To obtain a causal FIR filter, we will expand
in powers of
instead of
.
For simplicity of notation, define
. Then we obtain the Taylor series expansion
of
about
to be
where the derivatives are with respect to
, e.g.,
.
Our maximally flat
th-order Langrange FIR interpolation filter is obtain by truncating this expansion at order
:
where
This can be viewed as an FIR filter structure in which the usual delay elements are replaced by