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Analysis of Equal Gain Combining in Energy
Detection for Cognitive Radio over Nakagami
Channels
Sanjeewa P. Herath, Nandana Rajatheva
Telecommunications Field of Study, School of Engineering and Technology,
Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand
[email protected], [email protected]
AbstractThis paper addresses the problem of energy detec-tion of unknown deterministic signal of a primary user in acognitive radio environment. As an extension to the previousworks, we focus on equal gain combining technique when thewireless channel is modeled as Nakagami-m. We derive seriesform exact expressions for probability of detection and falsealarm when the number of diversity branches are 1, 2, 3 andL 4. Finally, performance variation is shown against thenumber of diversity branches and the time bandwidth productin decision statistic with the aid of numerical results.
I. INTRODUCTION
Detection of spectrum holes in cognitive radio environment
is a problem of significant interest. To best utilize the spectrum
holes by secondary users, the detection of such spectrum holes
quickly and accurately is of great importance. Mathematically
speaking, achieving higher probability of detection with very
low probability of false alarm by means of few number of
samples is the objective of a secondary user system designer.
Detection of a primary user while a secondary user experi-
ences a deep fade is also expected and hence the detectingmechanism should be able to tolerate the low signal to noise
ratio (SNR) conditions. Since the secondary user has very
little knowledge about the primary user, the detection of an
unknown deterministic signal of that user by means of an
energy detector is a proposed mechanism for a secondary user
to identify the presence of primary user.
The detection of an unknown deterministic signal in the
presence of noise through an energy detector is first studied
by Urkowitz [1]. In his original work, the receiver consists of
an energy detector which measures the energy of the received
signal over a period of time. Then the detection problem
reduces to a binary hypothesis test with the decision variable
having statistics of Chi-square (2) distribution under eachhypothesis [1]. Further, by means of the lowpass equivalent
representation of a bandpass process, statistics of the decision
variable of bandpass signal are also shown to follow the 2
distribution.
Kostylev [2] further analyzes the problem formulated in [1]
and extends the work for Rayleigh, Rician and Nakagami
fading channels. In his work, closed form expression for
probability of false alarm (Pf) and probability of detection(Pd) are derived for Rayleigh fading channel. However, the
calculation of Pd requires numerical integration for Nakagami
fading channel and an infinite summation for the Rician fading
channel.
An alternative approach for calculating Pd for aforemen-
tioned fading channels is presented in [3] and closed form
expressions are derived for Pd under each fading environment.
Further, derivations in [3] are cross checked with the work in[2]. More importantly, closed form expressions for Pd when
diversity techniques are used and channel is under Rayleigh
fading is introduced in [3] including maximal ratio, selection
and switch and stay combining. Recent results in [4] present
closed form expressions for Pd and Pf over AWGN channels
and Nakagami Fading channel. In this work, closed form
expressions for Pd and Pf are obtained for both correlated and
i.i.d. Rayleigh fading channels when Square Law Combining
(SLC) technique is used. Expressions for Pd and Pf over
Rayleigh fading channel for Square Law Selection (SLS)
combining is also presented therein. Authors in [5] derive the
expressions for Pd and Pf when maximum ratio and selection
processing diversity combining is used.In this paper, we discuss the same detection problem deriv-
ing series form expression for Pd and Pf when the wireless
channel is modeled by Nakagami-m fading and an equal
gain combining receiver, assuming the statistics of diversity
branches to be i.i.d. Our work considers the number of i.i.d.
branches (L) 1,2,3 and L 4. Finally numerical results arepresented showing the variation of Pd against fixed value of
Pf over various parameters of interest.
I I . DETECTION OVER AWGN CHANNEL
To be consistent with the previous works and to minimize
the consequences of different notations, we follow the nota-tions similar to [3] given below.
s(t) : Unknown deterministic signal waveformn(t) : Noise waveform - White Gaussian random processr(t) : Received signalT : Observation time interval
W : One sided bandwidth
u = T W: Time bandwidth productN01 : One sided noise power spectral density
Es : Signal energy over the interval T
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: Energy threshold of the receiver
L : Number of branches of the receiver combiner
H0 : Hypothesis 0; no s(t) presentH1 : Hypothesis 1; s(t) present2 : Central Chi-square distribution with
degrees freedom
2() : Non central Chi-square distribution with degrees freedom and non centrality parameter
Energy detector decision is a result of binary hypothesis
test. The test can be written as below [1].
y(t) =
n(t) : H0s(t) + n(t) : H1
Hence, the decision statistic Y, can be represented as 22uunder H0 and
22u() under H1 as given below,
fY(y) =
1
2u(u)yu1e
y2 : H0
12(
y2 )
u12 e
2+y2 Iu1(
2y) : H1
where (.) is gamma function and I(.) is the th ordermodified Bessel function of the first kind [1],[3]. Hence, Pdand Pf over AWGN channel can be given as in (1) and (2),
Pd = Qu
2,
(1)
Pf =
u, 2
(u)
(2)
where Qu(., .) is the generalized Marcum Q-function and(., .) is the incomplete gamma function [3].
