DA-MUSIC: Data-Driven DoA Estimation via Deep

Augmented MUSIC Algorithm

Julian P. Merkofer, Student Member, IEEE, Guy Revach, Student Member, IEEE , Nir Shlezinger, Member, IEEE,

Tirza Routtenberg, Senior Member, IEEE, and Ruud J. G. van Sloun, Member, IEEE

Abstract—Direction of arrival (DoA) estimation of multiple

signals is pivotal in sensor array signal processing. A popular

multi-signal DoA estimation method is the multiple signal clas-

sification (MUSIC) algorithm, which enables high-performance

super-resolution DoA recovery while being highly applicable in

practice. MUSIC is a model-based algorithm, relying on an

deep augmented MUSIC (DA-MUSIC) algorithm is thus a hybrid

model-based/data-driven DoA estimator, which leverages data

to improve performance and robustness while preserving the

interpretable flow of the classic method. DA-MUSIC is shown

to learn to overcome limitations of the purely model-based

method, such as its inability to successfully localize coherent

sources as well as estimate the number of coherent signal sources

present. We further demonstrate the superior resolution of the

DA-MUSIC algorithm in synthetic narrowband and broadband

scenarios as well as with real-world data of DoA estimation from

including radar, communications, image analysis, and speech

enhancement [2]–[4]. Over the last decades, a multitude of

different

DoA

estimation algorithms have been proposed, and

the problem is an active area of research [5], [6].

A leading scheme employed in many

DoA

popular multiple signal classification (

MUSIC

which can provide asymptotically unbiased estimates of the

number of incident wavefronts present, their approximate fre-

quencies, and their

MUSIC

e.g., conventional beamforming [8] and

beamform-

Merkofer is with the EE Dpt., Eindhoven University of Technology, Eindhoven,

The Netherlands (e-mail: [email protected]), this work was partially done

while being with the ETH Z

¨

urich. G. Revach is with the D-ITET, ETH Z

¨

urich,

Switzerland, (email: grev[email protected]). N. Shlezinger and T. Routtenberg are

with the School of ECE, Ben-Gurion University of the Negev, Beer Sheva,

Israel (e-mail:

{

nirshl; tirzar

}

@bgu.ac.il). T. Routtenberg is also with the ECE

Research, Eindhoven, The Netherlands (e-mail: r.j.g.v[email protected]).

information regarding the

angles at hand. Generally, these

processing methods, the covariance matrices of the observations

in different frequency bins are coherently combined using

certain transformation matrices. Most coherent techniques are

based on the coherent signal subspace (

CSS

) concept [11],

which focuses the transformations of the covariances at

utilizes different transformation matrices. Incoherent broadband

processing methods combine

DoA

estimates obtained separately

Net architecture, respectively. Other methods [18], [19] utilize

the knowledge of the spatial covariance matrices, where [18]

trained a

CNN

based on a classification task (allowing it to also

operate with unknown number of sources) and [19] analyzed

purely

DD

estimators and

DD

methods in combination with

maximum likelihood estimation (

MLE

also combined

by using multiple dense

angles. [22] extends the architecture of [21] to estimate the

number of sources with the ResNet. While such black-box

the work [25]and [26] proposed to estimate the discretized

MUSIC

spectrum from the (spatially smoothed) covariance

matrix of the measurements through the utilization of multiple

convolutional

NN

s. While these methods are more robust

to model inaccuracies compared with the original

algorithm, utilizing the

MUSIC

spectrum as a label for

training causes them to experience the same drawbacks as

1

arXiv:2109.10581v5 [eess.SP] 11 Jan 2023

their

MB

counterpart. Another

DD

considered systems with subarray sampling and uses

obtain a single estimated covariance matrix from incoherent

subarray measurements. This

NN

true covariance matrix as a label, without considering its

estimation of a single source. The two approaches, however,

utilize very different preprocessing methods and are trained

classification approach limits itself to a fixed resolution, the

regressor of [31] outputs the

can achieve an arbitrary resolution, yet it is limited to the

spatial covariance matrix as input. Another approach [32]

first decomposed the broadband signals into non-overlapping

narrowband components and used support vector regression to

obtain the DoA.

The limitations of both existing

and

DD DoA

estimation

algorithms in resolving multiple signals that are possibly

it with

NN

architecture overcomes the fundamental limitations of

enabling it to accurately detect the locations of coherent sources.

