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2012 International Conference on Computer Communication and Informatics (ICCCI-2012), Jan. 10 12, 2012, Coimbatore, INDIA

Spectrum Sensing: Approximations for Eigenvalue RatioBased Detection

Tata Jagannadha Swamy

Dept of Electronics and Communication Engineering GRIETHyderabad, India [email protected]

Srinivas.Avasarala, Thaskani Sandhya, and Garimella RamamurthyCommunication Research Centre

IIIT Hyderabad Hyderabad, India

[email protected]

AbstractIn Cognitive Radio technology, spectrum

sensing is a fundamental component. Secondary users

may utilize the frequency bands of primary users

when these bands are not being used. To support this

spectrum reuse functionality, secondary users are

required to sense the radio frequency environment.

Eigenvalue ratio based spectrum sensing is one among

the best achievable solutions. In this paper, we haveproposed a ratio distribution and an accurate decision

threshold to the eigenvalue ratio distribution. The

proposed analytical solutions are very useful in

spectrum sensing.

Keywords- Cognitive Radio, spectrum sensing,

Eigenvalue ratio based detection, Random matrix,

asymptotic eigenvalue statistics, moment based

approximation.

I. INTRODUCTIONOwing to the emergence of new wireless applications, andthe compelling need for broadband wireless access, the

consumer demand for radio spectrum has increaseddramatically. Conventional approach to spectrummanagement is very inflexible in the sense that eachoperator is granted an exclusive license to operate in acertain frequency band. However, with most of the usefulradio spectrum already allocated, it is becomingexceedingly hard to find vacant bands to either deploy newservices or enhance existing ones. On the other hand, thelicensed spectrum is rarely utilized continuously acrosstime and space. This observation has prompted theregulatory bodies to investigate a radically different accessparadigm where secondary (unlicensed) systems areallowed to opportunistically utilize the unused primary(licensed) bands. Cognitive Radio technology (CR) is a

paradigm for wireless communication in which either anetwork or a wireless node changes its transmission andreception parameters to communicate efficiently, avoidinginterference with licensed or unlicensed users. Thisalteration of parameters is based on the active monitoringof several factors in the internal and external environment,such as radio frequency spectrum, user behavior andnetwork state. The main functions of Cognitive Radio are

spectrum sensing, spectrum management, spectrummobility and spectrum sharing [1].

Spectrum sensing is one of the essential mechanisms ofcognitive radio. Detecting the unused spectrum andsharing it without interference with other users (primary).Detecting primary users is the most efficient way to detect

spectrum holes. Generally, spectrum sensing techniquescan be categorized as energy detection [1]-[6], matchedfilter detection [4], [7] cyclostationary feature detection[8] and eigenvalue ratio based detection [9], [10]. Amongthem, matched filter and cyclostationary feature detectorhave higher sensing ability but require prior knowledgeabout primary signal. Unlike them, energy detector isoften used to determine the presence of signals withoutprior knowledge. The main drawback of the energydetector is its inability to discriminate between primarysignal energy and noise energy at low SNR values.Asymptotically eigenvalue ratio based detector methoddoes not have noise uncertainty problem. This method canbe used for signal detection without prior knowledge of

the signal, and noise power. The performance ofeigenvalue ratio based detector depends on itsasymptotical distributions. In [20], the distribution of theratio of eigenvalues is simplified to a closed formexpression which may incur some approximation error fornon asymptotic cases.

In this work we focus on non asymptotic approximationfor the ratio of two independent Gaussian randomvariables. These variables are largest and smallesteigenvalues of the covariance matrix. The proposed ratiodistribution is new and good for optimal decision thresholdcalculations in spectrum sensing.

The rest of the paper is organized as follows. In Section II,the system model and some background information withvarious test statistics are provided. In Section III we deriveGaussian approximations to ratio of maximum andminimum eigenvalues. Standard Gaussian CumulativeDistribution Function (CDF) and Optimal threshold areanalytically derived in Section IV. Approximation toLargest eigenvalue distribution in section V. Simulation

978-1-4577-1583-9/ 12/ $26.00 2012 IEEE

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results will also be presented in this section. Conclusionsare given in section VI.

II. SYSTEM MODELWe assume that there are M1 antennas at the receiver.These antennas can be sufficiently close to each other toform an antenna array or well separated from each other.We assume that a centralized unit is available to processthe signals fromall the antennas. There are two

hypotheses:0H , signal absent, and 1H , signal present.

