The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

MAXIMUM-MINIMUM EIGENVALUE DETECTION FOR COGNITIVE RADIO

Yonghong Zeng and Ying-Chang LiangInstitute for Infocomm Research, A*STAR

21 Heng Mui Keng Terrace, Singapore 119613

ABSTRACT

Sensing (signal detection) is a fundamental problem in cogni-tive radio. In this paper, a new method is proposed based onthe eigenvalues of the covariance matrix of the received sig-nal. It is shown that the ratio of the maximum eigenvalue tothe minimum eigenvalue can be used to detect the signal exis-tence. Based on some latest random matrix theories (RMT), wecan quantize the ratio and find the threshold. The probabilityof false alarm is also found by using the RMT. The proposedmethod overcomes the noise uncertainty difficulty while keepsthe advantages of the energy detection. The method can beused for various sensing applications without knowledge of thesignal, the channel and noise power. Simulations based on ran-domly generated signals and captured ATSC DTV signals arepresented to verify the methods.

I. INTRODUCTION

Careful studies by FCC reveal that most of the allocated spec-trum experiences low utilization. This motivates the concept ofopportunistic spectrum access that allows secondary networksto borrow unused radio spectrum from primary licensed net-works. The core technology behind opportunistic spectrumaccess is "cognitive radio" [1]. Unique to cognitive radio isits ability to sense the environment over huge swath of spec-trum and adapt to it. That is, it is necessary to dynamicallydetect the existence of signals of primary users. In Decem-ber 2003, the FCC issued a Notice of Proposed Rule Makingthat identifies cognitive radio as the candidate for implement-ing negotiated/opportunistic spectrum sharing [2]. In responseto this, from 2004, the IEEE has formed the 802.22 WorkingGroup to develop a standard for wireless regional area networks(WRAN) based on cognitive radio technology.As discussed above, sensing is a fundamental component

of cognitive radio. There are several factors which makethe sensing difficult. First, the signal-noise-ratio (SNR) maybe very low (lower than -20dB). Secondly, fading and multi-path in wireless signal complicate the problem. Fading willcause the signal power fluctuates dramatically (can be lOdBor even higher), while unknown multipath will cause coherentdetection methods unreliable. Thirdly, noise/interference levelchanges with time (noise power uncertainty) and the noise canbe non-Gaussian. There are two types of noise uncertainty:receiver device noise uncertainty and environment noise uncer-tainty, which are caused by receiving devices or environment,respectively [3, 4, 5]. Due to noise uncertainty, in practice, it isvery difficult (virtually impossible) to obtain the accurate noisepower.

There have been some sensing algorithms including the en-ergy detection [6, 7, 3], the matched filtering (MF) [3] and cy-

clostationary detection [8, 9]. These algorithms have differentrequirements and advantages/disadvantages. Energy detectionis a major and basic method. Unlike other methods, energydetection does not need any information of the signal to be de-tected and is robust to unknown dispersive channel. However,energy detection is vulnerable to the noise power uncertainty[7, 3, 4], because the method relies on the knowledge of accu-rate noise power. In practice, it is very difficult (virtually im-possible) to obtain the accurate noise power. To overcome thisshortage, we propose a new method based on the eigenvalues ofthe covariance matrix of the received signal. It is shown that theratio of the maximum eigenvalue to the minimum eigenvaluecan be used to detect the signal existence. Based on some latestrandom matrix theories (RMT), we can quantize the ratio andfind the threshold. The probability of false alarm is also foundby using the RMT. The proposed method overcomes the noiseuncertainty difficulty while keeps the advantages of the energydetection. The method can be used for various signal detectionapplications without knowledge of the signal, the channel andnoise power. Simulations based on randomly generated signalsand captured ATSC DTV signals are presented to verify themethods.Some notations are used in the following: superscripts T

and t stand for transpose and Hermitian (transconjugate), re-spectively. Iq is the identity matrix of order q.

