Difference between revisions of "ECE438 Week10 Quiz" - Rhea

Revision as of 13:33, 27 October 2010

Quiz Questions Pool for Week 10

Q1. Consider the following difference equation

$ y[n]=ay[n-1]+x[n]-x[n-1]\,\! $

a. Compute the transfer function $ H(z) $, and find its poles and zeros.

b. Compute the impulse response $ h[n] $ using a ROC of $ |z|>a $. For what values of $ a $ is the system stable?

c. Compute the impulse response $ h[n] $ using a ROC of $ |z|<a $. For what values of $ a $ is the system stable?

• Solution.

Q2. The condition for the discrete-time signal $ x[n] $ to be real is

$ x[n]=x^{\ast}[n] $

Then, what is the condition of the frequency response $ X(w) $ for $ x[n] $ to be real?

(Hint: Apply DTFT to the above equation)

• Solution.

Q3.

The respective difference equation for each of these two systems in parallel above may be expressed as

y1[n] = ½ x[n] + ½ x[n-1]
y2[n] = ½ x[n] - ½ x[n-1]

Consider the respective outputs of these two systems as the inputs to a pair of length two FIR filters with impulse response, g1[n] and g2[n] respectively, as shown in the diagram.

a. Given that g1[0] = ½, determine the values of g1[n] and g2[n], where n = 0, 1, such that the difference equation for the overall system is simply

y[n] = x[n-1]

That is, determine length-2 FIR filters g1[n] and g2[n] so that output is the input delayed by one (for any input).

b. Let H(\omega) denote the frequency response of the overall system equal to the DTFT of h[n] below:

h[n] = h1[n] * g1[n] + h2[n] * g2[n]

Plot both the magnitude H($ \omega $) and the phase H($ \omega $) over -$ \pi $ < $ \omega $ < $ \pi $.

• Solution.

Q4.

• Solution.

Q5.

• Solution.

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