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MULTIANTENNA SPECTRUM SENSING: DETECTION OF SPATIAL CORRELATION

AMONG TIME-SERIES WITH UNKNOWN SPECTRA

David Ram´ ırez1, Javier V´ ıa1, Ignacio Santamar´ ıa1, Roberto L´ opez-Valcarce2and Louis L. Scharf3

1Communications Engineering Dept., University of Cantabria, Santander, Spain.

e-mail: {ramirezgd,jvia,nacho}@gtas.dicom.unican.es

2Dept. de Teor´ ıa de la Se˜ nal y las Comunicaciones, University of Vigo, Vigo, Spain.

e-mail: valcarce@gts.tsc.uvigo.es

3Depts. of Electrical and Computer Eng. and Statistics, Colorado State University, Ft. Collins, USA.

e-mail: scharf@engr.colostate.edu

ABSTRACT

One of the key problems in cognitive radio (CR) is the detection

of primary activity in order to determine which parts of the spec-

trum are available for opportunistic access. This detection task is

challenging, since the wireless environment often results in very low

SNR conditions. Moreover, calibration errors and imperfect analog

components at the CR spectral monitor result in uncertainties in the

noise spectrum, making the problem more difficult. In this work, we

present a new multiantenna detector which is based on the fact that

the observation noise processes are spatially uncorrelated, whereas

any primary signal present should result in spatial correlation. In

particular, we derive the generalized likelihood ratio test (GLRT) for

thisproblem, which isgiven by thequotient between thedeterminant

of the sample covariance matrix and the determinant of its block-

diagonal version. For stationary processes the GLRT tends asymp-

totically to the integral of the logarithm of the Hadamard ratio of the

estimated power spectral density matrix. Additionally, we present

an approximation of the frequency domain detector in the low SNR

regime, which results in computational savings. The performance of

the proposed detectors is evaluated by means of numerical simula-

tions, showing important advantages over existing detectors.

Index Terms— Cognitive radio, multiple-channel signal detec-

tion, generalized likelihood ratio test, Hadamard ratio, coherence

spectrum.

1. INTRODUCTION

In the last years the cognitive radio (CR) paradigm has emerged as

a key technology to improve spectrum usage [1]. The basic idea be-

hind CR is the opportunistic access of some users (secondary users)

to the wireless channel when the licensed (primary) users are not

transmitting. Therefore, any CR system necessarily relies on a spec-

trum sensing device for determining which parts of the spectrum

band are available (spectrum holes). Even when a spectrum hole is

found and exploited, secondary users must periodically check

whether it has been reclaimed by the primary network, in which case

the spectrum hole must be quickly vacated.

Detection of primary users inCR isa challenging problem, since

fading and shadowing may result in very weak received primary sig-

nals. This means that the spectrum monitor must operate in very

low SNR environments preventing synchronization to and/or decod-

ing of these signals, even if the modulation format and parameters

of primary transmitters were known. A number of detectors have

been proposed for CR applications, see [2] and references therein.

Perhaps the most popular (and computationally cheapest) one is the

energy detector (ED), which does not require any a priori informa-

tion about the primary system and does not need any sort of syn-

chronization. The main drawback of the ED resides in its sensitivity

to uncertainties in the background noise power, which may result in

undetectable primary signals if the SNR is below certain level, even

as the observation time goes to infinity [3]. Alternative approaches

to the ED exploit some features of primary signals, such as cyclo-

stationarity or the presence of pilots. However, these methods are

sensitive to synchronization errors [4], unavoidable in low SNR con-

ditions.

Another way to improve the detection performance of spectrum

monitors is to use multiple antennas. Intuitively, the presence of any

primarysignalshould resultinspatial correlationintheobservations;

a feature that can be used for detection since the noise processes at

different antennas can be safelyassumed statisticallyindependent. A

multiantenna ED extension that also exploits knowledge of the pri-

mary spectral emission mask was proposed in [5], but this scheme

remains sensitive to noise uncertainty. The multiantenna detector

suggested in [6] does not need knowledge of the noise variance, but

it implicitly assumes that the noise processes are white and with the

same power at all antennas. In practice, calibration errors become

unavoidable, and thus any deviation from these assumptions will re-

sult in performance degradation.

In this work we propose a multiantenna detector in which no as-

sumptions are made about the primary signal nor the spectral prop-

erties of the noise. Rather, it is exclusively based on the assumption

that, in the absence of primary transmissions, the observations are

spatially uncorrelated. We derive the generalized likelihood ratio

test (GLRT) for the block-diagonal structure of the space-time co-

variance matrix, which is asymptotically approximated by the inte-

gral of the log of the Hadamard ratio of the estimated power spectral

density (psd) matrix. In the low SNR regime, of particular inter-

est in CR applications, a computationally cheaper approximation of

the frequency domain detector can be derived. The benefits of the

proposed detectors are illustrated by means of some numerical sim-

ulations.

