IEEE Circuits and Systems Magazine - Q1 2020 - 52

Regrettably, a rapid scan of the recent literature on
CS related works reveals that among the overwhelming
number of works that can be found, only a negligible
fraction of them deals with the actual circuital implementation of the proposed algorithm or technique.
To the best of authors' knowledge, the first prototype
capable of implementing a CS-based system via the generic y = Ax product has been presented in [47]. The
circuit is a sub-Nyquist rate receiver for radar pulse
signal designed in 90 nm technology, capable to acquire
signals up to 2 GHz. In [48], the authors describe a CSbased data acquisition front-end for a radio frequency
(RF) communication system implemented in 90 nm
CMOS process. The work in [49] presents an analog
front-end for ECG signals designed in 180  nm CMOS
process, while [50] reports an area and power efficient
multi-electrode arrays acquisition system based on CS
designed in 180 nm CMOS process, outperforming previously presented works in terms of compression rate and
reconstruction quality by a run-time adaptation. The
work [51] describes a low-power sub-Nyquist sampler
for the multichannel acquisition of cortical intracranial
electroencephalographic (iEEG) signals. The peculiarity of this architecture, which has been fabricated in
180 nm CMOS process, is to consider the signal features
not only in the temporal domain, but also in the spatial
domain. The architecture presented in [37], designed
in 180 nm CMOS process, is an analog-to-information
converter for generic biomedical signals. It introduces a
smart saturation checking mechanism with which it is
possible to reconstruct the acquired signal even if many
measurements suffer saturation, and exploits the aforementioned pow-R approach introduced in Section III-B.
In [52], the authors propose a run-time signal evaluation
module (indicated as dynamic knob) designed in 130 nm
CMOS process, with the aim of improving the quality of
the following CS encoder by adapting a few parameters
towards the input biosignal dynamics. Finally, in [53],
the complexity of the VLSI implementation of the generator of the sensing matrix A is investigated, and the
hardware efficient generation of deterministic sparse
sensing matrices is considered.

A first group of works [37], [47]-[49], [51], [54] deal
with an analog input signal represented as a function
x (t ) of the time variable t. As an example, x (t ) may
be the output of a sensor (typically a voltage), or, more
frequently, x (t ) is differentially encoded, i.e., it is represented as the difference between two (voltage) quantities as x (t ) = x +(t ) - x -(t ). In a few cases, as in [51],
the authors consider a multi-channel scenario, and the
input signal x (t ) is actually an array of p signals coming
from different sensors.
For the sake of simplicity, in the following we limit ourselves to the scalar case regardless of the fact that the
actual implementation of x is differential or single-ended.
When dealing with an analog signal, two issues immediately arise if one thinks about the computation of
y = a < x : i ) according to the standard CS model, x is assumed to be a vector, while in a real implementation the
best fit for the input signal is a function of time x (t ); ii )
x is n-dimensional, while x (t ) is defined over the whole
R , i.e., it has an infinite dimensionality.
The standard solutions adopted to reconcile the different signal models are sketched in Figure 9. The common way to cope with dimensionality is by windowing
x (t ) [37], [47]-[49]. The input signal is sliced to get the
functions x (l )(t ), x (l + 1)(t ), x (l + 2)(t ), f, each of them defined on contiguous and non overlapping time intervals
I (l ), I (l + 1), I (l + 2), f, and such that x (l ) : I (l ) 7 R . The l-th
slice of the input signal x (l )(t ) gives rise to measurements
y (l ), that are used to reconstruct x (l )(t ). The complete
input signal can then be achieved by joining all the reconstructed slices.
Conversely, coping with the first issue has not a
unique solution. The most general approach is that adopted by [47], [48], where the generic measurement y
is achieved by a continuous time multiply-and-integrate
architecture as in the "analog continuous-time" case of
Figure 9. The l-th slice of the input signal x (l )(t ) is first
multiplied by a sensing function a(t ) (assumed defined
over I (l ) ) and then integrated over I (l ). Focusing for simplicity on the case l = 0 where I (0) = [0, Tw], the measurement is expressed as
Tw

y (0) =
A. Computation of Compressed Measurements
All aforementioned implementations share the same
philosophy: in order to compute m different measurements, the very same hardware block is either replicated m times or used in an interleaved way m times,
and driven by the input signal x and the m different
instances corresponding to the rows of A. For the sake
of simplicity, in the following we will focus on the scalar
product y = a < x, where a. is a generic row of A and y
the corresponding element of the measurement vector.
52

IEEE CIRCUITS AND SYSTEMS MAGAZINE

#

a(t ) x (0) (t ) dt

(9)

0

It is interesting to notice that also this case can
be easily incorporated into the standard framework
y = a < x under the reasonable assumption that a(t ) is
generated starting from n coefficients a k stored into a
local memory as the pulse-amplitude modulated (PAM)
function a (t) = R nk -= 10 a k g (t $ n/Tw - k), being g ($) a normalized pulse5.
5

A typically, but not necessary, choice for g(·) is the rectangular pulse.
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