II. SENSING MODEL

Assume that we are interested in the frequency band with cen-tral frequency fJ and bandwidth W. We sample the receivedsignal at a sampling rate higher than the Nyquist rate. Assumethat there are M > 1 receivers (antennas). The received dis-crete signal at receiver i is denoted by xi (n) (i = 1, 2, ... M).There are two hypothesises: (1) hypothesis 'Ho: there existsonly noise (no signal); (2) hypothesis X-t1: there exist both noiseand signal. At hypothesis 'Ho, the received signal at receiver iis

(1)wi(n) =t ri(n), n = a,1, rciei,

while at hypothesis 'Hi, the received signal at receiver i' is

p Nijn 1k 0h k s k) + T n

j=l k=o

where sj(n) (j = 1,2, ,P) are P > 1 source signals,hij (k) is the channel response from source signal j to receiveri, Nij is the order of channel hij(k), and r1j(n) is the noisesamples. Based on the received signals with little or no in-formation on the source signals, channel responses and noisepower, a sensing algorithm should make a decision on the exis-tence of signals. Let Pd be the probability of detection, that is,

(2)

1-4244-1144-0/07/$25.00(®)2007 IEEE

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

at hypothesis 'H1, the probability of the algorithm having de-tected the signal. Let Pfa be the probability of false alarm, thatis, at 'Ho, the probability of the algorithm having detected thesignal. Obviously, for a good detection algorithm, Pd shouldbe high and Pfa should be low. The requirements of the Pdand Pfa depend on the applications.

III. MAXIMUM-MINIMUM EIGENVALUE DETECTIONdefLetting Nj = max(Nij), zero-padding hij(k) if necessary,

and defining

def ]Tx(n) = [xi(n),X2(n), xM(n)

hj(n) df [hij(n),h2j(n), ,hhmj(n)T,def T4j(n) [71i(n),qr2(n), * ,tM(n)],

we can express (2) into vector form as

p Nj

x(n) = 0 hj(k)sj(n-k) + r(n), n = 0, 1,j=l k=o

(3)

(4)

(5)

* (6)

Considering L consecutive outputs and defining

def [T(n XT(n1:** XT 1n+)]Tr(n) [xT(n), T(n -1), ,xT(n- L+1)IT

s(n) d [s1(n), ,s1(n -N1 -L + 1), ...

sp(n), ,sp(n-Np-L + 1)]T: (7)

we get

x(n) = Hs(n) + 4(n), (8)

defwhere H is a ML x (N + PL) (N E Nj) matrix definedj=1

asdef

H = [H1,H2,...*,HP], (9)

def h(O) hj (Nj) h..0)

O . hi (O) ..hj (Nj) j

The following assumptions for the statistical properties oftransmitted symbols and channel noise are assumed.

(Al) Noise is white.(A2) Noise and transmitted signal are uncorrelated.Let R(N5) be the sample covariance matrix of the received

signal, that is,

R(N5) defI L-1+N,- 1N x(n)Xt (n),

n=L

where N, is the number of collected samples. If N, is large,based on the assumptions (A1-2), we can verify that

R(N5) ~dR E(x(n)xI(n)) = HRsT + 2IML, (12)

where R, is the statistical covariance matrix of the input signal,Rs = E(s(n)st(n)), ur2 is the variance of the noise, and IMLis the identity matrix of order ML.

Let Amax and Amin be the maximum and minimum eigen-value of R, and Pmax and Pmin be the maximum and mini-mum eigenvalue of HIRsTHt. Then Amax = Pmax + 97 and'7

Amin = Pmin + 7r . Obviously, Pmax = Pmin if and only ifHRsHLt = 6IML, where is d is a positive number. In practice,when signal presents, it is very unlikely that HIRsTHt = cIML(due to dispersive channel and/or oversampling and/or corre-lation among the signal samples). Hence, if there is no sig-nal, Amax/Amin 1; otherwise, Amax/Amin > 1. The ratioAmax/Amin can be used to detect the presence of signal. Thedetection algorithm is summarized as follows.

Algorithm 1 Maximum-minimum eigenvalue (MME) detec-tion

Step 1. Compute

L-1+NsR(Ns) 1 Ex1(n)x (n).

Nsn=LStep 2. Obtain the maximum and minimum eigenvalue of the

matrix R(NA), that is, Amax and Amin.Step 3. Decision: ifAmax > 7Amin, signal exists ("yes" de-

cision); otherwise, signal does not exist ("no" decision), wherey > 1 is a threshold (to be given in the next section).

