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The discussion page for this homework is [[hw5ECE438_discussion|here]]. Feel free to share your answers/thoughts/questions on [[hw5ECE438_discussion|that page]]. | The discussion page for this homework is [[hw5ECE438_discussion|here]]. Feel free to share your answers/thoughts/questions on [[hw5ECE438_discussion|that page]]. | ||
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==Question 1== | ==Question 1== | ||
Recall the expression of the Whitaker-Kotelnikov-Shannon expansion | Recall the expression of the Whitaker-Kotelnikov-Shannon expansion | ||
| − | <math>x_r(t)= \sum_{k=-\infty}^\infty x(kT) \text{ sinc } \left(\frac{t- | + | <math>x_r(t)= \sum_{k=-\infty}^\infty x(kT) \text{ sinc } \left(\frac{t-kT}{T}\right)</math> |
a) Show (mathematically) that, for any integer k, | a) Show (mathematically) that, for any integer k, | ||
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b) Illustrate graphically (i.e., sketch it for a specific signal) the relationship between the signal <math>x(t)</math> and its zero-order hold reconstruction <math>x_0(t)</math>. | b) Illustrate graphically (i.e., sketch it for a specific signal) the relationship between the signal <math>x(t)</math> and its zero-order hold reconstruction <math>x_0(t)</math>. | ||
| − | c) Under which conditions does there exist an LTI system that would output <math>x_0(t)</math> when the input is <math>x(t)\sum_{k=-\infty}^\infty \delta (t-kT)</math> | + | c) Under which conditions does there exist an LTI system that would output <math>x_0(t)</math> when the input is |
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| + | <math>x(t)\sum_{k=-\infty}^\infty \delta (t-kT)?</math> | ||
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| + | What is the unit impulse response of this system? | ||
d) True or false? If <math>x(t)</math> is band-limited, then <math>x_0(t)</math> is also band-limited. (Justify your claim.) | d) True or false? If <math>x(t)</math> is band-limited, then <math>x_0(t)</math> is also band-limited. (Justify your claim.) | ||
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==Question 3== | ==Question 3== | ||
| − | a) What is the relationship between the DT Fourier transform of x[n] and y[n]=x[ | + | a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[3n]? (Give the mathematical relation and sketch an example.) |
| − | + | b) What is the relationship between the DT Fourier transform of x[n] and that of | |
<math>z[n]=\left\{ \begin{array}{ll} | <math>z[n]=\left\{ \begin{array}{ll} | ||
| − | x[n/ | + | x[n/4],& \text{ if } n \text{ is a multiple of } 4,\\ |
0, & \text{ else}. | 0, & \text{ else}. | ||
\end{array}\right.</math> | \end{array}\right.</math> | ||
Latest revision as of 02:55, 24 September 2010
Homework 5, ECE438, Fall 2010, Prof. Boutin
Due in class, Wednesday September 29, 2010.
The discussion page for this homework is here. Feel free to share your answers/thoughts/questions on that page.
Question 1
Recall the expression of the Whitaker-Kotelnikov-Shannon expansion
$ x_r(t)= \sum_{k=-\infty}^\infty x(kT) \text{ sinc } \left(\frac{t-kT}{T}\right) $
a) Show (mathematically) that, for any integer k,
$ x_r(kT)=x(kT). $
b) Under what conditions is it true that, for any real number t,
$ x_r(t)=x(t)? $
(Justify your answer.)
Question 2
Recall the zero-order hold reconstruction you learned in ECE301. (See Sections 7.1 and 7.2 of Oppenheim-Willsky if you need to refresh your memory.)
a) Obtain the reconstruction formula corresponding to the zero-order hold reconstruction $ x_0(t) $. (Show mathematically how to obtain this formula).
b) Illustrate graphically (i.e., sketch it for a specific signal) the relationship between the signal $ x(t) $ and its zero-order hold reconstruction $ x_0(t) $.
c) Under which conditions does there exist an LTI system that would output $ x_0(t) $ when the input is
$ x(t)\sum_{k=-\infty}^\infty \delta (t-kT)? $
What is the unit impulse response of this system?
d) True or false? If $ x(t) $ is band-limited, then $ x_0(t) $ is also band-limited. (Justify your claim.)
Question 3
a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[3n]? (Give the mathematical relation and sketch an example.)
b) What is the relationship between the DT Fourier transform of x[n] and that of
$ z[n]=\left\{ \begin{array}{ll} x[n/4],& \text{ if } n \text{ is a multiple of } 4,\\ 0, & \text{ else}. \end{array}\right. $
(Give the mathematical relation and sketch an example.)
