Mathematics of Signal Representations
Math 4355 - Spring 2013 - Homework

Assignment 8, due Thursday, April 4, 2013

1. Consider the Butterworth filter with system function

             1
m (ω) = 1-+-√2-iω---ω2.
              Ω   Ω2
  1. Verify that it is a Butterworth filter by computing |m(ω)|2.
  2. The impulse response h for this (causal) filter is for t > 0 of the form
    h(t) = A(e-at - e-bt).
    Determine the (complex) constants A, a and b in terms of Ω.

2. You have obtained the sampled values {f(jΩπ)}j=-∞ for a continuous, square integrable function f which is Ω-bandlimited, Ω > 0.

To process f on your computer, you define a digital “Butterworth convolution” with (any) parameters , a and b,

        j
     ˜ ∑    -a(j-k)π∕Ω   - b(j- k)π∕Ω   kπ-
βj = A     (e          - e        )f(Ω )
      k=-∞
and reconstruct the corresponding Ω-bandlimited function
       ∑∞    sin(Ωt---jπ-)
g(t) =     βj  Ωt- jπ   .
      j=-∞

Compute the system function mΩ of an analog filter L on L2(ℝ) which vanishes outside of the bandlimits,

m Ω(ω) = 0, for all |ω| > Ω
and produces the result of the digital convolution with parameters , a, and b,
g = Lf .
Hint: Use the linearity of the sampling series and the properties of the Fourier transform on page 102.

3. If you choose a and b as in the first homework problem, which value of do you need to obtain that the resulting mΩ satisfies mΩ(0) = 1?

(If you couldn’t compute a and b, state a condition on those two parameters.)