm are essentially equivalent [21].

is generally very accurate. Finally, [21] showed that atomic norm minimization can be applied to

compressed sensing in a manner similar to `1-minimization to provide exact signal recovery. In sum-

mary TV/atomic norm based methods of SLE represent the state-of-the-art in terms of theoretical

guarantees.

Alongside TV/atomic norm based methods, several authors have attempted to “modernize”

classical methods and revisit sparse recovery. The motivation for this being that such methods

can oﬀer computational savings. The work of [28] provides empirical comparisons of OMP and

CoSaMP algorithms for compressed DOA estimation. It has been shown both theoretically and

empirically that single observation versions of MUSIC can be competitive with atomic norm based

methods [29], and have practical applications in realistic scenarios [30].

We note that an independent method based on ﬁnite rates of innovation was shown to be able

to recover Dirac trains from lowpass measurements in [31,32]. Though some theoretical guarantees

are presented, the method of recovery fundamentally relies on polynomial root ﬁnding. As a result,

this approach is generally considered to be ill-suited for use in the presence of noise [33].

To accompany this review of SLE methods, Section 5oﬀers a performance comparison of OMP,

CoSaMP, `1-minimization, and atomic norm minimization. Though it is readily apparent from

these results that atomic norm minimization provides the best performance, a key observation is

that the level of improvement is not particularly drastic. This is important when we consider how

we will transition to the broadband regime since modifying most of these algorithms to account

for bandwidth is generally diﬃcult. In particular, for atomic norm minimization, it is not even

clear how to deﬁne an atomic set for general broadband signals.3Even if such a set existed it is

not clear that it could be cast to a tractable framework [34]. On the other hand, we will see that

the iterative OMP and CoSaMP algorithms can be modiﬁed to account for bandwidth in a fairly

natural manner.

2.2 Beyond SLE

The problem of source localization in the broadband regime has been studied to a lesser extant than

its narrowband counterpart. Some formulations of the standard atomic norm based frameworks

allow for a known bandlimited point spread function (PSF) or kernel to be convolved with the

component signals [22]. However, this does not generalize to arbitrary bandlimited signals and it

is generally assumed that a de-convolution step has occurred prior to the optimization stage. The

advances in [35,36] can be interpreted as an improvement upon this by attempting to estimate

the sinusoidal components and the kernel that they have been convovled with. Both methods

hinge on said kernel lying in a known low dimensional subspace. As will be discussed in Section 3

this diverges from our general broadband signal model in which none of the component signals lie

exactly in a low dimensional subspace and merely reside close to one.

In terms of our approach and methodology, our work most closely resembles [3], which utilizes

a union of Slepian spaces model similar to our own. However, this work splits the spectrum into

ﬁxed intervals from which a dictionary can be deﬁned. Signals are then determined to be present

on these intervals or not, localizing the spectral support to a sub-band. While this accounts for

bandwidth, like most of the applications of sparse approximation techniques to spectral estimation

described above it again makes the ﬂawed assumption that the component signals are centered on

a (relatively coarse) grid of frequencies. While the proposed greedy algorithms of [3] are similar

3Even in the discrete case e.g. `1-minimization we must carefully devise some notion of group sparsity.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

IterativeBroadbandSourceLocalizationColemanDeLude,RakshithSharma,SanthoshKarnik,ChristopherHood,MarkDavenport,andJustinRombergOctober20,2022AbstractInthispaperweconsidertheproblemoflocalizingasetofbroadbandsourcesfromanitewindowofmeasurements.Inthecaseofnarrowbandsourcesthiscanbereducedtotheproblem...

展开>> 收起<<

Iterative Broadband Source Localization Coleman DeLude Rakshith Sharma Santhosh Karnik Christopher Hood Mark Davenport and Justin Romberg.pdf

共27页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Iterative Broadband Source Localization Coleman DeLude Rakshith Sharma Santhosh Karnik Christopher Hood Mark Davenport and Justin Romberg

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: