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Lossless Filter Examples
The simplest lossless filter is a unitmodulus gain

(12.82) 
where
can be any phase value. In the real case
can only be 0
or
, hence
.
A lossless FIR filter can only consist of a single nonzero tap:

(12.83) 
for some fixed integer
, where
is again some constant phase,
constrained to be 0
or
in the realfilter case.
We consider only causal filters here, so
.
Every finiteorder, singleinput, singleoutput (SISO),
lossless IIR filter (recursive allpass filter) can be written as

(12.84) 
where
,
, and
. The polynomial
can be obtained by reversing the order of the coefficients in
,
conjugating them, and multiplying by
. (The factor
above
serves to restore negative powers of
and hence causality.) Such
filters are generally called allpass filters.
The normalized DFT matrix is an
order zero paraunitary
transformation. This is because the normalized DFT matrix,
,
, where
, is a unitary matrix:

(12.85) 
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