IV. THEORETIC ANALYSIS AND THE THRESHOLD

In practice, we only have finite number of samples available.Hence, the sample covariance matrix R(N5) may be well awayfrom the statistical covariance matrices R. The eigenvalue dis-tribution ofR(N5) becomes very complicated [10, 11, 12, 13].This makes the choice of the thresholds very difficult. At lowSNR, the performance of a sensing algorithm is very sensitiveto the threshold. Since we have no information on the signal(actually we even do not know if there is signal or not), it isdifficult to set the threshold based on the Pd. Hence, usuallywe choose the threshold based on the PfaWhen there is no signal, R(Ns) turns to R,,(N5), the sample

covariance matrix of the noise defined as,

I L-1+N,

RI, (Ns) = E: (n)-t(n).n=L

(13)

R1 (N5) is a special Wishart random matrix [10]. The study ofthe spectral (eigenvalue distributions) of a random matrix is avery hot topic in recent years in mathematics as well as com-munication and physics. The joint probability density function(PDF) of ordered eigenvalues of the random matrix RI (Ns)has been known for many years [10]. However, since the ex-pression of the PDF is very complicated, no closed form ex-pression has been found for the marginal PDF of ordered eigen-values. Recently, I. M. Johnstone and K. Johansson have foundthe distribution of the largest eigenvalue [11, 12] for real andcomplex matrix, respectively, as described in the following the-orems.

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

t -3.90 -3.18 -2.78 -1.91 -1.27 -0.59 0.45 0.98 2.02Fi (t) 0.01 0.05 0.10 0.30 0.50 0.70 0.90 0.95 0.99

Table 1: Numerical table for the Tracy-Widom distribution of order 1

Theorem 1. Assume that the noise is real. Let A(Ns)(72 R(Ns),t ( N5 1+ 2ML)and v=(Ns 1+

I + 1) 1/3 Assume that liM N Y1N- VIM-L N, -c Ns

(O < 1). Then A... (A(N,)) "converges (with probabilityone) to the Tracy-Widom distribution of order 1 (W1) [14].

Theorem 2. Assume that the noise is complex. LetA(Ns) = N R (N ), uI ( + ML)2 and vl

( /Ns+ ML)( 1 + 1 )1 . Assumethat lim MLv/M-L N, cxDNs

y (O < y < 1). Then A atA(N)) " converges (with prob-ability one) to the Tracy-Widom distribution of order 2 (W2)[14].

Note that for large Ns, u and ,u', v and v' are almost thesame, that is, the mean and variance for the largest eigenvalueof real and complex matrix are almost the same. However, theirlimit distributions are different.

Bai and Yin found the limit of the smallest eigenvalue [13]as described in the following theorem.

Theorem 3. Assume that lim ML y (0 < y < 1).

Then lim Amin = y)2(I -)2

Based on the theorems, we have the following results:

tection is

Pfa = P (Amax >'YAmin)

P (1 Amax(A(Ns)) > -YAmin)Ns~~~~~~~~~~,- P (Amax (A(Ns)) > -) (V

(Amax (A(Ns))- >t y(N-

1 (N

Hence we should choose the threshold such that

1-F_(/N( - ML)2 A)Pfa

This leads to

F1 V,- M/L)2 _ 1)

or, equivalently,

(N5- ML)'2 AtV

1 -Pfa,

From the definitions of and v, we finally obtain the threshold

Amax NNS ML)2

Amin Nss ML)2

(14)

(15)

The Tracy-Widom distributions were found by Tracy andWidom (1996) as the limiting law of the largest eigenvalue ofcertain random matrices [14]. Let F1 and F2 be the cumulativedistribution function (CDF) (sometimes simply called distribu-tion function) of the Tracy-Widom distribution of order 1 andorder 2, respectively. There is no closed form expression for thedistribution functions. It is generally difficult to evaluate them.Fortunately, based on numerical computation, there have beentables for the functions [11]. Table 1 gives the values of F1 atsome points. It can be used to compute the F1 1 (y) at certainpoints. For example, F1 1(0.9) = 0.45, F1 1(0.95) = 0.98.

Using the theories, we are ready to analyze the algorithms.For real signal, the probability of false alarm of the MME de-

= N+ VML)2(Ns _ML)2

(\N~-v- VMbL) -2/3. + M) F (1(NsML) 1/6 1( Pfa)). (20)

For complex signal, the only difference is that the functionF1 should be replaced by F2, the CDF of the Tracy-Widomdistribution of order 2.

