IEEE Signal Processing - May 2018 - 44

of [21] proposed an iterative reweighted least-square (LS)-
based CS approach to solve (4) in a nonconvex approach as

0.8

st = arg min
/ w i s i subject to Hs = x,
s

0.6
Pd

^l -1h p -2

0.4
0.2

Random Demodulator
Gaussian Distributed Matrix

0
10-2

10-1
P/N

100

FIGURE 2. The detection probability versus the compression ratio with different measurement matrices. In this case, the signal is one-sparse and
the simulation iteration is 1,000 [73].

reconstruction problem in (4) is both numerically unstable and
NP-hard [3] when the , 0-norm is used.
So far, there are mainly two types of relaxations to problem
(4) to find a sparse solution. The first type is convex relaxation,
where the , 1-norm is used to substitute for the , 0-norm in (4).
Then, (4) can be solved by standard convex solvers, e.g., CVX.
A study has proved that the , 1-norm results in the same solu-
tion as the , 0 -norm when the RIP is satisfied with the constant
d 2k 1 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 comparison with convex relaxation, the greedy algo-
rithm usually requires a lower computational complexity and
time cost, which makes it more practical for wireless commu-
nication systems. Furthermore, recent results have shown that
the recovery accuracy achieved by some greedy algorithms is
comparable to convex relaxation but requires much lower com-
putational cost [20].

Reweighted CS
As mentioned previously, the , 1-norm is a good approxima-
tion for the NP-hard , 0-norm problem when the RIP holds.
However, the large coefficients are penalized more heavily
than the small ones in , 1-norm minimization, which leads to
the performance degradation of the signal recovery. To bal-
ance the penalty on the large and small coefficients, reweight-
ed CS is introduced by providing different penalties on those
coefficients. Previous work developed a reweighted , 1-norm
minimization framework [3] to enhance the signal recovery
performance with fewer compressed measurements by solving
st = arg min
Ws
s

subject to Hs = x,

(5)

where W is a diagonal matrix, with w 1, f, w n on the diago-
nal and zeros elsewhere.
Moreover, we can utilize the , p-norm, e.g., 0 1 p 1 1, to
lower the computational complexity of the signal recovery pro-
cess caused by the , 1-norm optimization problem. The authors
44

(6)

i =1

where they computed w i = s i
based on the result of
^l -1h
the last iteration s i .
Equation (4) becomes nonconvex when p 1 1. The existing
algorithms cannot guarantee reaching a global optimum and
may produce only local minima. However, other studies [22],
[23] have proven that, under some circumstances, the recon-
struction in (4) will reach a unique and global minimizer [24],
which is exactly st = s. Therefore, we can still exactly recover
the signal in practice.

Distributed CS
Distributed CS (DCS) [25] is an extension of the standard
version that considers networks with M nodes. At the mth node,
measurement x m can be given by
x m = H m s m, 6m ! M,

(7)

where M is the set of nodes in the network. As stated in (2), H m
is the sensing matrix deployed at the mth node, and s m is a sparse
signal of interest. DCS becomes standard CS when M = 1.
In applications of standard CS, the signal received at the
same node has its sparsity property because of its intracorrela-
tion, while, for networks with multiple nodes, signals received
at different nodes exhibit strong intercorrelation. The intra-
correlation and intercorrelation of signals from the multiple
nodes lead to a joint sparsity property. The joint sparsity level
is usually smaller than the aggregate over the individual sig-
nal's sparsity level. As a result, the number of compressed mea-
surements required for exact recovery in DCS can be reduced
significantly compared to the case where standard CS is per-
formed at each single node independently.
In DCS, there are two closely related concepts: distributed
networks and DCS solvers. The distributed networks refer to
those in which different nodes perform data acquisition in a dis-
tributed way and where standard CS can be applied at each node
individually to perform signal recovery. In contrast, for DCS
solvers, the data acquisition process requires no collaboration
among sensors, and the signal recovery process is performed at
several computational nodes, which can be distributed in a net-
work or locally placed within a multiple core processor [25]. In
DCS, it is generally of interest to minimize both the computa-
tional cost and communication overhead. DCS's most popular
application scenario is that all signals share the common sparse
support but with different nonzero coefficients.

Common sparse domains for CS-enabled 5G
and IoT networks
CS-enabled sub-Nyquist sampling is possible only if the sig-
nal is sparse in a certain domain. The common sparse
domains utilized in CS-enabled 5G and IoT networks include

IEEE Signal Processing Magazine

May 2018

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