The received signal at antenna/receiver i is given by

i iy (n) = (n).:0H (1)

where0H (absence of primary signal). The samples

contain only noise. In the other case:

i i iy (n) = s (n) + (n), i =1,..., M,=1H (2)

0, ..., N 1.n =

In hypothesis 1H (presence of a signal), ( )is n is the

received source signal at receiver/antenna i , which may

include the channel multipath and fading effects. In

general ( )i

s n expressed as

ipqP

ip pp=1 l= 0

= h (l)s (n l).i

s (n) % (3)

Where P denotes the number of primary user/antenna

signals. ( )ps n% denotes the transmitted signal from

primary user/antenna p. ( )iph l denotes the propagation

coefficient for the channel from the pth primary user to

the ith receiver, and ipq denotes the channel order

for .iph It is assumed that the noise samples 'si (n) are

independent and identically distributed (i.i.d) over both nand i. The objective of spectrum sensing is to make a

decision on the binary hypothesis testing (choose 0H

or 1H ) based on the received signal. The signal from the

Mantenna/receivers yields the followingM 1 vectors;T

1 M( ) [y (n).....y (n)] ,y n = (4)T

1 M( ) [s (n)......s (n)] ,s n = (5)T

1 M( ) [ (n)..... (n)] .n = (6)

The hypothesis testing problem based onNsignal samples

to M number of antenna/receiver represented as M N

data matrix Y in fusion centre is

1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

.

N

N

M M M N

y y y

y y y

y y y

=

Y

K

K

M M O M

K

(7)

The receiver signal covariance matrix R at the secondary

user signal is defined as R = YY where is theHermitian conjugate operator. The ordered eigenvalues of

R are 1 2 .... .M > > > This test statistic when compared

with the precalculated optimal decision threshold

opt value gives the presence or absence of primary user.

The sensing threshold opt must be calculated for a

required probability of false alarm. If the optimal

threshold is greater than test statistic i.e.,opt y

T > the

detector output is ,0H otherwise .1H Distributions of the

test statistics are required for optimal decision threshold

calculations. The complexity of mathematical calculationsis high when the number of sensors or samples per eachsensor is more. Asymptotically, from the ratiodistribution, the proposed distribution has to be evaluatednumerically.

III.ASYMPTOTIC THRESHOLDAPPROACH

In this section we derive approximations for the largestand smallest eigenvalue distributions of the covariancematrix. These results are utilized to analyze the eigenvalueratio based detector functionality. Under hypothesis H0,

consider a random matrix ( , )Y M N M N be Gaussian

Unitary Ensemble (GUE) with ,M N and

(0,1)M

uN

. From the statistical considerations,

signal at the receiver is arranged as a White Wishartmatrix R. For the optimal threshold observations,distributions of largest and smallest eigenvalues arerequired so that distribution of the largest random variable1 converges to the second order Tracy-Widomdistribution [14]. The distribution of this variable is givenby

1 1

1

(M,N).

(M,N)1

= (8)

where centering and scaling parameters to the largesteigenvalue random variable is

2

1( , ) ( M + N) ,M N = (9)1/2

1

1 1( , ) ( M + N) + .

M NM N

=

(10)

Similarly for the distribution of smallest eigenvalue andom

variable,M

is given by

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M M

M

(M,N)= .

(M,N)M

(11)

where centering and scaling parameters are smallesteigenvalue random variable is

2( , ) ( M N)M

M N = (12)1/2

1 1( , ) ( M N)

M NM M N

=

(13)

The distribution ofM converges to the Tracy-Widom

Distribution of order two [14], [15]. The CDF of the

largest eigenvalue is 1 ( )F y and smallest eigenvalue

( )MF y are obtained by the approximations to the linear

transformation of Tracy-Widom distribution of order two.The CDF of the largest eigenvalue is:

11 TW2

1

y (M,N)( ) F .

(M,N)F y

(14)

Similarly, the cumulative distribution function of thesmallest eigenvalue is:

MTW2

M

y (M, N)( ) F .

(M,N)M

F y

(15)

The distribution function of Tracy-Widom distribution oforder two can be defined from a certain Painlev IIequation [18].

2

2

y

( ) exp (s y)u (s)ds ,TW

F y

=

(16)

2( ) 2u (y) + yu(y).u y =% (17)

where ( )u y is the unique solution of the Painlev II

equation with the asymptotics ( ) ( )i

u y A y as y

where ( )iA y is the Airy function defined by [18]. For a

given y, 2 ( )TWF y has been calculated with RMT lab

software package [16]. From the observations, the signal

at the receiver is ordered by a M N matrix. IfM N

the largest eigenvalue distribution is close to Gaussiandistribution. From these observations, the closed-formGaussian approximation to the ratio distribution is derived.The closed form Gaussian approximation captures theGaussian moments, of the largest and smallest eigenvaluedistribution using asymptotic moments. The proposedapproximation is generally valid for any matrix dimension.

