Wireless Sensor Network, 2011, 3, 24-37
doi:10.4236/wsn.2011.31004 Published Online January 2011 (http://www.SciRP.org/journal/wsn)
Copyright © 2011 SciRes. WSN
Collaborative Spectrum Sensing for Cognitive Radio:
Diversity Combining Approach
Oscar Filio-Rodriguez1, V. Kontorovich2, Serguei Primak1, F. Ramos-Alarcon2
1Department of Electrical and Compu ting Engineering, University of Western Ontario, London, Canada
2Electrical Engineering Department, Research and Advanced Studies Center, Mexico City, Mexico.
E-mail: valeri@cinvestav.mx
Received August 5, 2010; revised September 2, 2010; accepted January 15, 2011
Abstract
In this paper it is shown that cyclostationary spectrum sensing for Cognitive Radio networks, applying mul-
tiple cyclic frequencies for single user detection can be interpreted (with some assumptions) in terms of op-
timal incoherent diversity addition for “virtual diversity branches” or SIMO radar. This approach allows
proposing, by analogy to diversity combining, suboptimal algorithms which can provide near optimal cha-
racteristics for the Neyman-Pearson Test (NPT) for single user detection. The analysis is based on the Gene-
ralized Gaussian (Klovsky-Middleton) Channel Model, which allows obtaining the NPT noise immunity
characteristics: probability of misdetection error (PM) and probability of false alarm (Pfa) or Receiver Opera-
tional Characteristics (ROC) in the most general way. Some quasi-optimum algorithms such as energetic re-
ceiver and selection addition algorithm are analyzed and their comparison with the noise immunity proper-
ties (ROC) of the optimum approach is provided as well. Finally, the diversity combining approach is ap-
plied for the collaborative spectrum sensing and censoring. It is shown how the diversity addition principles
are applied for distributed detection algorithms, called hereafter as SIMO radar or distributed SIMO radar,
implementing Majority Addition (MA) approach and Weighted Majority Addition (WMA) principle.
Keywords: Spectrum Sensing, Cognitive Radio, Diversity Combining, Collaborative Sensing, Majority
Diversity Addition, Sequential Analysis
1. Introduction
Spectrum sensing is one of the most important elements
for the functioning of Cognitive Radio (CR) networks.
As it is well known [1], CR networks are made up of
primary users (PU) which have “legal” use of certain
frequency bands and secondary or cognitive users (CU),
located in different space–distributed cells, which share
the same frequencies as the PU in a part-time fashion.
The CU produce undesired interferences to the PU and
so they are allowed to share the same spectrum with the PU
if and only if the Quality of Service (QoS) degradation
provoked to the PU does not reach a pre-established level.
The cognitive users have to make first a spectrum
sensing in order to determine whether the primary users
are “on” or “off”1 and then “adapt” their transmission rate
and transmission power in order to avoid producing
harmful interferences to the PU or take advantage of the
“spectrum holes” free of PU [1,2], etc.
The common approaches are based on the Interference
Temperature (IT) and power spectrum estimations, ener-
gy detection and cyclostationary feature detection (see
[1-4] and the references therein). The last one was first
proposed in [5] and generalized for multiple cyclic fre-
quencies at [4].
It is worth mentioning here that the cyclostationarity,
as phenomenon, is not a recent development at all (see for
example [6]) but effective tests for indication of second
order cyclostationarity using a Neyman-Pearson type test
was proposed not long ago. Its natural generalization for
multiple cyclic frequencies was recently proposed for PU
identification in CR networks [4].
As it was already mentioned in [4,5] the cyclostatio-
narity is present, practically, in many communication
signals: multiple cyclic frequencies may be related to
symbol rate, guard periods (as in the case of OFDM sys-
tems), etc.
1Actually it is not a necessary condition for CU to have access to the
frequencies allocated to PU [2].
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
25
In the following it will be shown that single-user detec-
tion algorithms (in the form of expected value estimation
of the cyclic autocorrelation) can be interpreted as an
specific form of the mixed frequency-delay incoherent
“diversity combining” block with the number of virtual
“branches” equal to the product numbers of cyclic fre-
quencies and time delays; it can be also called as a SIMO
radar. This is the main difference between this paper and
the material presented in [4,5].
Based on practical reasons, it is possible to assume
that these “branches” suffer from fading which in the
general case can be modeled with the help of generalized
Gaussian statistics, or Klovsky-Middleton model, (see
for example [7]).
Moreover, in the following, depending on the frequen-
cy and delay diversity parameters, fading in these
branches or antennas is assumed as non-homogeneous,
homogeneous and totally correlated (the latter was consi-
dered at [4] for flat Rayleigh fading) or statistically inde-
pendent. Certainly those last two marginal cases are as-
sumed in the following only in order to obtain tractable
analytical results; generalized analysis for correlated
branches, based on the statistical description of Gaussian
quadratic forms (see [8,9], etc.) will be considered else-
where later on, but one important special case of corre-
lated branches is considered in the following as well.
It is worth mentioning that the concept of “diversity
approach” for multiple cyclic detection is useful not only
for effective development of quasi-optimal approaches,
but also allows to consider the necessary “trade-off” be-
tween the number of delays and cyclic frequencies for the
detection procedure and the statistical dependency of the
corresponding “diversity branches” in order to fulfill the
noise immunity or “Receiver Operating Characteristics”
(ROC) requirements.
Moreover, in the following, it will be shown that the
“diversity” concept for spectrum sensing is rather con-
structive for the analysis of collaborative sensing as well
(see [9], etc.). In [9] the collaboration is tackled in a ra-
ther different way, than in the following. For the latter the
set of secondary users (SU) being collaborating between
themselves or operating through a Fusion Center (FC) can
be interpreted as virtual branches (antennas) of the distri-
buted detection system, which can apply NPT detection
technique or Sequential Analysis methods.
This distributed system is nothing else as a distributed
SIMO radar, where virtual receiving branches are af-
fected by statistically independent flat fading (the above
mentioned incoherent combining algorithm at SU is also
working as an optimum SIMO radar, but not in a distri-
buted fashion). See also some examples at [10,11], etc.
for calculation of its noise immunity properties.
Regularly proposed counting rules [2,12-14] for oper-
ating at FC can be also interpreted as a special case of
quasi-optimum incoherent diversity addition (see MA
algorithm in the following) and can be modified in order
to approach its ROC properties to the optimum SIMO
radar case (see the WMA algorithm in the following).
In this paper only the novel theoretical material, which
is the kernel of a deeper and original insight into the
Spectrum Sensing problem, is presented. Some simula-
tions related to this problem have been included in [15];
however, comprehensive and thorough simulations are
reserved for another work of the authors (a book chapter
already in process for publication).
The paper is organized as follows. Section 2 briefly
presents some fundamental results concerned to the Ge-
neralized Gaussian (GG) channel modeling. Section 3 is
totally dedicated to single user multiple cyclic frequency
detection and its relation to incoherent optimum diversity
combining. In section 4 the noise immunity of the NPT
for the GG channel is analyzed. In Section 5 some subop-
timal algorithms for multiple frequency cyclostationary
sensing are considered. Here some discussion of the re-
sults is presented as well. Section 6 is totally dedicated to
collaborative sensing issues. Conclusions will be pre-
sented at Section 7.
2. Generalized Gaussian (Klovsky-Middleton)
Channel Model
Basically most of the existing fading channel models are
based on the concept of the module and phase of the ran-
dom vector with Gaussian Probability Density Functions
(PDF) for orthogonal statistically independent quadrature
components “x” and “y2, i.e., [7,8]:



