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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
1
Characteristics of p-norm Signal Detection in
Gaussian Mixture Noise
V. I. Kostylev, I. P. Gres
Abstract—Performance of the p-norm detector has been
analyzed recently in additive white Gaussian noise and several
analytical solutions were obtained. We have developed these
solutions for the case of non-Gaussian noise. Expressions for the
false alarm probability PF and the detection probability PD were
found in series-form for an arbitrary index of power p using an
analysis based on the moment generating function (MGF). Seriesform expressions for the average detection probability PD in the
non-Gaussian environment for Nakagami-m, k-µ and η-µ fading
channels were derived as well. Recently proposed p-law selection
(pLS) and p-law combining (pLC) schemes for antenna diversity
reception were analyzed. The detection characteristics of a pnorm detector were compared with a traditional energy detector
and with an optimal detector. All analytical results were verified
with a statistic simulation.
Index Terms—Detection probability, diversity reception,
energy detector, non-Gaussian noise, p-norm detector.
I.
D
INTRODUCTION
of signals corrupted by noise is an important
problem for wireless communication, radar and
navigation. Often these signals have to be reliably detected
without any channel state information. In this case the energy
detector discussed in [1] is widely used in practice, e.g. for
detection of signals with random amplitude [2]. Despite the
fact that it was proposed many years ago it is still exploited in
the modern communication systems such as cognitive radio
[3], especially for spectrum sensing [4]. As a development of
the energy detector an improved energy detector was proposed
in the literature [5, 6].
A decision statistic of this detector is calculated by raising
an absolute value of an input mixture of noise and a signal in a
positive arbitrary power p, instead of a power two for a
conventional energy detector. The signal and noise under
consideration in [5] are Gaussian. Since the energy detector is
optimal under such conditions, it is unlikely to find any
detector working better than the energy detector. The
improved energy detector is also known as a p-norm and a Lpnorm detector; we will refer to it further as p-norm. In [7] the
same detector is analyzed and a variety of analytical and
numerical approaches was developed. However, this analysis
is limited to additive white Gaussian noise (AWGN), as well,
but fading channels and diversity reception schemes are also
considered.
Often in practice a detector has to operate in various types
ETECTION
of non-Gaussian noises such as atmospheric, a man-made
impulsive noise, and different kinds of interference. In this
case the performance of the conventional energy detector
degrades significantly. In [6] the central limit theorem
approach (CLT) is used for analyzing the distribution of the pnorm detector decision statistic for the non-Gaussian
environment. This approach is not always precise [7],
moreover, the equations are derived for a small signal-noise
ratio (SNR). Nevertheless, it was shown, that the p-norm
detector has a vast performance gain comparing with the
energy detector. Moreover, it is easy to show, that despite the
energy detector being optimal in case of the Gaussian signal
detection in Gaussian noise (for independent samples), it is no
longer optimal, if the noise is non-Gaussian.
In this paper, exploiting the MGF-based approach from [7]
and the non-Gaussian noise model (Gaussian mixture noise,
GMN) [6], we derived series-form equations for the false
alarm probability PF and the detection probability PD. These
detection characteristics were compared with the optimal and
energy detectors’ characteristics; the quasioptimal value of the
power index p (the value that provides the highest detection
probability if the false alarm probability is fixed) is also found
under given conditions. It is shown that the quasioptimal
detector can be close to optimal, but has as simple a structure
as the energy detector. Fading channels are usually evaluated
with the average detection probability PD , averaging over the
signal-noise ratio (SNR) distribution. For such cases seriesform expressions for the average detection probability PD are
obtained for the Nakagami-m, k-µ and η-µ fading channels.
The pLC and pLS diversity schemes proposed in [7] are also
analyzed and corresponding series-form expressions for the
average detection probability PD are derived. Statistic
simulation is carried out in order to validate all presented
expressions.
The organization of this paper is as follows. Section II
describes the decision statistics and the noise model. The false
alarm and detection probabilities for Gaussian mixture noise
are derived in Section III. The performance analysis for
Nakagami-m, k-µ and η-µ channels is carried out in Section
IV. The characteristics of pLC and pLS diversity schemes are
found in Section V. The numerical results and presented
figures are discussed in Section VI. Section VII concludes the
paper and gives directions for our future work.
0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
2
II. DECISION STATISTIC AND NOISE MODEL
The signal detection problem can be considered as a binary
hypothesis test. It is assumed that under hypothesis H0 a signal
is absent and it is present under H1. An i-th sample of a
detector input signal is given by
(1)
yi   hi si  xi ,
where λ is an a priori unknown parameter which is defined as
0, H 0 ;

