1 Three more Decades in Array Signal Processing Research An Optimization and

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Three more Decades in Array Signal

Processing Research: An Optimization and

Structure Exploitation Perspective

Marius Pesavento, Minh Trinh-Hoang and Mats Viberg

I. INTRODUCTION

The signal processing community currently witnesses the emergence of sensor array processing and

Direction-of-Arrival (DoA) estimation in various modern applications, such as automotive radar, mobile

user and millimeter wave indoor localization, drone surveillance, as well as in new paradigms, such as

joint sensing and communication in future wireless systems. This trend is further enhanced by technology

leaps and availability of powerful and affordable multi-antenna hardware platforms. New multi-antenna

technology has led to a widespread use of such systems in contemporary sensing and communication

systems as well as a continuous evolution towards larger multi-antenna systems in various application

domains, such as massive Multiple-Input-Multiple-Output (MIMO) communications systems comprising

hundreds of antenna elements. The massive increase of the antenna array dimension leads to unprecedented

resolution capabilities which opens new opportunities and challenges for signal processing. For example,

in large MIMO systems, modern array processing methods can be used to estimate and track the physical

path parameters such as DoA, Direction-of-Departure (DoD), Time-Delay-of-Arrival (TDoA) and Doppler

shift of tens or hundreds of multipath components with extremely high precision [1]. This parametric

approach for massive MIMO channel estimation and characterization beneﬁts from enhanced resolution

capabilities of large array systems and efﬁcient array processing techniques. Direction-based MIMO

channel estimation, which has not been possible in small MIMO systems due to limited number of

antennas, not only signiﬁcantly reduces the complexity but also improves the quality of MIMO channel

Marius Pesavento is with the Communication Systems Group, TU Darmstadt, Darmstadt, Germany (e-mail:

pesavento@nt.tu-darmstadt.de).

Minh Trinh-Hoang is with Rohde&Schwarz, Munich, Germany.

Mats Viberg is with Blekinge Institute of Technology, Sweden (e-mail: mats.viberg@bth.se).

current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new

collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other

works. DOI: 10.1109/MSP.2023.3255558

arXiv:2210.15012v2 [eess.SP] 3 Apr 2023

prediction as the physical channel parameters generally evolve on a much smaller time-scale than the

MIMO channel coefﬁcients.

The history of advances in super resolution DoA estimation techniques is long, starting from the

early parametric multi-source methods such as the computationally expensive maximum likelihood (ML)

techniques to the early subspace-based techniques such as Pisarenko and MUSIC [2]. Inspired by the

seminal review paper “Two Decades of Array Signal Processing Research: The Parametric Approach”

by Krim and Viberg published in the IEEE Signal Processing Magazine [3], we are looking back

at another three decades in Array Signal Processing Research under the classical narrowband array

processing model based on second order statistics. We revisit major trends in the ﬁeld and retell the story

of array signal processing from a modern optimization and structure exploitation perspective. In our

overview, through prominent examples, we illustrate how different DoA estimation methods can be cast

as optimization problems with side constraints originating from prior knowledge regarding the structure

of the measurement system. Due to space limitations, our review of the DoA estimation research in the

past three decades is by no means complete. For didactic reasons, we mainly focus on developments

in the ﬁeld that easily relate the traditional multi-source estimation criteria in [3] and choose simple

illustrative examples.

As many optimization problems in sensor array processing are notoriously difﬁcult to solve exactly

due to their nonlinearity and multimodality, a common approach is to apply problem relaxation and

approximation techniques in the development of computationally efﬁcient and close-to-optimal DoA

estimation methods. The DoA estimation approaches developed in the last thirty years differ in the

prior information and model assumptions that are maintained and relaxed during the approximation and

relaxation procedure in the optimization.

Along the line of constrained optimization, problem relaxation and approximation, recently, the Partial

Relaxation (PR) technique has been proposed as a new optimization-based DoA estimation framework

that applies modern relaxation techniques to traditional multi-source estimation criteria to achieve new

estimators with excellent estimation performance at affordable computational complexity. In many senses,

it can be observed that the estimators designed under the PR framework admit new insights in existing

methods of this well-established ﬁeld of research [4].

The introduction of sparse optimization techniques for DoA estimation and source localization in the

late noughties marks another methodological leap in the ﬁeld [5]–[9]. These modern optimization-based

methods became extremely popular due to their advantages in practically important scenarios where

classical subspace-based techniques for DoA estimation often experience a performance breakdown,

e.g., in the case of correlated sources, when the number of snapshots is low, or when the model order

is unknown. Sparse representation-based methods have been successfully extended to incorporate and

exploit various forms of structure, e.g., application-dependent row- and rank-sparse structures [10],

[11] that induce joint sparsity to enhance estimation performance in the case of multiple snapshots.

