ECE 534 RANDOM PROCESSES SPRING 2004
PROBLEM SET 7 Due Wednesday, December 8
Wiener Filtering of Random Processes
Assigned Reading: Chapter 7 of the notes.
1. A quadratic predictor
Suppose Xis a mean zero, stationary discrete-time random process and that nis an integer with
n≥1. Consider estimating Xn+1 by a nonlinear one-step predictor of the form
b
Xn+1 =h0+
n
X
k=1
h1(k)Xk+
n
X
j=1
j
X
k=1
h2(j, k)XjXk
(a) Find equations in term of the moments (second and higher, if needed) of Xfor the triple
(h0, h1, h2) to minimize the one step prediction error: E[(Xn+1 −b
Xn+1)2].
(b) Explain how your answer to part (a) simplifies if Xis a Gaussian random process.
2. Estimation of a filtered narrowband random process in noise
Suppose Xis a mean zero real-valued stationary Gaussian random process with the spectral density
shown.
1
f
8 Hz
S (2 f)π
X
8 Hz
10 Hz
4
10 Hz
4
(a) Explain how Xcan be simulated on a computer using a pseudo-random number generator that
generates standard normal random variables. Try to use the minimum number per unit time. How
many normal random variables does your construction require per simulated unit time?
(b) Suppose Xis passed through a linear time-invariant system with approximate transfer function
H(2πf ) = 107/(107+f2). Find an approximate numerical value for the power of the output.
(c) Let Zt=Xt+Wtwhere Wis a Gaussian white noise random process, independent of X, with
RW(τ) = δ(τ). Find hto minimize the mean square error E[(Xt−b
Xt)2], where b
X=h∗Z.
(d) Find the mean square error for the estimator of part (c).
3. Interpolating a Gauss Markov process
Let Xbe a real-valued, mean zero stationary Gaussian process with RX(τ) = e−|τ|. Let a > 0.
Suppose X0is estimated by b
X0=c1X−a+c2Xawhere the constants c1and c2are chosen to
minimize the mean square error (MSE).
(a) Use the orthogonality principle to find c1,c2, and the resulting minimum MSE, E[(X0−b
X0)2].
(Your answers should depend only on a.)
(b) Use the orthogonality principle again to show that b
X0as defined above is the minimum MSE
estimator of X0given (Xs:|s| ≥ a). (This implies that Xhas a two-sided Markov property.)
4. Filtering a WSS signal plus noise
Suppose Xand Nare jointly WSS, mean zero, continuous time random processes with RXN ≡0.
The processes are the inputs to a system with the block diagram shown, for some transfer functions
K1(ω) and K2(ω):
1
