Difference between revisions of "ECE438 Week11 Quiz" - Rhea

Revision as of 16:24, 3 November 2010

• Under construction --Zhao

Quiz Questions Pool for Week 11

Q1. Consider the two LTI systems, $ y[n]=T_1[x[n]] $ and $ y[n]=T_2[x[n]] $, with the following difference equations,

$ y[n]=T_1[x[n]]=x[n]-x[n-1]\,\! $

$ y[n]=T_2[x[n]]=\frac{1}{2}y[n-1]+x[n]\,\! $

Then, calculate the impulse response and difference equation of the combined system $ (T_1+T_2)[x[n]] $.

• Solution.

Q2. Consider a causal FIR filter of length M = 2 with impulse response

$ h[n]=\delta[n]-\delta[n-1]\,\! $

a) Provide a closed-form expression for the 8-pt DFT of $ h[n] $, denoted $ H_8[k] $, as a function of $ k $. Simplify as much as possible.

b) Consider the sequence $ x[n] $ of length 8 below,

$ x[n]=\text{cos}(\pi n)(u[n]-u[n-8])\,\! $

$ y_8[n] $ is formed by computing $ X_8[k] $ as an 8-pt DFT of $ x[n] $, $ H_8[k] $ as an 8-pt DFT of $ h[n] $, and then $ y_8[n] $ as the 8-pt inverse DFT of $ Y_8[k] = X_8[k]H_8[k] $.

Express the result $ y_8[n] $ as a weighted sum of finite-length sinewaves similar to how $ x[n] $ is written above.

• Solution.

Q3. Consider a causal LTI system with transfer function

$ H(z)= \frac{1-\frac{1}{2}z^{-2}} {1-\frac{1}{\sqrt{2}} z^{-1} +\frac{1}{4} z^{-2}} $

a. Sketch the locations of the poles and zeros.

b. Determine the magnitude and phase of the frequency response $ H(\omega) $, for

$ \omega =0,\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \text{ and }\pi $.

c. Is the system stable? Explain why or why not?

d. Find the difference equation for y[n] in terms of x[n], corresponding to this transfer function H(z).

• Same as HW7, Q4 available here.

Q4. Given the impulse response, compute the transfer function of filters.

$ x_1[n]=(\frac{1}{2})^nu[n]+2^nu[-n-1] $

$ x_2[n]=6(\frac{1}{2})^nu[n]-6(\frac{3}{4})^nu[n] $

Are the systems stable? Why or why not?

• Solution.

Q5. Define a two-zero band-stop filter such that

There is a gain 2 for constant input. The filter has a zero frequency response at $ \omega=\frac{\pi}{2} $. Express the system using a constant coefficient difference equation.

• Solution.

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