2. PROBLEM FORMULATION

We address the problem of detecting the presence of a primary user

in a cognitive radio node equipped with L antennas, without any

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prior knowledge about the primary transmission, the wireless chan-

nel, or the noise processes (beyond spatial independence). In partic-

ular, we test the covariance structure of the vector-valued time series

{x[n], n = 0,±1,...}, where x[n] = [x1[n],...,xL[n]]Tis a

vector of measurements at time n, or equivalently, {xi[n]} is the

time series at the i-th antenna. The detection problem is given by

H1 : x[n] = s[n] + v[n],

H0 : x[n] = v[n],

n = 0,...,N − 1,

n = 0,...,N − 1,

where s[n] is the vector with the samples of the primary signal at

the L antennas, and at time n; and v[n] = [v1[n],...,vL[n]]Tis

the additive noise vector, which is assumed to be zero-mean circu-

lar complex Gaussian and spatially white, i.e., E{vi[n]v∗

j[k]} = 0

for i = j and all n, k. No assumptions are made on the temporal

correlation of the noise processes, E{vi[n]v∗

Let us define the data matrix

i[k]}.

X =

⎡

⎢⎢⎢

⎣

x1[0]

x2[0]

...

xL[0]

x1[1]

x2[1]

...

xL[1]

...

...

...

...

x1[N − 1]

x2[N − 1]

...

xL[N − 1]

⎤

⎥⎥⎥

⎦=

⎡

⎢⎢⎢

⎣

xT

xT

...

xT

1

2

L

⎤

⎥⎥⎥

⎦,

where the i-th row, xT

N-samples of the i-th time series {xi[n]}, and the n-th column is

the n-th sample of the vector-valued time series {x[n]}. The vector

z = vec?XT?stacks the columns of XT, and its covariance matrix

⎡

⎢⎢⎢

The covariance matrices Rik = E?xixH

{xi}L

Inorder toproceed, we

{x[n]} under H1. We take it to be zero-mean, circular complex

Gaussian. In addition to resulting in tractable models and useful

detectors, this assumption is reasonable if the primary network em-

ploys orthogonal frequency division multiplexing (OFDM) as mod-

ulation format. Thus, the hypothesis testing problem becomes

i = [xi[0],xi[1],...,xi[N − 1]], contains

is

R = E

?

zzH?

=

⎣

R11

R21

...

RL1

RH

21

...

...

...

...

RH

RH

L1

R22

...

RL2

L2

...

RLL

⎤

⎥⎥⎥

⎦∈ CLN×LN.

?, 1 ≤ i,k ≤ L capture

needthedistribution

k

all space-time second-order information about the random vectors

i=1.

of

H1 : z ∼ CN (0,R1),

H0 : z ∼ CN (0,R0),

where CN (0,Rl) denotes the complex Gaussian distribution with

zero mean and covariance Rl. Under H0, R0 is an unknown pos-

itive definite block-diagonal matrix, i.e. R0 ∈ R0, where R0 =

{R | R = diag(R11,...,RLL)},withtheonly constraint that Rii

is Hermitian positive definite, and under H1, R1 ∈ R1, where R1

is the set of unknown positive definite covariance matrices with no

temporal or spatial structure, since we do not use any prior informa-

tion about the primary signals. The block-diagonal structure of the

covariance matrix under the null hypothesis is due to spatial uncor-

relation of the noise.

3. DERIVATION OF THE GLRT

Let us assume an experiment producing M independent realizations

of the data matrix X, or equivalently z. The joint probability density

function (pdf) for these measurements is the product of the pdfs, and

is given by

p(z[0],...,z[M − 1];R) =

M−1

?

n=0

p(z[n];R) =

=

1

πLNMdet(R)Mexp

?

−MTr

?

R−1ˆR

??

,

whereˆR is the sample covariance matrix given by,

ˆR =

1

M

M−1

?

n=0

z[n]zH[n] =

⎡

⎢⎢⎢

⎣

ˆR11

ˆR21

...

ˆRL1

ˆRH

ˆR22

...

ˆRL2

21

...

...

...

...

ˆRH

ˆRH

L1

L2

...

ˆRLL

⎤

⎥⎥⎥

⎦,

andˆRik ∈ CN×Nis the {i,k}-th block ofˆR, which represents the

estimated cross-covariance matrix between the N-sample windows

of the i-th and k-th time series.

To solve our hypothesis testing problem, we will use the gen-

eralized likelihood ratio test (GLRT). Although it is known that the

GLRT is not optimal in the Neyman-Pearson sense, it provides good

performance [7]. The GLRT for H0 : R ∈ R0vs. H1 : R ∈ R1is

based on the generalized likelihood ratio (GLR) [7]

λ =

max

R∈R0p(z[0],...,z[M − 1];R)

max

R∈R1p(z[0],...,z[M − 1];R)

?M

whereˆR1andˆR0are the maximum likelihood estimates of R under

hypotheses H1and H0, respectively.

Now, wewill obtainthe ML estimatesof thecovariance matrices

under both hypotheses, for which we need to assume M ≥ N. As

previously pointed out, under H0the correlation matrix R is block-

diagonal, withthe only constraint that RiiisHermitian non-negative

definite. That is, we force spatial uncorrelatedness but do not force

temporal stationarity. Then, it is easy to show that the ML estimate

ofˆR0isˆR0 = diag

?ˆR11,ˆR22,...,ˆRLL

For H1we take R1tobe the set of matrices R, withno temporal

or spatial structure imposed, with the only constraint being that R is

an Hermitian non-negative definite matrix. Then, the ML estimate

of R1 is given byˆR1 =ˆR. Taking the ML estimates into account,

the GLRT is

= det

?ˆR−1

0

ˆR1

exp

?

−Mtr

??ˆR−1

0

−ˆR−1

1

?

ˆR

??

,

?

.

λ

1

NM= det

?ˆR−1

0

ˆR1

? 1

N=

?

det

?ˆR

?ˆRii

?? 1

N

?

L ?

i=1det

??1

N,

(1)

which is a special case of a general result in [7]. Specifically, the

GLRT is a generalized Hadamard ratio. Interestingly, this statistic

is invariant to independent linear transformations of the time-series,

including any arbitrary filtering of the sequences {xi[n]}.

3.1. Frequency Domain Detector

The time domain detector in (1) was derived without any stationary

assumption. When the time series {xi[n]},i = 1,...,L, are jointly

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stationaryrandom vectorswhose dimensions increasewithout bound

(jointly stationary time series) and following an argument along the

lines of [8], the limiting form of (1) (L fixed and N → ∞) may be

approximated1by

l = λ

1

NM=

exp

⎧

⎩

⎨

π

?

−π

π

?

−π

logdet

?ˆS

?

ejθ??dθ

ejθ??

2π

⎫

⎭

⎬

exp

⎧

⎩

⎨

log

?

L

?

i=1

ˆSii

?

dθ

2π

⎫

⎭

⎬

,

(2)

whereˆS(ejθ) is a standard quadratic estimator of the psd matrix,

averaged over M realizations

ˆS(ejθ) =

⎡

⎢⎢⎢

⎣

ˆS11(ejθ)

ˆS21(ejθ)

...

ˆSL1(ejθ)

ˆS∗

ˆS22(ejθ)

...

ˆSL2(ejθ)

21(ejθ)...

...

...

...

ˆS∗

ˆS∗

L1(ejθ)

L2(ejθ)

...

ˆSLL(ejθ)

⎤

⎥⎥⎥

⎦.

Therefore, (2) can be rewritten as

l = exp

⎧

⎪

⎪

⎪

⎪

⎩

⎨

π

?

−π

log

⎡

⎢⎢

⎣

det

?ˆS?ejθ??

ˆSii(ejθ)

L ?

i=1

⎤

⎥⎥

⎦

dθ

2π

⎫

⎪

⎪

⎪

⎪

⎭

⎬

,

(3)

i.e., the GLRT in the frequency domain can be approximated by the

integral over the Nyquist band of the logarithm of a Hadamard ratio.

Finally, we must point out that in the case of L = 2 time series the

term inside the logarithm is just a function of the magnitude squared

coherence (MSC) spectrum [8].

3.2. Low SNR Approximation

In cognitive radio, the most interesting case is the low SNR regime.

In this scenario, and following the ideas of [9], the statistic in (3) can

be approximated by

l ≈ exp

?

−1

2

?π

−π

???ˆC(ejθ)

???

2

F

dθ

2π+L

2

?

,

(4)

whereˆC(ejθ) =ˆD(ejθ)−1/2ˆS(ejθ)ˆD(ejθ)−1/2andˆD(ejθ) is a

diagonal matrix formed from the main diagonal ofˆS(ejθ). This ap-

proximation, which can be seen as a generalization of [9] to vector-

valued time series, allows us to simplify the detector in the low SNR

regime, and it also results in a more robust test statistic when the

number of available samples is small.

4. SIMULATION RESULTS

In this section, we present some simulation results to illustrate the

performance of the proposed detectors (eq. (3) and eq. (4)) and

compare it to that of the following detectors:

• The energy detector (ED) using LN samples per realization

(the total number of samples is therefore MLN).

1Notice that the ML estimates of the covariance matrices in (1) are not

Toeplitz, in general; and consequently (2) is just an approximation of the

asymptotic GLRT for stationary processes.

