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.)
