ELEC 320, Fall 2006
Prof. Rich Kozick

## Homework 13

Date Assigned: Monday, November 20, 2006
Date Due: Friday, December 1, 2006

1. Reading: Please continue to study Chapter 4 in the Lathi text on the Fourier transform.  For next week, please study Section 5.1 on the sampling theorem.  You may also want to browse sections 5.2 and 5.3 on the DFT and FFT (we discussed these in lab).
2. The signal transmitted in an ultrasound imaging system is typically a "sine wave burst" of the form  . This is a 1 MHz cosine that is "on" for T seconds.
(a) For T=10-6 sec, sketch x(t), then find and plot .
(b) Repeat part (a) for T=10-5.
(c) Which case of signal would you call "narrowband", and which is "wideband"?  Look at the signal bandwidth relative to the center frequency, 1 MHz. Can you see why these names describe the "bandwidth" of the signal?

You may want to use MATLAB to produce the plots.

3. Please solve the following problems in the Lathi text:
Problems 4.3-7(a), 4.4-3, 4.6-2, 4.7-1 (a and b), and 4.7-3.

4. Suppose we model a telephone channel as an ideal low-pass filter with cutoff frequency 4000 Hz, with frequency response .

(a) What is the impulse response, h(t), of this filter?  Write the formula for h(t), and sketch h(t).  Is this a causal system?

(b) What is the filter output, y(t), when the input is ? Sketch x(t) and y(t).

Extra credit: What is the filter output y(t) when the input is a rectangular pulse, ?

Thank you, and have a Happy Thanksgiving!