EECS 451______________________PROBLEM SET #1______________________Fall 2009
ASSIGNED: Sep. 10, 2009. 1996 Text: Chapter 1 and Sections 2.1-2.3 on basic signals.
DUE DATE: Sep. 17, 2009. 2007 Text: Chapter 1 and Sections 2.1-2.3 on basic signals.
THIS WEEK: Periodic signals; Sampling theorem; aliasing.
    Review of cont.-time Fourier transform computation:
  1. [15] Compute the Fourier transform of signal shown at bottom.
    Use linearity and time delays. Simplify to sines and cosines.
    Your expression should be entirely real (how can you tell?).
    Periodic signals and periods:
  2. [15] (5@3) Determine the fundamental period (which may be infinity) of:
    (a) x(t)=3cos(5t+π/6) (b) x(n)=3cos(5n+π/6) (c) x(n)=2exp[j(n/6-π)]
    (d) x(n)=cos(n/8)cos(πn/8) (e) x(n)=cos(πn/2)-sin(πn/8)+3cos(πn/4+π/3)
    [1996 & 2007 Text #1.3]. Watch t vs. n and presence or absence of π.
  3. [5] Show that the fundamental period of exp(j2πkn/N) is
    N/(greatest common divisor of N and k).
    [1996 & 2007 Text #1.4a]. Np is period of just sk, not ALL of the sk.
    Sampling theorem, sampling and aliasing:
  4. [20] (4@5) x(t)=sin(480πt)+3sin(720πt) is sampled 600 times/second. Find:
    (a) Nyquist rate. (b) Folding frequency. (c) Discrete frequencies.
    (d) x(t)→IDEAL A/D→x(n)→IDEAL D/A→y(t). Compute y(t).
    [1996 & 2007 Text #1.9]. Draw pictures of the spectra; note aliasing.
  5. [15] (3@5) x(t)=3cos(600πt)+2cos(1800πt) is quantized using 10 bits.
    The sampled & quantized signal is transmitted at 10000 bits/second.
    Note: 10 bits=1024 levels and bit rate determines sampling rate.
    (a) Compute sampling and folding frequencies. (b) Compute Nyquist rate.
    (c) Compute discrete frequencies. Is the sampled & quantized signal aliased?
    [1996 & 2007 Text #1.10abc]. Note 1024 levels=10 bits and aliasing.
  6. [15] Compute output if x(t)=3cos(100πt)+2sin(250πt) is:
    (i) Sampled every 5 ms; (ii) reconstructed using 1 ms;
    (iii) Ideal-lowpass-filtered with cutoff frequency=(sampling frequency)/2.
    [1996 & 2007 Text #1.11]. Note different sampling rates of input and output.
    No antialias filter; the reconstruction lowpass filter has cutoff at 500 Hertz.
  7. [15] To see a more realistic example of aliasing on a real-life signal:
      Run the bare-bones Matlab program below (label the plots better).
    1. [5] Write the lyrics on top of the signal parts corresponding to them.
      Use sound(Y,8192) to listen to it. Use headphones if in a CAEN lab!
    2. [5] Explain how Z (interpolated signal) differs from Y (original signal)
      aside from the low-pass filtering. (Look carefully at the two spectra.)
    3. [5] Turn in an original printout of the figure.
      fft computes Fourier transform of periodic extension of signal.
      interpft interpolates downsampled signal with sincs (lowpass filter). 
      clear; load handel; Y=y(1:65536); 
subplot(3,2,1),plot(Y),axis tight
subplot(3,2,2),plot(abs(fft(Y))),axis tight
X=Y(1:2:length(Y)); subplot(3,2,3),plot(X),axis tight
subplot(3,2,4),plot(abs(fft(X))),axis tight
Z=interpft(X,length(Y));subplot(3,2,5),plot(Z),axis tight
subplot(3,2,6),plot(abs(fft(Z))),axis tight