Difference between revisions of "ECE438 Week9 Quiz" - Rhea

Revision as of 11:13, 20 October 2010

Quiz Questions Pool for Week 9

Q1. Consider the following second order FIR filter with the two zeros on the unit circle as shown below.

The transfer function for this filter is given by $ H(z) = (1-e^{j\theta}z^{-1})(1-e^{-j\theta}z^{-1})=1-2\cos\theta z^{-1}+z^{-2} $

a. Find the difference equation of this filter.

b. Find the frequency response $ H(w) $ from the difference equation by the following two approaches:

i. apply the input $ e^{jwn} $ to the difference equation describing the system,

ii. find the DTFT of the impulse response,

and verify that both methods lead to the same result.

c. Find the response of this system to the input

$ x[n]=\left\{ \begin{array}{rl} 1, & \text{ if }n=-1,\\ 1, & \text{ if }n=0,\\ 0, & \text{ else. } \end{array} \right. $

d. When $ \theta=\pi/2 $, is this filter a lowpass, highpass, bandpass or a bandstop filter?

e. An interference signal modulated at 2kHz and sampled at 8kHz is being inputted to this system and you want to eliminate this interference. What must be the value of $ \theta $ to eliminate this signal?

• Solution.

Q2. When we have a LTI system, the impulse response $ h[n] $ must be real

in order for $ y[n] $ to be real whenever $ x[n] $ is real.

The condition for $ h[n] $ to be real is

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

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

(Hint: Apply DTFT to the above equation)

• Solution.

Q3.

Consider a DT LTI system described by the following equation

$ y[n] = x[n] + 2x[n - 1] + 0.5y[n - 1] $

a. Compute the impulse response h[n] of the system.
b. Compute the output when x[n] = u[n].
c. Compute the output when $ x[n] = 0.25^nu[n] $.

• Solution.

Q4.

• Solution.

Q5.

• Solution.

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