III . AVERAGE DETECTION PROBABILITY (Pd) OVERNAKAGAMI-m FADING CHANNEL WITH EQUAL GAI N
COMBINING
If the number of branches in the equal gain combiner is
L, then the combined signal amplitude is given by [8, eqn.
6.32, pp.285],
=Li=1
i. (3)
The SNR of the composite signal () is defined by
= Es2
LN01[8, eqn.6.33, pp.285]. This follows, average SNR
per branch E() = =EsN01 where = E(
2i ) and E(.) is
the mathematical expectation.
By means of the alternative representation of Marcum Q-
function given in [7, eqn.4.63, pp.101], we can write
Qu (a
, b) = 1 ea2+ b2
2
n=u
b
a
nIn (ab
) (4)
where a =
2 and b =
. Hence, the probability of detec-
tion when i number of branches are used (Pd,i, i = 1, 2, . . ,L)
can be calculated averaging (4) over the probability density
function (PDF) of ( 0) i.e. gi(),
Pd,i =
0
Qu(a
, b) gi() d
= 1 eb2
2
n=u
b
a
n 0
n2 e
a22 In (ab
) gi()d
(5)
where we have used the fact that
0gi()d = 1.
A. Single Branch (L = 1)
When the received signal follows Nakagami-m distribution,
PDF of the received signal amplitude is given by,
f() =2mm2m1
(m)me
m2
, 0 (6)
where (.) is gamma function and m is fading figure. Hence,g1() can be written as in (7).
g1() =1
(m)
m
mm1e
m , 0 (7)
By (5) and (7), Pd,1 can be expressed as,
Pd,1 = 1e
b2
2
(m)
m
m n=u
b
a
n
0
em1n2 In(ab
)d
(8)
where = 1 + m
. Using [9, eqn.6.643-2, pp.709], we can
show Pd,1 for m > 0 as in (9),
Pd,1 = 1 e2
m
m n=u
2
n1
(n + 1)
1F1m; n + 1; 2(9)
where we have use the relation given in (10),
Mk,(x) = x+ 1
2 ex2 1F1
1
2+ k; 1 + 2; x
. (10)
Here, Mk,(.) is the Whittaker function and 1F1(., .; .) is theconfluent hypergeometric function of the first kind.
This result is numerically equivalent to the result [3,
eqn.(20)] for integer m. Since m is not restricted to integers in
(9), this can be treated as a more general result for Nakagami-
m fading channel compared to the result in [3, eqn.(20)].
B. Two i.i.d. branches (L = 2)
When the received signal follows Nakagami-m distribution,PDF of combined signal amplitude is given by [6, eqn.(4)].
Hence by following the same procedure for (7), g2() can bederived as given in (11),
g2() =2
2m1emL
24m1(2m)
2(m)(2m + 12)
mL
2m
1F1
2m; 2m +1
2;
mL
2
, 0.
(11)
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By (5) and (11), Pd,2 can be expressed as,
Pd,2 = 1 2(m)eb2
2
L
2m n=u
b
a
n 0
2mn21
emL+ a
2
2
In(ab
)1F1
2m; 2m +
1
2;
mL
2
d
(12)
where 2(m) defined by (13),
2(m) =2
(2m)m2m
2(m)(2m + 12)24m1
. (13)
The hypergeometric series expansion of the form 1F1(.; .; .) isgiven by (14) [10, pp.202],
1F1(a; b; x) =k=0
(a)k xk
(b)k k!. (14)
Here, (.)k is Pochhammer symbols [10, pp.202] which isdefined by (15).
(a)k = (a + k)(k)
(15)
Using [9, eqn.6.643-2, pp.709], (12),(13),(14) and (15), Pd,2for m > 0 can be evaluated as,
Pd,2 = 1
e2
n=u
k=0
2
nmL
2m+k
2(m,n,k)24m+k2 k!
1F1(2m + k; n + 1;
2)
(16)
where 2(m,n,k) is defined by (17) and = 1 +mL
.