V. SIMULATIONS

We require the probability of false alarm Pfa < 0.1. Thenthe threshold is found based on the formulae in Section IV.For comparison, we also simulate the energy detection for thesame system. Energy detection needs the noise power as a pri-ori. Unfortunately, in practice, the noise power uncertainty al-ways presents [7, 3]. Due to the noise uncertainty, the estimated(or assumed) noise power may be different from the real noisepower. Let the estimated noise power be uJ2 = aur2. We definethe noise uncertainty factor (in dB) as B = max{10 log10 a}.

It is assumed that a (in dB) is evenly distributed in an interval

_-_ ML)2)

ML/)2 _

At). (16)

v

(17)

(18)

F1 l(1 -Pfa). (19)

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

method EG-2 dB EG-1.5 dB EG-1 dB EG-0.5 dB EG-OdB MMEPf a 0.497 0.496 0.490 0.470 0.103 0.077

Table 2: Probabilities of false alarm (M -4,P = 2)

00auau

0

0

0~n

-20 -15 -10SNR (dB)

0.8 1 1.2 1.4 1.6 1.8 2 2.2Number of samples x 104

Figure 1: Probability of detection (M 4, P = 2) Figure 2:dB)

Probability of detection (M = 4, P

[-B, B] [3, 5]. In practice, the noise uncertainty factor of re-

ceiving device is normally 1 to 2 dB [3, 5]. The threshold forthe energy detection is given in [3]. In the following, all theresults are averaged over 2000 Monte Carlo realizations.

First, randomly generated signals are used. We considera system with 4 receivers/antennas (M 4) and 2 signals(P = 2). The channel orders are N1 N2 = 4 (5 taps).Assume that all the channel taps are independent with equalpower. The smoothing factor is chosen as L = 6. 50000 sam-

ples are used for computing the sample covariance matrix. Theprobabilities of detection (Pd) for the MME method and energy

detection (with or without noise uncertainty) are shown in Fig-ure 1, where and in the following "EG-x dB" means the energy

detection with x-dB noise uncertainty. If the noise variance isexactly known (B = 0), the energy detection is pretty good.However, if there is noise uncertainty, the detection probabilityof the energy detection is much worse than that of the proposedmethod.The Pfa results are shown in Table 2 (note that the Pfa is

not related to the SNR because there is no signal). The Pfa forthe proposed method and the energy detection without noiseuncertainty meet the requirement (Pfa < 0.1), but the Pfafor the energy detection with noise uncertainty far exceeds thelimit. This means that the energy detection is very unreliablein practical situations with noise uncertainty.To test the impact of the number of samples, we fix the SNR

at - 15dB and vary the number of samples from 7000 to 22000.Figure 2 and Figure 3 show the Pd and Pf,a respectively. Itis seen that the Pd of the proposed algorithm and the energy

detection without noise uncertainty increases with the numberofsamples, while that of the energy detection with noise uncer-

tainty almost has no change. This means that the noise uncer-

tainty problem cannot be solved by increasing the number ofsamples. For the Pf,a all the algorithms do not change muchwith the varying number of samples (note that the threshold isset based on Pf,a N, and L).

Secondly, we test the algorithms based on the capturedATSC DTV signals [15]. The real DTV signals (field mea-

surements) are collected at Washington D.C., USA. The datarate of the vestigial sideband (VSB) DTV signal is 10.762 sam-

ples/s. The sampling rate at the receiver is two times that rate.The multipath channel and the SNR of the received signal are

unknown. In order to use the signals for simulating the algo-rithms at very low SNR, we need to add white noises to obtainthe various SNR levels [5]. In the simulations, the smooth-ing factor is chosen as L = 16. The number of samples usedfor each test is N, = 100000 (corresponding to 4.65 mili sec-

onds sampling time). The results are averaged over 1000 tests(for different tests, different data samples and noise samplesare used). Figure 4 gives the Pd results based on the DTV sig-nal file WAS-003/27/01 (the receiver is outside and 48.41 milesfrom the DTV station) [15]. The Pfa results are shown in Ta-ble 3. The Pfa ofMME is well below 0.1. But the Pfa for theenergy detection with noise uncertainty far exceeds 0.1.

In summary, all the simulations show that the proposedmethod works well without using information of the signal,the channel and noise power. The energy detection is not reli-able (low probability of detection and high probability of falsealarm) when there is noise uncertainty.

