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. 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
Determine the (complex) constants A, a and b in terms of Ω.

2. You have obtained the sampled values {f()}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 Ã, a and b,

and reconstruct the corresponding Ω-bandlimited function

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

and produces the result of the digital convolution with parameters Ã, a, and b,
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 Ã 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.)