1. ECE 7251: Signal Detection and Estimation
Spring 2002
Prof. Aaron Lanterman
Georgia Institute of Technology
Lecture 35, 4/12/02:
Continous-Time Detection of Deterministic
Signals in White Gaussian Noise

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
Signals normalized:
Not necessarily orthogonal; define correlation coefficient

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

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

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 the previous lecture) is

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

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

10. Alternate Method of Computation
Normalized version of the difference signal
Exercise: Confirm has unit energy
New test statistic computed by

11. Moments of Statistic l
Exercise: Confirm that

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

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