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First Exam, 2nd semester, 2003-2004 (January 8, 2004)
1. Find 5 possible values for a sampling frequency Fs such that the analog signals
cos( 2 p · 6500 Hz · t) and cos( 2 p · 8000 Hz · t)
are aliases of each other when sampled at Fs. Also find the largest possible value for Fs.
2. Use z-transforms to determine the impulse response of the system described by
y(n) = 0.7y(n-1) - 0.1y(n-2) + 2x(n) - x(n-2).
3. Sketch direct form I and direct form II realizations for the causal linear time-invariant system described by
y(n) = 0.7y(n-1) - 0.1y(n-2) + 2x(n) - x(n-2).
4. A relaxed causal linear time-invariant system with impulse response { 0.5, 1, -1, -0.5} is given an input
{ 1, 1.5, 0.5, -1, -0.5, 1, 2, 0, 0.5}. Determine the output.5. [10 pts] A causal linear time-invariant system has an infinite impulse response given by
h(n) = d(n) + 0.8 d(n-20) + 0.6 d(n-40) + (0.6)(0.9) d(n-46) + (0.6)(0.9)2 d(n-52) + (0.6)(0.9)3 d(n-58) + ...+ (0.6)(0.9)i d(n-40-6i) + ...
Find a difference equation describing the system.
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