(2)
1, H1 ,
hi is a i-th sample of the fading gain, si and xi are i-th samples
of the detecting signal and the noise, i = 1,..,n, n is the sample
size. The samples of fading gain, signal and noise are
independent and identically distributed (i.i.d). The signal
samples are assumed to be samples of real Gaussian process
with zero mean and variance  s 2 .
A detector is intended for a decision statistic calculation.
The decision statistic of the p-norm detector can be written as
p
n  y 
p   i  ,
(3)
i 1   n 
where p is an arbitrary positive power and σn is the noise
standard deviation. The energy detector is a special case of the
p-norm detector when p = 2.
The noise samples are assumed to be Gaussian mixture
noise samples (GMN). The GMN model describes man-made
impulsive and atmospheric interferences [8, 9]. Also, it can be
used for the Middleton class A interference approximation [9].
The probability density function (PDF) of the GMN is given
by
V
 x2 
bv
g ( x)  
exp   2  ,
(4)
v 1
2 v2
 2 v 
where bv > 0 is the weight of the noise component,

V
b 1
v 1 v
,  v is a standard deviation of the noise component, V is a
constant representing the number of the components in the
GMN. The variance of the GMN is given by
V
 n2   bv v2 .
(5)
v 1
The simplest GMN model consists of V = 2 components, it is
also referred as ε-mixture noise [9].
The decision statistic of the optimal detector can be derived
exploiting a maximum likelihood ratio and taking into account
the signal and the noise models. It can be written as
P( y1 , y2 ,..., yn | H1 )
op 
,
(6)
P( y1 , y2 ,..., yn | H 0 )
where P( y1 , y2 ,..., yn | H w ) is the joint probability density
function (JPDF) of the signal-noise mixture under given
hypothesis; w = 0 or 1, for H0 or H1 respectively. Since
samples are i.i.d. this JPDF can be written as
n
P( y1 , y2 ,..., yn | H w )   g ( yi | H w ) ,
i 1
where g ( yi | H w ) is the PDF of the i-th sample.
(7)
Whereas the optimal detector provides the best performance
in comparison with the other detectors, calculation of the
decision statistic  op
requires high computational
complexity. Moreover, the information about the noise and the
signal distributions and parameters, which might be difficult to
quantify and changes with time in practice, is required. The
locally optimal (LO) detector [10] does not require any
information about the signal distribution and signal energy,
but it exploits noise parameters and has quite a complicated
decision statistic. This motivates the usage of quasioptimal
detectors that have a lower performance, but they do not need
channel state information and have a low implementation
complexity.
Detectors compare the decision statistic ((3) or (6) for pnorm or optimal respectively) with a predefined decision
threshold T. The decision rule is given by
H1
 T,
(8)

H0
where  is the decision statistic of the optimal detector  op or
p-norm detector  p . According to the Neyman-Pearson
criterion the threshold is defined by the value of the fixed false
alarm probability PF.
III. PERFORMANCE ANALYSIS FOR NON-FADING CHANNEL
The false alarm probability PF and the detection probability
PD are common metrics which are widely used for the
detectors performance evaluation. These probabilities can be
written in the form
(9)
PF  P{  T | H0 }  1  F (T | H 0 ) ,
PD  P{  T | H1}  1  F (T | H1 ) ,
(10)
where P{}
 is the conditional probability of an event under
given hypothesis ( H 0 or H1 ) and F (T | H 0 ) , F (T | H1 ) are
the cumulative distribution functions (CDF) of the decision
statistic under hypotheses H0 and H1.
In this section we consider a non-fading channel, it means
that all hi are equal to 1 and the SNR is given by
 
 s2
.
 n2
(11)
First, using (4), (6) and (7) we can finally write the decision
statistic of the optimal detector as
V


bv
xi2
exp  

2
2 
2
2
n v 1
2 ( v   s )
 2( v   s ) 
op  
.
(12)
V
 xi2 
bv
i 1
exp



2 
v 1
2 v2
 2 v 
Second, utilizing the MGF-based approach [7], the
conditional MGF of the decision statistic of the p-norm
detector can be written as
n


