EECS 451______________________PROBLEM SET #9_____________________Fall 2009

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ASSIGNED: Nov. 19, 2009. 1996 Text: Sections 08.3 (IIR design); 10.1-4 (multirate filtering).
DUE DATE: Dec. 03, 2009. 2007 Text: Sections 10.3 (IIR design); 11.1-4 (multirate filtering).

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IIR filter design by transformation of analog filters:
1. [20] (4@5) From EECS 216, integrator y(t)=∫^tx(v)dv has cont.-time transfer function H_a(s)=1/s.
   1. Compute the gain of an analog integrator (using H_a(s)=1/s). What are H(0) and H(∞)?
   2. Use bilinear transform with T=2 to design a digital integrator. Write difference equation.
   3. Compute the gain of a digital integrator (using H(z) of (b)). What are H(0) and H(π)?
   4. Compare your answers to (a) & (c). For what range of frequencies do these agree closely?

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2. [25] (5@5) From EECS 216, a 1-pole filter has continuous-time transfer function H_a(s)=a/(s+a).
   1. Compute the gain of analog 1-pole filter (using H_a(s)=a/(s+a)). What are H(0) and H(∞)?
   2. Use bilinear transform with T=2 to design a digital 1-pole filter. It will also have a zero.
   3. Compute the gain of digital 1-pole filter (using H(z) of (b)). What are H(0) and H(π)?
   4. Compare your answers to (a) & (c). What kind of filter are both of these filters?
   5. Prewarp "a" and use bilinear transform to design a digital 1-pole filter with its pole at -a.
      Recall "prewarp" means use Ω=(2/T)tan(ω/2) and solve for Ω if ω=a.

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3. [20] FIR filter design of digital differentiators by windowing and frequency sampling
   
   Use subplot to put the 4 plots for #3 and #5 on a single page. Also turn in Matlab code.
   1. [10] (2@5) Use (Hamming) window method to design a 21-tap differentiator from H_a(s)=s.
      Plot magnitude response (gain) of resulting digital filter. Compare to ideal differentiator.
   2. [10] (2@5) REPEAT #3 using frequency sampling (so you only need real(ifft) here).
      Plot magnitude response (gain) of resulting digital filter. Compare answers to (a) and (b).

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4. [4] Use firpm to design a causal linear phase FIR digital differentiator of length 60.
   It should pass up to 0.2π, have transition band 0.2π to 0.3π, and reject above 0.3π.
   Confirm you get the same coefficients listed on p.654[1996] & plotted on p.694[2007].
   All you need to give for this problem is a single long Matlab command and its output.
   Isn't it nice knowing that you got the correct answer? Gives you a warm feeling inside...

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5. [15] (3@5) Define y(n) in Matlab as: filter(1,[1 0 0 0 0 0 0 0 0.99],cos([1:1024].^2/1000));
   Pretend you don't know where this signal came from (as if you had downloaded it).
   (I was going to make you download it, but then I decided to make it easier for you.)
   1. Plot y(n) and its spectrum. You should note some periodic spikes in the spectrum.
   2. From the locations of those spikes, discern how the original signal was filtered.
   3. Determine the inverse filter, and filter y(n) to recover the original signal x(n).

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6. [16] (8@2) So what is the original signal x(n) anyway? Its spectrum wasn't much help.
   Segment (chop up) x(n) into 8 segments each having length 128 [x(1:128), x(129:256), etc].
   Compute the 128-point DFT of each segment, using a Hamming window. Plot the 1^st 64 points.
   Explain what the signal is. Use subplot to put the 8 plots in a 3× 3 array on a single page.