Download Random Processes: ECE 534 University of Illinois at Urbana-Champaign Final Exam, Fall 2006 and more Exams Electrical and Electronics Engineering in PDF only on Docsity! 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) 1 1. Consider a random process {Z(t) : t ∈ (−∞,∞)} made up of a se- quence of pulses as in the figure below. Z0 0 Z1 Z2 t1 2 3 We model Z(t) as Z(t) = ∞∑ i=−∞ Zip(t− i) where {. . . , Z−1, Z0, Z1, . . .} 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. 2 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) = v0 and Θ = θ. 5 f) Using your result in part e), find the conditional distribution of V (0.5) given V (0) = v. g) Is V (t) a Gaussian process? Explain. 6 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). 7 10 b) Now suppose in general that M > k + 1. Find {Ck+1, . . . , CM} so that x̂k+1|M , given by (1), is indeed the MMSE linear estimate of xk+1 given {y1, . . . , yM}. You can express each Ci in terms of expectations involving xi and ỹi. Hint: use the a similar type of reasoning from part a). Remember to justify that this is indeed the MMSE linear estimator. 11 12 Figure 2 below shows one possible decoder architecture. Here G(ω) is the transfer function of a stable and causal LTI system. e(t) X̂(t) G(ω)+ + + e−jωT P (ω) R(t) Figure 2: decoder c) A plausible objective for the design of G(ω) is to minimize E [( X̂(t)−X(t) )2] . Find the stable and causal G(ω) that meets this objective. Hint: first attempt to understand what R(t) is. 15 16 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) X̂(t) H(ω) Figure 3: general decoder architecture d) Show that if P (ω) is causal, then 1 1−e−jωT P (ω) is causal. 17
