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University of Illinois at Urbana-Champaign
ECE 534: RANDOM PROCESSES
Fall 2006
Final Exam
Thursday, December 14, 2006
Name:
- This is a closed-book exam. You may consult both sides of three sheets of notes, typed in font size 10 or equivalent handwriting size.
- Calculators, laptop computers, Palm Pilots, two-way email pagers, etc. may not be used.
- Write your answers in the space provided.
- Please show all of your work. Answers without appropriate justification will receive very little credit.
Score: 1: (20 points) 2: (10 points) 3: (20 points) Total: (50 points)
- Consider a random process {Z(t) : t ∈ (−∞, ∞)} made up of a se- quence of pulses as in the figure below. Z 0
Z 1
Z 2
1 2 3 t
We model Z(t) as
Z(t) =
∑^ ∞
i=−∞
Zip(t − i)
where {... , Z− 1 , Z 0 , Z 1 ,.. .} is a sequence of i.i.d. N (0, 1) random variables and p(t) = 1 for t ∈ [0, 1) and 0 elsewhere.
a) Is Z(t) a Gaussian process? Explain.
Now suppose that we have a random process {X(t) : t ∈ (−∞, ∞)} that is input to the channel and let Y (t) = X(t) + Z(t) be the output. Suppose that X(t) is stationary and a.s. ergodic.
d) Select a linear (possibly time-varying) filter h(t, τ ) so that by shov- ing Y (t) through the filter, the mean of X(t) can be recovered with probability one.
Now suppose that Z(t) is modified by introducing a random time lag Θ which is uniformly distributed on [0, 1] and is independent of all the {Zi}. In other words, the new process V (t) is defined as
V (t) =
∑^ ∞
i=−∞
Zip(t − i − Θ).
e) Find the conditional distribution of V (0.5) given that V (0) = v 0 and Θ = θ.
h) Consider the random variable T (t, t′) given by
T (t, t′) =
∑^ ∞
i=−∞
p(t − i − Θ)p(t′^ − i − Θ).
Calculate P (T (t, t′) = 1).
i) Calculate the covariance function for V (t).
j) Is V (t) WSS?
b) Now suppose in general that M > k + 1. Find {Ck+1,... , CM } so that ˆxk+1|M , given by (1), is indeed the MMSE linear estimate of xk+ given {y 1 ,... , yM }. You can express each Ci in terms of expectations involving xi and ˜yi. Hint: use the a similar type of reasoning from part a). Remember to justify that this is indeed the MMSE linear estimator.
- Figure 1 below shows a model for the encoder in a differential mod- ulation system. The signal of interest is X(t) and W (t) represents additive noise W (t). P (ω) is the transfer function of a stable and causal LTI system.
X(t) V (t) e(t)
W (t)
+^ +
+^ + −
e−jωT^ P (ω)
Figure 1: encoder
We model X(t) and W (t) as uncorrelated zero-mean WSS random pro- cesses. Suppose the power spectral densities of X(t) and V (t) = X(t) + W (t) are given by
SX (ω) =
4 + ω^2 (9 + ω^2 )(25 + ω^2 )
SV (ω) =
16 + ω^2 (36 + ω^2 )(49 + ω^2 )
a) Find an anticausal and invertible whitening filter A(ω) for V whose inverse is causal.
The structure of this encoder allows for more efficient signal representa- tions in terms of e(t) as compared to X(t). Hence, a plausible objective of the design of P (ω) is to reduce the energy in the encoded signal e(t).
b) Find the stable and causal system function P (ω) that minimizes E[e^2 (t)].
More generally, our decoder architecture could have the form as shown in Figure 3, where H(ω) is the transfer function of another stable, causal LTI system.
e(t) Xˆ(t) H(ω)
Figure 3: general decoder architecture
d) Show that if P (ω) is causal, then (^1) −e−jωT^1 P (ω) is causal.
