GLRT for Cooperative Spectrum Sensing: Threshold Setting in Presence - - PowerPoint PPT Presentation

GLRT for Cooperative Spectrum Sensing: Threshold Setting in Presence of Uncalibrated Receivers

Andrea Mariani, Andrea Giorgetti, and Marco Chiani

University of Bologna, Italy DEI - Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi” Cesena Campus

4th Int. Workshop of COST Action IC0902 Rome, Italy October 9–11, 2013

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 1 / 18

Outline

Study of GLRT in presence of uncalibrated receivers (SUs with different noise power) Statistical description of the test Approximated expressions for setting the decision threshold (Neyman-Pearson approach)

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 2 / 18

Cooperative sensing scenario

PUNetworks

... Channel matrix ... ...

SU N t k SUNetwork

Fusioncentre Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 3 / 18

System model

Given nR cooperative SUs and nT PUs the output of the receiving antennas at the i-th time instant is yi = H xi + ni where ni ∈ CnR AWGN vector xi ∈ CnT PU transmitted symbol vector; xi ∼ CN(0, Rx) H ∈ MnR×nT(C) channel gain matrix Observation matrix from nS snapshots Y = (y1| · · · |ynS) .

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 4 / 18

GLRT derivation

Under the Hj hypothesis, with j = 0, 1, the likelihood function of Y is L(Y|Σj) = 1 πnRnS |Σj|nS exp

• −nS tr
• Σj

−1S

• where the sample covariance matrix (SCM) is defined as S =

1 nS YYH.

The GLR to detect the hypothesis H0 is T =

• Σ1
• Σ0
• H0

≷ H1 ξ where 0 ≤ ξ ≤ 1. Σ1 and Σ0 are the ML estimates of Σ0 and Σ1, respectively.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 5 / 18

Sphericity Test

Assuming H0: Σ0 = E

• yiyi H|H0
• = σ2 InR, i.e. the SU have the same noise power

H1: No assumptions on Σ1 = E

• yiyi H|H1
• The GLRT is

T(sph) = |S| (tr{S}/nR)nR H0 ≷ H1 ξ where S is the sample covariance matrix (SCM) S =

1 nS YYH.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 6 / 18

Test of Independence

Assuming H0: Σ0 is diagonal; ”unbalanced receivers” case H1: No assumptions on Σ1 In this case the GLRT is T(ind) = |S| nR

k=1 sk,k

H0 ≷ H1 ξ where si,j is the (i, j) element of S.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 7 / 18

Detection with uncalibrated receivers

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 T(sph), ∆ = 0 dB T(sph), ∆ = 0.5 dB T(sph), ∆ = 1 dB T(sph), ∆ = 1.5 dB T(ind), ∆ = 0 dB T(ind), ∆ = 0.5 dB T(ind), ∆ = 1 dB T(ind), ∆ = 1.5 dB PD PFA

Figure: ROC comparison between T(ind) and T(sph) in presence of 4 SUs and a single

• PU. We assume that the noise power levels in dB at the SU receivers equal

σ2

ref, σ2 ref + ∆, σ2 ref − ∆, σ2 ref

. The reference level σ2

ref correspond to SNR = −10 dB.

nS = 500.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 8 / 18

Threshold setting

To design the threshold for both tests we need to know the distribution

• f T(ind) and T(sph) under H0.

Unfortunately, such statistical distributions are: known in closed-form for T(ind), but expressed as the Meijer G-function multiplied by a normalizing constant1 not known in closed-form for T(sph), but expressed as an infinite sum2 In both cases, such expressions cannot be easily inverted for threshold setting! Hence, we propose a moment matching approach.

• 1M. D. Springer and W. E. Thompson, “The distribution of products of beta, gamma and gaussian random

variables,” SIAM Journal on Applied Mathematics, vol. 18, no. 4, pp. 712-737, Jun. 1970.

