Mathematics of Signal Representations

Math 4355 - Spring 2013 - Homework

Assignment 8, due Thursday, April 4, 2013

1. Consider the Butterworth filter with system function

- Verify that it is a Butterworth filter by computing |m(ω)|
^{2}. - The impulse response h for this (causal) filter is for t > 0 of the form

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,

Compute the system function m_{Ω} of an analog filter L on L^{2}()
which vanishes outside of the bandlimits,

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 m_{Ω} satisfies m_{Ω}(0) = 1?

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