1. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
International Journal of Electronics and Communication
IJECET
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
Engineering & Technology (IJECET)
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online)
Volume 2, Number 1, Jan – April (2011), pp. 17-23 ©IAEME
© IAEME, http://www.iaeme.com/ijecet.html
COGNITIVE RADIO: SPECTRUM SENSING AND PERFORMANCE
EVALUATION OF ENERGY DETECTOR UNDER
CONSIDERATION OF RAYLEIGH DISTRIBUTION OF THE
RECEIVED SIGNAL
SRIJIBENDU BAGCHI
Dept. of Electronics and Communication,
RCC Institute of Information Technology, Kolkata, India
ABSTRACT
Cognitive radio is proposed as a solution to rationalize the concept of recycling the
spectrum in today’s spectrum hungry scenario. Here unlicensed users utilize the licensed
frequency band when that particular band is not in use. Spectrum sensing is important to
sense the arrival of primary user. In this paper, sensing is done by energy detector and
two figures of merit namely probability of false alarm and probability of detection are
calculated by treating the received signal as Rayleigh distributed.
Keywords: Cognitive radio, probability of false alarm, probability of detection
1. INTRODUCTION
Available unlicensed spectrum has become a scarce resource for communication
due to recent advances in wireless technology [1]. Fixed spectrum allocation to different
services also causes limited usage of frequency bandwidth. However, recent studies of
Federal Communications Commission (FCC) have shown that 70% of allocated spectrum
in US has no proper utilization and in time domain also spectrum is utilized
insignificantly [6]. This brings the concept of opportunistic spectrum usage approach,
where unlicensed spectrum users utilize the unused licensed frequency bands by finding
spectrum holes. Cognitive radio is proposed as a reconfigurable device of the unlicensed
spectrum user that finds the spectrum holes under dynamic spectrum changing situation
[2, 6].
In this paper, licensed users are declared as primary users whereas unlicensed
users as secondary users. It is already said that secondary users make use of unused
primary spectrum with tolerable interference to primary network [4] but they are to
vacate the frequency band as primary users appear. Secondary users search for
appropriate spectrum hole to run the communication process.
Spectrum sensing is one of the important features in cognitive radio concept for finding
an appropriate spectrum hole as well as realizing a primary user’s appearance while
utilizing a licensed frequency band [3]. Primary user sends a pilot signal of low
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2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
bandwidth and low power during its arrival and the secondary user senses this within a
specific sensing time [7] and vacates the band. Here, suboptimal as well as non-coherent
energy detection scheme by radiometer is considered where any prior knowledge of pilot
signal is not required [4].
In this paper, section 2 deals with the entire system model where the decision regarding
the arrival of primary user is specified by binary hypotheses. An appropriate test statistic
is formed and two figures of merit namely probability of false alarm and probability of
detection are calculated. These calculations are justified by simulation results. Section 3
concludes the paper.
2. THE SYSTEM MODEL
2.1 Framing of hypotheses and calculations of the figures of merit
The total process of primary signal sensing can be described by the following binary
hypotheses [null hypothesis H0 and alternative hypothesis H1] as follows:
H0: y(n) = w(n) decide pilot signal is absent
H1: y(n) = x(n) + w(n) decide pilot signal is present
2
where x(n) ~ CN ( 0, σ x ) is the transmitted primary signal within the sensing time and
2
w(n) ~ CN( 0, σ w ) is the white Gaussian noise. Here CN denotes circularly symmetric
complex Gaussian (CSCG) distribution. The fading effect of the channel is neglected.
This immediately follows that
2
y(n) ~ CN( 0, σ w ) under H0
2 2
y(n) ~ CN( 0, (σ x + σ w ) ) under H1
Since y(n) is a complex variable, the modulus of y(n) is taken into account neglecting the
phase part. Thus it is obtained that
2
y (n) ~ Rayleigh ( 1 σ w )
2
under H0
2 2
y (n) ~ Rayleigh ( 1 (σ x + σ w ))
2
under H1
The test statistic (T) can be found from energy-detection scheme as
N
T = ∑ [ y ( n) ]
2
(1)
n =1
(Proof of (1) is shown in Appendix A)
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3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
Now, since Rayleigh distribution is a chi-square distribution with 2 degrees of freedom, it
is found that T follows chi-square distribution with 2N degrees of freedom under both H0
and H1, that is
2
T ~ χ 2N
under both H0 and H1 but with different parameters. T is compared with a pre-settled
threshold value γ th and decision is taken as follows:
Decide in favour of H0 if T < γ th
Decide in favour of H1 if T > γ th
In this decision procedure two types of errors generally occur.