VI. CONCLUSIONS

Method based on the maximum and minimum eigenvalues ofthe sample covariance matrix of the received signal has been

.2

X 0670

04

2, SNR=-15

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

method EG-2 dB EG-1.5 dB EG-1 dB EG-0.5 dB EG-OdB MMEPfa 0.495 0.494 0.490 0.479 0.107 0.081

Table 3: Probabilities of false alarm (for real DTV signal detection, M = P = 1)

proposed. Latest random matrix theories have been used to setthe thresholds and obtain the probability of false alarm. Themethod can be used for various signal detection applicationswithout knowledge of the signal, the channel and noise power.

o 55-e- EG-2dB

0.5 .EG-15dB REFERENCES-'-EG-ldB

0.45t---- -*-.... EG-05dB [1] J. Mitola and G. Q. Maguire, "Cognitive radios: making software radios-EG-0dB TmknE MME more personal," IEEE Personal Communications, vol. 6, no. 4, pp. 13

0 .354.- 18, 1999.

[2] FCC, "Facilitating opportunities for flexible, efficient, and reliable spec-4-- 0.3 --- - - -- - - - - --------------- -------------------trum use employing cognitive radio technologies, notice of proposed

rule making and order," in FCC 03-322, Dec. 2003.0.25 .. . .............

0.2 [3] A. Sahai and D. Cabric, "Spectrum sensing: fundamental limits andn practical challenges," in Proc. IEEE International Symposium on New

0.15 .............Frontiers in Dynamic Spectrum Access Networks (DySPAN), (Balti-0.1 ,.more, MD), Nov. 2005.

0.05 _ -.-.-.-.-.-.-.-.-.-.-.----- [4] R. Tandra and A. Sahai, "Fundamental limits on detection in low SNR0.8 1 1.2 1.4 1.6 1.8 2 2.2 under noise uncertainty," in WirelessCom 2005, (Maui, HI), June 2005.

Number of samples x 104[5] S. Shellhammer, G. Chouinard, M. Muterspaugh, and M. Ghosh, Spec-

Figure 3.Probability of false alarm (M 4, P 2)trum sensing simulation model. http://grouper.ieee.org/groups/802/22

Figure 3: Probablity of false alarm (M = 4, P 2) /M\4eeting documents/2006 July/22-06-0028-07-0000-Spectrum-Sensing-Simulation-Model.doc, July 2006.

[6] H. Urkowitz, "Energy detection of unkown deterministic signals," Pro-ceedings of the IEEE, vol. 55, no. 4, pp. 523-531, 1967.

[7] A. Sonnenschein and P. M. Fishman, "Radiometric detection of spread-spectrum signals in noise of uncertainty power," IEEE Trans. onAerospace and Electronic Systems, vol. 28, no. 3, pp. 654-660, 1992.

[8] W. A. Gardner, "Spectral correlation of modulated signals: part i-analogmodulation," IEEE Trans. Communications, vol. 35, no. 6, pp. 584-595,t S Ct ~~~~~~~~~~~~~~~~1987.-e- EG-2dB197

0.9 -~~~~~~y'- ~~~EG-1.5dB- EG-ldB [9] W. A. Gardner, W. A. Brown, III, and C.-K. Chen, "Spectral correlation/ - *A/ //^ EG-0.5dB of modulated signals: part ii-digital modulation," IEEE Trans. Commu-

0.8 - EG-OdB.MME nications, vol. 35, no. 6, pp. 595-601, 1987.

0

a,0.7 -;---- -f --------------[10]A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Coimmunications. Hanover, USA: now Publishers Inc., 2004.

[11] I. M. Johnstone, "On the distribution of the largest eigenvalue in prin-0. - --------ciple components analysis," The Annals of Statistics, vol. 29, no. 2,

2 u.4 pp. 295-327,2001..4 ... ..

[12] K. Johansson, "Shape fluctuations and random matrices," Comm. Math.0.3 Phys., vol. 209, pp. 437-476, 2000.

0.2 [13] Z. D. Bai, "Methodologies in spectral analysis of large dimensional ran-dom matrices, a review," Statistica Sinica, vol. 9, pp. 611-677, 1999.-20 - 15 - 10 -5 o 5

SNR (dB) [14] C. A. Tracy and H. Widom, "On orthogonal and symplectic matrix en-sembles," Comm. Math. Phys., vol. 177, pp. 727-754, 1996.

Figure 4: Probability of detection (based on DTV signal WAS- V. Tawil, 5] captured DTVsignal. http:Hgrouper.ieee.org/groups/802/22003/27/0 M =P 1) /M\4eeting documents/2006 May/InformalDocuments, May 2006.-,-- .7