ELG5377: Adaptive Signal Processing
(Fall 2008)
Final Exam
Instructor: Dr. Claude D’Amours
Date: Dec 12, 2008
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Duration: 3 hours
Open book examination. Non-programmable calculators only are allowed.
All questions must be answered in the booklets provided.
Be sure to write your name and student number on this question paper
Please include final answers as well as any required computations. All symbols used must be defined
(except when they are given in the problem statement).
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Question 1
(10 points = 4+6)
An M tap adaptive uses the LMS algorithm to update its tap weights. The tap input vector u(n) = [u(n), u(n1), …, u(n-M+1)]T and the filter weight vector is w(n) = [w0(n), w1(n), …, wM-1(n)]T. Answer the following
questions:
(a) If we use  = 2/E[|u(n)|2], will the algorithm converge? Justify your answer.
(b) Suppose that max of R = E[u(n)uH(n)] is much greater than all other eigenvalues of R. Assuming that
the value of  used in (a) leads to convergence of the algorithm, discuss the use of this value of  on
the misadjustment and mean square deviation of the algorithm. Consider both the stationary and
nonstationary cases.
Question 2 (35 points: all subquestions = 5 points)
At time n, the input to a 2 tap filter is u(n) = d(n)-0.3d(n-1)+I(n)+N(n), where d(n) is the desired output, I(n)
is an interference signal and N(n) is white noise with 0 mean and variance 0.2. Let u(n) = [u(n), u(n-1)]T, and
Ru = E[u(n)uH(n)]. Let E[d(n)]= E[I(n)] = E[N(n)] = 0 and E[d(n)d*(n-i)] = 2e-i and E[I(n)I*(n-i)] = 10.25|i| for |i|<4 and 0 otherwise. Furthermore d(n), I(n) and N(n) are all drawn from mutually independent
stochastic processes.
(a) Design a two-tap Wiener filter for this system.
(b) If yo(n) is the output of your filter, what is E[|d(n)-yo(n)|2]?
(c) Suppose we select a fixed filter wx = [0.6 -0.2]T for which the output y(n) = wxHu(n). What is E[|d(n)y(n)|2] for this filter?
(d) Rather than using the Wiener filter, let us use an adaptive filter that uses the LMS algorithm. What is
the maximum value for  that we can use?
(e) Let 1 = 0.1max and 2 = 0.5max, where max is the answer found in (c). Which value of  will allow
a faster convergence? Which will provide a better misadjustment? Justify your answer.
(f) If we used the RLS algorithm with  = 0.9, will that improve convergence and/or misadjustment
compared to the LMS based filters of (e)?
(g) Let  = 0.05max. What is the mean square error between the desired output and the filter output,
J(∞)?
Question 3 (8 points)
Under what circumstances is it beneficial to use the fast block LMS algorithm instead of the conventional
LMS algorithm?
Question 4 (7 points)
Consider the correlation matrix (n) = u(n)uH(n) + I, where  is a small positive constant. Use the matrix
inversion lemma to evaluate P(n) = -1(n).
Question 5 (10 points)
The inputs to a two tap filter for n=1, 2, … 5 are u(1) = 3, u(2) = 0.5 u(3) = -1, u(4) = 0.9, u(5) = -0.8. The
desired outputs for n=2, 3, …5, are d(2) = 1, d(3) = -1, d(4) = 1 and d(5) = 1. Find the two tap Least Squares
solution to this problem as well as the minimum sum of error squares.
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