In particular array geometries, additional structure, such as Vandermonde and shift-invariance, can be

used to obtain efﬁcient parameterizations of the array sensing matrix that avoid the usual requirement of

sparse reconstruction methods to sample the angular Field-of-View (FoV) on a ﬁne DoA grid [12], [13].

Despite the success of sparsity-based methods, it is, however, often neglected that these methods also

have their limitations such as estimation biases resulting from off-grid errors and the impact of the sparse

regularization, high computational complexity and memory demands as well as sensitivity to the choice

of the so-called hyperparameters. In fact, for many practical estimation scenarios, sparse optimization

techniques are often outperformed by classical subspace techniques in terms of both resolution of

sources and computational complexity. From the theoretical perspective, performance guarantees of sparse

methods are generally only available under the condition of minimum angular separation between the

source signals [9]. Therefore, it is important to be aware of these limitations and to appreciate the beneﬁts

of both traditional and modern optimization-based DoA estimation methods.

The narrowband far-ﬁeld point source signals with perfectly calibrated sensor arrays and centralized

processing architectures have been fundamental assumptions in the past. With the trend of wider reception

bandwidth on the one hand, and larger aperture and distributed array on the other hand, the aforementioned

assumptions appeared restrictive and often impractical. Distributed sensor networks have emerged as a

scalable solution for source localization where sensors exchange data locally within their neighborhood

and in-network processing is used for distributed source localization with low communication overhead

[14]. Furthermore, DoA estimation methods for partly calibrated subarray systems have been explored

[15], [16].

Model structure, e.g., in the form of favorable spatial sampling pattern, is exploited for various purposes:

either to reduce the computational complexity and to make the estimation computationally tractable,

or to generally improve the estimation quality. In this paper, we revisit the major trends of structure

exploitation in sensor array signal processing. Along this line, we consider advanced spatial sampling

concepts designed in recent years, including minimum redundancy [17], augmentable [18], nested [19] and

co-prime arrays [20], [21]. The aforementioned spatial sampling patterns were designed to facilitate new

DoA estimation methods with the capability of resolving signiﬁcantly more sources than sensors in the

array. This is different from conventional sampling pattern, e.g., ULA, where the number of identiﬁable

sources is always smaller than the number of sensors.

II. SIGNAL MODEL

In this overview paper, we consider the narrowband point source signal model1. Under this signal

model, we are interested in estimating the DoAs, i.e., the parameter vector θ= [θ1, . . . , θN]T, of Nfar-

ﬁeld narrow-band sources impinging on a sensor array comprised of Msensors from noisy measurements.

We assume that the DoA θnlies in the FoV Θ, i.e., θn∈Θ. Let x(t) = A(θ)s(t) + n(t)denote the

linear array measurement model at time instant twhere s(t)and n(t)denote the signal waveform vector

and the sensor noise vector, respectively. The sensor noise n(t)is commonly assumed to be a zero-mean

spatially white complex circular Gaussian random process with covariance matrix νIM. The steering

matrix A(θ)∈ ANlives on a N-dimensional array manifold AN, which is deﬁned as

AN=A= [a(ϑ1),...,a(ϑN)] ϑ1< . . . < ϑNand ϑn∈Θfor all n= 1, . . . , N.(1)

In (1), the steering vector a(θ) = [e−jπd1cos(θ), e−jπd2cos(θ), . . . , e−jπdMcos(θ)]Tdenotes, e.g., the array

response of a linear array with sensor positions d1, . . . , dMin half-wavelength for a narrow-band signal

impinging from the direction θ. The steering matrix A(θ) = [a(θ1),...,a(θN)] must satisfy certain

regularity conditions so that the estimated DoAs can be uniquely identiﬁable up to a permutation

from the noiseless measurement. Mathematically, the unique identiﬁability condition requires that if

A(θ(1))s(1)(t) = A(θ(2))s(2)(t)for t= 1, . . . , T then θ(1) is a permutation of θ(2). Generally, this

condition must be veriﬁed for any sensor structure and the corresponding FoV. Speciﬁcally, it can be

shown that if the array manifold is free from ambiguities, i.e., if any oversampled steering matrix

