1
Sparse Representation for Wireless
Communications
Zhijin Qin
1
, Jiancun Fan
2
, Yuanwei Liu
3
, Yue Gao
3
, and Geoffrey Ye Li
4
1
Lancaster University, Lancaster, UK
2
Xi’an Jiaotong University, Xi’an, China
3
Queen Mary University of London, London, UK
4
Georgia Institute of Technology, Atlanta, GA, USA
Abstract—Sparse representation can efficiently
model signals in different applications to
facilitate processing. In this article, we
will discuss various applications of sparse
representation in wireless communications, with
focus on the most recent compressive sensing
(CS) enabled approaches. With the help of the
sparsity property, CS is able to enhance the
spectrum efficiency and energy efficiency for
the fifth generation (5G) networks and Internet
of Things (IoT) networks. This article starts
from a comprehensive overview of CS principles
and different sparse domains potentially used
in 5G and IoT networks. Then recent research
progress on applying CS to address the major
opportunities and challenges in 5G and IoT
networks is introduced, including wideband
spectrum sensing in cognitive radio networks,
data collection in IoT networks, and channel
estimation and feedback in massive MIMO
systems. Moreover, other potential applications
and research challenges on sparse representation
for 5G and IoT networks are identified. This
article will provide readers a clear picture of
how to exploit the sparsity properties to process
wireless signals in different applications.
Keywords: Wireless communications, com-
pressive sensing, sparsity property, 5G, Internet
of Things.
I. INTRODUCTION
Sparse representation expresses some signals
as a linear combination of a few atoms from a
prespecified and over-complete dictionary [1].
This form of sparse (or compressible) structure
arises naturally in many applications [2]. For
example, audio signals are sparse in frequency
domain, especially for the sounds representing
tones. Image processing can exploit a sparsity
property in the discrete cosine domain, i.e.
many discrete cosine transform (DCT) coeffi-
cients of images are zero or small enough to be
regarded as zero. This type of sparsity property
has enabled intensive research on signal and
data processing, such as dimension reduction
in data science, wideband sensing in cognitive
radio networks (CRNs), data collection in large-
scale wireless sensor networks (WSNs), and
channel estimation and feedback in massive
MIMO.
Traditionally, signal acquisition and trans-
mission adopt the procedure with sampling
and compression. As massive connectivity is
expected to be supported in the fifth generation
(5G) networks and Internet of Things (IoT) net-
works, the amount of generated data becomes
huge. Therefore, signal processing has been
confronted with challenges on high sampling
rates for data acquisition and large amount of
data for storage and transmission, especially in
IoT applications with power-constrained sensor
nodes. Except for developing more advanced
sampling and compression techniques, it is
natural to ask whether there is an approach
to achieve signal sampling and compression
simultaneously.
As an appealing approach employing sparse
representations, compressive sensing (CS) tech-
nique [3] has been proposed to reduce data
acquisition costs by enabling sub-Nyqusit sam-
pling. Based on the advanced theory [4], CS
has been widely applied in many areas. The
2
key idea of CS is to enable exact signal re-
construction from far fewer samples than that
is required by the Nyquist-Shannon sampling
theorem provided that the signal admits a sparse
representation in a certain domain. In CS, com-
pressed samples are acquired via a small set
of non-adaptive, linear, and usually randomized
measurements, and signal recovery is usually
formulated as an l
0
-norm minimization prob-
lem to find the sparsest solution satisfying the
constraints. Since l
0
-norm minimization is an
NP-hard problem, most of the exiting research
contributions on CS solve it by either approx-
imating it to a convex l
1
-norm minimization
problem [4] or adopting greedy algorithms,
such as orthogonal match pursuit (OMP).
It is often the case that the sparsifying trans-
formation is unknown or difficult to determine.
