- The simplest allpass filter is a unit-modulus gain

where 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:
for some fixed integer , 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
where ,
and
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
where ,
.
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
which is infinite order. Given a pole (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
The frequency response of such a term is
which is obviously unit magnitude.

**Next Section:** Paraunitary
FiltersC.4**Previous Section:** Elementary Filter Problems