2
2
22
1
,exp
222
y
x
xy xy
ym
xm
Wxy
 

(1)
where 22 ,yx

and mx, my are variances and expectations
of the “x” and “y” quadrature components respectively.
Then, defining the module 22 yx 
and the
phase of the random vector
x
y
arctan
, one can get:
 





2
0
2
2
2
2
2
sin
2
cos
exp
2d
m
m
W
y
y
x
x
yx
(2)
From (2) it is possible to obtain various representations
for W(), which actually depend on four parameters:
,
x
y
mm and
22 ,yx

, [7,8]. For this reason in the
following the term “four parametric distribution” is used,
and the rest of this section corresponds to [8].
2Those issues were tackled comprehensively in the 60th-70th of the las
t
century by many authors. Here we would like to distinguish D. Klovsk
y
[
8
],
D. Middleton
[
16
],
P. Beckman
[
17
],
etc. Details can be found at
[
7
]
.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
26
Hereafter the following two equivalent forms for the
four-parametric distribution W(
)3 will be used:

2
2
2
22
22
exp
x
x
y
y
yx
m
m
W




0
222
2
211
2!!2
ky
y
k
k
yx
k
y
y
k
k
km
I
mk
H

(3a)

0
2
2
2
!
)(
kk
II
k
I
k
k
k
mm
k
R
W


22
2
0
2
222
2
exp III
III mmI
mm
(3b)
where I0(z) is the modified Bessel function of order zero
[18],
2
yx
I
mm
m
, 2
yx
II
mm
m
, 2
22
2yx

,
22
22
yx
xy
R


,
2
22
11
2y
yx
x
m


22
x
y
.
Other forms (see for example [7]) follow from the way
the integrand in (2) is calculated, but they are not applied
in the following.
Beckman, Hoyt, Rice, Rayleigh and truncated Gaus-
sian distributions follow from (3) directly.
Let us introduce the new parameters:
22
22
2
yx
yx mm
q

, 2
2
2
y
x
, 222
0yx mm 
,
222
0
2yx

 ,
x
y
m
m
arctan
0
Beckman distribution follows from (3) when my = 0, 0
=
x
m, while Hoyt PDF appears, when 22 yx

0 =
mx = my = 0. Rayleigh PDF follows when 0 = 0 and mx =
my = 0; truncated Gaussian when additionally to the latter
0
2
x
.
That is why (3) is named as Generalized Gaussian
model. Next it is easy to find the parameter “m” for
equivalent Nakagami distribution [19]:

 
22
22
42 2222
00
11
212 1cossin
q
q
m



 
(4)
It is worth mentioning here that Nakagami distribution
is only an approximation for the four-parameter case, but
mainly it adequately represents the “dynamics” of the
variation of the four-parameter PDF functional form.
3. Single User Multiple Cyclic Frequency
Detection
The cyclostationary (CL) properties of the communica-
tion signals have been already widely investigated and
applied (see [4,5,20] etc.).
For the case of PU the signal shapes are known a- pri-
ori, and so their cyclic frequencies of interest are known
as well. Following here the material of [4], let us intro-
duce the set
P
n
A1
for cyclic frequencies of interest
and let
P
nn
NN
1
be the numbers of integers for time
delays for the autocovariance function calculus for each
cyclic frequency from A (here P denotes the number of
cyclic frequencies).
Thus, the estimation of the autocovariance function is
[4]:
  