Our design builds upon the insight that the sensitivity to model

mismatch and inability to handle coherent and broadband

sources of

MUSIC

lie in its estimation of the noise and

DA-MUSIC

crucial step by obtaining a surrogate, pseudo covariance matrix

through a recurrent

RNN

) from the measurements, which

classifier. Numerical results show an estimation accuracy of

98% in comparison to

MUSIC

with 89% accuracy in the same

non-coherent scenario. We further demonstrate

’s

capabilities in various

SNR

domains, three different broadband

scenarios, as well as real-world seismic signals.

DA-MUSIC

not only manages to focus frequency components but is also

to receive a stable azimuth estimation.

describes the assumed system model and surveys some of

the related work of

details of the

MB MUSIC

the

DA-MUSIC

the simulations; and Section V provides concluding remarks.

vectors; e.g.,

, and boldface uppercase letters, e.g., for

i

th entry of a vector

x

commas represent column vectors. We use calligraphic letters

the identity matrix.

II. SYSTEM MODEL AND PRELIMINARIES

velop the proposed

aim, we first discuss the signal model in Subsection

the conventional narrowband setting and its more general

broadband setup. For more details regarding the two models,

we refer the reader to the textbooks [10], [33], and [34].

signal processing literature concerning

DoA

estimation is the

narrowband setting. Here, the time it takes for the waves to

propagate the array is assumed to be negligible such that the

occurring delays are sufficiently small. Therefore, the signal

of

M

sensor elements measuring

D

2

Consequently, the steering vectors (and, in turn, the matrix

A(θ)

of the measurements at the array elements over multiple time

instances is defined as

, (4)

where T is referred to as the number of snapshots.

2) Broadband: In practice, many signals are broadband,

where

represents the time delay of the

measuring source

at the reference element

. Transformed to the frequency

domain, the relationship in (5) at frequency ω becomes

e

For a

, the broadband steering vectors are defined for

frequency ω and angle of interest ψ as



, . . . , e

, (8)

where c is the propagation velocity.

x(t) = [x

Fig. 1. The following additional assumptions are imposed upon

the observation model as well as the derived synthetic data.

The signals are uncorrelated to the noise (though signals may

possibly be correlated with each other) and generated in the far-

field region. And there is uniform propagation in all directions

in an isotropic and non-dispersive medium. We further assume

that we have knowledge (though possibly mismatched) of the

underlying array geometry, implying that we can compute an

approximation of

. We consider a data-driven

DoA

s are not obtainable at all, the data-driven

DoA

DoA

methods based on the above problem formulation, motivating

the need for our proposed

detailed in Section III.

An extensive overview of various

estimators can

be found in [5], [6], and a recent literature review of

DoA

estimation approaches can be found in [35]. We thus

only discuss some representative

methods, followed by

reviewing relevant broadband extensions, and conclude with

DD

[9] and other adaptive beamformers [36]. An overview of

beamforming techniques can be found in [37].

methods, which aim at recovering the

s by identifying

the noise and signal subspaces. The

algorithm [7]

is a highly popular subspace-based method and has been

researched extensively. Being the focus of this paper, more

information and a detailed description of the algorithm can

be found in Section

MUSIC [38], a polynomial-rooting version, as well as spatially

signal parameters via rotational invariance techniques (

[41] and its variations [42]. These methods rely heavily on the

sensitive to array aperture perturbations, and are inherently

derived for narrowband signals.

Broadband Estimators: Broadband

DoA

algorithms can be categorized into two groups: incoherent

methods and coherent methods. Generally, incoherent methods

3

Similarly to the above, the methods in [25], [26] propose

CNN

estimation by taking the same channels of

the sample covariance matrix as input (spatially smoothed

covariance of single snapshot for [26]). However, the

are trained as a regression task estimating a discretized segment

of the

MUSIC

drawbacks and imitations of

MUSIC

but are, besides increased

with

K

being the covariance of the incident signals

s(t)

The matrix

A(θ)K

A

is singular and has a rank of less

4

Fig. 3: Block diagram of the DA-MUSIC algorithm.