M  p ( s | H w )    exp(  s ( x /  n ) p ) f ( x | H w )dx  ,
0

(13)
0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
3
where f ( x | H w ) is the PDF of the absolute value of the input
sample yi.
The PDF of the absolute value of the input sample is given
by
V
bv
f ( x | H w )  2
2 Dw,v
v 1
 x2 
exp  
 2D  ,
w, v 

(14)
 D0,v   ,
H0 ;
Dw,v  
(15)
2
2
 D1,v   s   v , H1 .
Substituting the PDF (14) into (13) and then integrating we
obtain the conditional MGF of the decision statistic  p as
2
v
 2  (1)k   (2k  1) / p  V bv n2 k 1 
M  p (s | H w )  
, (16)

k  0.5 
 

2k k ! s (2 k 1)/ p
k 0
v 1 Dw, v


where ( x) is the gamma function [11, 8.310] .
Using [11, 0.314], the conditional MGF can be written in
the series-form as
n
n
 2 2
M  p (s | H w )   
 
where
C0  a0 n ,
1
ja0

 Ck s

2k  n
p
,
(17)
k 0
j
 (kn  j  k )
k 1
(18)
(1)   (2k  1) / p 
C j k
p 2k k !
k
b  2 k 1
 v k n 0.5 ,
v 1 Dw, v
V
a0 
Due to multipath propagation in real wireless channels
hi  1 . In this section results from the previous section for the
non-fading channels are extended to the case of random SNR
whose distribution depends on the fading channel type.
The instantaneous SNR is given by
2
 i  hi2
where
Cj 
IV. QUASIOPTIMAL DETECTOR FOR FADING CHANNELS
(1 / p) V bv n
.

p
Dw, v
v 1
(19)
(20)
Therefore, the CDF F p (T | H w ) can be obtained by using
the inverse Laplace transform on the M  p ( s | H w ) / s as
n
Ck T (2 k  n)/ p
 2 2 
.
(21)
F p (T | H w )    
   k 0   (2k  n) / p  1
Substituting (21) into (9) and (10) we obtain the series-form
expressions for the false alarm probability PF and the detection
probability PD as
n
Ck0T (2 k  n )/ p
 2 2 
,
PF  1    
   k  0   (2k  n) / p  1
(22)
n
Ck1T (2 k  n )/ p
 2 2 
.
PD  1    
   k  0   (2k  n) / p  1
(23)
where C k0 and C k1 are defined by (18)-(20) with D0,v and
D1,v respectively.
The detection characteristics (22) and (23) can be
approximated by the partial sum. For calculation Python and
mpmath library were used. This library contributes to very
precise calculation of the series coefficients.
 s2
.
 n2
(24)
The instantaneous detection probability can be calculated
using (23). The average detection probability PD is widely
used as a performance metric of the detector in the fading
channel. It can be calculated by averaging the conditional
MGF over the SNR distribution.
Exploiting (15) and (11) we rewrite (16) in the form
 2  (1) k   (2k  1) / p 
M  p ( s,  | H w )  
 2k k !s(2k 1)/ p
  k 0
(25)
n
V

bv

.
2
2 k  0.5 
v 1 ( n   v )

Therefore, the average conditional MGF over the SNR
distribution is given by
 2  (1) k   (2k  1) / p 
M  p ( s | H w )  
 2k k !s(2k 1)/ p
  k 0
(26)
n
V 
 n2 k 1bv f ( )d  
 