Type I error: Decision is taken in favour of H1 when H0 is true
Type II error: Decision is taken in favour of H0 when H1 is true
Since both of these errors cannot be simultaneously reduced, one error is specified to a
fixed value, whereas the other error is reduced. A common approach is to specify the
probability of Type I error and the probability of Type II error is decreased.
Two figures of merit are generally proposed for any signal detection scheme – probability
of false alarm (Pfa) and probability of detection (Pd) . Pfa is the probability of Type I error
i.e. the probability of misjudging the arrival of primary signal under H0, i.e.
Pfa = P (T > γ th | H0 )
1 ∞
∫σth2 exp(−t )t dt
N −1
i.e. Pfa = γ (2)
Γ( N ) w
Probability of Type II error is known as probability of misdetection ( P ). Pd is the
md
probability of correctly identifying the arrival of primary signal under H1 i.e. it is the
complementary of P md
This means,
Pd = P (T > γ th | H1 )
1 ∞
∫σ 2γ+thσ 2 exp(−t )t dt
N −1
i.e. Pd = (3)
Γ( N ) w x
(Proofs of (2) and (3) are shown in Appendix B)
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4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
2.2 The simulation results
2 2
In the simulation results, it is assumed that σ w = 1 as well as σ x = 1 . Here Pfa and Pd are
plotted with γ th .
Figure 1 shows the Pfa vs. γ th (dB) plot. It can be easily found that Pfa falls rapidly with
increasing value of γ th . As a small value of Pfa is desirable, so from this point of view,
γ th should be a large value.
Figure 1: Pfa vs. γ th (dB) plot
Figure 2 shows the Pd vs. γ th (dB) plot. Here it is also found that Pd value falls with the
increasing value of γ th . As high value of Pd is desired, so from this point of view,
γ th should be a small value.
So, from both the figures, it is found that obtaining lower value of Pfa and higher value of
Pd simultaneously is not possible because both of these objectives are self-contradictory.
For this reason, we are to compromise between the two and choose an appropriate
threshold value so that the system performance can be optimized. For example, if we
choose the range 0 < Pfa < 0.5 and 0 < Pd < 0.5 as the system constraints, a proper
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5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
threshold value may be chosen satisfying both of these constraints and maximizing the
throughput.
Figure 2: Pd vs. γ th (dB) plot
3. CONCLUSION
Primary signal sensing is an important feature in cognitive radio domain.
Suboptimal detection scheme by energy detector is the most popular scheme in this area.
Detection should be robust even for weak pilot signal i.e. in the low SNR regime.
Generally pilot signal contains 1-10% of the total primary signal power. Treating the
received signal as Rayleigh distributed also provides improved system performance. An
appropriate threshold value may be chosen to meet the desired specifications of the
system.
REFERENCES
[1] S. Haykin, “Cognitive Radio: Brain-empowered wireless communications”, IEEE J.
Selected Areas in Communications, vol. 23, no. 2, pp. 201-220, Feb 2005
[2] A. Ghasemi, E.S. Sousa, “Collaborative Spectrum Sensing for Opportunistic Access
in Fading Environments”, In proc. of DySPAN’05, November 2005.
[3] D. Cabric, A.Tkachenko and R.W.Brodersen, “Spectrum sensing measurements of
pilot, energy and collaborative detection”, IEEE Military Communications Conference
(MILCOM), 2006
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6. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
[4] R. Tandra and A.Sahai, “SNR Walls for Signal Detection”, IEEE J. Selected Topics
in Signal Processing, vol 2, no. 1, pp. 4-17, February 2008
[5] D. Cabric, A. Tkachenko, R.W. Brodersen, “Experimental Study of Spectrum Sensing
based on Energy Detection and Network Cooperation”, in Proc. of 1st Intl. Workshop on
Technology and Policy for Accessing Spectrum (TAPAS 2006), Boston, August 2006.
