EECS 451______________________PROBLEM SET #5______________________Fall 2009

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ASSIGNED: Oct. 15, 2009. 1996 Text: Sections 4.3-4.4.5.
DUE DATE: Oct. 22, 2009. 2007 Text: Sections 4.4,5.1-2.

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THIS WEEK: Basics of Discrete-Time Fourier Transform (DTFT)

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Discrete-Time Fourier Series (DTFS) (for periodic discrete-time signals)
1. [15] x(n)={...1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2...} has period=6 and x(0)=3. Compute:
   (a) (6@2) DTFS of x(n). (b) [3] Average power in time and frequency domains.
   [1996 & 2007 Text: #4.4]. Just compute the DTFS; don't sketch or plot it or x(n).
   x(n) real and even implies DTFS coefficients c_k real and even. Use e^jω+e^-jω=2cos(ω).

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2. [15] (16@1) Compute x(n) having period=8 and DTFS coefficients (also period=8):
   (a) c_k=cos(πk/4)+sin(3πk/4) (b) Skip. (c) c_k={...0,¼,½,1,2,1,½,¼,0...} with c₀=2.
   [1996 & 2007 Text: #4.7ac]. How can you tell x(n) to part (a) is complex-valued?
   HINTS: (a) Do cos (even) and sin (odd) parts separately. (c) Use e^jω+e^-jω=2cos(ω).
   

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   Discrete-Time Fourier Transform (DTFT) (for non-periodic discrete-time signals)
3. [20] (4@5) DTFTs: (a) u(n)-u(n-6) (b) (2)ⁿu(-n) (c) (¼)ⁿu(n+4) (d) [αⁿsin(ωn)]u(n).
   [1996 & 2007 Text: #4.9a-d]. DTFT drill. Don't sketch. Try z-transforms and z=e^jω.
   For (d): Use Matlab to plot |X(ω)| for 0 ≤ ω ≤ 2π and α=0.99 and ω=1. Use subplot.

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4. [20] (5+5+10) Inverse DTFTs of these 4 periodic (with period=2π) functions X(ω):
   (a) X(ω)=1 for ω_o < |ω| ≤ π and X(ω)=0 for 0 ≤ |ω| ≤ ω_o; (b) X(ω)=cos²(ω);
   (c) X(ω)=1 for |ω±ω_o| < δω/2 and X(ω)=0 for |ω±ω_o| > δω/2 for constants ω_o,δω.
   [1996 & 2007 Text: #4.10a-c]. Inverse DTFT drill. Note for each part:
   1. High-pass filter; write as X(ω)=1-(low-pass filter) so x(n)=impulse-sinc.
   2. Use cos²x=(1+cos(2x))/2, set e^jω=z and use inverse z-transform.
   3. Modulation principle of the DTFT on bandpass filter; note real and even.

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5. [15] (5@3) x(n)={-1,2,-3,2,-1} has duration=5 and x(0)=-3. JUST using DTFT
   properties, WITHOUT explicitly computing X(ω), compute the following:
   (a) X(0); (b) Phase[X(ω)]; (c) ∫_-π^πX(ω)dω; (d) X(π); (e) ∫_-π^π|X(ω)|²dω;
   (b) has one of two values; answer should be "? or ?" for 2 numbers "?" and "?."
   [1996 & 2007 Text: #4.14]. Can do VERY quickly (relief after doing #3 and #4, I'll bet!)

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6. [15] A voice scrambler (the digital version of a technique used in World War II): Run:
   1. load handel;X=y(27001:35192);FX=fft(X);FY=fftshift(FX);Y=real(ifft(FY));
      subplot(3,2,3),plot(X);subplot(3,2,4),plot(abs(FX));
      subplot(3,2,5),plot(Y);subplot(3,2,6),plot(abs(FY))
      Plot #3d using subplot(3,2,1) in the same figure (total of 5 plots on 1 page).
   2. Explain how DTFT of Y differs from DTFT of X: what have we done?
   3. Listen to Y using sound(y). Can you tell what X was from its scrambled Y?
   4. Explain how to get Y directly and easily from X. HINT: DTFT modulation property.
   5. Explain how to recover original signal X directly and easily from scrambled signal Y.

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