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[[Category:2010 Fall ECE 438 Boutin]] | [[Category:2010 Fall ECE 438 Boutin]] | ||
| + | [[Category:Problem_solving]] | ||
| + | [[Category:ECE438]] | ||
| + | [[Category:digital signal processing]] | ||
== Quiz Questions Pool for Week 11 == | == Quiz Questions Pool for Week 11 == | ||
---- | ---- | ||
| − | Q1. | + | Q1. Consider the two LTI systems, <math>y[n]=T_1[x[n]]</math> and <math>y[n]=T_2[x[n]]</math>, with the following difference equations, |
| + | :<math>y[n]=T_1[x[n]]=x[n]-x[n-1]\,\!</math> | ||
| + | :<math>y[n]=T_2[x[n]]=\frac{1}{2}y[n-1]+x[n]\,\!</math> | ||
| + | |||
| + | Then, calculate the impulse response and difference equation of the combined system <math>(T_1+T_2)[x[n]]</math>. | ||
* [[ECE438_Week11_Quiz_Q1sol|Solution]]. | * [[ECE438_Week11_Quiz_Q1sol|Solution]]. | ||
---- | ---- | ||
| − | Q2. | + | Q2. Consider a causal FIR filter of length M = 2 with impulse response |
| + | :<math>h[n]=\delta[n]-\delta[n-1]\,\!</math> | ||
| + | a) Provide a closed-form expression for the 8-pt DFT of <math>h[n]</math>, denoted <math>H_8[k]</math>, as a function of <math>k</math>. Simplify as much as possible. | ||
| + | |||
| + | b) Consider the sequence <math>x[n]</math> of length 8 below, | ||
| + | :<math>x[n]=\text{cos}(\pi n)(u[n]-u[n-8])\,\!</math> | ||
| + | <math>y_8[n]</math> is formed by computing <math>X_8[k]</math> as an 8-pt DFT of <math>x[n]</math>, <math>H_8[k]</math> as an 8-pt DFT of <math>h[n]</math>, and then <math>y_8[n]</math> as the 8-pt inverse DFT of <math>Y_8[k] = X_8[k]H_8[k]</math>. | ||
| + | |||
| + | Express the result <math>y_8[n]</math> as a weighted sum of finite-length sinewaves similar to how <math>x[n]</math> is written | ||
| + | above. | ||
* [[ECE438_Week11_Quiz_Q2sol|Solution]]. | * [[ECE438_Week11_Quiz_Q2sol|Solution]]. | ||
---- | ---- | ||
| − | Q3. | + | Q3. Consider a causal LTI system with transfer function |
| − | * [[ | + | <math> |
| + | H(z)= \frac{1-\frac{1}{2}z^{-2}} | ||
| + | {1-\frac{1}{\sqrt{2}} z^{-1} +\frac{1}{4} z^{-2}} | ||
| + | </math> | ||
| + | |||
| + | :a. Sketch the locations of the poles and zeros. | ||
| + | :b. Determine the magnitude and phase of the frequency response <math>H(\omega)</math>, for | ||
| + | |||
| + | ::<math>\omega =0,\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \text{ and }\pi</math>. | ||
| + | |||
| + | :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 [[ECE438_HW7_Solution|here]]. | ||
---- | ---- | ||
| − | Q4. | + | Q4. Given the impulse response, compute the transfer function of filters. |
| + | |||
| + | <math>x_1[n]=(\frac{1}{2})^nu[n]+2^nu[-n-1]</math> | ||
| + | |||
| + | <math>x_2[n]=6(\frac{1}{2})^nu[n]-6(\frac{3}{4})^nu[n]</math> | ||
| + | |||
| + | Are the systems stable? Why or why not? | ||
* [[ECE438_Week11_Quiz_Q4sol|Solution]]. | * [[ECE438_Week11_Quiz_Q4sol|Solution]]. | ||
---- | ---- | ||
| − | Q5. | + | 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 <math>\omega=\frac{\pi}{2}</math>. | ||
| + | Express the system using a constant coefficient difference equation. | ||
* [[ECE438_Week11_Quiz_Q5sol|Solution]]. | * [[ECE438_Week11_Quiz_Q5sol|Solution]]. | ||
Latest revision as of 09:42, 11 November 2011
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]] $.
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.
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?
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.
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