noise uncertainty in cognitive radio analytical modeling and detection performance

Noise Uncertainty in Cognitive RadioAnalytical Modeling and Detection Performance

Marwan A. Hammouda

Supervisor: Prof. Jon WallaceJacobs University Bremen

June 19, 2012

Marwan A. Hammouda Noise Uncertainty in Cognitive Radio

Outlines

MotivationIntroduction

Cognitive RadioPrimary Sensing

Noise Uncertainty NUSystem Model

General AssumptionsNoise Uncertainty Model.

Detection with NUCase 1: Uncorrelated SignalsCase 1: Correlated Signals

Noise Calibration MeasurmentsConclusionFuture WorksPublished WorkReferences


Motivation

Methods for primary user detection in cognitive radio may be severelyimpaired by noise uncertainty (NU) and the associated SNR wallphenomenon.

Propose the ability to avoid the SNR wall by detailed statistical modelingof the noise process when NU is present.

Derive closed-form pdfs of signal and energy under NU, allowing anoptimal Neyman-Pearson detector to be employed when NU is present.

Explore energy detector at low SNR in a practical system.


IntroductionCognitive Radio

Cognitive Radio is an interesting emerging paradigm for radio networks.

Basically aims at improving the spectrum utilization where radios cansense and exploit unused spectrum

Allow networks to operate in a more decentralized fashion.

Challenge: Require low missed detection at low SNR


IntroductionPrimary Sensing

Usually treated using classical detection theory.

The decision is made among two hypothesis:

H0 : xn = wn, n = 1,2, . . . ,N

H1 : xn = wn + sn, n = 1,2, . . . ,N(1)

Neyman-Pearson (N-P) test statistic:

L(x) =fH1(x)

fH0(x), (2)

where fH(x) is the joint pdf of the observed samples for hypothesis H

Provides optimal detection if pdfs in (2) are known.

Some famous detectors: Energy detector, Cyclostationary detectors, CAVdetectors, Corrsum, and others.


Noise Uncertainty

Given a perfect noise information, detection is possible at any SNR withenergy detector.

Practical systems will only have a estimate of the noise variance σ2. Thisimperfect knowledge is refereed to as noise uncertainty (NU).

The NU concept was identified and studied in detail in [2].

In [2], σ2 is assumed to be confined in the interval [σ2lo,σ

2hi ], but otherwise

unknown.

Worst-case detector assumes

σ2 =

{σ2

hi under H0

σ2lo under H1

}For some value of SNR, the detector exhibits Pd < Pfa, regardless of thenumber of samples⇒ SNR wall

Below SNR wall, no useful detection is possible for the model above.


Noise Uncertainty

So, the main idea behind this work is to find out a good statistical model for theNU and investigate if we can avoid the SNR wall be detailed statisticalmodeling.


System ModelGeneral Assumptions

Define random noise parameter α = 1/σ, where σ2 is the variance.

Assume noise/signal Gaussian

f (xn|α) =α√2π

exp{−α2x2

n/2}, (3)

Assuming i.i.d. process, the marginal pdf of sample vector x is

f (x) =1

(2π)N/2

∫∞

0f (α)α

N exp

{−α2

2

N

∑n=1

x2n

}dα, (4)

where f (α) is the distribution of the noise parameter α


System ModelNoise Uncertainty Model

Popular Log Normal Model:

fLN(α) =1

ασLN√

2πexp

{−1

2(logα + µLN)2/σ

2LN

}(5)

Fit to truncated Gaussian with

µ = E{α}= exp{−µLN + σ2LN/2}, (6)

σ = Std{α}= [exp(σ2LN)−1]exp(−2µLN + σ

2LN), (7)


System ModelNoise Uncertainty Model

Log Normal vs. Gaussian Approximation

NU = 0.5 dB

NU = 1.0 dB

0

2

4

6

0.7 0.9 1 1.1 1.2 1.3

LogNorm

f(α

)Gauss

0.8α

f(α

)

0

1

2

3

0.6 0.8 1 1.2 1.4 1.6

LogNormGauss

α


Detection with NUCase I: Uncorrelated Signal Samples

In (4), see that p = ∑n x2n sufficient statistic.

Pdf of p conditioned on noise parameter

f (p|α) =α2

2N/2Γ(N/2)(α

2p)N/2−1 exp{−α2p/2}, (8)

Required marginal distribution on p only:

f (p)=1

2N/2Γ(N/2)

∫∞

0f (α)α

2(α2p)N/2−1 exp{−α2p

2} dα. (9)



Using the Gaussian model for f (α), we can derive the closed-form f (p)as follows:

f (p)=c0e−c3

2

N

∑k=0

(Nk

)ck

2

cLk1

Γ(Lk )[1+(−1)N−k Γ

(Lk ,c1c2

2

)](10)

where Lk = (N + 1− k)/2 and

c0 =pN/2−1

2N/2Γ(N/2)√

2πσα

c1 =p2

+1

2σ2

c2 = µα/(σ2αp + 1) c3 =

12

µ2α

σ2α

[1− 1

σ2αp + 1

]



Example Detection Performance

Parameters: SNR=0 dB, NU=1 dB, N = 20 samples

Proposed detector knows σα but not realizations of α

For robust (worst-case) detector let α ∈ [µα−1.5σα,µα + 1.5σα]

00.10.20.30.40.50.60.70.80.91

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pd

Pfa

Modeled NUWorst Case NU


Detection with NUCase II: Correlated Signal Samples

Assume a correlated primary user signal with a covariance matrix Σs.Consider the following assumptions:

Σs = σ2s .Σ́s, where σ2

s is the signal variance.σ2

s = σ2.γ, where σ2 is the noise variance and γ is the SNR.SNR is constant, one can think about it to be the worst SNR.