Therefore, projecting a signal to its proper
sparse domain is quite essential in many appli-
cations that invoke CS. In 5G and IoT networks,
the identified sparse domains mainly include
frequency domain, spatial domain, wavelet do-
main, DCT domain, etc. CS can be used to
improve spectrum efficiency (SE) and energy
efficiency (EE) for these networks. By enabling
the unlicensed usage of spectrum, CRNs exploit
spectral opportunities over a wide frequency
range to enhance the network SE. In wideband
spectrum sensing, spectral signals naturally ex-
ploit a sparsity property in frequency domain
due to low utilization of spectrum [5], [6],
which enables sub-Nyquist sampling on cog-
nitive devices. Another interesting scenario is a
small amount of data collection in large-scale
WSNs with power-constrained sensor nodes,
such as smart meter monitoring infrastructure in
IoT applications. In particular, the monitoring
readings usually have a sparse representation
in DCT domain due to the temporal and spatial
correlations [7]. CS can be applied to enhance
the EE of WSNs and to extend the lifetime
of sensor nodes. Moreover, massive MIMO is
a critical technique for 5G networks. In mas-
sive MIMO systems, channels corresponding to
different antennas are correlated. Furthermore,
a huge number of channel coefficients can be
represented by only a few parameters due to
a hidden joint sparsity property caused by the
shared local scatterers in the radio propagation
environment. Therefore, CS can be potentially
used in massive MIMO systems to reduce the
overhead for channel estimation and feedback
and facilitate precoding [8]. Even though vari-
ous applications have different characters, it is
worth noting that the signals in different sce-
narios share a common sparsity property even
though the sparse domains can be different,
which enables CS to enhance the SE and EE
of wireless communications networks.
There have been some interesting surveys
on CS [9] and its applications [10]–[12]. One
of the most popular articles on CS [9] has
provided an overview on the theory of CS as
a novel sampling paradigm that goes against
the common wisdom in data acquisition. CS-
enabled sparse channel estimation has been
summarized in [10]. In [11], a comprehen-
sive review of the application of CS in CRNs
has been provided. A more specific survey on
compressive covariance sensing has been pre-
sented in [12] that includes the reconstruction
of second-order statistics even in the absence
of prior sparsity information. These existing
surveys serves different purposes. Some cover
the basic principles for beginners and others
focus on specific aspects of CS. Different from
the existing literature, our article provides a
comprehensive overview of the recent contri-
butions on CS-enabled wireless communica-
tions from the perspective of adopting different
sparse domain projections.
In this article, we will first introduce the
basic principles of CS briefly. Then we will
present the different sparse domains for signals
in wireless communications. Subsequently, we
will provide CS-enabled frameworks in various
wireless communications scenarios, including
wideband spectrum sensing in CRNs, data col-
lection in large-scale WSNs, and channel es-
timation and feedback for massive MIMO, as
they have been identified to be critical to 5G
and IoT networks and share the same spirit by
exploiting the sparse domains aforementioned.
Within each identified scenario, we start with
projecting a signal to a sparse domain, then in-
3
troduce the CS-enabled framework, and finally
illustrate how to exploit joint sparsity in the CS-
enabled framework. Moreover, the reweighted
CS approaches for each scenario will be dis-
cussed, where the weights are constructed by
prior information depending on specific appli-
cation scenarios. The other potential applica-
tions and research challenges on applying CS
in wireless networks will also be discussed and
followed by conclusions.
This article gives readers a clear picture on
the research and development of the applica-
tions of CS in different scenarios. By identi-
fying the different sparse domains, this article
illustrates the benefits and challenges on apply-
ing CS in wireless communication networks.
II. SPARSE REPRESENTATION
Sparse representation of signals has received
extensive attention due to its capacity for ef-
ficient signal modelling and related applica-
tions. The problem solved by the sparse rep-
resentation is to search for the most compact
representation of a signal in terms of a linear
combination of the atoms in an overcomplete
dictionary. In the literature, three aspects of
research on the sparse representation have been
focused:
1) Pursuit methods for solving the optimiza-
tion problem, such as matching pursuit
and basis pursuit;
2) Design of the dictionary, such as the K-
SVD method;
3) Applications of the sparse representa-
tion, such as wideband spectrum sensing,
channel estimation of massive MIMO,
and data collection in WSNs.