For the largest eigenvalue 1 the expected value isexpressed in-terms of centering and scaling parameters is

1 1 1 1[ ] E[ (M,N)+ (M,N) ],E = (18)

1 1 1 1[ ] [ (M,N)+ (M,N)E[ ].E = (19)

and the variance to 1 is

1 1 1 1[ ] V[ (M,N)+ (M,N) ],V = (20)2

1 1 1[ ] ( (M,N)) V[ ].V = (21)

From the asymptotic behavior, the distribution of the

mean and variance converges to the CDF of Tracy-Widom distribution. Asymptotically the mean and

variance [13] of 1 will converge to the Tracy-Widom

variable [19];

1 2[ ] [ ] 1.7711,TWE E y = (22)

1 2[ ] [ ] 0.8132.TWV V y = (23)

Fitting these moments to corresponding Gaussianmoments gives a closed form of Gaussian approximation.

1 1 1 TW2 (M,N)+ (M, N)E[y ], = (24)

2

V[ ]= ( (M, N)) V[y ],1 1 TW2

2

=1(25)

M M TW2 (M,N)+ (M, N)E[y ],M = (26)2 2

M M TW2V[ ] = ( (M, N) V[y ].M = (27)

Asymptotically the mean and variance of the ratiodistribution for the largest and smallest eigenvalues are

1 ,2

1 , M and2

M . From the above approximations,

the ratio 0distribution of extreme eigenvalues is easilycalculated and these results are suitable to any matrixdimensions.

IV. THRESHOLD VALUE WITH

SIMULATION RESULTS

The Eigenvalue ratio based detector is one of the efficientmethods in spectrum sensing applications. From theasymptotic approximation with two independent extreme

eigenvalues 1, and ,M a CDF is derived from the

standard Gaussian distribution [11], [12] as follows;

* M 1

2 2 2

M 1

( ) ,

F

=

(28)

y0

* M

M

( ) F () + 2 g(x, y)dxdy.

F

=

(29)

where ( , )g x y is the joint density of largest and smallesteigenvalues. ( ) is the CDF of the standard Gaussian

random variable with the ratio distribution and ( )F is

approximated as * ( )F in [20]. For non asymptotic cases,

(29) can be approximated with addition of the term

M

M

then the equation is;

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M 1 M

2 2 2MM 1

( ) + .

+ F

(30)

Decision threshold can be derived analytically from the

above equations. To achieve a given false alarmprobability, the optimal decision threshold opt is obtained

from

fa( ) 1 P ,F = (31)

2 2 2 2 2 2 2

1 M 1 M M 1 1 M

2 2 2

M M

+ + .

opt

=

(32)

Where 1 Mfa

M

1 P .

=

The closedform optimal decision threshold to the

eigenvalue ratio based detector is numerically calculatedfrom the inverse of the Tracy-Widom distribution [11],[12], [17]. In Fig. 1 we compare the CDF of theeigenvalue ratio distribution (30) to the exact and otherdistributions [20] of the Eigenvalue ratio. From thesimulation results, approximation with correction term isvery much closer to the exact distribution than asymptoticapproximation [20] with large M and small N values. InFig. 2 we plot the decision threshold as a function of thefalse alarm probability for various M and N values. Theobtained statistics are good for small values ofM andlarge values ofN. The Tracy-Widom distribution [14],[15] gives the exact distribution for eigenvalue ratio baseddetector calculations.

Fig. 1 Distribution of the test statistics

Fig. 2 Performance comparison

V. APPROXIMATION TO LARGEST

EIGENVALUE DISTRIBUTION

Apart from the sensing methods considered above, a newdetection scheme based only on largest eigenvalue isproposed in [21]. In [20], a Gaussian approximation to thedistribution of largest eigenvalue is proposed. In thissection we propose a truncated Gaussian approximation tothe exact distribution of the test statistic (largesteigenvalue). This approximation is motivated from theobservation that eigenvalues are always positive and aGaussian approximation is not appropriate for a non

negative random variable. In this sense results in [20] arewrong. However, the Gaussian approximation is valid for

values of 4M > and 4N> since the truncation error Q

(mean/variance) is as low as for 4.M N= =

VI. CONCLUSIONS

Spectrum sensing is the fundamental element in Cognitiveradio environment; especially, when the secondary useroccupies the primary user spectrum. In this paper, wederived the approximate decision threshold as a functionof the desired probability of false alarm for spectrumsensing in cognitive radio. This is based on the actualdistribution of the ratio of the extreme eigenvalues ofComplex Wishart matrix. The approach is based onasymptotically Gaussian approximation to the eigenvalueratio distribution. The decision threshold for the detectorwas derived and the results are validated against exactdistribution. The obtained statistical observations give

accurate results in spectrum sensing applications.

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