EE313 Linear Systems and Signals - Homework #7 Hints

• Prof. Evans' Wednesday office hours will be shifted to 7:30-9:00 pm and take place in his office of ENS 433B.

• Computing phase responses: One can compute the phase a complex number with angle in Matlab/Mathscript. The angle also works for a vector of complex values, e.g. samples of a frequency response taken at different frequencies. The angle command returns a phase between [-pi/2, pi/2], which is known as wrapped phase.

  One could compute the unwrapped phase by applying the phase command to a vector of complex values.

  Here is a simple example illustrating unwrapped phase. Consider an ideal delay. The transfer function is H[z] = z^-1. The frequency response is H[e^{j w}] = e^{-j w}. The phase is -w. If we were to plot the phase response as arctan( e^{-j w} ), then we would see jumps of discontunity at ..., -3 pi/2, -pi/2, pi/2, 3 pi/2, ... instead of a straight line.

• Problem 7.1. The Web link and Matlab code from the question are repeated here:
  • Western Musical Scale Frequencies
  • (a) Matlab code.

    load gong
    n = 1 : Fs;
    f0 = 440;
    w0 = 2*pi*f0/Fs;
    noteSound = cos(w0*n);
    sound(noteSound);
    

    The command load gong loads the samples in the gong sound to the variable y and sets the sampling rate Fs for the sound card.
    Note: For LabVIEW Mathscript users, you can replace load gong with Fs = 8192; or load a different sound file of a similar format.

    The following Matlab code assumes that the variable noteSound holds the samples of the sound and plots the magnitude of the frequency content in y frequencies -Fs/2 to Fs/2. The amplitude A is shown on a deciBel (dB) scale, i.e. 20 log₁₀ A.

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

  • (b) Matlab code to play individual notes for one second each.

    %%% Set the sampling rate, Fs, of sound card
    load gong
    
    wholeNotesWesternScale = [ 440 494 523 587 659 698 784 880 ];
    n = 1 : Fs;
    
    %%% Play one note at a time
    for f0 = wholeNotesWesternScale
      w0 = 2*pi*f0/Fs;
      noteSound = cos(w0*n);
      sound(noteSound);
    end
    

    Note: For LabVIEW Mathscript users, you can replace load gong with Fs = 8192; or load a different sound file of a similar format.

    Download homework7sound.mat.

    Here is Matlab code to load the homework7sound file, play it and plot its frequency content:

    load homework7sound.mat
    sound(homework7sound);
    y = homework7sound;
    freqPoints = (-1 + 2*cumsum(ones(1,length(y)))/length(y))*Fs/2;
    magValuesIndB = 20*log10(fftshift(abs(fft(y))));
    plot( freqPoints, magValuesIndB );
    

• Problem 7.2: Compute the Fourier series coefficients using a₀, a_n and b_n as we did on slide 11-7 (which is mistakenly numbered 19-7 but appears in the place where lecture 11 slides should be).

  The function x(t) is odd symmetric about the origin. Hence, the terms a₀ and a_n will be zero.

  Here is example Matlab code that could be used to generate x(t) and the Fourier expansion for n=1, x₁(t), for two periods. The Matlab expression rectpuls(t) is a rectangular pulse centered at the origin with non-zero extent over -0.5 ≤ t ≤ 0.5. We need two rectangular pulses, i.e. one for each period being plotted. The example code uses the hold command to superimpose plots in Matlab on the same figure.

  T0 = 2;
  f0 = 1/T0;
  w0 = 2*pi*f0;
  t = -1 : 0.01 : 3;
  x = sin(2*pi*t) .* ( rectpuls(t) + rectpuls(t - T0) );
  b1 = 1;                  %%% Use your calculated value
  x1 = b1 * sin(w0*t);
  hold on;
  plot(t, x, 'b-');        %%% Plot in blue solid lines
  plot(t, x1, 'r--');      %%% Plot in red dashed lines
  hold off;
  

  If you experience a division by zero in computing one of more the b_n terms, then you might try to simplify the expression of b_n for that value of n using L'Hopital's Rule.

• Problem 7.4. The impulse train in the time domain provides a way to express the sampling of a signal f(t) using a sampling period of T_s by multiplying f(t) and the impulse train.

  The impulse train is even symmetric about the origin. Hence, the b_n terms will be zero.

  The Fourier series expansion of a periodic signal will be exact if an infinite number of terms is used.

  Here is Matlab code to visualize what the Fourier series expansion is converging to. The code arbitrarily sets the fundamental period to 1 s. Try out N = 10 and then N = 100 and finally N = 1000.

  T0 = 1;
  f0 = 1/T0;
  w0 = 2*pi*f0;
  
  tmin = -0.5*T0;
  tmax = 0.5*T0;
  t = tmin : (tmax-tmin)/1000 : tmax;
  
  %%% Fourier series expansion
  N = 10;
  a0 = 1;
  fourierSeriesExp = a0*ones(1,length(t));
  for n = 1:N
    an = 2;
    fourierSeriesExp = fourierSeriesExp + an * cos(n*w0*t);
  end
  
  plot(t, fourierSeriesExp);
  

bevans@ece.utexas.edu