,
2
2 k  0.5 
v 1 0 (   v /  n )

where f ( ) is the SNR distribution. Further we derive the
average detection probabilities for different fading channels
such as Nakagami-m, k-µ and η-µ [12]. It should be pointed
out that the k-µ and η-µ distributions include Nakagami-m as a
special case, but we analyze it separately because resulting
expressions for this model has a lower complexity (the
equations for the coefficients do not include an infinite sum)
and can be calculated faster.
A. Nakagami-m distribution
The PDF of the Nakagami-m distribution is given by
m
f Nak ( ) 
1  m  m 1   
,
   e
 ( m)   
m
(27)
where m and  are distribution parameters.
Substituting (27) into (26) and calculating the integral, we
obtain
 2  m m  (1) k   (2k  1) / p 
M  p (s | H w )  
     
2k k ! s (2 k 1)/ p
k 0

(28)
n
m  k  0.5
V
  v2 

m v2  
  bv  2 
U  m, m  k  0.5,
 ,
 n2  
v 1
n 


0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
4
Fig. 1. Detection probability PD as a function of the power index p of the
Fig. 2. Detection probability PD as a function of the SNR  for different
p-norm detector for GMN with V = 2, V = 3 components and for Gaussian
noise, the SNR   3 and the various sample sizes n.
values of the power index p. The same characteristics of the optimal and LO
detectors are included for comparison. The noise parameters are following:
V = 2, b1  0.9985 , b2  0.0015 ,  22 / 12  100 ; the sample size n = 15.
Fig. 3. Detection probability PD as a function of the sample size n for the 1-
Fig. 4. Average detection probability PD vs. average SNR
norm and energy detectors for the different values of the SNR. The noise
parameters are following: V = 2, b1  0.9985 , b2  0.0015 ,  22 / 12  100 .
(curves at left) and the energy (curves at right) detectors in the different
fading channels and noise environments, namely: k-µ with k = 1 and µ = 3,
V = 3, n = 15; Nakagami with m = 1, V = 3, n = 15; η-µ with η = 0.99 and
µ = 2.5, V = 2, n = 20.
where U(a,b,z) is the confluent hypergeometric function of the
second kind [13, 13.1.3].
The series-form expression for the MGF can be written as
C0Nak  (a0Nak )n ,
n
 2 2  m 
M  p (s | H w )     
    
where
mn

C
k 0
Nak
k

s
2k  n
p
,
(29)
C jNak 
1
ja0Nak
 
 bv 

v 1
 
V
2
v
2
n
j
k 1
for the 1-norm
(30)
 (kn  j  k )
m  k  0.5

(1)   (2k  1) / p  Nak
C j k
p 2k k !
k

m v2
U  m, m  k  0.5,
 n2


,

(31)
0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
5
Fig. 5. Average detection probability PD vs. average SNR

for the 1-norm
(curves at left) and the energy (curves at right) detectors in the different
fading channels, namely: Rayleigh, V = 3, n = 15; Rice with k = 1, V = 3,
n = 15; Rice with k = 2, V = 3, n = 15.
m  0.5

m v2 
(1 / p ) V   v2 
bv  2 
U  m, m  0.5,
(32)
.

p v 1   n 
 n2 

Therefore, the average detection probability after the
inverse Laplace transform application is given by
a0Nak 
PD
 2 2  m 
 1    
    
mn
CkNak T p
.

k  0   (2k  n) / p  1
(33)
B. k-µ distribution
The k-µ distribution describes small scale variations of a
fading signal with line-of-sight conditions [12]. The k-µ
distribution includes Nakagami-m and Rice distributions as
special cases. The PDF of the k-µ distribution can be written
as
f k   ( ) 
 (1  k )
 1
1 k
2
 1

 1   (1 k ) 

2
e

k (1  k )
I  1  2



 , (34)

k 2  2 e k
where I t ( x) is the modified Bessel function of the first kind
and order t [13, 9.6],  is the average SNR.
Exploiting representation of I t ( x) [13, 9.6.10], substituting
it into (34) and using (26), we obtain the average series-form
MGF. Then, using the inverse Laplace transform we derive the
average detection probability PDk   as
n
 2  2   (1  k ) 
PDk    1     k

   e  
where
C0k    (a0k   )n ,
n
2l  n
Clk   T p
,

l  0   (2l  n) / p  1

(35)
(36)
for the 1-norm
b2  0.002 , b3  0.0015 ,  32 / 12  100 ,  22 / 12  50 , n = 15) for no
diversity case, pLC and pLS with L = 2 branches.
C kj   