2(m,n,k) = 2(2m + k)
2(m)(n + 1)(2m + k + 12)(17)
C. Three i.i.d. branches (L = 3)
When the received signal follows Nakagami-m distribution,
PDF of combined signal amplitude is given by, [6, eqn.(8)] and
by following the same procedure used for (7), g3(), 0can be expressed as,
g3() =4
(2m)emL
3(m)24m1
n=0
(2m + n)
(2m + n + 12)
(4m + 2n) 3m+n1
(6m + 2n)(n + 1) 2nmL
3m+n
2F2
2m, 4m + 2n; 3m + n +1
2, 3m + n;
mL
2
.
(18)
The Hypergeometric series expansion of generalized hyper-
geometric function [10, pp.202] is given in (19).
pFq(a1,...,ap; b1,...,bq; x) =
n=0
(a1)n...(ap)n(b1)n...(bp)n
xn
n!(19)
Using the form of 2F2(a1, a2; b1, b2; x) in (19) and using (5)and (18), Pd,3 can be shown as in (20),
Pd,3 = 13(m)eb2
2
n=u
p=0
k=0
b
a
n m2
p+k L
3m+p+
(2m +p)(4m + 2p)(6m + 2p)(p + 1)(2m +p + 12)
(2m)k (4m + 2p)k(3m +p)k
3m +p + 12
k
k!
0
3m+p+kn21 e
mL+ a
2
2
In(ab
)d
(20)
where 3(m) is defined by (21),
3(m) =4
(2m)m3m
3(m)24m1. (21)
Using [9, eqn.6.643-2, pp.709] and the identity given in (10),
Pd,3 for m > 0 can be computed as,
Pd,3 = 1e2
n=u
p=0
k=0
2n
mL
3m+p+k
3(m,n,p,k)24m+p+k3 k!
1F1
3m +p + k; n + 1;
2
(22)
where 3(m,n,p,k) is defined by (23) below and = 1+mL
.
3(m,n,p,k) =(2m +p)(2m + k)(3m +p)
3(m)(n + 1)(p + 1)(2m +p + 12)
(3m +p +12)(4m + 2p + k)
(3m +p + k + 12)(6m + 2p)(23)
IV. FOUR OR MORE I.I.D. BRANCHES (L 4)When the received signal follows Nakagami-m distribution,
PDF of combined signal amplitude of four branch EGC is
given by [6, eqn.(9)] and by following the same procedure used
for (7), g4() can be expressed as given in (24). Following asimilar procedure in deriving (22), Pd,4 for m > 0 can beevaluated as given in (25) where 4(m ,n,p ,q ,k) is definedby (26) and = 1 + mL
.
The PDF of combined signal amplitude of Nakagami-m
faded L 4 branch EGC is given in [6, eqn.(10)]. Byfollowing the same line of arguments for (7), gL() can be
evaluated as given in (27). Following similar procedure inderiving (22), Pd,L for m > 0 can be evaluated as given in(28) where = 1 + mL
. Replacing k1 by p, k2 by q, p by k
and L = 4 in (28), we can easily verify (25).Probability of false alarm depends only on the hypothesis
H0. Hence the statistics of each sample taken from combined
signal follows central Chi-square distribution and further de-
cision variable also follows the central Chi-square distribution
with 2u degree freedom (22u) [1],[3]. This leads to the samePf,i for i = 1, 2,...,L as given by (2).
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g4() =8
(2m)m4memL
4(m)24m1
p=0
q=0
(2m +p)(2m + q)(4m + 2p + q)
(p + 1)(q + 1)(2m +p + 12)(8m + 2p + 2q)4m+p+q1 2qp mp+q
L
4m+p+q2F2(2m, 6m + 2p + 2q; 4m +p + q +
1
2, 4m +p + q;
mL
2)
(24)
Pd,4 = 1
e2
n=u
p=0
q=0
k=0
2
nmL
4m+p+q+k4(m ,n,p ,q ,k)
24m+p+kq4 k!1F1
4m +p + q + k; n + 1;
2
(25)
4(m ,n,p ,q ,k) =(2m +p)(2m + q)(2m + k)(4m +p + q)(4m + 2p + q)(4m +p + q + 12)(6m + 2p + 2q + k)
4(m)(q + 1)(p + 1)(n + 1)(2m +p + 12)(4m +p + q + k +12)(6m + 2p + 2q)(8m + 2p + 2q)
(26)
gL() =2L1
(2m) mLm e
mL
24m1 L(m)
k1=0
k2=0
...
kL2
L2i=1
(2m + ki)
(1 + ki)
(4m + 2k1 + k2) (6m + 2k1 + 2k2 + k3) ...
