Log-Normal Approximation of Chi-square Distributions for. Signal Processing. Wassim Jouini (Ph.D.), Daniel Le Guennec, Christophe MOY. And Jacques Palicot ...
Log‐Normal Approximation of Chi‐square Distributions for Signal Processing Wassim Jouini (Ph.D.), Daniel Le Guennec, Christophe MOY And Jacques Palicot (Speaker) q ( p )
SUPELEC Signal Communication Embedded Electronic lab.
XXX URSI General Assembly and Scientific Symposium of International Union of Radio Science. URSI 2011, August 2011, Istanbul
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Presentation Outline
1 Motivation 1. M ti ti 2. Mathematical Model 3. Main results and simulations 4. Conclusion
URSI 2011, August 2011, Istanbul
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I – Motivation
II – Mathematical Model
III – Main Results
Motivation: Energy Detection under uncertainty (1)
• Neyman Person ‐ N P E Energy Detection: Statistic D t ti St ti ti H. Urkowitz. Energy detection of unknown deterministic signals. Proceedings of the IEEE, vol. 55, no. 4, pp. 523-531, 1967.
Noise (known noise level) Signal g + Noise Chi-Square Distribution with M degrees of freedom
Gaussian Samples
• Neyman Person ‐ Energy Detection: Threshold based Detection Detection Policy
Threshold for a Pfa ≤ α
Decision o tcome outcome
Performance criteria
Receiver Operating Characteristic
Probabilities of False Alarm and Right Detection URSI 2011, August 2011, Istanbul
Cumulative Distribution Function of Chi-Square Distributions 3
I – Motivation
II – Mathematical Model
III – Main Results
Motivation: Energy Detection under uncertainty (2)
• Log‐Normal Noise Uncertainty L N lN i U t i t A. Sonnenschein and P.M. Fishman. Radiometric detection of spread-spectrum signals in noise of uncertain power. IEEE Transactions on Aerospace and Electronic Systems, vol. 28, pp. 654-660, July 1992. W. Jouini Energy Detection Limits under Log-Normal Approximated Noise Uncertainty Signal Processing Letters, Volume 18, Issue 7, Pages: 423426, July 2011.
1.
Environment Model = same as NP‐Energy Detection
2. Unknown Noise level
Noise
3. Estimated Noise level follows Log‐Normal Distributions
Signal + Noise
Suggested by A. Sonnenschein d b h i and P.M. Fishman in d ih i 1992 based on empirical measures Noise uncertainty Model:
where Uncertainty: u
URSI 2011, August 2011, Istanbul
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I – Motivation
II – Mathematical Model
III – Main Results
Motivation: Energy Detection under uncertainty (3) W. Jouini Energy Detection Limits under Log-Normal Approximated Noise Uncertainty Signal Processing Letters, Volume 18, Issue 7, Pages: 423-426, July 2011.
To alleviate the lack of knowledge on noise i l level, l we analyze l th the following statistic’s distribution (that usually has no simple form):
We seek a closed form approximation i ti off its it distribution. Chi-Square Chi Square to Log Log-Normal Normal ratio
In this presentation: We argue that Chi-Square distributions can be accurately approximated by Log-Normal distributions. Thus, the suggested statistic can be approximated by a normal distribution!
To that purpose: 1. We model the approximation error functions with parameter M 2. Compute their maximum values and show their quick convergence to 0 as M grows large URSI 2011, August 2011, Istanbul
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I – Motivation
II – Mathematical Model
III – Main Results
Mathematical Model (1): Definitions
• Distributions Probability Density Functions Di t ib ti P b bilit D it F ti Chi-Square q Distribution Log-Normal Distribution Normal Distribution
Such that all distributions have the same mean and variance.
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I – Motivation
II – Mathematical Model
III – Main Results
Mathematical Model (2): Definitions
• Analytical Approximation A l ti l A i ti
Approximation order: n Polynomial Components Residual R id l function: f ti very smallll compared to the rest of the expression. Tends to zero as x tends to x0 x0 = M and Large M approximation
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I – Motivation
II – Mathematical Model
III – Main Results
Mathematical Model (3): Approximation errors
• Approximation errors
Error due to the approximation of Chi-Square distributions by Normal distributions ((usuallyy exploited p in signal g p processing) g) Error due to the approximation of Chi-Square distributions by Log-Normal distributions
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I – Motivation
II – Mathematical Model
III – Main Results
Main results (1)
• Approximation errors (3rd order approximation)
Error due to Normal Approximation of C o Chi-Square Squa e distributions d st but o s
Error due to Log-Normal Approximation of Chi-Square distributions
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I – Motivation
II – Mathematical Model
III – Main Results
Main results (2)
• Approximation errors (Extrema: position and amplitude)
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I – Motivation
II – Mathematical Model
III – Main Results
Main results (3)
• Approximation errors (Extrema: position and amplitude)
Notice that: In both cases, maximum distribution approximation error decrease as 1/M Moreover: The maximum error due to Log-Normal approximation of Chi-Square distributions is smaller than the maximum error due to a normal approximation of Chi-Square distribution (Cf. Figure in next Slide) URSI 2011, August 2011, Istanbul
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I – Motivation
II – Mathematical Model
III – Main Results
Main results (4)
Error due to Log-Normal Approximation of Chi-Square distributions Error due to Normal Approximation of Chi-Square q distributions
URSI 2011, August 2011, Istanbul
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I – Motivation
II – Mathematical Model
III – Main Results
Conclusion
1. We showed that Log‐Normal Distributions offer accurate approximations of Chi‐Square Distributions
2. We showed that Log‐Normal approximation of Chi‐Square distributions offer a better approximation than Normal approximations of Chi‐square distributions
3 This study is motivated and support further analysis on Energy Detection 3. Thi t d i ti t d d t f th l i E D t ti under Log‐Normal noise uncertainty: W. Jouini Energy Detection Limits under Log‐Normal Approximated Noise Uncertainty, Signal Processing Letters Volume 18 Issue 7 Pages: 423‐426 Signal Processing Letters, Volume 18, Issue 7, Pages: 423 426, July 2011. July 2011
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