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.)



Back to ECE438, Fall 2010, Prof. Boutin

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett