Homework Assignment 14
Due Dec. 8, 2004

1. Problem 7.15 (a) (b). Do these by hand.
2. Problem 7.15 (a) (b) (c) (d). Do these by ifft(fft(x).*fft(v)). Show that (a) and (b) agree with the hand calculations.
3. Problem 7.21. Do signals (i), (ii), (iii), (vi)
4. Problem 7.29 (a) (b) (c)
5. Download the sound file falling.wav. Read the file into MATLAB with the command
   x = wavread('falling.wav');
   The data is recorded at an 8 kHz rate. Make a time vector t with the command
   t = (0:length(x)-1)/8000;
   1. Plot the sound signal x vs the time vector t.
   2. As in problem 7.21 (a) of the text, find the unit pulse response h_lp[n] of an ideal lowpass filter which has a cutoff frequency of 400 Hz. (First find the discrete-time frequency which corresponds to 400 Hz. Note the the discrete time frequency 2 π corresponds to the continuous time sampling frequency 8 kHz.)
   3. Let hd_lp be the sequence h_lp[n] for n = -100 to 100. Use a stem plot to plot hd_lp.
   4. Let N = length(x) + length(hd). Plot the magnitude of the N-point DFT of hd_lp vs. continuous-time frequency. Verify that this filter will pass low frequencies and block high frequencies.
   5. To find the output of the filter, let x_lp be the linear convolution of x and hd_lp. (To do the linear convolution in MATLAB, take the inverse DFT of the N-point DFT of x times the N-point DFT of hd_lp.)
   6. Take the N-point DFT of x using the MATLAB FFT command. Plot the magnitude of the DFT vs. the continuous time frequency.
   7. Plot the magnitude of the N-point DFT of x_lp. Verify that the low frequencies are gone.
   8. Repeat (b), (c), (d), (e) and (g) to find x_hp using a high-pass filter with a cutoff frequency of 400 Hz. Note: Once you have h_lp[n], it is very easy to find h_hp[n]. Just use the fact that H_hp(Ω) = 1 - H_lp(Ω).
   9. Save your output sounds using the MATLAB wavewrite command:
      wavwrite(xlp,'falling_low.wav');
      wavwrite(xhp,'falling_high.wav');
      Listen to the three sound files. Did the filters separate the low-frequency sounds from the high-frequency sounds?
   10. If you want to, listen to the effects of the above filters on some of your favorite music.