Then, the marginal pdfs of the received signal for both hypothesis are:H0

f (x) =1

(2π)N/2|Σ́s + I|1/2

∫∞

0f (α)α

N exp

{−α2

2XTX

}dα, (11)

H1

f (x) =1

(2π)N/2|Σ́s + I|1/2

∫∞

0f (α)α

N exp

{−α2

2XT(Σ́s + I)−1X

}dα,

(12)



Now, consider the following:Make integration for the exponential parts since they assume to have themost effect.Take the Eigendecomposition of the signal covariance matrix.

Then, the N-P detector can be derived as follow:

L(Y) = exp

{µ2

αB0

1 + 2σ2αB0− µ2

αB1

1 + 2σ2αB1

}√1 + 2σ2

αB0

1 + 2σ2αB1

erfc

[−√

µ2α

2σ2α(1+σ2

αB1)

]erfc

[−√

µ2α

2σ2α(1+σ2

αB0)

](13)

whereY is the uncorrelated version of the received signal X with

Σy =

{σ2I for H0

σ2(γΛ + I) for H1

}where Λ = diag(λ1...λN) and λn is the nth eigenvalue of Σ́s



Continue ..B0 = 1

2 ∑Nn=1 y2

n and B1 = 12 ∑

Nn=1 λany2

n

where λan is the nth eigenvalue of the matrix A = (γΣ́s + I)−1

Using the identity (Q + ρM)−1 ' Q−ρQ−1MQ−1, we haveA' I− γ.Σ́s. Note this identity is used for small values of γ

Then, B1 ' B0− 12 γ.∑N

n=1 λny2n = B0−R

Note B0 represents the signal energy, where R is seen to be acorrelation-based value.

Taking the logarithm of the NP detector in (13), rewriting it in terms of B0

and R and considering only the exponential term:

l(y) =

{µ2

αB0

1 + 2σ2αB0− µ2

αB0−µ2αR

1 + 2σ2αB0−2σ2

αR

}(14)



Assuming a covariance matrix with an exponential correlation model, asfollows:

cov(xi ,xj) = σ2.γ.

{1 for i = jρ|i−j| for i 6= j

}i, j = 1,2, ..,N and ρ is the correlation coefficient

The inverse of the covariance matrix is then known to be tridiagonal matrix,and the a closed form for the eigenvalues of this tridiagoal matrix can beobtained. Then, a closed form for the eigenvalue λn could be as follows:

λn =γ.(1−ρ2)

1 + ρ2 + 2ρcos( πnN+1 )

(15)



At this point, I don’t have clear results to show for the next steps. I trying tostudy more the detector in (14) by applying Taylor series expansion andperforming sensitivity analysis to investigate how dominant B0 and R are forwith respect to the number of samples, NU level and SNR


Noise Calibration Measurment

Since most of the noise in a true receiver comes from the front-end LNA, asimple architecture depicted below can be used for noise calibration


Noise Calibration Measurment

Parameters: f = 2.55GHz,BW = 20MHz,Ns = 100,L = 110dB,M =800Realizations,(SNR =−6dB)

TX1 RX1TX0 RX1TX1 RX0TX0 RX0

0

1

2

0 0.5 1 1.5 2 2.5 3Norm. Energy

1.5

Prob.Density

0.5

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Pfa

Pd

6400

Ns =100

800

Parameters: f = 2.55GHz,BW = 20MHz,Ns = 6400,L = 120dB,M =600Realizations,(SNR =−16dB)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Norm. Energy

Prob.Density

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Pfa

6400Pd 800

Ns =100

4

8

12

16


Conclusion

Noise uncertainty limits robust detection at low SNR.

SNR can be relaxed by simple NU modeling.

Experiment demonstrates useful detection to -16 dB


Future Work

More analysis on the detector in case of a correlated signal

Study the importance of signal energy and correlation-based value on thedetection in case of a correlated signal.

Make more measurements with longer integration times and lower gradeamplifiers.


Published Work

Hammouda, M. and Wallace, J., ”Noise uncertainty in cognitive radio sensing:analytical modeling and detection performance,”, the 16th International ITGWorkshop on Smart Antennas WSA,2012


References

Mitola, J., III and Maguire, G. Q., Jr., ”Cognitive radio: Making software radiosmore personal,”, IEEE Personal Commun. Magazine, vol. 6, pp. 1318, Aug. 1999.

R. Tandra and A. Sahai, ”SNR walls for signal detection,”, IEEE J. SelectedTopics Signal Processing, vol. 2, pp. 417, Feb. 2008. 1318, Aug. 1999.

S. M. Kay, ”Fundamentals of Statistical Signal Processing: Detection Theory,”,Prentice Hall PTR, 1998.

F. Heliot, X. Chu, and R. Hoshyar, ”A Tight closed-form approximation of theLog-Normal fading channel capacity,”, IEEE Transaction on EirelessCommunications, vol. 8, No. 6 , June. 2009. 1318, Aug. 1999.


noise uncertainty in cognitive radio analytical modeling and detection performance

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