• The GLRT for white time series [9], which is equivalent to

the generalized coherence (GC) proposed in [10] and is given

by 1−det(ˆC[0]), where

ˆ

C[0] =ˆD[0]−1/2ˆR[0]ˆD[0]−1/2, is

the L × L spatial coherence matrix in the time domain,

ˆR[0] =

1

NM

M−1

?

n=0

X[n]XH[n],

andˆD[0] is a diagonal matrix formed from the main diagonal

ofˆR[0].

• A modification of the detector [11] to handle noises with dif-

ferent powers at each antenna. The detector is based on the

ratio of largest to smallest eigenvalues of the spatial coher-

ence matrix

ˆ

C[0].

For the simulations, we have used an OFDM-modulated DVB-T

signal2with a bandwidth of 7.61 MHz. The signal undergoes propa-

gation through a spatially uncorrelated frequency-selective Rayleigh

fading channel with exponential power delay profile and unit power;

at the spectrum monitor, it is downconverted and asynchronously

sampled at 16 MHz. The additive noises at each antenna are gener-

ated by filtering independent zero-mean and complex white Gaus-

sian processes with common variance σ2with finite impulse re-

sponse (FIR) filters with 4 i.i.d. random taps distributed as ai[n] ∼

CN(0,1/4),n = 0,...,3; i = 1,...,L, and the common SNR for

all antennas is defined as SNR(dB) = 10log10(1/σ2).

Figures1 and 2show thereceiver operating characteristic(ROC)

curve for a typical rural area (delay spread of 0.097 μsec) and for a

typical urban area (delay spread of 0.779 μsec) [12]. The remaining

parameters are: L = 3 antennas, N = 100 samples3, the number

of realizations is M = 10, the signal-to-noise ratio is SNR = 0 dB

and the psd matrix is estimated using the Welch’s approach. As can

be seen in the figures, the proposed detectors present the best results,

mainly for the most selective channel (Fig. 2), which indicates that

exploiting the frequency structure of the time series significantly im-

proves the performance of the detectors. These examples also show

that the Frobenius norm approximation (denoted as F-GLRT in the

figures) presents good results, and it even outperforms the logdet de-

tector in some cases. Obviously, the GC and the detector based on

the eigenvalue spread perform poorly because they were designed

for temporarily white processes, and never intended for correlated

time series.

Finally, Fig. 3 shows the miss probability as a function of the

SNR for a fixed value of the false alarm probability (pFA = 0.01)

using the same parameters of the second example. Contrary to the

energy detector, the threshold of the proposed detectors does not de-

pend on the actual value of the SNR. Therefore, following the ideas

of [13], it can be calculated in advance by simulations. In the figure,

we can see that the proposed detectors obtain the highest slopes and

that the Frobenius approximation performs well for low and moder-

ate SNRs.

5. CONCLUSIONS

Inthiswork wehavepresented anew multiantenna detector for spec-

trum sensing in cognitive radio. This detector does not require syn-

chronization at any level with the primary signal, and is based on the

fact that under the noise-only hypothesis, the observations should be

28K mode, 64-QAM, guard interval 1/4 and inner code rate 2/3.

3For the energy detector the total number of samples is MLN = 3000.

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0 0.20.40.60.81

0

0.2

0.4

0.6

0.8

1

Pfa

Pd

GLRT

F−GLRT

GC

λmax(C[0]) /λmin(C[0])

ED

Fig. 1. ROC for the rural area. We have considered L = 3 antennas,

M = 10 realizations of length N = 100 and the SNR = 0 dB.

0 0.20.40.60.81

0

0.2

0.4

0.6

0.8

1

Pfa

Pd

GLRT

F−GLRT

GC

λmax(C[0]) /λmin(C[0])

ED

Fig.2. ROCfor theurban area. Wehaveconsidered L = 3antennas,

M = 10 realizations of length N = 100 and the SNR = 0 dB.

spatially uncorrelated. The GLRT, and a frequency domain approx-

imation were derived under a Gaussian signal model. Since no as-

sumptions are made on the power and spectra (nor even stationarity)

of the signal and/or the noise, this scheme is robust to uncertainties

in this regard, commonly found in practice due to imperfect analog

components and calibration errors.

6. ACKNOWLEDGMENTS

This work was supported by the Spanish Government, Ministerio

de Ciencia e Innovaci´ on (MICINN), under project MULTIMIMO

(TEC2007-68020-C04-02), SPROACTIVE (TEC2007-68094-C02-

01/TCM),projectCOMONSENS(CSD2008-00010, CONSOLIDER-

INGENIO 2010 Program) and FPU grant AP2006-2965.

7. REFERENCES

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−10−505 10

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−2

10

−1

10

0

SNR (dB)

1 − Pd

GLRT

F−GLRT

GC

λmax(C[0]) /λmin(C[0])

ED

Fig. 3. Detection performance for the urban area as a function of the

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