IEEE Signal Processing - July 2018 - 58
exact relation between D (number of parameters), M (dimension of the compressive measurements), and N (dimension of
the unknown signal) that ensures perfect recovery of H. We will
illustrate the importance of sampler design through the following three classes of inverse problems 1) underdetermined source
localization, 2) compressive structured covariance estimation, and
3) SBL. Throughout the article, M denotes the dimension of the
compressed measurements or samples y [l], and the sample covariance matrix R yy = E ^y [l] y H [l]h acts as a compressive sketch
of R xx (H). This is equivalent to acquiring M 2 scalar measurements of N 2 entries of R xx (H). Since N 2 M (or equivalently,
N 2 2 M 2), recovering R xx (H) from these M 2 measurements is
an underdetermined problem.
Underdetermined source localization using
antenna arrays
Signal processing algorithms for antenna arrays typically utilize
a space-time model where both spatial and temporal samples are
collected and processed simultaneously. The signal model
depends on the physics of electromagnetic (EM) wave propagation as well as the stochastics of the impinging signals. We first
consider a one-dimensional antenna array, and the ideas can be
extended to higher dimensions. Referring to the general model
introduced in (1), y [l] represents the lth temporal snapshot of
the measurements collected at an array of M antennas. The
unknown signal x [l] comprises D narrow-band plane waves
(with a carrier wavelength of m c) impinging on the array from
elevation angles given by H = [i 1, i 2, f, i D] T , and it has the
following algebraic structure:
x [l] = A (H) s [l] .
Underdetermined source localization and difference sets
In contrast to classical DOA esimtation, the problem of underdetermined source localization has gathered significant interest in
recent times, where the number of active sources can exceed the
number of physical sensors, i.e., D 2 M [11], [21]-[24]. This is
very different from traditional compressive sensing where the
angle space is discretized into N candidate directions, but the
number of active sources is still required to be less than M [37].
In this setting, we will mostly restrict ourselves to the case of
statistically uncorrelated sources, i.e., the source waveforms s i [l]
and s j [l] satisfy E (s i [l] s )j [l]) = v 2i d [i - j]. An important
observation is that when D 2 M, the data covariance matrix
R yy will no longer be low-rank, and traditional subspace-based
methods that utilize the null space of R yy would fail to resolve
more sources than sensors. However, a closer examination of the
algebraic structure of R xx (H) reveals an important insight
regarding the combinatorial structure of the antenna locations.
Given the set of antenna locations S = {d 1, d 2, f, d M}, define
its difference set, or difference coarray [38], [39] as
S diff = {d m - d n, 1 # m, n # M} .
[a (i i)] m = e jrd m sin ii,
assuming that the mth antenna is located at a distance of
d m m c /2 from the origin of reference, where d m is an integer. As
such, the algebraic structure of the steering vector fully captures
the geometry of the array and plays an important role in determining how many angles can be simultaneously resolved using
an array of M antennas.
The problem of localizing the DOA of narrow-band sources
(i.e., estimating H) has been intensely studied in statistical array
processing for more than three decades [28]-[36]. However, most
works (classical and modern) only consider an overdetermined
signal model where the number of sensors (M) is larger than the
number (D) of sources (i.e., M 2 D). Referring to (1), this represents the uncompressed setting where M = N and the subsampling matrix S is simply an identity matrix. In this case, the low
rank of the signal covariance R xx (H) is retained in the data covariance matrix R yy = E ^y [l] y H [l]h. Traditional subspace-based
(3)
It can be shown that each entry of R yy is actually a function of
this difference set, i.e.,
6R yy@m, n =
(2)
Here A (H) = [a (i 1), a (i 2), f, a (i D)] denotes the so-called
array manifold matrix and a (i i) is known as the steering vector
corresponding to the ith directional source. The vector s [l] represents D (baseband) source signals. The elements of the steering vector are given by
58
high-resolution algorithms such as MUSIC and ESPRIT [31], [32],
[36] exploit the null space of low-rank R yy to uniquely determine
the parameter vector H (assuming we have large enough temporal
snapshots L).
D
/ e j (d
i= 1
r
m-
d n)sin i i
v i + v w d 6m - n@ .
2
2
Here v 2w represents noise power. The number of elements in
S diff determines the distinct values in the data covariance matrix
R yy . It can be easily seen that the maximum possible cardinality
of a difference set is M 2 - M + 1. Using these distinct correlation terms judiciously in different ways, it is possible to substantially increase the number of active sources that can be detected
by the array. In particular, the vectorized covariance matrix has
the form [11]
vec ^R yyh = A diff (H) p + v 2w vec (I),
where A diff (H) is an M 2 # D matrix that behaves like the array
manifold of a (longer) virtual array whose sensor locations are
given by the difference set S diff . Hence, instead of the physical
array, one can think of applying DOA estimation algorithms to
the difference coarray and resolve more sources than sensors. For
well-designed arrays such as nested and coprime arrays, the size
of S diff can be O (M 2), and therefore the number of resolvable
sources can also potentially be as large as D = O (M 2). It is
important to understand what kind of array geometries produce
S diff with O (M 2) consecutive distinct integers [i.e., S diff is a uniform linear array (ULA), with O (M 2) elements]. The constraint
on the elements to be consecutive integers is important, since it
IEEE Signal Processing Magazine
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July 2018
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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
Contents
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