Prof. Rich Kozick

ELEC 320, Fall 1997

## Homework 20

**Date Assigned: **Monday, October 27, 1997

**Date Due: **Friday, October 31, 1997

**Quiz 4** will be on Monday, November 3, 1997.
**Reading:** Chapter 4 and Section 5.4 of Chapter 5.
**Lab Project:** Please submit a description of your lab project to me by email by Friday,October 31 at 9 AM. You will work on the projects next week.
- Find the steady-state output
*y(t)* of the RC circuit below when the input is

*x(t) = 1 + 0.5 cos (1000 t) + 0.25 cos (2000 t)*

*
*- A (low-speed) 100 bits/sec modem operates as follows. Bits are transmitted by sending for a "1" and for a "0", where for and for other values of
*t*. That is, consists of 10 cycles of a 1000 Hz sine wave, so it lasts for 0.01 seconds. A sequence of bits is encoded as the signal where *n* is an integer and each is either +1 or -1.

The receiver uses a matched filter that integrates over 0.01 seconds:

and whose output is sampled every 0.01 seconds.

What is the matched filter output when a "1" is transmitted and when a "0" is transmitted?

Now suppose that we want to send 200 bits/sec. We can define another pulse

for and for other values of *t*. Then a different bit sequence can be transmitted at the same time as using the signal

How should *x(t)* be processed to recover both bit sequences and ? Can you define a second matched filter that will detect ? Explain your reasoning.

Hint: What is ?
- Use the table of Fourier transform pairs and theFourier transform properties to answer items 6-9. Do problem 4.12(a), and find the Fourier transform of the signals
*x(t)* and *y(t)* shown below.

- 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 seconds.

(a) For sec, sketch *x(t)*, then find and plot .

(b) Repeat part (a) for .

(c) Which case of signal would you call "narrowband", and which is "wideband"?

You might want to use MATLAB to produce the plots. An example is on page 164 of the text.
- Problem 5.28. Hints: and .
- Suppose we model a telephone channel as an ideal low-pass filter with cutoff frequency 4000 Hz, as shown in the frequency response shown below.

(a) What is the *impulse response* of this filter?

Is this a *causal* system?

(b) What is the filter output *y(t)* when the input is

? Find and sketch *y(t)*.

**Extra credit:** What is the filter output *y(t)* when the input is a rectangular pulse
Thank you.