Signal Processing - January 2016 - 82

sampling, this approach advocates the acquisition of a small
domain-only a subset K of antennas was used to estimate R x;
number of samples indexed by a subset of the Nyquist grid:
the remaining antennas can be disconnected, or, more simply,
they need not be deployed. Broadly, acquisition hardware represents the bottleneck of many current signal processing systems,
y [i] = x (k i Ts), K = {k 0, f, k K - 1} .
(6)
whose designs aim at meeting an everincreasing demand for processing rapAs we will soon see, this average
ACquISItIon hArdwArE
idly changing signals. In practice,
rate reduction has led to the techrEPrESEntS thE bottLEnECk
Nyquist acquisition of wideband signals
nology of compressive ADCs
oF MAny CurrEnt SIGnAL
becomes prohibitive in many applica(C-ADCs), conceived to circumvent
ProCESSInG SyStEMS,
tions since the sampling rate drastically
the aforementioned hardware
affects power consumption and hardtradeoffs. Before exploring this
whoSE dESIGnS AIM At
ware complexity. The ambition to break
topic, let us expand the families of
MEEtInG An EvEr-InCrEASInG
this bandwidth barrier has prompted a
samplers we are about to consider.
dEMAnd For ProCESSInG
growing interest in innovative acquisiBy forming x = [x [0], f,
rAPIdLy ChAnGInG SIGnALS.
tion hardware architectures that
x [L - 1]] T and y = [y [0], f,
replace traditional equipment, such as
y [K - 1]] T, the operation in (6)
the slow and power-hungry ADCs. In this section, we delve into
can be equivalently represented as a row-selection operation
compression methods that can be applied not only for compresr x.
sive acquisition of spatial signals but also for time signals and
y=U
(7)
more general classes of signals.
r ! C K # L, which contains ones at the positions
In particular, suppose that we are interested in estimating the
The matrix U
second-order statistics of x (t), indexed by the continuous-time
(i, k i) and zeros elsewhere, is, therefore, a sparse matrix with at
index t. A traditional ADC ideally produces the sequence
most one nonzero entry at each row or column. Rather than
restricting ourselves to matrices of this form, there are certain
applications where the usage of dense compression matrices has
x [l] = x (lTs), l = 0, f, L - 1,
(5)
proven to be successful, both in the time domain (see, e.g., [4]
and [5]) and in the spatial domain (see, e.g., [23]). In correspondwhere 1/Ts is the sampling rate, a number that must exceed the
ence with this terminology, we talk about dense samplers when
Nyquist rate of x (t) to avoid aliasing. Unfortunately, power conr is dense and about sparse samplers when U
r is sparse.
sumption, amplitude resolution, and other parameters dictated by
U
the application establish stringent upper bounds on the values
As opposed to most applications in array processing, it is comthat the sampling rate can take on. These limitations conflict with
mon in time-domain applications to observe just a single realizathe constantly increasing need for larger bandwidths and, hence,
tion of the signal of interest, i.e., T = 1. This is why we dropped
higher Nyquist rates.
the subscript x from x and y in (7) when compared to x x and
A compression approach similar to the one described for the
y x in the previous section. For simplicity, we omit this subscript
spatial domain may potentially alleviate these limitations by reduthroughout when possible, keeping in mind that several snapshots
cing the average sampling rate. Generally known as nonuniform
may be available.

S = 2L − 1 Real Unknowns
WSS Processes
Direction-of-Arrival Estimation
Incoherent Imaging

S = 2d − 1 Real Unknowns
WSS with d Limited Lags
MA(d ) Time Series

(a)

(b)

S = L Real Unknowns
Orthogonal Frequency-Division
Multiplexing Signals
Multiband Signals
Incoherent Imaging
(c)

[FIG2] Some common covariance structures, along with their main applications: (a) toeplitz, (b) d-banded, and (c) circulant.

IEEE SIGNAL PROCESSING MAGAZINE [82] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