General sparse representation methods, such
as principal component analysis (PCA) and
independent component analysis (ICA), aim to
obtain a representation that enables sufficient
reconstruction. It has been demonstrated that
PCA and ICA are able to deal with signal
corruption, such as noise, missing data, and out-
liers. For sparse signals without measurement
noise, CS can recover the sparse signals exactly
with random measurements. Furthermore, the
random measurements significantly outperform
measurements based on PCA and ICA for the
sparse signals without corruption [13]–[15]. In
the following, we will focus on the principles of
CS and the common sparse domains potentially
used in 5G and IoT scenarios.
A. Principles of Standard Compressive Sensing
The principles of standard CS, such as to be
performed at a single node, can be summarized
into the following three parts [3]:
1) Sparse Representation: Generally speak-
ing, sparse signals contain much less infor-
mation than their ambient dimension suggests.
Sparsity of a signal is defined as the number of
non-zero elements in the signal under a certain
domain. Let f be an N-dimensional signal of
interest, which is sparse over the orthonormal
transformation basis matrix Ψ ∈ R
N×N
, and s
be the sparse representation of f over the basis
Ψ. Then f can be given by
f = Ψs. (1)
Apparently, f can be the time or space domain
representation of a signal, and s is the equiva-
lent representation of f in the Ψ domain. For
example, if Ψ is the inverse Fourier transform
(FT) matrix, then s can be regarded as the
frequency domain representation of the time
domain signal, f. Signal f is said to be K-sparse
in the Ψ domain if there are only K (K N )
out of the N coefficients in s that are non-zero.
If a signal is able to be sparsely represented
in a certain domain, the CS technique can be
invoked to take only a few linear and non-
adaptive measurements.
2) Projection: When the original signal f
arrives at the receiver, it is processed by the
measurement matrix Φ ∈ R
P ×N
with P < N,
to get the compressed version of the signal, that
is,
x = Φf = ΦΨs=Θs, (2)
where Θ = ΦΨ is an P ×N matrix, called the
sensing matrix. As Φ is independent of signal
f, the projection process is non-adaptive.
Fig. 1 illustrates how the different sensing
matrices Θ influence the projection of a signal
from high dimension to its space, i.e., mapping
4
s ∈ R
3
to x ∈ R
2
. As shown in Fig. 1,
s =
s s 0
is a three-dimensional signal.
When s is mapped into a two-dimensional
space by taking Θ
1
=
1 −1 0
0 0 1
!
as the
sensing matrix, the original signal s cannot be
recorded based on the projection under Θ
1
.
This is because that the plane spanned by
the two row vectors of Θ
1
is orthogonal to
signal s as shown in Fig. 1(a). Therefore, Θ
1
corresponds to the worst projection. As shown
in Fig. 1(b), we can also observe that the
projection by taking Θ
2
=
1 0 0
0 0 1
!
is not
a good one. It is noted that the plane spanned
by the two row vectors of Θ
2
can only contain
part of information of the sparse signal s, and
the sparse component in the direction of s
2
is missed when the signal s is projected into
the two-dimensional space. When the sensing
matrix is set to Θ
3
=
1 1 0
0 0 1
!
, as shown
in Fig. 1(c), the signal s can be fully recorded
as it falls into the plane spanned by the two row
vectors of Θ
3
. Therefore, Θ
3
results in a good
projection and s can be exactly recovered by
its projection x in the two-dimensional space.
Then it is natural to ask what type of projection
is good enough to guarantee the exact signal
recovery?
The key of CS theory is to find out a stable
basis Ψ or measurement matrix Φ to achieve
exact recovery of the signal with length N
from P measurements. It seems an undeter-
mined problem as P < N. However, it has
been proved in [4] that exact recovery can be
guaranteed under the following conditions:
• Restricted isometry property (RIP): Mea-
surement matrix Φ has the RIP of order
K if
1 − δ
K
≤
kΦf k
2
`
2
kf k
2
`
2
≤ 1 + δ
K
(3)
holds for all K-sparse signal f , where δ
K
is the restricted isometry constant of a
matrix Φ.
• Incoherence property: Incoherence prop-
erty requires that the rows of measure-
ment matrix Φ cannot sparsely represent
the columns of the sparsifying matrix Ψ
and vice verse. More specifically, a good
measurement will pick up a little bit in-
formation of each component in s based
on the condition that Φ is incoherent with
Ψ. As a result, the extracted information
can be maximized by using the minimal
number of measurements.
