EE313 Linear Systems and Signals - Homework #9 Hints

• Problem 9.1. Here's the Matlab code to generate and play the cosine at fixed frequency of 660 Hz:

  Fs = 8192;
  numberOfSeconds = 2;
  n = 1 : numberOfSeconds*Fs;
  f0 = 660;
  w0 = 2*pi*f0/Fs;
  noteSound = cos(w0*n);
  sound(noteSound);
  

  The following Matlab code will plot the spectrum of the signal defined by y:

  y = noteSound;
  freqPoints = (-1 + 2*cumsum(ones(1,length(y)))/length(y))*Fs/2;
  magValuesIndB = 20*log10(fftshift(abs(fft(y))));
  plot( freqPoints, magValuesIndB );
  

• Problem 9.2. This problem is an example of single sideband (SSB) amplitude modulation (AM), which is used to save transmission bandwidth.

  Tthe transmitter consists of (1) sinusoidal amplitude modulation due to multiplication by cos(2 pi f_c t) and (2) bandpass filtering represented by its frequency response |H(f)|. You can assume an amplitude of one for the bandpass filter over the passband frequencies of -f_c - f_m < f < -f_c and f_c < f < f_c + f_m. Also, you can assume that the phase is zero over the passband of the bandpass filter so that the plot of |H(f)| is also the plot of H(f).

  The communication channel, which is ideal, is represented by the arc that connects the transmitter output y_t to the receiver input x_r. That is, |H(f)| does not represent the frequency response of the communication channel as suggested in the note for problem 9.2 on the homework assignment.

  The receiver consists of sinusoidal amplitude demodulation, which consists of (1) sinusoidal amplitude modulation due to multiplication by cos(2 pi f_c t) and (2) lowpass filtering.

  In this problem, you should choose f_m so that y_f(t) = x_t(t).

  It is easier to work this problem in the Fourier domain (frequency domain) by drawing pictures. In particular, the following two Fourier transform properties will be very helpful:

  • Multiplication in the time domain is convolution in the Fourier domain.
  • Convolution in the time domain is multiplication in the Fourier domain.

bevans@ece.utexas.edu