1
ja0k  
j
 (un  j  u )
u 1
(1)u   (2u  1) / p 
p 2u u !



and the energy detectors in the Nakagami fading channel ( b1  0.9965 ,
2k  n
n
Nak
Fig. 6. Average detection probability PD vs. average SNR
lk    
i 0
k 
u
C kj u
(37)
,
(  2 k (k  1))i
 ii!
 2 
bv  v2 

v 1
n 
V

 (1  k ) v2
U    i,   i  l  0.5,
 n2

(1/ p) k  
a0k   
0 .
p
  l  i  0.5

,

(38)
(39)
C. η-µ distribution
The η-µ distribution is another general fading model which
is in contrast to the k-µ distribution describes small scale
variations of a signal with non-line-of-sight conditions [12].
The η-µ distribution includes Hoyt (Nakagami-q) and
Nakagami-m distributions as special cases. The PDF of the ηµ distribution can be written as
2  h


2     0.5 h 

  0.5

f   ( ) 

e
I   0.5  2 H  , (40)
  0.5   0.5
 
(  ) H


where
2   1  
,
(41)
h
4
 1  
.
(42)
H
4
By analogy with the derivation of the average detection
probability for the k-µ fading channel PDk   we can obtain
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Transactions on Vehicular Technology
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6
TABLE I
DETECTION PROBABILITIES FOR DIFFERENT VALUES OF INDEX OF POWER
AND FIXED SNR
Index of power p
Detection Probability PD
0.7
1
2
2.5
0.985
0.946
0.605
0.342
PD  
 
C0
Cj  

2

  h 
 1 2 2 
2 
 (  ) 
 (a0   )n ,


1
  
ja0
n
j
n
 

Cl
p 2u u !
µ; and W    , Ck   for η-µ. These coefficients are given by
T
Ln
2l  n
p
(43)
W
Nak
l 0
 2  2 m
   
    
Lmn
,
Ln
k 
 2  2   (1  k ) 
   k

   e  
(44)
W
(45)
W    2 2


u
,

(50)
L n
,
(51)
Ln
Ln
  2 h 
,

2 
 (  ) 
(52)
C0Nak  (a0Nak ) Ln ,
V
 
(  H ) 2i
bv 


i  0  i !(   k  0.5) v 1
 

l    
(49)
channel type: W Nak , CkNak for Nakagami; W k   , Ckk   for k-
   (2l  n) / p  1 .

j u
(1)u   (2u  1) / p 
PDpLC
CkfcT p
 1  W fc 
,
k  0   (2k  Ln) / p  1
where W fc and Ckfc are the coefficients for the given fading
 (un  j  u )C   
u 1
2 k  Ln

2
v
2
n
2i
2   2 i  l  0.5
C jNak 

 h 
(2  2i)U  2  2i, 2  2i  l  0.5, 2
,
 

(1/ p)   
a0   
0 .
p
(46)
2
v
2
n
C kj   
V. DIVERSITY RECEPTION
It is apparent that multipath propagation decreases the
performance of a detector, but modern wireless
communication systems require a reliable signal detection and
high performance. Diversity-combining schemes facilitate to
mitigation of the performance loss. Two recently proposed
non-coherent diversity-combining techniques (pLS and pLC)
showed better performance than the classical maximal ratio
combining and selection combining [7]. So we analyze these
schemes for the fading channels in GMN in order to assess the
performance gain in comparison with the no diversity
reception.
A. p-Law Combining
The p-law combining scheme consists of L branches each of
them includes the p-norm detector. The outputs of the
detectors then added together and form the pLC decision
statistic
p
L
n  y

i ,l
 ,
(48)
 pLC   
 n 
l 1 i 1


where yi,l is an i-th sample form a l-th branch. The decision
rule is still described by (8).
pLC
The false alarm probability PF
can be found using
expressions (18-20, 22) and replacing n by Ln into them. The
pLC
D
average detection probability P
scheme is given by
for the pLC diversity
 ( Lkn  j  k )
k 1
(1)   (2k  1) / p  Nak
C j k
p 2k k !
k
m  k  0.5
 

m v2 
 bv 
U  m, m  k  0.5,

,
 n2 
v 1
 

C0k    (a0k   ) Ln ,
2
v
2
n
V
(47)
1
ja0Nak
(53)
j

1
ja0k  
j
 ( Lun  j  u )C
u 1
(1)u   (2u  1) / p 
p 2u u !
k 
j u
(55)
uk  
(56)
,
C0    (a0   ) Ln ,
Cj   