 M
l
xx ljlxlx
M
R
1
*2exp*
1
,
ˆ

,
(5)
where the time delay is an integer and is fixed, the cyc-
lic frequency is fixed as well, M is the number of ob-
servations at (5) and x(l) is an input complex sample, with
x*(l) being its complex conjugate.
Representing the complex exponent in (5) in a trigo-
nometric form and assuming that x(l) is a sample of the
ergodic stochastic process, one can easily see that when
M1 or the time of analysis T is much more than one,
the estimations *
ˆxx
R are nothing else but estimations of
the complex Fourier coefficients for fixed and (see
also [6]).
If one forms a complex vector of (5) for different
and , the Generalized Maximum Likelihood Ratio
(GMLR) for its estimation (assuming asymptotic Gaus-
sianity of the observation) is well known (see for exam-
ple [4,8,21]):
0*
1
*ˆ
ˆ
ˆT
xxxx rr, (6)
where *
ˆxx
r is a complex vector of estimations of the
Fourier coefficients (F-Coefficients);
ˆ is a 2N × 2N
covariance matrix of *
ˆxx
r (in the non-asymptotic case
generally those coefficients are correlated),
P
nn
NN
1
(7)
Let us define an estimation of each “j” complex F- coeffi-
cient as
jjj VVV
~
ˆ , (8)
where jj VV
~
, are real and imaginary parts of j
V
ˆ; here it
is assumed that in the estimation process necessarily takes
place n(t) – the additive white Gaussian noise (AWGN)
with intensity N0 equal for all j.
3Taking into account, that in this paper the incoherent diversity combining
will be applied, the PDF of the phase is not presented hereafter.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
27
It is well known that “true” F-Coefficients are not cor-
related, but their estimations for finite “M” and corrupted
by the noise are not correlated only asymptotically, when
M or T (or both) are much more than one. With this as-
sumption, applied systematically in the following, (6) can
be significantly simplified taking into account the total
Gaussianity of the terms in (8) [8], (see also [4]):


Q
iii
NP VV
1
1
2
11
21
,ˆ
),(
ˆ

 
, NPQ ,
where
22
1
1
2
1
,,
2
1
Q
ihh
diag
.
Finally the left side in (6) can be represented in the way:
Q
ii
iih
VV
1
2
22
2
~
, (8a)
where 2
0
2
ii
hEN, 22
2~
ˆiii VVV , Ei = PiT and Pi is
the average power of each F-coefficient (fading is not
considered here).
One can see that the algorithms (8), (8a) are nothing
else but an optimum incoherent quadratic diversity com-
bining of Q total virtual “branches”, where 2
2
1
i
h are
weighting coefficients for each branch, generally related
to inhomogeneous conditions for combining.
Note that quadratic combining to obtain the NPT can
be presented in the way:
0
1
2
2
2
ˆ
Q
ii
i
h
V, (9)
where 0 is a detection threshold.
Formula (9) is not only a formal analogy with diversity
addition or SIMO radar test: it is an essential reflection of
the analogy between the autocovariance estimation and
diversity combining of statistically independent data (see
also [22]).
So, in absence of fading, all branches are asymptoti-
cally statistically independent.
In presence of fading 2
ˆi
V can be statistically inde-
pendent as well, but also might be totally correlated in
scenarios of flat fading both in frequency and time do-
mains. Both cases will be considered while noise immun-
ity of this single user multiple cyclic frequency algorithm
will be analyzed (see next section).
4. Noise Immunity of the Algorithm (9) in
Generalized Gaussian Channels
It is well known [23] that the Neyman-Pearson Test (NPT)
in terms of hypothesis testing, can be formulated as fol-
lowing:
Q
ii
i
Q
ii
i
tn
h
H
tn
h
V
H
1
2
2
1
1
2
2
0
)(
2
:
)(
2
ˆ
:
, (10)
here 2
i
are “true” F-coefficients, n(t) – white Gaussian
noise with intensity N0.
For simplicity, in the following let us suppose that all
2
2
1
i
h are the same and inhomogeneous features of the
virtual branches will be addressed to different 2
i
222
0ii yx

 . It means that in (10) one has to consider
only the routine form for quadratic combining4:
Q
i
i
Vz
1
2
ˆ (11)
As it is well known, the NPT is characterized by Pfa
and PM which are respectively the probability of false
alarm and the probability of misdetection error [23].
In absence of fading, the “z” is formed by squares of
the normally distributed components and its PDF for dif-
ferent hypothesis can be defined in the way [8,23]:
PDF square-chi central-non
ˆ
,:
PDF square-chi central )(:
1
22
21
2
20
Q
iiQ
Q
VzH
zH
(11a)
where
Q
ii
V
1
2
ˆ is the expectation of the sum of 2
ˆi
V
and is a parameter of the noncentral chi-square distribu-
tion [11].
It is worth to notice that in presence of fading, the
functional forms for these distributions will differ de-
pending on the scenarios for GG channel model and will
be considered in the following.
4.1. Statistically Independent Virtual Branches
with Flat Generalized Gaussian Fading in
Each Branch
Let us assume that each 2
ˆi
V, see (8, 8a), is: 222 ˆˆ
ˆiii yxV  ,
where, iiiVxx
ˆ, iii Vyy ~
ˆ and

,
ii
x
y are quadrature
Gaussian components of the GG fading model. Here we
have to notice that for both hypothesis each quadrature
components in 2
ˆi
V now are Gaussian as well, as before, but
their means are not equal and their variances are arbitrary.
Now, if
 Q
iii VVz
1
22 ~, (11b)
the routine procedure for calculus of the noise immunity
can be applied [7,8,23], etc.
4In the following the module sign will be omitted.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
28
Particularly for hypothesis H0:




2
2
2
2
2
2
2
0
2
2
2
0
21
2
~
,
21
2
21
2
2
1
2
~
21
2
2
1
2
i
i
i
i
i
i
i
i
i
i
i
i
y
y
i
x
x
i
y
y
y
i
yi
x
x
x
i
xi
h
h
VD
h
h
VD
h
h
h
N
E
mVM
h
h
h
N
E
mVM
(12)
For hypothesis H1:




22
2
2
0
2
2
0
2
~
,2
21
2
2~
21
2
2
ii
i
i
i
i
i
i
yixi
y
y
i
yi
x
x
i
xi
hVDhVD
h
h
N
E
mVM
h
h
N
E
mVM

(13)
where M{} and D{} are symbols of mean and variance
respectively; 2
0
22
PN
E
hi
xx ii
; 2
0
22
PN
E
hi
yy ii
. Here it
is taken into account that for the frequency diversity case
there are P out of Q virtual branches and the transmitted
power has to be divided between them [24].
From (12) and (13) it follows that for both hypothesis
the PDF W(z) is always a non-central chi-square distribu-
tion.
For analytical evaluation the special cases of “high re-
liability detection” 1,22 ii yxhh , are considered. Then for
these conditions, from (12) it follows that for H0, }{ i
VM
and }
~
{i
VM are close to zero, while the variances are
close to one, then 0
)( H
zW is tending to 2
2Q
central
distribution.
For this case, Pfa is well known [23]:

 


1
0
0
00 !
exp,
!1
1
~
Q
q
q
fa q
Q
Q
P

(14)
Fixing the level of Pfa one can find
0 and once more,
applying the conditions 1,22  ii yx hh from (13) it fol-
lows that for the hypothesis H1, the variances }{ i
VD and
}
~
{i
VD are going to be extremely large.
In this case PM is:

22
0
2
1
11
~!2
QQii
Miii
q
PQh



 ii i
i
ii
q
0
22
0
2
2
22
sincos
2
1
exp

(15)
where

2222
2
0
22
iiii yxyx
i
imm
PN
E
h

; the remaining
parameters were introduced at (3). For one sided- Gaus-
sian distribution (15) becomes:

0
2
2
1
2!
Q
Q
i
M
Q
PQ
hQ

(16)
Now let us repeat the same analysis as before, but for
Nakagami fading channels (see Section 5). Assuming
non-correlated homogeneous conditions for the fading in
all “virtual branches” one can get:
Qm
Q
Mhm
m
Q
P
2
0
2
2
! (17)
where can be found from (14) and “m” from (4).
4.2. Totally Dependent Virtual Branches (Flat
Fading) in GG Channel
In this case the fading processes at all the virtual branches
are totally correlated. Obviously it means that the result-
ing SNR after combining is:
Q
ii
h
P
h
1
2
2
21
; thus the
problem can be transferred to the quadrature addition
algorithm for one equivalent branch, i.e. without diversity
but, with the GG model of flat fading:
0
22
~
 oo VV (18)
where
Q
ii
VV
1
22
0,
Q
ii
VV
1
22
0
~~ . Here formulas (12)
and (13) are valid but for conditions of single channel, i.e.,
without index “i”.
Then Qeqv 1 and from (14)
0
exp
fa
P,
fa
P
1
ln
0
and



2
22
2222
00
2
1
ln 11
1
2expcos sin
2
fa
M
i
i
q
P
Pq
h


(19)
Dependence between Pfa and PM is usually called as
“Receiver Operational Characteristic-ROC” and they are
presented at Figures 1-3, where the continuous lines cor-
respond to case a and the dotted lines to case b.
Comparison of (19) and (15) deserves some comments.
1) When in both scenarios 2
i
h are equal and Pfa is
fixed, then PM from (19) is much more than PM from (15).
The latter can be explained by the diversity effect at (19),
see also [8,25].
2) Then it is reasonable to choose a small set of delays
and multiple frequencies (Q 5 [10,25]) in order to pro-
vide (if the channel conditions allow it) statistically inde-
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
29
Figure 1. ROC, continuous (15), dotted (19).
Figure 2. ROC (15) and (19) for another set of parameters.
Figure 3. ROC (15) and (19) for a different set of parameters.
pendent fading in those virtual branches, i.e. it is rea-
sonable to “sacrifice” the numbers of P and N by big-
ger intervals between
and
so as to artifi-
cially create independent fading in the frequency and
delay domains, which certainly leads to better noise
immunity after “diversity combining”. So, appropriate
choice of cyclostationary features Q = NP of the de-
sired signals of PU can significantly improve their
ROC properties.
4.3. One Special Case of Covariance Matrix for
Correlated Branches at Quadratic Incohe-
rent Addition Algorithm
Let us consider in the following one special case of the
covariance matrix for quadrature components
Q
l
xx 1
and
Q
l
yy1
: assume that across the branches all xl or yl
Gaussian components are correlated with coefficients Rx
or Ry and there is no cross-correlation at all between xl
and yl Gaussian components. One can see, that this as-
sumption restricts (in general) the type of the covariance
matrix of the GG channel model but might be useful for
the first step examination of the influence of the cova-
riance between virtual (but not only virtual!) branches at
the noise immunity characteristics of the SU: consider,
for example, SU which applies multi-antenna receiving
system, etc.
It is well known that for each pair of x or y Gaussian
variables, by the well known angle rotation linear trans-
form it is possible to obtain a new set of statistically in-
dependent Gaussian variables: rotating of the coordinate
system (linear transform) by the angle
2
2
2
1
21
2arctan


R, where R is a correlation co-
efficient; 2
1
, 2
2
-are variances of two correlated Gaus-
sian quadrature components, while new Gaussian va-
riables are statistically independent [17,11].
In order to provide tractable analytical results, in the
following only the case Q = 2 for the algorithm (11) will
be considered.
Then noise immunity analysis can be done in the same
way as it was done at IIIa, but the means and variances
for hypothesis H0 and H1 have to be calculated by the
formulas:

 
22 2
12
2
,22
222 12
12 22
12
21
4
111
III
R
R


 

where 2
1
, 2
2
correspond to the variances of the qua-
drature components, calculated for different hypothesis
H0 and H1 (see (12), (13)); 2
,III
are new variances of the
quadrature components after angle rotation (for each two
branches).
Moreover, assuming in the following for simplicity Rx
= Ry = R, in the same way as before one gets:
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
30




cossin
sincos
21
21
mmm
mmm
II
I


where m1, m2 are expectations of the initial quadrature
components (see also (12),(13) ); mI, mII are new means
after angle rotation.
Now all the set of these parameters can be considered as
new parameters of the GG model with the statistically in-
dependent branches. So the noise immunity (ROC) can be
calculated in the same way as in IIIa (see formulas (14),
(15)). This calculation, (in general) is rather cumbersome,
because the new parameters of the GG model come from
rather complex expressions (see above).
Therefore assuming that Pfa and PM. are much less than
one as it was done earlier, it is possible to apply the
asymptotic calculus (see [10,11]).
Particularly, for the hypothesis H0 (see IIIa) all means
will be close to zero and all variances will be equal to
(1-R2). So Pfa can be calculated by (14) but with the new
threshold = 0/(1-R2), depending on R, but the method
of calculus is the same as in IIIa.
Then for the asymptotic case PM << 1, one can get:



 
2
22
2
22 2
222
222
2
~exp
221 2121
11
21 1
21
y
xx
xx
y
yxx
xxx
y
I
III
M
IIII
II II
III
II
m
mm
P
mm


 







 
 
 
22
22
222 222
22
222
11
211 21 1
1.
21 1
yy
xx
xxx yyy
yy
yyy
II
II II
IIII IIIII
II II
IIII II
m
m
m
 








 










This formula is for the GG channel model and is rather
general in the sense that it does not provide a “transparent
picture” about the dependence of PM, for example, on R
etc; it requires implicitly numerical calculus.
Let us consider a special case:  222
yxx IIII

22

y
II ; next, introducing 222
0yxI II mm 
and
222
0yxII IIII mm 
one can get:


4
2
0
4
2
0
2
2
0
2
2
0
4
2
22
3
22
exp
2
~
IIIIII
M
P.
For this special case it can be found that:
 
12
12
12
12
222
00
2
22
00
22 2
00 222
00
21
4
11(1 )
hh R
hh
hh Rhh





when 2
2
0
2
I and 2
2
0
2
II are much less than one (strong
fading), then:


12
22
4222
00
33 1
~2
21
M
Phh R



12
22
4222
00
33 1
~2
21
M
Phh R

The last formula shows that, losses related to correla-
tion between diversity branches depend mainly on

2
1
1R; this result was in some sense predictable (see [7,
8] for example).
When the fading follows the truncated Gaussian PDF then:
 
22
2
22
22
13
~22
2121
11
yy
yy
yy
III
M
III
III
P











when PM 1, then:


12
2
22 2
00
5
~281
M
PhhR
One can see that losses once more depend on

2
1
1R
as well.
The same character of losses can be found for signifi-
cantly Rician character of the GG model; so, it can be
considered as a rather “universal” dependence of losses
on the correlation coefficient value.
Of course changes of the threshold, which depends on
R”, influence the character of the dependence of ROC on
the correlation properties of the GG model in a nonlinear
way, but this will be discussed elsewhere in the future.
Concluding the material of this section, it is worth to
mention, that from the theory of diversity combining it is
well known [7,8] that correlation between branches has
influence, mainly, on the noise immunity characteristics
(ROC, in our case) when resulting SNR is rather high, i.e.
PM is much less than one.
5. Suboptimal Algorithms and Their Noise
Immunity
The first suboptimal algorithm considered hereafter will
be an energetic receiver where the desired signal is repre-
sented in the way:
B
iiittx
1
)()(

; here B
it1
)}({
-are
orthonormal functions.
According to [22,23] the corresponding algorithm
(NPT) can be represented in the way:
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
31
 
2
0
10
:
T
B
i
j
H
zt tdtnt



(20)
 
2
1
10
:
T
B
i
j
H
zt tdt



, i.e. =

*
ˆ
xx
rnt,
where B is the number of orthonormal functions B
it1
)}({
applied for the expansion of the desired signal x(t); the
received signal is

ztxt nt, T is the time of
analysis.
Now, for the representation of

x
t and
nt let us
apply the F-basis in the same way as it was done at [10],
(see also [22] and references therein).
Then:

 

 
00
0
00
0
cos sin
cos sin
B
kk
k
B
kk
k
x
taktbkt
ntk tk t




(21)
where 02T
, B = 2FT, 1
12
F
kk
F – fre-
quency bandwidth, k2, k1 are upper and lower indexes
taken into account here for the F-series expansion.
Then:





B
kkkkk
B
kkk
baH
tnH
0
0
22
1
0
22
0
2
1
:
2
1
:


(22)
As all

k
a,

k
b,

k
and
k
are Gaussian
distributed coefficients, the left side in (22) has central or
non-central 2
2B
distributions respectively. Defining those
left sides in (22) as 1 and 2 (see [10,22] and Section 4
one has to apply:


/2 1
/2 1
11
1/2
/2 1
22
22
2
1exp 2
2(/2)
1exp
22
B
B
B
B
WDD
DB
WI
DDD


 

 
 


 


 
 
(23)
where xk
B
kkPba 2
2
0
2
is the average power of x(t)
and the parameter 2
0TN
D.
Then the threshold 0 can be easily found from (14)
where QB and Pfa are fixed.
The detection probability PM is:

2
exp
2
1
,,
2
exp
2
12
2
0
2h
Bh
D
F
h
PM
(24)
where 1,,0 2
0
Bh
D
F is the Cumulative Distri-
bution Function (CDF) of the non-central 2
2B
PDF.
The upper bound of PM for the GG channel model with
flat fading is known from [8]:



22
0
2
1
11cos
1exp 21 1
M
q
qh
PCqh






22
0
11sin
1
q
h


(25)
where 22
2
0
2E
hPN
,
and

2
12
21 11
h
Cq


An exact tractable analytical expression of PM for the
GG model is not available.
In absence of fading it is possible to obtain an analyti-
cal result in the following way.
First, representing the Bessel function as in [18] in the
way:


21
22
2
21
0!2
Bk
Bk
hy
IhykB k

, (26)
then the PM from (24) is:


1
2
042
2
1
0
2,
12
exp
22 !22
Bk
Mk
k
Bk h
hD
PkB k




 

