IEEE Circuits and Systems Magazine - Q1 2020 - 57

Yet, as extensively shown in this paper, CS performance strongly depends on the choice of the acquisition sequences. We have seen that sensing sequences
need to be randomly drawn or generated by adopting
one of the discussed approaches. While the complexity
of the second solution is obvious, the first case may
present drawbacks when using a simple linear feedback shift register (LFSR) due to the low quality of the
generated stream. The problem is relevant, in particular, if the number of channels is high, and so the number of different elements in a to be generated at the
same time.
While in [48], [51] the generation of sequences a is
achieved by a simple LFSR with no additional details,
in [49] a complex Fibonacci-Galois 384-bit LFSR is designed. Basically, 64 6-bit Fibonacci LFSRs have been integrated into the circuit, each one generating a different
a. Then, the 64 LFSRs are further randomized by dithering their less significant bits in a Galois fashion, each
LFSR using the most significant bits of another stage. An
external trigger signal enables a 384-bit seed load at the
beginning of each integration window7.
A completely different problem was instead faced in
[47]. The proposed integrated circuit is a sub-Nyquist
sampler for a 2 GHz bandwidth input signal. The circuit
has a continuous-time analog architecture, and uses
PAM sampling signal a(t ) obtained from antipodal sequences a, where the multiplication by a(t ) is simply
achieved by exchanging the differential lines of the
differential input signal. Due to the 2 GHz input signal
bandwidth, the a k symbols must be generated at a rate
equal to 4 Gbit/s. The use of an internal serial memory
for storing the a vector, built upon a programmable
shift register, even with all its implementation drawbacks, has been found to be the only solution allowing
versatility at this speed.
The problem of the efficient generation of the elements of a has also been considered in [53]. The authors propose a simple and deterministic algorithm to
generate binary vectors a (i.e., a k ! {0, 1}) that, once
collected into the sensing matrix A, ensures that i ) A
satisfies the theoretically requirements for input signal
reconstruction at the decoder side; and ii ) A is easily
obtained with a finite state machine.
The method proposed in [53] relies on the quasicyclic array code based binary matrix framework. In
particular, the authors aim to utilize the parity-check
matrices of array codes and their submatrices to construct A. In more details, let us indicate with I q the
7
Note that, even if [49] is the only work where the elements of each a
are the approximations of real quantities, the authors considers only
uniformly distributed random values due to the complexity of generating a Gaussian distribution at hardware level.

FIRST QUARTER 2020

q # q identity matrix, and Pq the q # q cyclic permutation matrix defined as
R0 1 0 g 0V
S
W
S0 0 1 g 0W
Pq = SS h h h j h WW
S0 0 0 g 1W
S1 0 0 g 0W
T
X
Given an integer r, the sensing matrix A is the rq # q 2
binary matrix given by
R
SI q I q
SI q Pq
A=S
h
Sh
SI q P qr - 1
T

V
Iq
g
W
P qq - 1 W
g
W
j
h
W
(r - 1)(q - 1)W
g Pq
X

According to [53], the proposed approach shows comparable recovery performance for EEG and spike data
compression with respect to standard approach at a reduced hardware complexity.
F. The Spatio-Temporal Approach
In the works [50], [51] authors consider a multichannel
EEG recording as input. A multichannel signal may be
modeled as an array x (t ) composed of p real functions,
with x : R 7 R p. By windowing it and sampling it at rate
Tw /n, we get for each I (l ), a slice of signal represented by
a p # n matrix, that can be easily unrolled to get a p # n
vector. The standard CS framework can then be applied
to this vector with no other modification.
Indeed, a spatio-temporal approach may lead to several advantages with respect to a standard approach,
since it make possible to exploit the input signal features
in two different domains (i.e. spatial and temporal one).
This is an open research topic. Yet, by limiting ourselves to a pure circuital level consideration, it interesting to see how in [51] this model is used to reduce the
hardware complexity of y = a < x. In fact, at each time
step, a number of measurements is computed as a linear
combination of the samples coming from all input signal
channels sampled at that time step. Mathematically, at
time step k, we get the generic measurement
p-1

/ a j x ( j ) c k Tnw m

(12)

j=0

Then, all measurements generated at the k-th time
step are collected by the reconstruction algorithm and
joined to all measurements generated for all other time
steps belonging to I (l ). During reconstruction, both
sparsity (on the time domain) and correlation (on the
spatial domain) are used to improve the input signal reconstruction quality.
Computing measurements with a multiply-and-accumulate operation in the spatial domain only as in (12)
has a twofold advantage.
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IEEE Circuits and Systems Magazine - Q1 2020

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