1
ja0  
(57)
j
 ( Lun  j  u )C

j u
u 1
(1)u   (2u  1) / p 
p 2u u !
(54)
u  
(58)
,
where a0Nak , uk   , a0k   , u   , a0   are defined by (32),
(38), (39), (46) and (47) respectively.
B. p-Law Selection
The p-law selection scheme also consists of L branches, but
only the largest branch output is selected. The decision
statistic of the pLS diversity scheme is given by


 . (59)

For independent and identically distributed branches
expressions for the false alarm PF pLS and the average
 pLS
 n  y p n  y

i ,1
i ,2
 , 
 max  





 i 1  n  i 1  n
p
n  y

i,L
 ,...,  



i 1

 n




p
pLS
detection PD probabilities are given by
PFpLS  1  (1  PF ) L ,
(60)
PDpLS  1  (1  PD ) L ,
(61)
where PF is given by (22) and PD is the average detection
probability for the Nakagami-m, k-µ or η-µ fading channels
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(equations (33), (35) and (43) respectively).
VI. NUMERICAL RESULTS AND DISCUSSION
In this section, the analysis of the derived expressions is
done. For all the presented curves we set the false alarm
probability PF = 0.01 and found a corresponding detection
threshold. For the false alarm probability PF, the detection
probability PD and the average detection probability PD
calculations Python and mpath library were used. Since
expressions for these probabilities are written in series-form, it
is required to truncate the infinite sums. We carried out a
sufficient number of numerical experiments and found out that
these infinite sums are readily converging and can be assessed
by the partial sums. Furthermore, all results were proven by
the statistic simulation with 106 iterations.
A. Non-fading channel performance
In Fig. 1, the detection probability PD vs. the index of
power p curves are presented. The curves are plotted using
(22) and (23) for the following parameters: GMN with V = 2
components ( b1  0.9985 , b2  0.0015 ,  22 / 12  100 ), the
sample sizes n = 8, 10, 15; and with V = 3 components (
 32 / 12  50 ,
b1  0.9965 ,
b3  0.0015
b2  0.002 ,
 22 / 12  30 ), the sample size n = 15. Relying on the achieved
results it can be said that a quasioptimal value of pqo (value
providing the maximal detection probability PD with the fixed
false alarm probability PF) depends on the number of the
GMN components, the sample size and the noise parameters.
It has been found that the value pqo is contained in [0.5, 2].
The quasioptimal index of power pqo is close to 2 if the noise
is close to Gaussian (the noise pulses are weak or absent),
otherwise if the noise has powerful pulses it tends to be close
to 0.5. For the presented curves (Fig. 1, 1-5) the indices of
power are pqo1 = 1.7, pqo2 = 1.2, pqo3 = 0.7, pqo4 = 0.7, pqo5 = 2.
In order to evaluate the performance of the quasioptimal
detector in comparison with the p-norm, optimal and energy
detectors, the detection probability PD vs. the SNR  curves
are plotted in Fig. 2. These results (curves 1-5) are obtained
for GMN with the V = 2 components ( b1  0.9985 ,
b2  0.0015 ,  22 / 12  100 ) and the sample size n = 15. The
curve 1 in Fig. 2 corresponds to the optimal detector and
establishes an upper bound for the performance, there is no a
detector which operates better. The quasioptimal p-norm
detector (curve 2, p = 0.7) has lower performance, but its
detection probability is sufficiently higher in comparison with
the energy detector (curve 4, p = 2), for the SNR   4 the
detection probabilities are given by table I. As is shown by
Fig. 2, the detection probabilities decrease if the index of
power grows, e.g. the detection probability of the 2.5-norm
detector is lower than detection probability of the energy
detector.
As it was said in Section II the GMN model describes
impulse noise, which can be considered as a train of randomamplitude and randomly occurring narrow pulses in a
7
background of Gaussian noise. The amplitude of a pulse raised
in the power p > 1 increases if it more than 1. In other words,
the energy detector (the index of power p = 2) increases the
amplitude of powerful pulses, but the 1-norm detector does
not. Therefore, the detection threshold T grows rapidly for the
p-norm detectors with p > 1. High detection threshold limits
the detection probability. It means that the p-norm detectors
