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Next: Comparison With a Rectangular Up: Maximally Flat Impulse Response Previous: Maximally Flat Impulse Response

Properties of the Maximally Flat Impulse Response

After calculating tex2html_wrap_inline1178 (see the Appendix for this), the maximally flat impulse response is
eqnarray228
which expands in the neighborhood of t=0 to
eqnarray238
which shows quite clearly that this function has the desired properties.

We will now introduce the notation tex2html_wrap_inline1182 which will be convenient in the following.

The maximally flat impulse response is related to the Poisson distribution
equation247
since
equation253
Note that this means that tex2html_wrap_inline1184, tex2html_wrap_inline1186 is strictly decreasing as a function of t.

We also have
eqnarray259
hence
equation271
which is a form of the incomplete gamma function. Using this form, we can compute the moments of the maximally flat impulse response as
eqnarray280

We can also compute the transfer function of the maximally flat impulse response (i.e. the Laplace transform) as
eqnarray313



Fri Jun 27 03:10:38 EDT 1997

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