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Properties of the Maximally Flat Impulse Response After calculating (see the Appendix for this), the maximally flat impulse response is which expands in the neighborhood of t=0 to which shows quite clearly that this function has the desired properties. We will now introduce the notation which will be convenient in the following. The maximally flat impulse response is related to the Poisson distribution since Note that this means that , is strictly decreasing as a function of t. We also have hence which is a form of the incomplete gamma function. Using this form, we can compute the moments of the maximally flat impulse response as We can also compute the transfer function of the maximally flat impulse response (i.e. the Laplace transform) as

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