(27)
where B = 2F, (,x) – is the lower incomplete gamma
function.
Analysis of (27) shows that influence of B can be sig-
nificant and it can be shown that for fixed Pfa or D
0
,
while B grows, PM also grows. To the best of our know-
ledge, influence of B and not only of 2
h on the noise
immunity of the energetic (autocovariance) receiver was
first stressed in [22].
Then for the multiple cyclic frequency case, when the
number of frequencies P is rather large while T is fixed, F
is large as well and PM grows.
Therefore the energetic detector is not definitely a good
candidate for spectrum sensing for this scenario, as its PM
is much worse than for the optimum detector (see the
previous section). In some sense this comment coincides
with the simulations [4], besides that there the energetic
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
32
detection was not implemented in the same way as men-
tioned above.
Another option for suboptimal detection is to take ad-
vantage of the analogy between multiple cyclic frequency
detection and quadratic diversity combining and apply a
suboptimal variant of incoherent diversity addition (see [7,
8]). Hereafter a selection (switching) combining method
was chosen, assuming that fading has a Nakagami PDF
(see [4]). See formula (4) to adjust parameters of Naka-
gami PDF and four–parameter distribution.
There are several different approaches for switching
combining but in the following we will analyze only the
algorithm of selection of the “virtual branch” with
i
max , 0
,1Qi .
Let us assume here, for simplicity, the homogeneous
fading conditions then, the distribution of the maximum
value of the identically distributed random values is [11]:
 
1
0
()
Q
WQW Wydy



(28)
If W() is [19]:
 
21 2
22
2exp
mm
m
my y
Wm
m



 

(29)
Then it is necessary to average PM for one virtual
branch without fading through (28) with the help of (29)
while Pfa is:

 exp
fa
P, (30)
where is a threshold.
The PM of the channel without fading is:
,hQPM (31)
Following the above mentioned procedure for the case
Q = 2 and h > 1 one can get from (28)-(31) an approx-
imate formula:




1
02
2
2
2
2
2
2)!1(!
1)!1(
1
2
1
m
iim
m
h
m
m
h
m
Mmi
im
m
h
P
(32)
for m–integers.
One can compare this method of switching combining
(with fixed 2
h and Q = 2) with the optimum approach
(see (14), (17)).
Please notice that in fading channel conditions when
the number of virtual branches is growing, one encounters
the so-called “hardening effect”, i.e. while Q is increasing,
the increment of noise immunity might be low.
Therefore, with Q = 2 there is a good option to com-
pare the effectiveness of the selection combining method
with the optimum one.
Figure 4. Comparison of the ROC for the optimum (17), and
quasioptimum (32), cases.
In Figure 4 the ROC for this method is presented,
where for comparison some of the “optimum” ROC’s,
see (17), are presented. One can see that the energetic
losses for PM = 10-4 are rather small and for m = 1 are
negligible.
In the same manner as above, the well known set of
sub-optimum combining algorithms can be applied: other
methods of switching combining, linear (weighted and
non-weighted) addition, etc. Their application is rather
straightforward and is not presented here.
Some discussion regarding the obtained results
One can ask: if both algorithms (11) and (12) rely on
quadratic addition of the F-coefficients, then why their
noise immunity is so different, particularly with the GG
channel fading? What is going on? The answer is rather
straightforward.
At (11) the object of the quadratic addition are the F-
coefficients, but from the autocovariance function of the
output of the multiple cyclic frequency optimum detector,
i.e. after optimum processing of the quadrature compo-
nents of the input signals. It is also possible to provide
statistically independent fading of the virtual branches for
incoherent addition by properly choosing the cyclic fre-
quencies and delays, etc. which drastically increase the
noise immunity (through the diversity effect).
In contrary, the energetic receiver, as it is in (20)-(22),
does not apply specific properties of the cyclic frequen-
cies and just extracts the total energy of the aggregate
input signal. It is often hardly possible for this case to the
F-coefficients of the input signal to exhibit statistical in-
dependency in fading conditions.
Moreover, for the energetic receiver (22) the noise
immunity, even in the case of a constant channel (without
fading), goes down while the bandwidth F grows (B = 2
FT, with T fixed) as the noise power grows. Therefore the
energetic receiver for multiple cyclic frequency signals
might be useless when FT1.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
33
For a quasi-optimum alternative for optimum quadrat-
ic combining it is possible to consider all the set of
switching combining algorithms (see (30) for example),
as well as a whole set of quasi-optimum algorithms of
non-coherent diversity combining such as a set of linear
combining methods with rather low energetic losses for
the fixed Pfa and PM.
6. Collaborative Spectrum Sensing with
Censoring
Here as in previous section it is supposed that spectrum
sensing is based on the cyclostationary properties of the
signals of the primary users (PU) and the secondary users
(SU) are spatially distributed within certain area. All the
set of SU can sense the whole frequency band of interest,
or each SU may sense just a partial band. Hereafter it
will be assumed that all SU are sensing the same fre-
quency band. In both cases of spectrum monitoring, SU
have to share the sensing information between them or
might be coordinated by a Fusion Center.
It seems reasonable that, no matter what kind of ex-
change is used, the local decision information has to be
obtained by a minimum set of observations M in (5), while
N and P are fixed5. In other words, the time of analysis in
(5) has to be reduced as much as possible; meanwhile the
amount of transmitted data has to be reduced as well.
Then it is opportunistic to apply the sequential analy-
sis of A. Wald [26] where ML test, in contrary to NPT,
has to be compared with two thresholds related to re-
quirements of Pfa and PM.
Let us suppose that for the latter, highly reliable final
results for the test are predefined, so Pfa and PM have to
be rather low. This might be a rational way to make cen-
soring for the local test as only reliable information has
to be forwarded to the FC or other SU.
One has to notice that each SU will obtain those relia-
ble final results (whether PU exists or not) at different
time instants. This information has to be sent to other
fellow SU or to FC in binary way.
Next let us consider several rather general but differ-
ent scenarios of collaborative spectrum sensing.
- Each n-th SU, Kn ,1, passes, after time “T”, the
information of “zn” (not binary) to the system of qu-
adratic addition at the FC. So,
KQ
jj
K
nn
zZ
zZ
1
1
or
(33)
Then Z might be analyzed by the NPT or by sequential
analysis (see below) assuming hereafter that the channel
SU FC is error free. So after final addition, the result
of quadratic diversity addition of KQ virtual branches (or
of K SU) is analyzed, assuming statistically independent
fading along all summations (see (10) and (11) above).
This scenario can be called as a distributed optimum in-
coherent “SIMO passive radar” and its characteristics are
equal to (14), (15) with the number of virtual branches KQ.
- Each of the n-th SU make an individual decision re-
garding to the presence of PU and then send the bi-
nary decision to the FC by error free channels. As-
suming that all those decisions are statistically inde-
pendent, the final result at the FC can be obtained
according to the majority rule (see for example [27])
with the majority not- weighted (or weighted) diver-
sity addition method. This case can be also called
“SIMO radar” but in contrary to the first one it is
non-optimum. In the following the topics related to
those issues will be thoroughly considered.
Majority diversity addition and weighted majority ad-
dition (WMA) in collaborative spectrum sensing
If the majority principle is applied at FC, then the de-
cision is made by analysis of the partial decisions at each
SU (here SU acts as a “virtual” diversity branch) and the
decision which takes place at the majority of the
branches is favored. This method is called “majority di-
versity addition”6.
If partial solutions are binary and the number of virtual
branches is odd, there cannot be any collision in the final
decisions for such method.
Let K = 2q-1 and “P0” denote the existence of PU after
q” tests on the branches.
So if after “m-1” probes on the virtual branches one
gets “q-1” results of existence of PU and “m-th” probe
gives the same, then for the “q” test one gets the proba-
bility of this event as