A(θ)∈ AKof dimension M×Kwith K≥Mhas a Kruskal rank qA(θ)=M, then N

DoAs with N < M can be uniquely determined from the noiseless measurement [3]. Equivalently,

any set of Mcolumn vectors {a(θ1),...,a(θM)}with Mdistinct DoAs θ1, . . . , θM∈Θare linearly

independent. In the so-called conditional signal model, the waveform vector s(t)is assumed to be

deterministic such that x(t)∼ NCA(θ)s(t), νIM. The unknown noise variance νand the signal

waveform S= [s(1),...,s(T)] are generally not of interest in the context of DoA estimation, but they

are necessary components of the signal model. In contrast, in the unconditional signal model, the waveform

is assumed to be zero-mean complex circular Gaussian such that x(t)∼ NC0M,A(θ)P AH(θ)+νIM,

where the noise variance νand the waveform covariance matrix P=Es(t)sH(t)are considered as

unknown parameters. We assume, if not stated otherwise, that the signals are not fully correlated, i.e.,

Pis nonsingular.

1In practical wireless communication or radar applications, the receive signal may be broadband. Such scenarios require

extensions of the narrowband signal model, e.g., to subband processing or the multidimensional harmonic retrieval, which is

however out of scope of this paper.

III. COST FUNCTION AND CONCENTRATION

Parametric methods for DoA estimation can generally be cast as optimization problems with multivari-

ate objective functions that depend on a particular data matrix Yobtained from the array measurements

X= [x(1),...,x(T)] through a suitable mapping, the unknown DoA parameters of interest θand

the unknown nuisance parameters, which we denote by the vector α. Hence, the parameter estimates

are computed as the minimizer of the corresponding optimization problem with the objective function

f(Y|A(θ),α)as follows

A(ˆ

θ) = arg min

A(θ)∈AN

min

αf(Y|A(θ),α).(2)

Remark that in (2), we make no restriction how the data matrix Yis constructed from the measurement

matrix X. For example, in the most trivial case, the data matrix Ycan directly represent the array

measurement matrix, i.e., Y=X. However, for other optimization criteria, the data matrix Ycan

be the sample covariance matrix, i.e., Y=ˆ

R=1

TXXHas a sufﬁcient statistics, or even the signal

eigenvectors Y=ˆ

Us(or the noise eigenvectors Y=ˆ

Un) obtained from the eigendecomposition ˆ

Usˆ

Λsˆ

s+ˆ

Unˆ

Λnˆ

nwhere ˆ

Λs= diagˆ

λ1,...,ˆ

λNcontains the N-largest eigenvalues of ˆ

R. In Table

I, some prominent examples of multi-source estimation methods are listed: Deterministic Maximum

Likelihood (DML) [2, Sec. 8.5.2], Weighted Subspace Fitting (WSF) [22] and COvariance Matching

Estimation Techniques (COMET) [23].

As we are primarily interested in estimating the DoA parameters θ, a common approach is to concentrate

the objective function with respect to all (or only part of) the nuisance parameters α. In case that a

closed-form minimizer of the nuisance parameters w.r.t the remaining parameters exists, the expression

of this minimizer can be inserted back to the original objective function to obtain the concentrated

optimization problem. More speciﬁcally, let ˆ

α(θ)denote the minimizer of the full problem for nuisance

parameter vector αas a function of θ, i.e., ˆ

α(θ) = arg min

f(Y|A(θ),α). The concentrated objective

function gY|A(θ)=fY|A(θ),ˆ

α(θ)then only depends on the DoAs θ. Apart from the reduction

of dimensionality, the concentrated versions of multi-source optimization problems often admit appealing

interpretations. In Table I, the concentrated criteria corresponding to the previously considered full-

parameter multi-source criteria are provided. We observe, e.g., in the case of the concentrated DML and

the WSF criteria that at the optimum, the residual signal energy contained in the nullspace of the steering

matrix is minimized.

Due to the complicated structure of the array manifold ANin (1), the concentrated objective function

gY|A(θ)is, for common choices in Table I, highly non-convex and multi-modal w.r.t. the DoA

parameters θ. Consequently, the concentrated cost function contains a large number of local minima in

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1ThreemoreDecadesinArraySignalProcessingResearch:AnOptimizationandStructureExploitationPerspectiveMariusPesavento,MinhTrinh-HoangandMatsVibergI.INTRODUCTIONThesignalprocessingcommunitycurrentlywitnessestheemergenceofsensorarrayprocessingandDirection-of-Arrival(DoA)estimationinvariousmodernapplicatio...

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