It has been pointed out that verifying both
the RIP condition and incoherence property is
computationally complicated but they could be
achieved with a high probability simply by
selecting Φ as a random matrix. The com-
mon random matrices include Gaussian matrix,
Bernoulli matrix, or almost all others matrices
with independent and identically distributed
(i.i.d.) entries. Besides, with the properties of
the matrix with i.i.d. entries Φ, the matrix
Θ = ΦΨ is also random i.i.d., regardless of the
choice of Ψ. Therefore, the random matrices
are universal as they are random enough to be
incoherent with any fixed basis. It has been
demonstrated that random measurements can
universally capture the information relevant for
many compressive signal processing applica-
tions without any prior knowledge of either
the signal class and its sparse domain or the
ultimate signal processing task.
Moreover, for Gaussian matrices the num-
ber of measurements required to guarantee
the exact signal recovery is almost minimal.
However, random matrices inherently have two
major drawbacks in practical applications: huge
memory buffering for storage of matrix ele-
ments, and high computational complexity due
to their completely unstructured nature [16].
Compared to the standard CS that limits its
scope to standard discrete-to-discrete measure-
ment architectures using random measurement
matrices and signal models based on standard
sparsity, more structured sensing architectures,
named structured CS, have been proposed to
implement CS on feasible acquisition hardware.
So far, many efforts have been put on the design
of structured CS matrices, i.e., random demod-
ulator [17], to make CS implementable with ex-
pense of performance degradation. Particularly,
5
0
s
S s
=
1
s
2
s
3
s
0
s
S s
=
1
s
2
s
3
s
1
1 1 0
0 0 1
−
Θ =
2
1 0 0
0 0 1
Θ =
3
1 1 0
0 0 1
Θ =
2
s
0
s
S s
=
1
s
3
s
(a) Worst projection
(b) Bad projection (c) Good projection
Fig. 1: Projection of a sparse signal with one non-zero component with different sensing matrices.
the main principle of random demodulator is
to multiply the input signal with a high-rate
pseudonoise sequence, which spreads the signal
across the entire spectrum. Then a low-pass
anti-aliasing filter is applied and the signal is
captured by sampling it at a relatively low rate.
With the additional digital processing to reduce
the burden on the analog hardware, random
demodulator bypasses the need for a high-rate
analogue-to-digital converter (ADC) [17]. A
comparison of Gaussian sampling matrix and
random demodulator is provided in Fig. 2 in
terms of detection probability with different
compression ratios P/N. From the figure, the
Gaussian sampling matrix performs better than
the random demodulator.
3) Signal Reconstruction: After the com-
pressed measurements are collected, the orig-
inal signal should be reconstructed. Since most
of the basis coefficients in s are negligible, the
original signal can be reconstructed by finding
out the minimal set of coefficients that matches
the set of compressed measurements x, that is,
by solving
ˆs = arg min
s
ksk
`
p
subject to Θs = x, (4)
where k·k
`
p
is the `
p
-norm and p = 0 cor-
responds to counting the number of non-zero
elements in s. However, the reconstruction
10
-2
10
-1
10
0
0
0.2
0.4
0.6
0.8
1
P/N
P
d
Random demodulator
Gaussian distributed matrix
Fig. 2: Detection probability versus compres-
sion ratio with different measurement matrices.
In this case, the signal is one-sparse and the
simulation iteration is 1000.
problem in (4) is both numerically unstable and
NP-hard [3] when `
0
-norm is used.
So far, there are mainly two types of relax-
ations to problem (4) to find a sparse solution.
The first type is convex relaxation, where `
1
-
norm is used to substitute `
0
-norm in (4).
Then (4) can be solved by standard convex
solvers, e.g., cvx. It has been proved that `
1
norm results in the same solution as `
0
norm
when RIP is satisfied with the constant δ
2k
<
√
2 − 1 [18]. Another type of solution is to
use a greedy algorithm, such as OMP [19], to
find a local optimum in each iteration. In com-
