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.)

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