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Chernoff Bounds (Gaussian Examples)

Chernoff Bounds (Gaussian Examples). ECE 7251: Spring 2004 Lecture 26 3/19/04. Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>. Main object of interest:. Both representations will be useful.

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Chernoff Bounds (Gaussian Examples)

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  1. Chernoff Bounds(Gaussian Examples) ECE 7251: Spring 2004 Lecture 26 3/19/04 Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>

  2. Main object of interest: • Both representations will be useful • Discussion based on Van Trees, pp. 126-129 Material from Last Lecture • Consider the loglikelihood ratio test

  3. Ex. 1: Gaussian, Equal Variances

  4. Ex. 1: Completing the Square

  5. Ex. 1: Finish Computing the Mu

  6. Ex. 1: What Does Mu Look Like? (Graph from p. 126 of Van Trees Vol. I)

  7. Ex. 1: Basic Bound on PFA

  8. Ex. 1: Basic Bound on PM

  9. Ex. 1: Where are the Bounds Meaningful? • Recall we need

  10. Ex. 1: The Refined Bound for PFA • Recall the refined asymptotic bound: • In this case, since L is a sum of Gaussian random variables, the expression is exact:

  11. Again, since L is Gaussian, the expression is exact: Ex. 1: The Refined Bound for PM Typical thing to try:Using MATLAB, plot basic Chernoff bound and true probs. of false alarm on the same graph vs. d for various choices of 

  12. Ex. 1: Minimum Prob. of Error • For minimum prob. of error test, • Recall approximate expression for Pe from last slide of last lecture

  13. Ex. 1: Min. Prob. of Error Con’t • Recall the exact expression is: • Van Trees’ rule of thumb: “approximation is very good for d>6”

  14. The Bhattacharyya Distance • If the criterion is the minimum prob. of error and (s) is symmetric about s=1/2, then • -(s)is called the Battacharyya distance

  15. A common special case: EFTR: Ex. 2: Gaussian, Equal Means EFTR:

  16. Ex. 2: What Does Mu Look Like? (Graph from p. 128 of Van Trees Vol. I)

  17. Ex. 2: Gaussian, Equal Means EFTR: EFTR:Find an expression for the s which gives the tightest bound in terms of 

  18. Ex. 2: Gaussian, Equal Means Typical thing to do:Using MATLAB, plot four different ROC curves on the same graph: • The true curve, computed using expressions from previous lectures and built in MATLAB commands like chi2cdf, etc. • The curve given by the basic Chernoff upper bound (this gives an upper bound ROC curve) • The curve given by the asymptotic Chernoff expression using the Q functions • The curve given by the asymptotic Chernoff expression using the approximation to Q Try it for different values of n and

  19. Ex. 2: ROC Curve Comparison (Graph from p. 129 of Van Trees Vol. I)

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