Communications and Network, 2013, 5, 386-389
http://dx.doi.org/10.4236/cn.2013.53B2070 Published Online September 2013 (http://www.scirp.org/journal/cn)
Signal Detection Based on Walsh Transform for Spectrum
Sensing
Guoxiang Yang, Guangliang Ren, Kun Wu
State Key Lab. Of Integrated Services Networks, Xidian University, Xi’an China
Received July, 2013
ABSTRACT
Spectrum sensing is a key technology to improve spectrum efficiency. In this paper, we propose a novel signal detectio n
method based on Walsh transform for spectrum sensing. The main idea behind is that the received signal is transformed
into another domain by W alsh transform and the test statistic is obtained by exploiting the feature of the useful signal in
the new domain. The new method can perform well at low signal-to-noise rate (SNR). Simulation results show that the
proposed method has better performance than the spectral feature detection based on power spectrum.
Keywords: Cognitive Radio; Walsh Transformation; Feature Detector; Low SNR
1. Introduction
Due to the rapid development of wireless communication,
spectrum scarcity has become an urgent problem with the
increasing proliferation of wireless devices and services.
Cognitive radio (CR) is an emergent technology to solve
this problem, which can allow the unlicensed users to
opportunistically access the spectrum assigned to the
licensed users without harmful interference to the current
users. Spectrum sensing is one of the most important
functionalities of CR, which can reliably detect the exis-
tence of the user on the current bands.
In general, there have been several popular signal de-
tection methods for spectrum sensing such as energy
detection[3], coherent detection (matching filter) and
feature detection[4,5]. Energy detection is commonly
used, as it doesn’t require prior knowledge about the
correlation structure of the primary signal. However, the
energy detector requires perfect information for the noise
variance to properly perform the detection. If a pre-set
pattern of primary signals is known to the receiver, the
coherent detector is usually applied. The above two ap-
proaches are simple and effective, but they cannot detect
weak signals at very low signal-to-noise rate (SNR).
Feature detector is usually used when some information
of the transmitted signal is priori known. The spectral
feature detector based on power spectrum of the signal is
one example of the feature detectors, which exploits the
unique spectral pattern of a specific signal to detect its
presence. The spectral feature detector outperforms the
energy detector and the coherent detector.
In this paper, we study a feature detector based on
Walsh transform. Firstly, the received signal is processed
by Walsh transform, that is, the sign al is transfo r med into
another domain. Then, the test statistic is obtained by
exploiting the feature of the useful signal in the new do-
main, which is compared with the thresho ld to decide the
presence of the useful signal. The new method is appli-
cable to signals with rectangle envelope and robust to
white noise. The main contribution of this paper is that
we introduce the principle of how to improve the per-
formance of signal detection and propose a novel practi-
cal method which has much better performance than the
spectral feature detector.
The paper is organized as follows. In section 2 the
system model is introduced. In section 3 our method is
described, and the performance is analyzed. In section 4
the new detector is simulated and compared with the
spectral feature detector. Section 5 concludes the paper.
2. System Model
As the goal of detecting primary user (PU) signals is to
ascertain their presence or absence in the licensed fre-
quency band, it is usually formulated as a binary hy-
pothesis testing problem. It is assumed that the modula-
tion type and the symbol rate are known,
*This work was supported in part by the State Natural Science Founda-
tion of China, Grant No.61072102 and National Major Specialized
Project of Science and Technology, Grant No.2011ZX03001-007-01.
The authors are with the State Key Laboratory of Integrated Service
N
etworks, Xidian University, Xi’an 710071, China.
0
1
:()()
:()()(
Hrtnt
)
H
rtxt nt

 (1)
C
opyright © 2013 SciRes. CN

G. X. YANG ET AL. 387
where 0,
H
represents the absence of the PU signal
()
x
t and the received signal contains only addi-
tive white Gaussian noise (AWGN) with zero mean and
variance
()rt
2

. 1
H
stands for the presence of the PU
signal in the band, and consists of the primary sig-
nal ()rt
()
x
t corrupted by noise . The baseband signal ()nt
()
x
t is formulated as
() ()(),
nT b
n
x
tat agtnT


 
 (2)
where n is the symbol sequence, a()
T
g
t is the rectan-
gle envelope of symbols, is the symbol period.
b
T
3. Proposed Algorithm
Feature detectors usually have good performance for the
reason that they exploit th e typical feature of the PU sig-
nal which is different from that of noise. For example,
the spectral feature detector makes use of the feature of
the signal in frequency domain. In order to improve the
detection performance, the received signal needs to be
transformed into other domain, in which the feature of
the PU signal is more obvious. When the transformation
is orthogonal, if the obtained nonzero coefficients of the
PU signal after transformation are less, i.e., the PU signal
can be represented by less components of orthogonal
functions, the feature of the PU signal in the new domain
is more obvious. Then for the detection of signals whose
baseband signals have rectangle envelope, we introduce
the Walsh transform to process the signal, which is also
an orthog onal transformation.
It is assumed the duration time of the received signal
is T, but in order to making the derivation simple,
we regard T as 1. The signal which is integrable is
capable of being represented by a Walsh series defined
over the open interval (0, 1) as
()rt
()rt
01 2
( )(1,)(2,)...,rtccWAL tcWALt  (3)
where the coefficients are given by
1
0()( ,),
k
crtWALktdt
k
(4)
and the Walsh series is derived from
(,)WAL kt
11
1()
0
2, (,)(1)
1,2,...,,
prrr
pktt
p
r
NWALkt
N
 




