Maximally flat low-pass transfer function synthesis using continuous and discrete filters

Discrete linear uniformly sampled digital filters are often employed in situations where there exists some preliminary linear analog continuous filtering operation. The latter continuous filter transfer function
P(s)
is usually a low-pass function that is required for such purposes as reducing the effects of aliasing, limiting the dynamic range of the signal prior to digitization, etc. The realization of
P(s)
is usually achieved quite independently of the realization of the subsequent discrete transfer function
H(z)
, where
z = e^{sT}
. In this contribution, it is shown that maximally flat solutions exist for the composite transfer function
P(s) \cdot H(z)
, so that it is possible to employ a simple baseband prefiliter
P(s)
which, combined with
H(z)
, results in an overall maximally flat low-pass response.