1
0111
1mq
qq
m
PCPP
 (34)
The probability of P(P0) is a sum of statistically inde-
pendent probabilities of probes (34) through all “m”
from m-q up to m = 2q-1, i.e.:


21
1
0111
1
qmq
qq
m
mq
PCPP
 
, (35)
where P1 can be Pfa or PM, so P(P0) is a final probability
of false alarm or error detection (see for example [11,27]
as well), depending which one of hypothesis is consi-
dered.
If one defines “n” in the way n = mq, it yields:


1
1
0111
0
1
qn
q
qn
n
PCPP

 
(36)
5Meanwhile one has to notice, that asymptotic conditions for M (T) are
assumed to be valid here in order to preserve the uncorrelated
conditions for F-coefficients in (5).
6Some modifications of the majority diversity addition can be found a
t
[13,28], etc.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
34
At the same time, from the theory of diversity com-
bining it is known that majority addition is equivalent to
the optimal incoherent addition with the number of
branches (here virtual ones) “q”, i.e., to incoherent (qua-
dratic) addition with almost twice less branches.
So, comparing the characteristics of majority addition
with those of the optimum SIMO radar one can see sev-
eral limitations of the former:
- Optimum SIMO radar with incoherent addition ac-
tually operates with almost twice more virtual bran-
ches and therefore provides significantly better de-
tection characteristics (ROC’s);
- The majority addition operates successfully only with
odd number of virtual branches, while optimum SI-
MO radar operates with any number of branches.
The price one has to pay for the advantages of the op-
timum SIMO radar is a more complex data transmission
scheme: in the majority addition, simply binary results
are transmitted, and for SIMO radar the information of
the “z” value for each SU has to be transmitted to FC
through error free channels.
Is it possible to improve the ROC properties of the
majority addition in order to make them approach to
those of the optimum SIMO radar?
In order to approach the noise immunity properties of the
majority addition to those of the optimum incoherent addi-
tion, some modifications of the former were proposed.
One of them, the so-called “weighted majority addi-
tion” was proposed at [29,30]. The idea of this method is
rather simple: introduce in the majority addition algo-
rithm information of the channel gains for each partial
solution, or in other words introduce “weights” in the
procedure of the majority addition algorithm. In this way
the channel gains for the diversity “branches” work as
weighting coefficients in the process of majority selec-
tion. It was shown that this suboptimal method provides
results very close to those of the optimum incoherent
addition [29,30] if the communication scenario allows
taking advantages of channel gains.
One can see that it is not the case for one of the scena-
rios at FC: each result of detection at SU was obtained
through the optimum quadratic addition by the SU itself,
so the resulting fading at SU has a very low variance
when Q is rather large (hardening effect) (see, for exam-
ple [10,25]). Therefore, it is hardly possible to improve
the results of majority addition by introducing weighting
coefficients as all the weights might be practically equal.
But it is known that if the channels are sufficiently hete-
rogeneous, the hardening effect does not even appear or
it appears very slowly, while Q at the SU. So, let us
consider another extreme special case. Let us assume that
the fading at the SU’s are so heterogeneous, that practi-
cally all quadrature addition algorithms do not work as
the diversity combining algorithm and each SU have Q
1 (single reception) with m-distributed fading7 and the
fading is generally heterogeneous.
With this assumption one can see that the problem is
converted to the case of SIMO Radar: SU are sending to
the FC binary information of partial decisions together
with the information of their weights in order to provide
to the FC with weighted addition (the channel SU FC
is supposed to be error free).
Let us formulate here an assumption: if the final deci-
sions are taken at FC by applying the technique of
weighted addition of partial decisions, then the SU’s
have to transmit to the FC not only the information of
partial decisions, but information of their reliability as
well and all the system (PU, SU, FC) is working as a
distributed quasi-optimum SIMO Radar.
Returning back to the above mentioned scenario, one
can see that the decision of PU existence in the majority
of “branches” can be obtained by the algorithm:
10
11
q
K
j
HjH
jq j

 

, (37)
where
1
K
j
– are magnitudes of the channel gains;
hypothesis H1 and H0 have the same sense as in (10).
From Bayes theorem each of the summands in (37)
have the following PDFs:




0
1
1
1
j
j
jfa
jH fa
jd
jH d
WP
WP
WP
WP
(38)
and the error PM after addition, finally will take place if
the sign of the inequality in (37) changes to the opposite
one. Conditions for false alarm are defined in a similar
way when PU really does not exist in observations at SU.
Let us assume that [19]:
 
