Signal Processing - January 2016 - 86
Normalized MSE
H 0: R x = R w
H 1: R x = R r + R w ,
(16)
10−1
10−2
10−3
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Normalized Sampling Rate (M/N )
LSR, L = 1,786
Nyquist, L = 1,786
DS, L = 1,786
LSR, L = 5,952
Nyquist, L = 5,952
DS, L = 5,952
[FIG5] the mean-squared error of the estimate of the leastsquares algorithm when R x is 168-banded. (Figure adapted
from [19].)
compression, which is to reduce the average sampling rate-a
parameter that critically affects the hardware cost.
However, note that this approach does not exploit the fact that
R x is positive semidefinite. This constraint can be enforced to
improve the estimation performance at the expense of greater complexity. For instance, one may attempt to minimize the least
r *7U
r ) Svt | | 2 subject to the constraint
squares cost || vt y - (U
R x $ 0, which is a convex problem. Other constraints can also be
imposed if more prior information is available. For instance, the
elements of at might be nonnegative [30], in which case one would
introduce the constraint at $ 0. It can be known that at is sparse
either by itself or on a linearly transformed domain, in which case
one may impose the constraint || Fs at | | 0 # S 0, where S 0 is the
number of nonzero entries and Fs takes at to the domain where it
is sparse. For instance, the elements of Fs at may be samples of the
power spectrum [37]. Since the zero-norm in this constraint is not
convex, it is typically relaxed to an , 1 -norm. For example, an , 1
-norm regularized least-squares formulation can be adopted as
follows:
minimize
t
a
r *7U
r ) S at
vt y - (U
2
+ m F s at 1 .
(15)
In (15), signal compression is induced by the statistical structure
of R x beyond sparsity, while the additional sparsity structure can
lead to stronger compression at the expense of increased computational complexity compared to the closed-form solution in (14).
DETECTION
In detection theory, we are interested in deciding whether a signal
of interest is present or not. This operation is typically hindered by
the presence of noise and other waveforms, such as clutter in
radar or interference in communications.
In many cases, this problem can be stated in terms of the second-order statistics of the signals involved, so the goal is to decide
one of the following hypotheses:
where R r and R w, respectively, collect the second-order statistics
of the signal of interest and noise/interference. Our decision must
r x,
be based on the observation of the compressed samples y = U
r R wU
r H under H 0 and
whose covariance matrix R y is given by U
r (R r + R w) U
r H under H 1 . A most powerful detection rule
by U
exists for this simple setting and can be found using the Neyman-
Pearson lemma [11]. If p (y; H i) denotes the density under
hypothesis H i, this rule decides H 1 when the ratio
p (y; H 1) /p (y; H 0) exceeds a certain threshold set to achieve a
target probability of false alarm [11].
More general problems arise by considering basis expansions
like the one in (4). In this case, the goal may be to decide whether
one of the a i, say a 0, is positive or zero, while the others are
unknown and treated as nuisance parameters [30]. Since in these
cases no uniformly most-powerful test exists, one must resort to
other classes of detectors, such as the generalized likelihood ratio
test, which makes a decision by comparing p (y; at H1) /p (y; at H0)
against a threshold, where at Hi is the maximum-likelihood estimate of a under hypothesis H i [30].
ModAL AnALySIS
As mentioned in the section "Main Applications," the problem of
estimating the frequency of a number of noise-corrupted sinusoids and the problem of estimating the direction of arrival of a
number of sources in the far field are instances of the class of
sparse spectrum estimation problems, which allow a common
formulation as modal analysis [11].
Suppose that the observations are given by
x=
R-1
/ s i a (i) + w = As + w,
(17)
i=0
where a (i) = [1, e .~ i, f, e .~ i (L - 1)] T are the so-called steering
vectors, A = [a (0),f, a (R - 1)] is the manifold matrix, w is noise and
the coefficients s i, collected in the vector s, are uncorrelated random variables. The structure of a (i) stems from the fact that each
antenna receives the signal s i with a different phase shift. Because
the antennas are uniformly spaced in a ULA, the relative phase shift
between each pair of antennas is an integer multiple of a normalized
quantity ~ i, which is a function of the angle of arrival.
The covariance matrix of x is given by
R x = AR s A H + v 2w I L,
(18)
where v 2w is the power of the noise process, assumed white for
simplicity, and R s is the covariance matrix of s, which is diagonal
since the sources are uncorrelated. Note that these assumptions
result in R x having a Toeplitz structure.
rx=A
r s,
The compressed observations can be written as y = U
r A, and have covariance matrix
r =U
where A
IEEE SIGNAL PROCESSING MAGAZINE [86] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
Signal Processing - January 2016 - Cover1
Signal Processing - January 2016 - Cover2
Signal Processing - January 2016 - 1
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Signal Processing - January 2016 - Cover3
Signal Processing - January 2016 - Cover4
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