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?
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)
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 $.
Q4.
Q5.
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