IEEE Signal Processing - July 2018 - 61
+
+ H
(H) P (A diff
)
The signal covariance matrix R xx (H) = A diff
(here P is a diagonal matrix containing the source powers) is
Toeplitz structured, i.e., [R xx (H)] m, n is only a function of the
difference m - n. Furthermore, it is also rank deficient with
rank D as long as D 1 M ca . Owing to the structure of the nested array, all distinct entries of R xx (H) are retained in the measurement covariance matrix R yy. Therefore, it is possible to
recover R xx (H) from R yy and estimate H by applying standard
subspace-based algorithms on R xx (H). The exact procedure for
estimating R xx (H) from a finite number of measurement
vectors y [l], 1 # l # L is revisited in the section "Algorithms and Fundamental Limits for Correlation-Aware LowRank Estimation."
Toeplitz-structured covariance compression
The concept of difference set-inspired samplers can be actually generalized beyond array processing and used to
compressively sample a WSS process x [l] and yet perfectly reconstruct its second-order statistics or power spectrum
density; see [6] for a recent survey. Such WSS processes
arise in a large number of applications such as power spectrum estimation for communication and cognitive radio
[8], [20], [41], [42]. When x [l] is WSS, its autocorrelation
satisfies the property E (x [n] x ) [n - k]) = R xx [k], i.e., it only
depends on the lag k and not on the instant n. Let x [l] =
[x [l], x [l - 1], f, x [l - N + 1]] T . In this case, the covariance
matrix R xx = E (x [l] x H [l]) will be Toeplitz structured for
any N.
Owing to inherent redundancies present in a Toeplitz matrix,
it is possible to acquire compressive measurements of x [l] and
yet reconstruct R xx from the second-order statistics of the compressed measurements. This has been popularized as CCS and
compressive power spectrum estimation [6]-[8], [17], [18], [23].
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Given the blocked version x [l] (with block length N ) of the WSS
process x [l], the compressive measurements can be acquired as
y 6 l @ = Sx 6 l @ .
(5)
The compression factor is given by
C= M.
N
(6)
The covariance matrix R yy = E ^y [l] y H [l]h of the measurements serves as a compressive sketch of R xx and the goal of
CCS is to design a sampler S that can recover the uncompressed Toeplitz matrix from the compressive sketch. This
problem has been studied and families of sparse and dense samplers have been examined [6], [7], [18]. While dense sampling
matrices S mostly correspond to random matrices with independent and identically distributed (i.i.d.) entries, the design of
sparse S again relies upon the combinatorial idea of difference
sets. A sparse S can be readily implemented using conventional
A/D converters and also offers computational benefits, and
deterministic guarantees. A comparative study of compression
ratios attained by random and sparse-ruler based samplers is
provided in [6].
The optimal sparse S corresponding to a sparse ruler or circular sparse ruler (if the matrix is also circulant) of length N
does not have an analytic closed form. However, the idea of a
nested array can be generalized to design suitable sampling matrices for compressing WSS signals for arbitrary N. This gives
rise to a generalized nested sampling (GNS) matrix S GNS of
size M # N, where M = O ( N ) [43]. For the special case of
N = M 2 /4 + M/2, GNS coincides with the nested sampler S nest
introduced previously. Figure 3 shows how the GNS compresses
symmetric Toeplitz matrices by rearranging the N distinct
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Principal Minors
Toeplitz
GNS Sketch
Toeplitz Matrix
Figure 3. A GNS of length N compresses a Toeplitz matrix of size N # N to produce a sketch of size M # M, where M = O ( N ) . Shown are the individual entries of the Toeplitz matrix and how they are rearranged in the compressed sketch.
IEEE Signal Processing Magazine
|
July 2018
|
61
Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018
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