with p < 1 are more resistant to the powerful noise pulses as it
was shown by Fig. 1-2.
It is noteworthy that the 1-norm detector (Fig. 2, curve 3)
has almost equal performance with the 0.7-norm detector (Fig.
2, curve 2), but has a less complex structure. Its decision
statistic has the lowest complexity (no power operation
required), so it can be easily implement in practice. Another
argument in favor of the 1-norm detector is a slight decrease in
the detection probability when p growing for the curve 5 in
Fig. 1, which is plotted for Gaussian noise and has maximum
at p = 2. Noise parameters tend to change in time, so let us
assume, that during the first period of time the noise is close to
Gaussian and during the second period is non-Gaussian. The
detection probability of the 1-norm detector during the first
period will be slightly lower than the energy detector, but it
will be considerably higher during the second period. Thus,
the 1-norm detector can be a kind of a versatile solution for
the GMN environment with parameters that change in time.
In Fig 3, the detection probability PD vs. the sample size n
dependences are presented for the 1-norm and the energy
detectors (separate dots are connected to form curves for
illustration purposes). The curves are plotted using (22) and
(23) for the following parameters: GMN with V = 2
components ( b1  0.9985 , b2  0.0015 ,  22 / 12  100 ) and
the SNR   3 and   6 . As is shown by Fig 3, the detection
probabilities PD of the 1-norm detector increase if the sample
size n grows (curves 3 and 4 for SNR   3 and   6
correspondingly). Concerning the energy detector, as it can be
seen from Fig. 3 the curves 1 and 2 have a peak at n = 10 and
further growth of the sample size does not lead to the detection
performance increase for the given noise parameters. The
growth of the sample size n increases the probability to find
the noise pulse in the sample. It means that for the sample size
n > 10 and given noise parameters the detection threshold of
the energy detector starts to grow rapidly, that is why the
detection probability decreases in spite of the sample size
growth. The PD vs. n curves start to grow again after their
minimum. This happens because a contribution of one pulse
energy decreases with the sample size growth.
B. Fading channels performance
The performances of the 1-norm detector and the energy
detector are compared in the different fading channels
exploiting expressions (33), (35) and (43). The curves
presented in Fig. 4 and Fig 5 (the 1-norm detector curves at
left and the energy detector curves at right) characterize the
performance with the average detection probability PD vs. the
average SNR  for the two noise environments (1. V = 2
0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Transactions on Vehicular Technology
VT-2017-00673.R1
8
TABLE II
AVERAGE DETECTION PROBABILITIES FOR DIFFERENT FADING CHANNEL
TYPES AND TWO VALUES OF POWER INDEX
Nakagami
Channel
k-µ
η-µ
Power index
k = 1, µ = 3
η = 0.99, µ = 2.5
1
0.848
0.896
0.97
2
0.462
0.488
0.443
TABLE IV
AVERAGE DETECTION PROBABILITIES FOR D IFFERENT DIVERSITY SCHEMES
AND TWO VALUES OF POWER INDEX
No diversity
Scheme
pLC
pLS
Power index
1
0.874
0.989
0.984
2
0.539
0.645
0.787
TABLE III
AVERAGE DETECTION PROBABILITIES FOR DIFFERENT FADING CHANNEL
TYPES AND TWO VALUES OF POWER INDEX
Rayleigh
Rice
Channel
Rice
Power index
k = 1, µ = 1
k = 2, µ = 1
1
0.848
0.867
0.884
2
0.462
0.473
0.482
channels are found. Obtained results reveal that the p-norm
detector having almost as simple structure as the energy
detector has characteristics which are very close to the optimal
detector ones and are significantly higher than the energy
detector characteristics. Exploiting the GMN model, the
components
b1  0.9985 ,
b2  0.0015 ,
 22 / 12  100 ;
2. V = 3, b1  0.9965 , b2  0.002 , b3  0.0015  32 / 12  50 ,
 22 / 12  30 ). The following parameters of the fading
channels are set for Fig. 4: 1. k-µ distribution, k = 1, µ = 3
(solid line); 2. Nakagami distribution m = 1 (coincides with
Rayleigh distribution, dashed line), 3. η-µ distribution,
η = 0.99, µ = 2.5 (dotted line) and for Fig. 5: 1. Rayleigh
distribution (solid line), 2. Rice distribution (a special case of