2m(L 2) + 2L3i=1 ki + kL22m(L 1) + 2L2i=1 ki
(2m + k1 +12) (6m + 2k1 + 2k2) (8m + 2k1 + 2k2 + 2k3) ...
2Lm + 2
L2i=1 ki
2m(L 2) + 2L3i=1
ki + kL2
m(
L2i=1 ki)2(
L2i=2 kik1)(Lm+
L2i=1 ki1)
L
(Lm+L2i=1 ki)
2F2
2m, 2m(L 1) + 2 L
2i=1
ki; Lm +L
2i=1
ki +12
, Lm +L
2i=1
ki;mL
2
, L 4
(27)
Pd,L = 1
e2
n=u
k1=0
k2=0
...
kL2
p=0
L2i=1
(2m + ki)
(1 + ki)
(4m + 2k1 + k2)(6m + 2k1 + 2k2 + k3) ...
2m(L 3) + 2L4i=1 ki + kL32 2m(L 2) + 2L3i=1 ki + kL2
(2m + k1 +12)(6m + 2k1 + 2k2)(8m + 2k1 + 2k2 + 2k3) ... 2Lm + 2L2i=1 ki
(2m +p)
Lm +
L2i=1 ki
Lm +L2
i=1 ki +12
2m(L 1) + 2L2i=1 ki +pL(m) (n + 1)
mL +
L2i=1 ki +p +
12
2
n mL
(Lm+L2i=1 ki+p)2(4m+k1
L2i=2 kiL+p) p!
1F1
p + Lm +
L2i=1
ki; n + 1;
2
, L 4
(28)
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0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average SNR(dB)
Pd
Fading m=1.0
EGC L=2 m=0.5
EGC L=2 m=1.0
EGC L=2 m=1.5
EGC L=2 m=2.0
EGC L=2 m=2.5
EGC L=2 m=3.0
Fig. 1. Pd,i for i=1,2 when Pf = 0.001 and u = 1.
V. NUMERICAL RESULTS
We describe the performance of the receiver by means of
Pd,i vs. SNR curves for pre-specified Pf,i value. First, energy
threshold of the receiver () is calculated for the specified Pf,ivalue. Then for that , Pd,i is evaluated.
Fig.1 shows the Pd,i performance improvement of dual
branch EGC receiver over different values of m when u = 1and Pf,i = 0.001. As it is expected, higher values of mand SNR result in better performance. However, the highest
Pd,i improvement is observed over lower range of m values.
Fig.2 plots the similar analysis including the three branch EGC
receiver.
Fig.3 shows the Pd,i performance variation for u = 1, 4
when m = 2, L = 1, 2, 3 and Pf,i = 0.01. This plot shows theperformance reduction for higher value of u in each case, i.e.no-diversity fading channel and L = 2, 3. Comparing suitablecurves of Fig.1, Fig.2 and Fig.3, it is possible to observe the
performance reduction of Pd,i for higher Pf,i requirement.
However, the SNR requirement for higher detection probability
is moderate in all cases.
VI . CONCLUSION
We consider the problem of primary user detection in
cognitive radio over the Nakagami-m fading channel with
the equal gain combining diversity receiver. Expressions are
derived for exact probability of detection when the number
of diversity branches are 1, 2, 3 and L 4. Interestingly,all the expressions could be expressed in terms of confluenthypergeometric function of the first kind 1F1(.; .; .). Theseresults could be readily used in deciding the number of
diversity branches and the energy threshold value to achieve
a specified false alarm rate of equal gain combiner energy
detector receiver in cognitive radio.
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[1] H. Urkowitz, Energy detection of Unknown Deterministic Signals,Proc IEEE, vol. 55, no. 4, pp.523-531,Apr. 1967.
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average SNR(dB)
Pd
Fading u=1 m=1
Fading u=1 m=3
EGC L=2 u=1 m=1
EGC L=2 u=1 m=3
EGC L=3 u=1 m=1
EGC L=3 u=1 m=3
Fig. 2. Pd,i for i=1,2,3 when Pf = 0.01 and u = 1.
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average SNR(dB)
Pd
Fading u=1 m=2
Fading u=4 m=2
EGC L=2 u=1 m=2
EGC L=2 u=4 m=2
EGC L=3 u=1 m=2
EGC L=3 u=4 m=2
Fig. 3. Pd,i for i = 1, 2, 3 when Pf = 0.01.
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