than

M

when the number of array elements

is strictly

larger than the number of signals

D

is an

eigenvalue of

K

for additive white Gaussian noise (

denoted

λ

λ

MUSIC

of the signals. The algorithm takes as input

the waveforms at

M

in

and uses them to obtain an empirical estimate of

where

is the estimated multiplicity of the minimal eigenvalue

of

. The eigenvectors corresponding to the

eigenvalues form the noise subspace

E

and the

D

dominant peaks of

P (ψ)

are set as the estimated

DoA angles

MUSIC

method that is reasonably efficient and statistically consis-

entries within the covariance matrix and can, therefore, become

indistinguishable from noise [55]. Furthermore, the

MUSIC

algorithm is inherently a narrowband approach due to the

assumptions imposed on the system model. Nonetheless, it can

be extended to broadband, i.e., signal models as in

(7)

is also adopted in our proposed DA-MUSIC.

for each frequency, transform the covariances, and obtain a

focused covariance matrix by some form of averaging. In

CSS

signal subspaces to align the signal subspaces associated with

the

along all frequency bins. To formulate this, let

K(ω)

be the covariance of the frequency domain observations

(7)

and divide the spectrum into

where

T

is the focusing matrix and frequency bins are

prioritized by the weighting

α

. The focusing matrices can be

determined by attempting to focus the spectral components at

some frequency ω

as well as a lower resolution threshold than non-coherent

methods [52]. However, the

CSS

methods require initial values

for the focusing matrix

T

, the reference frequency

ω

, and the

relevant focusing angles

ψ

to find the focusing matrices with

(13)

, and are typically sensitive towards these initial values.

Additionally, it is not guaranteed that the alignment of the

signal and noise subspaces exists to form a viable general

covariance matrix without disarranging the noise subspace

[51].

estimation algorithm

derived from the classic

MUSIC

and model mismatch sensitive elements of the model-based

structure with specific

NN

s. Fig. 3 outlines the resulting

structure of the

DA-MUSIC

remaining similarities to the original

algorithm, de-

picted in Fig. 2.

5

Fig. 4: Detailed network structure of the

algorithm.

Principally,

improve the categorization of the noise and signal subspaces,

the correlation of the received measurements is learned from

temporal data by employing a dedicated

RNN

, which is

augmented into the overall flow of

on the architecture in Subsection

III-A

III-B

discussion in Subsection III-C.

The proposed

algorithm preserves the structure

of the

Next,

DA-MUSIC

s from the spatial spec-

trum

using an additional

followed by a single dense layer with linear activation. The

input to the

MLP

P (ψ)

taken uniformly

in

MLP

noise eigenvector. The selection is performed by computing

E = Vdiag(q) =

suitable noise subspace. Note that this setting specializes

in the conventional approach of assigning based on the

multiplicity of the minimal eigenvalues; in the conventional

approach, the entries of

q

are either ones or zeros and

coincides with the corresponding subspace. Consequently, the

6

Fig. 5:

DA-MUSIC

the noise subspace and facilitates coping with settings where

distinguishing between the eigenvectors is challenging due, for

example, to low SNRs.

Outputting varying number of DoAs:

strategy to overcome the dynamic output dimensions of

NN

to scale the output dimension to the maximum occurring value.

In our case, since

DA-MUSIC

is set to

. The only additional modification required with

D

module that estimates the number of sources, and then using

the first

D outputs as the recovered DoAs.

DA-MUSIC

the training set is comprised of

tuples of sequences of

measurements and their corresponding

DoA

tuples

as a preliminary step for discussing how the training procedure

is carried out in the general case where the model does not

have knowledge D.

1) Known Number of Sources: Given a sequence of mea-

angles

that are compared to the true

DoA

. Gradient-

based optimization is possible because every element of the

architecture is differentiable, allowing backpropagation through

the complete structure. Derived from the root mean squared

periodic error (

RMSPE

) [57], [58] the following loss function

RMSPE(θ,

ˆ



ˆ



where

is the set of all

permutations and

mod

denotes the element-wise modulus operation regarding the

angle range of interest, e.g.

previous subsection,

is designed to resolve a

represents the parameters of the remaining trainable modules

of

selector). The loss used to train

RMSPE

loss of

which is altered during training to account for varying sources

by computing,

,



,

θ

only the first

angles of the estimated

ˆ

with the true

DoA θ

while the remaining angles of