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A Fuzzy Hypothesis Testing Based Sample
Covariance Matrix for Spectrum Sensing in Cognitive
Radio Systems
Swahum Mukherjee
Dept. of Electronics and Telecommunication Engineering,
Jadavpur University
Kolkata, India.
[email protected]
2S. Pattanayak, 3P. Venkateswaran, 4R.Nandi2ECE Dept , 3,4ETCE
Dept
2Narula Institute of Technology, 3,4Jadavpur University
Kolkata , India
[email protected], [email protected],
[email protected]
AbstractSpectrum Sensing is a key technology in cognitive
radio systems. Since orthogonal frequency division
multiplexing
(OFDM) is one of the major wideband transmission techniques,
spectrum sensing of OFDM based primary user signals isconsidered
in this work. This paper proposes a two-stage scheme-
Fuzzy Hypothesis Testing based Sample Covariance Matrix
(FHTSCM) for primary user signal detection. Firstly, a
hypothesis testing scheme with p-value determination from
the
observed samples is used along with a fuzzy inference scheme
for
detecting primary user signals with SNR as low as -2dB. We
recognize the equivalence of the probability of Type-I error
and
the probability of false alarm (Pfa) and use an important
property of the p-value to form the fuzzy rule base. In the
low
SNR regime (less than -2dB), the second stage computes the
sample covariance matrix from a limited number of observed
samples and an appropriate test statistic is extracted and
tested
using Roys detector. Since the statistical covariances of
the
received signal and noise are usually different, they can be
used
to differentiate the case where the primary user signals
arepresent from the one where there is only noise. The
simulations
are performed for the 2K mode of the DVB-T transmission and
the probability of detection for different SNR levels is
presented
to verify the methods.
Keywordscognitive radio; orthogonal frequency division
multiplexing (OFDM); spectrum sensing; hypothesis testing;
p-
value; Neyman-Pearson lemma; fuzzy inference systems; sample
covariance matrix; test statistic; toeplitz matrix; maximum
eigenvalue.
I.
INTRODUCTION
Wireless communication systems depend on their efficient
use of scarce resources, most notably the radio
frequencyspectrum. The drastic rise in the number of
wirelesssubscribers, the emergence of new applications and
thecontinuous quest for higher data rates call for dynamic
andefficient utilization of the frequency spectrum. CognitiveRadio,
which was first proposed in [1] is a promisingtechnology for
exploiting the underutilized spectrum in anopportunistic manner.
One application of cognitive radios isspectral reuse which allows
the unsubscribed/secondary usersto access and use the spectrum
originally allocated to
subscribed/primary users when they are not active[2].
Thesecondary users frequently perform spectrum sensing i.e.
thedetection of primary user signals and use the spectrum for
their
applications if the primary users are detected to be
inactive.However if the primary users become active, the
secondaryusers have to detect the presence of those users in
highprobability and relinquish the spectrum immediately
withoutinterfering with the primary user transmission.
Spectrumsensing based on cognitive radio must be capable of
detectingvery weak primary user signals. Concurrent use of
spectrumhas been introduced and regulated by the
FederalCommunications Commission (FCC) and has led to the use ofthe
frequency reuse concept in the IEEE 802.22 wirelessregional area
networks (WRAN) [3] which operate on the veryhigh
frequency/ultra-high frequency bands currently allocatedfor TV
broadcasting services. Spectrum sensing methods basedon energy
detection [4-5], cyclostationarity [6] and goodness-
of-fit tests [7-8] have been proposed.Orthogonal Frequency
Division Multiplexing (OFDM) is
known to be one of the most effective multi-carrier
modulationtechniques due to its capability to combat multi-path
fadingand inter-symbol interference (ISI). It has evolved into
apractical scheme for wideband digital wireless communications[9]
and hence it is imperative to assume that many of theprimary users
will adopt this technology.
In this paper, we propose a new scheme Fuzzy HypothesisTesting
with Sample Covariance Matrix (FHTSCM) fordetecting the primary
user signals over a wide range of thesignal-to-noise ratio (SNR).
In the first stage, the scheme uses agoodness-of-fit test based on
statistical hypothesis testingtogether with a fuzzy inference
system for detecting theprimary user signal having SNR as low as
-2dB. If the result ofthis stage is inconclusive, which it is for
very low values ofSNR, we use the statistical covariances or
autocorrelations ofthe received signal samples to differentiate the
signalcomponent from the background noise. This is based on thefact
that the statistical covariance matrices of the signal andnoise are
generally different. We assume the noise covariancematrix to be
unknown and use the Roys detector [10] which is
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known to be robust in realistic scenarios with a single
primaryuser and arbitrary but unknown noise correlations.
The rest of the paper is organised as follows. In section II,the
OFDM signals model is proposed. Section III introducesthe
hypothesis testing scheme and the use of a fuzzy inferencesystem
for deciding on whether or not the primary user signalis
detectable. If the test in section III fails, the samplecovariance
matrix based detection scheme is proposed in
section IV. Simulations are shown in section V and section
VIconcludes the paper.
II. SYSTEM MODEL
We assume that the OFDM signals are modulated byquadrature
amplitude keying (QAM) or PSK and the primaryOFDM system employs N
sub-carriers. An OFDM signal isconstructed by serial to parallel
conversion over the N sub-carriers followed by an Inverse Fast
Fourier Transform (IFFT).Thus N is also the number of symbols in an
OFDMsample[11].
If ( ) ( ) ( )1,....,1,0 NSSS are N complex QAMsymbols, the
outputs of the IFFT in each OFDM sample are
( ) ( ) 1,.....,0,1 1
0
2
==
=
NnkSN
nSN
k
N
nkj
e (5)
Fig. 1.
OFDM primary signal with IFFT in Cognitive Radio
Each OFDM symbol consists of N-length data and a cyclic
prefix (CP) of length Nc which is added to the symbols in
all
the OFDM samples. The mthOFDM sample is thus
[ ]T
NNNcNm SSSSS 1,,.........0,1,......., =. (1)
Assuming the transmitted OFDM frame to consist of M
samples, we get the primary signal vector as
[ ]MSSSS ..,,........., 21= ..(2)
The corresponding noise vector www MW ,....,, 21= isassumed to
be i.i.d. zero mean circularly symmetric complex
Gaussian with variance 2 i.e. W~CN(0,
2I), where CN(.)
denotes the multivariate Gaussian distribution for complex
random variable.
III.
HYPOTHESIS TESTING AND FUZZY INFERENCE
SCHEME
A.
Goodness-Of-Fit Test for Null hypothesis and p-value
determination.
There are two hypotheses
1.
H0(The null hypothesis) : The signal does not exist.
2.
H1(The alternate hypothesis): The signal exists.The received
signal samples under the two hypotheses are
given by [12],[13],[14], and [15]
( ) ( )mWmXH =:0 (3)
( ) ( ) ( )mWmSmXH +=:1 (4)Two probabilities are of interest in
spectrum sensing
Probability of Detection(Pd) and Probability of False Alarm
(Pfa)
Corresponding to an observed value of a test statistic, the
p-
value is defined as the lowest level of significance at
which
the null hypothesis would have been refuted.
The steps involved in finding the p-value are
Let TS be the test statistic.
Compute the value of TS from the sample
X1,X2,,Xn.Say it is a.
The p-value is given by
p-value =
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