Noise Uncertainty in Cognitive Radio Analytical Modeling and Detection Performance

Noise Uncertainty in Cognitive Radio
Analytical Modeling and Detection Performance

Marwan A. Hammouda

Supervisor: Prof. Jon Wallace
Jacobs University Bremen

June 19, 2012

Marwan A. Hammouda Noise Uncertainty in Cognitive Radio

Outlines

Motivation
Introduction
Cognitive Radio
Primary Sensing
Noise Uncertainty NU
System Model
General Assumptions
Noise Uncertainty Model.
Detection with NU
Case 1: Uncorrelated Signals
Case 1: Correlated Signals
Noise Calibration Measurments
Conclusion
Future Works
Published Work
References

Motivation

Methods for primary user detection in cognitive radio may be severely
impaired by noise uncertainty (NU) and the associated SNR wall
phenomenon.
Propose the ability to avoid the SNR wall by detailed statistical modeling
of the noise process when NU is present.
Derive closed-form pdfs of signal and energy under NU, allowing an
optimal Neyman-Pearson detector to be employed when NU is present.
Explore energy detector at low SNR in a practical system.


Introduction
Cognitive Radio

Cognitive Radio is an interesting emerging paradigm for radio networks.
Basically aims at improving the spectrum utilization where radios can
sense and exploit unused spectrum
Allow networks to operate in a more decentralized fashion.
Challenge: Require low missed detection at low SNR


Introduction
Primary Sensing

Usually treated using classical detection theory.
The decision is made among two hypothesis:

H0 : xn = wn , n = 1, 2, . . . , N
(1)
H1 : xn = wn + sn , n = 1, 2, . . . , N

Neyman-Pearson (N-P) test statistic:

fH1 (x )
L (x ) = , (2)
fH0 (x )

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, CAV
detectors, Corrsum, and others.


Noise Uncertainty

Given a perfect noise information, detection is possible at any SNR with
energy detector.
Practical systems will only have a estimate of the noise variance σ2 . This
imperfect 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 [σ2 , σ2 ], but otherwise
lo hi
unknown.
Worst-case detector assumes

σ2 under H0
σ2 = hi
σ2
lo under H1

For some value of SNR, the detector exhibits Pd < Pfa , regardless of the
number 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 the
NU and investigate if we can avoid the SNR wall be detailed statistical
modeling.


System Model
General Assumptions

Deﬁne random noise parameter α = 1/σ, where σ2 is the variance.
Assume noise/signal Gaussian
α
f (xn |α) = √ exp{−α2 xn /2},
2
(3)
2π

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

N
1 ∞ α2
f (x ) = f (α)αN exp − ∑ xn2 d α, (4)
(2π)N /2 0 2 n =1

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


System Model
Noise Uncertainty Model

Popular Log Normal Model:

1 1
fLN (α) = √ exp − (log α + µLN )2 /σ2
LN (5)
ασLN 2π 2

Fit to truncated Gaussian with

µ = E {α} = exp{−µLN + σ2 /2},
LN (6)
2 2
σ = Std{α} = [exp(σLN ) − 1] exp(−2µLN + σLN ), (7)


System Model
Noise Uncertainty Model

Log Normal vs. Gaussian Approximation

LogNorm
6 NU = 0.5 dB Gauss
f (α)
4

2

0
0.7 0.8 0.9 1 1.1 1.2 1.3
α
NU = 1.0 dB LogNorm
3
Gauss
f (α)

2

1

0
0.6 0.8 1 1.2 1.4 1.6
α


Detection with NU
Case I: Uncorrelated Signal Samples

2
In (4), see that p = ∑n xn sufﬁcient statistic.
Pdf of p conditioned on noise parameter

α2
f (p|α) = (α2 p)N /2−1 exp{−α2 p/2}, (8)
2N /2 Γ(N /2)

Required marginal distribution on p only:

1 ∞ α2 p
f (p)= f (α)α2 (α2 p)N /2−1 exp{− } d α. (9)
2N /2 Γ(N /2) 0 2


Detection with NU

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

c0 e−c3 N
N k
c2
f (p)= ∑ L
Γ(Lk ) 1+(−1)N −k Γ Lk , c1 c2
2
(10)
2 k =0 k c1 k

where Lk = (N + 1 − k )/2 and

pN /2−1 p 1
c0 = √ c1 = +
2N /2 Γ(N /2) 2πσα 2 2σ2
1 µ2 1
c2 = µα /(σ2 p + 1)
α c3 = α
2
1− 2
2 σα σα p + 1