• 2B. N. Nagarsenker and M. M. Das, “Exact distibution of sphericity criterion in the complex case and its

percentage points,” Communications in statistics, vol. 4, no. 4, pp. 363-374, 1975. Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 9 / 18

Moments of T(ind)

Theorem (On the distribution of the independence test) Consider the test statistic T(ind) = |S| / nR

k=1 sk,k, where S = {si,j} is the

SCM of a nR-variate complex Gaussian population with zero mean and diagonal covariance matrix. Then T(ind) can be expressed as T(ind) = TnRTnR−1 · · · T2 =

nR

• k=2

Tk. where {Tk}k=2,...,nR are independent beta distributed r.v.s with p.d.f.

• 1

B(nS−k+1,k−1) tnS−k (1 − t)k−2,

0 ≤ t ≤ 1 0,

• therwise.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 10 / 18

Moments under H0

Moments of T(ind)

Based on the previous theorem the moments of T(ind) can be derived as m(ind)

p

=

nR

• k=2

E

• Tp

k

• m(ind)

p

=

• Γ(nS)

Γ(nS + p)

• nR−1 nR−1
• k=1

Γ(nS − k + p) Γ(nS − k) .

Moments of T(sph) are given by [Nagarsenker and Das, 1975]

m(sph)

p

= nnRp

R

Γ(nSnR) Γ(nSnR + nRp)

nR

• i=1

Γ(nS − i + 1 + p) Γ(nS − i + 1) .

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 11 / 18

MM-based approximation (1)

We approximate T(ind) and T(sph) to beta distributed r.v.s. Thus the approximated p.d.f. is given by fT (t) ≃

• 1

B(a,b) ta−1 (1 − t)b−1 ,

0 ≤ t ≤ 1 0,

• therwise

where B(a, b) = 1

0 xa−1 (1 − x)b−1 dx is the beta function with

parameters a and b: a = m1 (m2 − m1) m2

1 − m2

, b = (1 − m1) (m2 − m1) m2

1 − m2

.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 12 / 18

MM-based approximation (2)

Thus PFA Pr{T< ξ|H0} can be expressed as PFA ≃ ξ 1 B(a, b) ta−1 (1 − t)b−1 dt = B(a, b, ξ) where B(a, b, ξ) =

1 B(a,b)

ξ

0 xa−1 (1 − x)b−1 dx is the regularized beta

function. The decision threshold can be easily calculated as ξ = B−1 a, b, PDES

FA

• .

Note that B−1(·, ·, ·) can be easily computed using standard mathematical software.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 13 / 18

MM-based approximation - T(ind)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 nR =4 nR =6 nR =8

simulated β approx. CDF (x) x

Figure: Comparison between the CDF based on the moment matching strategy

and numerically simulated curve for T(ind) under H0 with nS = 20.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 14 / 18

MM-based approximation - T(sph)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 nR =4 nR =6 nR =8

simulated β approx. CDF (x) x

Figure: Comparison between the CDF based on the moment matching strategy

and numerically simulated curve for T(sph) under H0 with nS = 20.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 15 / 18

Chi squared approximation comparison

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

nS =100

nS =200

β approx. χ2 approx. estimated

CDF (x) x

(a)

0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

nS =500 nS =1000

β approx. χ2 approx. estimated

CDF (x) x

(b)

Figure: Comparison among the CDF and the empirically estimated curve for T(ind) under H0. nR = 20.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 16 / 18

Conclusions

We studied the generalized likelihood ratio test (GLRT) for cooperative spectrum sensing The most proper assumption is that every SU experience a different noise power level (noise unbalances) T(ind) is robust to noise unbalances ⇒ T(ind) should be adopted in place of T(sph) Both tests can be very well approximated as beta r.v.s. (under null hyp.) We provided easy-to-use expressions for setting the decision threshold under the Neyman-Pearson framework.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 17 / 18

Thank you!

andrea.giorgetti@unibo.it

• A. Mariani, A. Giorgetti, and M. Chiani, “Test of Independence for

Cooperative Spectrum Sensing with Uncalibrated Receivers,” in Proc. IEEE Global Commun. Conf. (GLOBECOM 2012), Anaheim, CA, USA,

• Dec. 2012.

Andrea Mariani, Andrea Giorgetti, and Marco Chiani GLRT for Cooperative Spectrum Sensing 18 / 18