[6] W.Y.Lee and I.F.Akyildiz, “Optimal Spectrum Sensing Framework for Cognitive
Radio Networks”, IEEE Transactions on Wireless Communications, vol 7, no. 10, pp.
3845-3858, October 2008
[7] Y.C.Liang, Y.Zeng, E.C.Y. Peh and A.T.Hoang, “ Sensing-Throughput Tradeoff for
Cognitive Radio Networks”, IEEE Transactions on Wireless Communications, vol 7,
no.4, pp. 1326-1336, April 2008
[8] A.Sahai and D.Cabric, ” A tutorial on spectrum sensing: Fundamental limits and
practical challenges”, Proc. IEEE Symp. New Frontiers in Dynamic Spectrum Access
Networks (DySPAN), Baltimore, MD, Nov. 2005
[9] J. Hillenbrand, T.A.Weiss and F.K.Jondral, “Calculation of Detection and False
Alarm Probabilities in Spectrum Pooling Systems”, IEEE Communication Letters, vol 9,
no. 4, pp.349-351, April 2005
[10] Z. Quan, S.Cui, H.V.Poor and A.H.Sayed, ”Collaborative Wideband Sensing for
Cognitive Radios”, IEEE Signal Processing Magazine, pp. 60-72, November 2008
[11] U.Madhow,” Fundamentals of Digital Communication”, Cambridge University
Press, 2008
[12] H. Arslan, “Cognitive Radio, Software Defined Radio, and Adaptive Wireless
Systems”, Springer, 2007
[13] D. Cabric. S.M. Mishra. R.W. Brodersen, “Implementation Issues in Spectrum
Sensing”, In Asilomar Conference on Signal, Systems and Computers, November 2004.
[14] K.C.Chen and R.Prasad, “Cognitive Radio Networks”, John Wiley & Sons. Ltd,
2009
[15] S.L.Miller and D.G.Childers, “Probability and Random Processes”, Academic Press,
2007
APPENDIX A
Derivation of the test statistic (T) and its distribution under H0 and H1
According to Neyman-Pearson lemma, a test statistic will be most powerful if it is
formulated in such a way so that inside the critical region (i.e. the region where H0 is to
be rejected) it satisfies
f(Y|H1) ≥ K f(Y|H0) (A1)
where Y = [|y(1)|, |y(2)|,……., |y(N) |] and f(Y) denotes the joint density function of all
| y (n) |, K being an arbitrary constant.
Considering the |y(n)|s are i.i.d. random variables, we can write from (A1)
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7. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976
– 6464(Print), ISSN 0976 – 6472(Online) Volume 2, Number 1, Jan - April (2011), © IAEME
N y ( n)  [ y ( n) ] 2  N y ( n)  [ y ( n) ] 2 
∏ 2
σ2
exp−

2
2σ 2 
 ≥ K ∏ 2 exp−
n =1 σ 1  2σ 12 

n =1    
where σ 12 = 1 σ w and σ 2 = 1 (σ w + σ x )
2
2 2
2
2 2
2
N
K/ σ2N
i.e. ∑ [ y ( n) ] 2
≥ ln N
σ1
/
where K = K 2 N
n =1  1 1 
2  2 − 2
N
σ 
 1 σ2 
N
This follows that T = ∑ [ y (n) ]
n =1
2
Since Rayleigh distribution is a chi-square distribution with 2 degrees of freedom, T
follows chi-square distribution with 2N degrees of freedom under both H0 and H1, that is
2
T ~ χ 2N
with parameters σ 12 under H0 and σ 2 under H1
2
APPENDIX B
Derivation of Pfa and Pd
1 ∞
∫γ th exp(−u / 2σ 1 )u du
2 N −1
By definition, Pfa = P (T > γ th | H0) = 2 N
(2σ ) Γ ( N )
1
1 ∞
∫σ 2
N −1
Substituting t = u / 2σ 12 , we get Pfa = γ th exp(−t )t dt
Γ( N ) w
Following the same approach, we get ,
1 ∞
Pd =
Γ(N ) ∫σ γ σ
2
th
+ 2
exp( − t ) t N −1 dt
w x
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