Detection with NU

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 ]

1
0.9
0.8
0.7
0.6
Pd

0.5
0.4
0.3
0.2 Modeled NU
0.1 Worst Case NU
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pfa


Detection with NU
Case II: Correlated Signal Samples

Assume a correlated primary user signal with a covariance matrix Σs .
Consider the following assumptions:
s
´
Σs = σ2 .Σs , where σ2 is the signal variance.
s
σ2 = σ2 .γ, where σ2 is the noise variance and γ is the SNR.
s
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

1 ∞ α2
f (x ) = f (α)αN exp − XT X d α, (11)
´
(2π)N /2 |Σs + I |
1/
2 0 2

H1

1 ∞ α2
f (x ) = f (α)αN exp − XT (Σs + I )−1 X d α,
´
´
(2π)N /2 |Σs + I|
1/
2 0 2
(12)


Detection with NU

Now, consider the following:
Make integration for the exponential parts since they assume to have the
most effect.
Take the Eigendecomposition of the signal covariance matrix.
Then, the N-P detector can be derived as follow:

µ2
2 2 erfc − α
2σ2 (1+σ2 B1 )
µα B0 µα B1 1 + 2σ2 B α 0 α α
L(Y) = exp −
1 + 2σ2 B0
α 1 + 2σ2 B1
α 1 + 2σ2 B1
α µ2
erfc − α
2σ2 (1+σ2 B0 )
α α
(13)
where
Y is the uncorrelated version of the received signal X with

σ2 I for H0
Σy =
σ2 (γΛ + I ) for H1
´
where Λ = diag (λ1 ...λN ) and λn is the nth eigenvalue of Σs

Detection with NU

Continue ..
B0 = 1 ∑N=1 yn and B1 = 1 ∑N=1 λan yn
2 n
2
2 n
2

´
where λan is the nth eigenvalue of the matrix A = (γΣs + I )−1
Using the identity (Q + ρM)−1 Q − ρQ−1 MQ−1 , we have
´
A I − γ.Σs . Note this identity is used for small values of γ
Then, B1 B0 − 2 γ. ∑N=1 λn yn = B0 − R
1
n
2

Note B0 represents the signal energy, where R is seen to be a
correlation-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:

µ 2 B0
α µ2 B0 − µ2 R
α α
l (y ) = − (14)
1 + 2σ2 B0 1 + 2σ2 B0 − 2σ2 R
α α α


Detection with NU

Assuming a covariance matrix with an exponential correlation model, as
follows:
1 for i = j
cov (xi , xj ) = σ2 .γ.
ρ|i −j | for i = 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 be
obtained. Then, a closed form for the eigenvalue λn could be as follows:

γ.(1 − ρ2 )
λn = (15)
1 + ρ2 + 2ρ cos( Nπn1 )
+


Detection with NU

At this point, I don’t have clear results to show for the next steps. I trying to
study more the detector in (14) by applying Taylor series expansion and
performing sensitivity analysis to investigate how dominant B0 and R are for
with 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, a
simple 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 )
2 1
TX1 RX1 800 6400
TX0 RX1
TX1 RX0 0.8
1.5

Prob. Density
TX0 RX0
0.6 Ns =100

Pd
1
0.4
0.5
0.2

0 0
0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1
Norm. Energy Pfa

Parameters: f = 2.55GHz , BW = 20MHz , Ns = 6400, L = 120dB , M =
600Realizations, (SNR = −16dB )
1
16
0.8
Prob. Density

12 6400
0.6
800
Pd

8 Ns =100
0.4

4 0.2

0
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.2 0.4 0.6 0.8 1
Norm. Energy Pfa


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 the
detection in case of a correlated signal.
Make more measurements with longer integration times and lower grade
ampliﬁers.


Published Work

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


References

Mitola, J., III and Maguire, G. Q., Jr., ”Cognitive radio: Making software radios
more personal,”, IEEE Personal Commun. Magazine, vol. 6, pp. 1318, Aug. 1999.
R. Tandra and A. Sahai, ”SNR walls for signal detection,”, IEEE J. Selected
Topics 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 the
Log-Normal fading channel capacity,”, IEEE Transaction on Eireless
Communications, vol. 8, No. 6 , June. 2009. 1318, Aug. 1999.


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