 (5)
where t and k are the argu men t s of th e fun c tio n ex pr e ss ed
in binary notation.
The obtained coefficients form a vector C,
{}
k
c
012
[ , ,,...]Cccc
which can represent the signal in the new domain.
Figure 1 shows an example of BPSK baseband signals
with rectangle envelope and Figure 2 shows the obtained
vector of the BPSK baseband signal after Walsh trans-
formation. From the figures it is illustrated that the BPSK
signal can be represented be limited components of the
Walsh functions.
()rt
Then, assuming that the vector of the signal ()
x
t
()
is
ideal which is normalized, the following correlation
formula can be applied to detect the presence of
C
x
t:
1
0
,
H
T
ideal
H
SCC t

 (6)
where S is the teat statistic, is transposition opera-
tion and t is the decision threshold. If S is greater than the
threshold, 1
()
T

H
is decided, i.e., presence of the signal
()
x
t; otherwise, 0
H
is decided, i.e., absence of the
primary signal.
In practical application, only discrete signal can be
dealt by computer, so we consider the performance of the
detector based on discrete Walsh transformation. As-
suming that the total sample points of the receiver signal
is and that the oversample ratio is m, the ob-
tained points after discrete Walsh transformation is also
N. When there is no noise in the sign al, the signal can be
represented by
()rt N
Nm components and the coefficients
are 01
[...]
N
m. However, when there is only noise
in the signal, the total N coefficients are nonzero. Since
the transformation is orthogonal, the energy after the
transformation is the same as that before the calculation,
so we can obtain
cc c
2
0
(|)~ (0,)SH Nm  and
2
1
(|)~ (,)SH NEm  ,
where E is the power of the PU signal ()
x
t in the re-
ceived signal. Then, the probability density function
(PDF) of S can be expressed as equation (7),
t
1

1

Figure 1. Example of BPSK signal.
index
c
0 1
n
. .....
Figure 2. Obtained after Walsh transformation. {}
k
c

G. X. YANG ET AL.
388
2
022
2
122
(| )exp()
22
()
(| )exp()
22
mmS
pS H
mmSE
pS H
 
 



(7)
Then the false alarm probability can be derived,
10 0
2
(|)(|)
1exp( )
22
[],
t
mt
PHHpSHdS
ldl
m
Qt









 (8)
where
0
2
01
[]exp( ).
22
u
u
Qu du



 (9)
Similarly, the detection probability can also be ob-
tained,
11 1
.
(|) (|)
[()
t
PHHpSHdS
m
QtE





]
(10)
From the above equations, the probability of false
alarm can be calculated for a given decision threshold
from equation (8), which can also be used to obtain the
threshold if the probability of false alarm is constrained.
4. Simulation Results
In this section, the performance of the proposed detector
in the present paper is compared with the spectral feature
detector in [4] thro ugh simulation.
The baseband signal used in the simulation is BPSK
signal with rectangle envelope. The symbol rate is
2
M
B and the sample frequency is 32
M
Hz . A couple
of numerical experiments are carried out and a block of
received data with 10240 samples is generated to calcu-
late a decision result for every experiment.
Figure 3 shows the variation of probability of detec-
tion with probability of false alarm for BPSK when SNR
is -18dB. The probability of detection decreases as the
probability of false alarm falls. When the false alarm
probability is given, the proposed method has higher de-
tection
Figure 4 illustrates the effect of SNR on the probabil-
ity of detection for BPSK when the probability of false
alarm is 0.01. It is showed that the probability of detec-
tion decreases as SNR falls when the probability of false
alarm is fixed. When SNR is given, the proposed method
has higher detection probability.
Figure 3. Probability of detection versus probability of false
alarm for BPSK when SNR is -18 dB.
Figure 4. Probability of detection versus SNR for BPSK
when the probability of false alarm is 0.01.
5. Conclusions
In this paper, a feature detector based on Walsh trans-
form is proposed, which exploits the feature of the sign al
after Walsh transformation to decide the presence of the
useful signal. It has been proved that the proposed detec-
tor performs better than the spectral feature detector at
low SNR by the simulation results. In addition, the pro-
posed method is low in complexity, which makes it typi-
cally suitable for real time spectrum sensing in cognitive
radio system.
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