IEEE Circuits and Systems Magazine - Q1 2020 - 41

prior on x to pick the right element among the infinite
set of solutions in p of the linear systems of equalities
y = ADp. The key point in decoding is that the search
for a sparse solution can be translated into a convex optimization problem that can be tackled either exactly or
by means of some iterative approximation [2].
Compressed sensing is no longer an utterly new topic
as the seminal papers of Donoho, Candes and Tao [2]-[4]
that shed light upon this double path date back not less
than 12 years.
The amount of theoretical development in this field
is impressive as is the number of techniques used for
reconstructing the so called original signal x from the
overly famous small number of linear measurements in
the vector y [5]-[9].
Among all the methodological results, adaptation
has always been given a secondary role. The very main
theory of CS centers on words such as universal, democratic, non-adaptive, etc. Actually, the pairing of sparsity
with linear encoding is such a powerful concept that all
the theoretical guarantees that ensure CS to work are
basically independent of the specific features of the signals involved in the process.
This is a key observation. In fact, a direct consequence
of this approach is that typical theoretical guarantees
are upper bounds on reconstruction errors, and upperbounding means that a worst-case analysis has been
carried out, and worst-case analysis implicitly considers
also the worst possible signals, whatever it may mean in
each specific context.
From a more applicative point of view, guarantees
are fundamental, but signals are not as bad as one is
forced to assume in the worst-case analysis. On the contrary, most of the times they have features in addition
to sparsity that can be leveraged through adaptation to
increase system performance. This is especially important in case of hardware implementations that, by their
nature, must maximize performance while complying
with possibly severe resource constraints.
Hence, while universal theories deal with asymptotic
trends, adaptation allows the tuning of the constant coefficients hidden in the asymptotic trend formulas that
are irrelevant in the run to infinity, but largely affect the
cost and the performance of real-world systems.
This playing with constants can be a key factor in expressing the practical potential of CS. In fact, the ultimate
simplicity of the encoding step hints at applications in
which resources at the acquisition side are scarce while
computational power is available at the receiver of the
sensed information. This perfectly fits within the general framework that most of the information engineering

community is working on, i.e., the grand view implied by
"memes" like Internet of Things (IoT) or Cyber-Physical
Systems (CPS).
Networks of ubiquitous sensors and actuators whose
activity entails an intense exchange of data between
them and to local hub infrastructures, that act as a
gateway to cloud-based processing and decision, and in
which acquired signals follow a path that goes from extremely simple sensing units, to concentrators, to server
farms. The ability to compress with a limited resource
budget is appealing both at individual nodes and in gateways, and CS can be a winning option.
This review article aims at collecting the most widespread adaptation techniques presented in literature
which proved capable of improving CS performance on
real-world signals and of making the implementation of
the acquisition subsystems of IoT networks more effective. We concentrate on adapted CS and not on adaptive
CS, meaning that adaptation is performed at design-time
considering the class of signal to acquire and not at runtime on each signal instance [10]. Roughly speaking,
adapted CS methods do not require any resource tradeoff. Encoding procedures are still based on the matrix
multiplication Ax, with the advantage to (strongly)
reduce the number of rows in A. With respect to the
adoption of the standard CS theory, the produced benefits are both higher data compression and lower computational burden of the encoder. Though we tried to be
as exhaustive as possible in bibliographic search, not
every contribution is reported here as we concentrate
on those that appear to give better performance.
As a teaser on what can be done following this path,
consider a case leveraging on the simplicity and flexibility of CS, whose performance can be substantially boosted by adaptation. Electro Cardio Graphics (ECG) signals
are of interest in both healthcare and wellness oriented
applications as they give information on the status of
the hearth as well as on the activity of the subject.
The general effectiveness of CS in ECG monitoring
is discussed, for example, in [11], that shows a ≈ 40%
improved battery lifetime compared to state-of-the-art
compression techniques for an embedded ECG monitor
although no adaptation is considered.
The approach discussed in [12] uses CS as a basic
building block for ECG compression simultaneously supporting both health care and wellness applications in
a dual-mode wearable monitor. More specifically, CS is
employed as low-resource scalable lossy compression
stage working immediately after signal digitalization
and before data dispatching. From y = Ax one gets that
each measurement in the vector y contains information

Mauro Mangia and Riccardo Rovatti are with the University of Bologna, Italy. Fabio Pareschi and Gianluca Setti are with Politecnico di Torino, Italy.
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