21 2
2
2
2exp
j
j
m
mj
jjj
jj
m
i
ji
m
Wm
m










 


(39)
where mj and the corresponding parameters for four-
parametric distribution are related by 1
2
j
m.
Then introducing the new variable
2
2
j
jj j
h
m
xm
7In relation to the fading model assumed here for simplicity, see
formula (4) for the definition of the parameter m through the parameters
of the Generalized Gaussian model.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
35
one can get:




0
02
21 2
2
2
21 2
2
22
2
2exp
2exp
2
j
j
j
j
j
m
mj
jj j
jH j
m
i
ji
m
mj
jj j
jH j
mhi
m
ji
j
mx x
Wx m
m
mx x
Wx m
h
mm









 












(40)
where Pfa 1, PM < 1.
From (40) it can be seen that

0
jH
Wx and

1
jH
Wx have the Nakagami PDF form.
Getting back to (37) and introducing the variables
[29,30]:
0
1
1
1
1
q
qjH
j
Q
Qqj H
jq
x
x
 
(41)
one can formally calculate the error probability in (37).
In the general case of the heterogeneous scenarios,
according to [31] it is possible to find distributions of
q
and 1Qq
 in a Nakagami PDF form after rather
cumbersome calculus. In the most tractable way, ac-
cording to (77)-(89) in [19], it is possible to provide the
error analysis for the following special case, when8:
2
2
1
1
n
n
mm
here n is the number of Nakagami variables at (41).
Then the sums in (41) will have equivalent Nakagami
parameters 0mmn and
22
0n


.
Note, that for the general case [31], the calculus of 0m
and
2
0
can be done mainly numerically. Then from
(37) is possible to get (see [30]) for the conditional error
probability (with q fixed):






0( )
0
0
2
00
0
()
12
mK q
mK q
qmK
Pq hKq
mK qmKm




(42)
But the number of virtual branches “q” with errors,
both for PM and Pfa, is a random variable with Bernoulli
PDF, when the virtual branches have statistically inde-
pendent fading [29,30].
Then:
 
1
1
()
K
error KK
q
PPqPqPK

(43)
where
 
11
1
K
q
qq
KK
Pq CPP
; P1 – is Pfa or PM from
(14), (15), when Q = 1.
Note that at (14), (15) the four parameters have to be
previously adjusted by (4) with the value of 0m and
2
0
at
2
h
.
The formula (43) is “universal” in the sense that the
final PM and Pfa can be calculated through it, because as
it was mentioned above, the inequality of the (37) type
can be applied for calculus of false alarm as well.
The ROC’s for WMA is presented at Figures 5-7. For
comparison purposes we have included the plots corres-
ponding to sections IVa and IVb denoted with conti-
nuous lines and with dotted lines the plots corresponding
to (43). One can see that for PM = 10-4 energetic losses
are less than 1.5 - 2 dB.
Finally let us compare the “ideology” of the weighted
majority addition with some of the approaches mentioned
at [32], see also the references therein, (in [32] it is also
assumed an error free channel between SUi FC).
The “simple counting” approach [32] is nothing else
than selecting for FC decision only “highly weighted” SU.
For sure this addition is “less optimum” than the approach
in [29] because some of the SU’s with small weights do not
participate in the decision-taking process at the FC.
Other two methods, namely the Partial Agreement
Counting and the Collision Detection, assume the exis-
tence of a feedback channel between SU and FC which
can be used for comparing partial decisions at the SU
and final decisions at the FC in order to select the “true”
final decision. This option was not considered in the
current analysis.
Figure 5. ROC for WMA (15) and (19) –––, (43) ---.
8Formally the following analysis is valid for the general case [30] as well.
O. FILIO-RODRIGUEZ ET AL.
Copyright © 2011 SciRes. WSN
36
Figure 6. ROC for WMA with different parameters.
Figure 7. ROC for WMA for different parameters.
For sure, application of the feedback channel opens
the possibility to improve the reliability of the final deci-
sion at the FC and taking into account that weighted ma-
jority addition is a practically optimum incoherent addi-
tion, the final characteristics might be better than what it
has been mentioned at [32].
7. Conclusions
In this paper we have shown that cyclostationary spec-
trum sensing as well as collaborative spectrum sensing
for Cognitive Radio networks can be interpreted as a
special case of the concept of optimum or sub-optimum
incoherent diversity combining approach (SIMO radar).
It was shown, that as sub-optimum algorithms for this
purpose it is possible to apply the whole “gamma” of
well known algorithms such as all types of switching
combining, as well as linear combining and counting
rules (discrete addition), etc.
The concrete detection algorithms (distributed or not)
utilizing NPT or sequential tests leads to the so-called SI-
MO radar algorithms and their ROC’s were analyzed here
for GG channel fading models in the most general way.
It is worth mentioning here that, application of the
cyclostationary properties of the PU signals (through the
estimation of the F-coefficients of the autocovariance
function) is a convenient but obviously not the unique
approach that allows construction of statistically inde-
pendent virtual diversity branches for the spectrum sens-
ing detection algorithms. For example, for the broad
band GG communication channels, virtual branches can
be constructed through channel orthogonalizations in the
frequency and time domains (in the same way as it was
done in [33]), or by choosing statistically independent
fading sub-carriers of OFDMA signals (see [34]), etc.
Other emerging problems, such as detailed analysis for
correlated virtual branches of the sensing algorithms,
adaptive methods of sensing for unknown parameters of
GG channels, application of the ideas of the feedback
algorithms for collaborative sensing, etc. will be pre-
sented by the authors elsewhere.
8. Acknowledgements
With the material presented above, the authors would
like to acknowledge the outstanding contribution of Prof.
D. D. Klovsky, who survived to the Holocaust and
passed away almost ten years ago, to the theory of diver-
sity combining in fading channels which as it was shown
above is fully operating for the new challenges in com-
munications.
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