Allpass Examples
- The simplest allpass filter is a unit-modulus gain
can be any phase value. In the real case
can only be 0 or
, in which case
.
- A lossless FIR filter can consist only of a single nonzero tap:
, where
is again some constant phase, constrained to be 0 or
in the real-filter case. Since we are considering only causal filters here,
. As a special case of this example, a unit delay
is a simple FIR allpass filter.
- The transfer function of every finite-order, causal,
lossless IIR digital filter (recursive allpass filter) can be written
as
,
We may think of
as the flip of
. For example, if
, we have
. Thus,
is obtained from
by simply reversing the order of the coefficients and conjugating them when they are complex.
- For analog filters, the general finite-order allpass
transfer function is
,
. The polynomial
can be obtained by negating every other coefficient in
, and multiplying by
. In analog, a pure delay of
seconds corresponds to the transfer function
(root of
at
), the polynomial
has a root at
. Thus, the poles and zeros can be paired off as a cascade of terms such as
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Paraunitary FiltersC.4
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Elementary Filter Problems