IEEE Signal Processing Magazine - January 2018 - 159

Channel Fading and Shadowing
Analog signals transmitted over physical channels are
affected by two main phenomena: Rayleigh fading, or
small-scale fading, and log-normal shadowing, or large-
scale fading [35], [36], [95]. The received signal is gen-
erally described in terms of the transmitted signal s i (t )
convolved with the impulse response of the channel
h ij (t ), i.e.,
(S13)

rij (t ) = s i (t ) ) h ij (t ),

where rij (t ) is the received signal corresponding to the ith
transmission at the jth cognitive radio and * denotes the
convolution. Fading and shadowing affect the channel
response h ij (t ).
Rayleigh fading
For most practical channels, the free-space propagation
model, which only accounts for path loss, is inadequate to
describe the channel. A signal typically travels from trans-
mitter to receiver over multiple reflective paths, which is
traditionally modeled as Rayleigh fading. This implies that
the amplitude and phase of the channel response
h ij (t ) = R (t ) e jz (t) are stochastically independent and identi-
cally distributed processes. The amplitude R(t ), for t ! R,
follows the Rayleigh distribution, given by
r e -r 2 /2v2 r $ 0
p R (r ) = * v 2
0
otherwise,

We consider the following collaborative model. A network
of N rec CRs receives the N sig transmissions, such that the
received signal at the jth CR is given by
N sig

N sig

i =1

i =1

(30)

The channel response h ij (t) is determined by fading and shadowing effects. Typical models are Rayleigh fading or smallscale fading and log-normal shadowing or large-scale
fading [35], [36], [95], as described in "Channel Fading and
Shadowing." In the frequency domain, the Fourier transform
of the jth received signal is given by
N sig

X (j) ( f ) = / S i ( f ) H ij ( f ).

PL = PL 0 + 10c log d + X v.
d0

(31)

i =1

Therefore, the support of x (j) (t) is included in the support of
the original signal x (t). Because the transmissions are affected

(S15)

Here, the reference distance d 0 corresponds to a point
located in the far field of the antenna (typically, 1 km for
large cells). The PL to the reference point PL 0 is usually
found through field measurements or calculated using free-
space PL. The value of the PL exponent c depends on the
frequency, antenna heights, and propagation environ-
ment. Finally, X v denotes a Gaussian random variable (in
decibels) with the variance v 2 determined heuristically as
well [95]. The shadowed received signal is thus given by
rij (t ) = 10 -PL ij /20 ·s i (t ),

(S14)

where 2v 2 is the mean power [95]. The phase z (t ), for
t ! R, is uniformly distributed over the interval [0, 2r).

x (j) (t) = / rij (t) = / s i (t) ) h ij (t).

Log-normal shadowing
Large-scale fading represents the average signal power
attenuation or path loss due to motion over large areas.
The resulting channel frequency response is, therefore, a
constant. This phenomenon is affected by prominent ter-
rain contours between the transmitter and receiver.
Empirical measurements suggest that this type of fading,
or shadowing, follows a normal distribution in decibel
units [96], or, alternatively, the linear channel gain may
be modeled as a log-normal random variable [36].
Therefore, the path loss (PL) measured in decibels is
expressed as

(S16)

where PL ij denotes the PL between the ith transmitter
and the j th receiver and the channel r e s p o n s e
h ij (t ) = 10 -PL ij /20 d (t ).

differently by fading and shadowing from each transmitter to
each CR, we can assume that the union of their respective supports is equivalent to the frequency support of x (t). The goal
here is to assess the support of the transmitted signal x (t)
from sub-Nyquist samples of the received x (j) (t), 1 # j # N rec,
by exploiting their joint frequency sparsity.
A simple and naive approach is to perform support recovery at each CR from its low-rate samples and combine the
local binary decisions, either in a fusion center for centralized
collaboration or in a distributed manner. In this hard-decision
strategy, the combination can be performed using several
fusion rules, such as AND, OR, or a majority rule. Although
this method is attractive due to its simplicity and low communication overhead, it typically achieves lower performance
than its soft-decision counterpart. To mitigate the communication overhead, soft-decision-based methods may rely on
sharing observations based on the low-rate samples, rather
than the samples themselves. In the next section, we review
such techniques both in centralized and distributed contexts.

IEEE SIGNAL PROCESSING MAGAZINE

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January 2018

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159
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