A new approach to causal filter design by Padé approximants

In recursive digital filter design, the only linear technique available is probably the method of Padé approximants. Unfortunately, to obtain a Padé approximant, a formal power (Maclaurin) series must be given. If an ideal amplitude response
|H(e^{j\omega})|
is given, the usual method is to approximate its truncated delayed Fourier series,
H_{N}(e^{j\omega}) = \Sigma\min{0}\max{2N}h_{-N+k}e^{-jk\omega}
. This procedure is not desirable especially when the Padé approximant method is applied, since the first few terms in the power series (that is,
h_{-N}, h_{-N+1}
, ... in H
N
) play the most important role in the characteristics of its Padé approximants. In this paper, we apply the idea of Hilbert transformations to obtain a complete complex frequency response H(e^{j\omega}) whose Fourier expansion gives rise to a power (Maclaurin) series. A method is given to compute this series, so that the Padé approximant technique can be applied readily.