Homework 17

Date Assigned: Wednesday, November 4, 1998
Date Due: Monday, November 9, 1998

1. Quiz 4 will be on Wednesday, November 11, 1998.

*** QUIZ 4 CHANGED TO FRIDAY, NOVEMBER 13, 1998
2. Reading: Chapter 4, Sections 4.3-4.6, and Section 5.4 of Chapter 5.
3. 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)
   
   







4. Use the table of Fourier transform pairs and the Fourier transform properties to answer items 4-7.

   Do problem 4.12(a), and find the Fourier transform of the signals x(t) and y(t) shown below.
   
   
   
   
   
   
   
   

5. 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"? Can you see why these names describe the "bandwidth" of the signal?
   
   You might want to use MATLAB to produce the plots. An example is on page 164 of the text.
6. Problem 5.28. Hints: and .
7. 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 ? Sketch y(t).
   
   Extra credit: What is the filter output y(t) when the input is a rectangular pulse ?

Thank you.