the k-µ distribution when µ = 1), k = 1 (dotted line), 3. Rice
distribution, k = 2 (dashed line). As is illustrated by the figures
the 1-norm detector has higher average detection probability
than the energy detector for the all considered fading
conditions. For the average SNR   4 the average detection
probabilities are given by table II and III for the given noise
and channel parameters.
C. Diversity-combining schemes performance
Two diversity schemes are compared using the average
detection probability PD vs. the average SNR  curves
plotted in Fig. 6. The curves were obtained for the 1-norm and
energy detectors for the following noise parameters: V = 3,
 32 / 12  100 ,
b1  0.9965 ,
b3  0.0015
b2  0.002 ,
 /   50 , for the Nakagami fading channel with m = 1
using (49), (50), (54), (61), and (33). In this example, the
impulse components of the noise are more powerful than in
the previous examples, that is why the average detection
probabilities for the no diversity case is quite low. In order to
improve the performance, the pLC and pLS diversity schemes
with L = 2 branches were analyzed. The results of the
calculations for the average SNR   8 are given by table IV.
So, it is shown for the given noise parameters that, firstly,
the 1-norm detector provides the higher average detection
probability than the energy detector and, secondly, the pLC
scheme yields slightly better performance than pLS for the 1norm detector, but the pLS scheme is better than the pLC for
the energy detector.
2
2
average detection probability PDk   for Nakagami-m, k-µ, η-µ
fading channels and antenna diversity schemes (pLS and pLC)
were found for an arbitrary index of power p in order to assess
the detection performance in more realistic environment. It has
been found that the p-norm detector with p ≤ 1 has significant
performance gain comparing with the energy detector for the
considered fading channels and diversity schemes.
The p-norm detector with p > 1 in GMN rapidly increases
the amplitudes of the powerful noise pulses and, therefore, has
an extremely high detection threshold. This leads to low
detection probability in non-fading channels and low average
detection probability in fading conditions regardless of the
fading channel type. The p-norm detector with p ≤ 1 appears
to be more resistant to powerful noise pulses and provides
higher performance in the GMN environment.
Future work directions may include investigations of the pnorm detector performance for non-Gaussian models which
are different from the GMN model.
REFERENCES
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[10]
VII. CONCLUSION
[11]
In this paper, the performance analysis of the p-norm signal
detection is carried out for non-Gaussian noise described by
the GMN model. Analytical expressions for the false alarm
probability PF and the detection probability PD for non-fading
[12]
[13]
H. Urkowitz, “Energy Detection of Unknown Deterministic Signals,”
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0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2762244, IEEE
Transactions on Vehicular Technology
VT-2017-00673.R1
Professor Vladimir Kostylev graduated
from Voronezh State University (VSU),
Voronezh, Soviet Union, in 1981, with
an honor MSc in Radiophysics. His
academic career continued at VSU with
the awarding of a Ph.D. in 1985, Doctor
of Science in 2002 and Professor in
2007. Now he is a professor of the
Electronics department of VSU.
Professor Kostylev is a co-author of the
book “Bistatic Radar: Principles and Practice”, Wiley, 2007.
His research interests include cognitive radio, signal
processing and radar theory.
9
Ivan Gres received the Bachelor of
Radiophysics Degree in Computer
Electronics from the Voronezh State
University,
Voronezh,
Russian
Federation in 2012 and Master of
Radiophysics Degree in Information
Systems from the Voronezh State
University,
Voronezh,
Russian
Federation in 2014.
He is currently working towards the
Ph.D. degree in System Analysis and Information Processing
at the Voronezh State University. His research interests
include wireless signal detection and digital signal processing.
0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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