IEEE Signal Processing - May 2018 - 42
This article gives readers a clear picture of the research
and development of CS applications in different scenarios.
By identifying the different sparse domains, we illustrate the
benefits and challenges in applying CS in wireless commu-
nication networks.
of the N coefficients in s that are nonzero. If a signal is able
to be sparsely represented in a certain domain, the CS tech-
nique can be invoked to take only a few linear and nonadap-
tive measurements.
Sparse representation
When the original signal f arrives at the receiver, it is pro-
cessed by the measurement matrix U ! R P # N with P 1 N, to
get the compressed version of the signal, i.e.,
Sparse representation of signals has received extensive atten-
tion because of its capacity for efficient signal modeling and
related applications. The method solves a problem by search-
ing for the most compact representation of a signal in terms of
a linear combination of the atoms in an overcomplete diction-
ary. The literature has focused on three aspects of sparse rep-
resentation research:
1) pursuit methods for solving the optimization problem, such
as matching pursuit and basis pursuit
2) design of the dictionary, such as the K-SVD method
3) applications of sparse representation, such as wide-band
spectrum sensing, channel estimation of massive MIMO,
and data collection in WSNs.
General sparse representation methods, such as principal com-
ponent analysis (PCA) and independent component analysis
(ICA), aim to obtain a representation that enables sufficient
reconstruction. Research has demonstrated that PCA and ICA
are able to deal with signal corruption, such as noise, missing
data, and outliers. For sparse signals without measurement
noise, CS can recover the sparse signals exactly, using random
measurements. Furthermore, the random measurements sig-
nificantly outperform measurements based on PCA and ICA
for the sparse signals without corruption [13]-[15]. In the fol-
lowing, we will focus on CS principles and the common
sparse domains potentially used in 5G and IoT scenarios.
Principles of standard CS
The principles of standard CS, when it is performed at a single
node, can be divided into three parts [3]: sparse representation,
projection, and signal reconstruction.
Projection
x = Uf = UWs = Hs,
where H = UW is a P # N matrix, called the sensing matrix.
As U is independent of signal f, the projection process
is nonadaptive.
Figure 1 illustrates how the different sensing matrices
H influence the projection of a signal from a high dimension
to its space, i.e., mapping s ! R 3 to x ! R 2 . As shown in Fig-
ure 1, s = ^s s 0h is a three-dimensional signal. When s is
mapped into a two-dimensional space by taking
H1 = c
f = Ws.
(1)
Apparently, f can be the time or space domain representation of
a signal, and s is the equivalent representation of f in the W
domain. For example, if W 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 W domain if there are only K ^K % N h out
42
1 -1 0
m
0 0 1
as the sensing matrix, the original signal s cannot be recorded
based on the projection under H 1. This is because the plane
spanned by the two row vectors of H 1 is orthogonal to signal
s, as shown in Figure 1(a). Therefore, H 1 corresponds to the
worst projection. As shown in Figure 1(b), we can also
observe that the projection by taking
H2 = c
100
m
001
is not a good one. We note that the plane spanned by the two
row vectors of H 2 can contain only part of the 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 (2-D) space. When the sensing matrix is set to
H3 = c
Sparse representation
Generally speaking, sparse signals contain much less infor-
mation than their ambient dimension suggests. The sparsity
of a signal is defined as the number of nonzero elements in
the signal under a certain domain. Let f be an N-dimension-
al signal of interest, which is sparse over the orthonormal
transformation basis matrix W ! R N # N , and let s be the
sparse representation of f over the basis W. Then, f can be
given by
(2)
110
m,
001
as shown in Figure 1(c), the signal s can be fully recorded,
as it falls into the plane spanned by the two row vectors of
H 3 . Therefore, H 3 results in a good projection, and s can be
exactly recovered by its projection x in the 2-D space. Then,
it is natural to ask what type of projection is good enough to
guarantee exact signal recovery.
The key in CS theory is to find out a stable basis W or mea-
surement matrix U to achieve exact recovery of the signal with
length N from P measurements. It seems an undetermined
problem, as P < N. However, it was proved in [4] that exact
recovery can be guaranteed under the following conditions:
■ Restricted isometry property (RIP): The measurement
matrix U has the RIP of order K if
IEEE Signal Processing Magazine
1 - dK #
|
May 2018
|
Uf ,22
# 1 + dK
f 2, 2
(3)
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