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Continuous-Time Detection of Deterministic Signals in White Gaussian Noise

Continuous-Time Detection of Deterministic Signals in White Gaussian Noise. ECE 7251: Spring 2004 Lecture 35 4/9/04. Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>. Weirdness of White Noise.

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Continuous-Time Detection of Deterministic Signals in White Gaussian Noise

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  1. Continuous-Time Detection of Deterministic Signals inWhite Gaussian Noise ECE 7251: Spring 2004 Lecture 35 4/9/04 Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>

  2. Weirdness of White Noise • Continuous-time white noise noise has covariance function Dirac delta • Integral equation for K-L expansion is • Any othonormal set will work! (only true for white noise; K-L expansions are usually unique)

  3. A Basic Hypothesis Testing Problem • Discussion based on Van Tress, Vol. I, • pp. 254-257 • Signals normalized: • Not necessarily orthogonal; define correlation coefficient

  4. As , the set of observablesYkbecomes equivalent to the original random process Y(t) Ulf Grenander calls these random variables observables Grand Strategy • Trouble: Can’t directly write a density defined on a continuous time process Y(t) • Solution: Expand in an orthonormal basis

  5. Remaining chosen to form a complete set and be orthogonal to and (we won’t need these explicitly) A Choice of Basis Functions • In our basic problem, the noise is white, so anyorthonormal basis will do. Try

  6. Statistics of the Observables

  7. Loglikelihood Ratio • Only Y1 and Y2 matter; the remaining do not depend on which hypothesis is true and are independent of Y1 and Y2, so the loglikelihood ratio test (from a previous lecture) is

  8. A Rearrangement of Terms • EFTR:Derive this alternate test form: Just a projection onto a distance vector

  9. v u Say H0 Say H1 Decision line Can simplify by expressing in a new coordinate system (u,v); coordinates still independent, but only v matters A Graphical Interpretation

  10. Alternate Method of Computation • Normalized difference signal: EFTR: Confirm has unit energy • New test statistic computed by

  11. Moments of Statistic V • EFTR: Confirm that

  12. Performance • Detectability index given by • For a Bayesian test • For a minimum probability of error test for equal priors • (Minimum distance receiver)

  13. Signal Selection • Suppose each signal has same energy E • For othogonal signals, • For antipodal signals, i.e, ,

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