Problem Set II: Optimum Filtering

The Cooper Union Department of Electrical Engineering
ECE416 Adaptive Filters
Problem Set II: Optimum Filtering

Note: Problems refer to Haykin 5th ed, or 4th ed. (problem numbers are the same). Also,
here use MATLAB as a ìcalculatorî and do not use built-in functions to compute FPEF
coe¢ cients, AR models and the like. For example, code up the Levinson-Durbin recursion on
your own, donít use a canned routine. Also, some of these problems require referencing the
notes given out that give exact formulas for the correlation, power spectrum and innovations
Ölters for an AR(2) process.
1. Problem 2.18. Some comments:
(a) For the Wiener Ölter, assume the signal u [n] has the form as given under hypothesis H1, so ~u = ~s +~v where ~s is deterministic. The desired signal is s [k] for some
Öxed k. By Önding "R" and "p" you can Önd the Wiener Ölter, and it will have
the form as given in the text. You will note regardless of the choice of s [k], all
these vectors are the same up to a scaling factor. And that is the point of the
problem: the solutions to (b) and (c) yield the same vector, up to a scaling factor.
(b) Haykin gives you a hint. Use it to transform the SNR into a Rayleigh quotient of
a matrix. Please donít compute gradients- you should know how to maximize a
Rayleigh quotient!
(c) You have (in my notes or in the book) the formula for the complex Gaussian
pdf. Remark: The maximum-likelihood (ML) decision rule for choosing the
hypothesis would be to compare  to 1 (or log  to 0). If you assume this,
you will get a speciÖc value for the threshold  involving quantities that are
presumed know (e.g., Rv). However, if you use say a Neyman-Pearson test, then
the threshold  will be an adjustable parameter that determines the probability
of false alarm (probability of choosing H1 when H0 is true) and probability of
detection (probability of choosing H1 when H1 is true). Either way, the point is
the form of the test will be to compare w
Hu to a threshold, where w again has
the same form as the other parts of the problem.
2. Consider an AR(2) process where the poles are 0:8 and 0:6. The exact model is:
x [n] = v [n]  a1x [n  1]  a2x [n  2]
where v is unit variance white noise.
(a) Find a1 and a2, and compute the PSD S (!) (you will graph it later, superimposed
with certain estimates). Also compute r [m] (exact values) up to order 10.
(b) Generate 103
samples of x, estimate r^[m] up to order 10.
(c) Find the maximum absolute error between the exact and estimated correlation.
Also compute the spectral norm of the di§erence between the actual correlation
matrix and the estimated one (for order 10)