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- The simplest lossless filter is a unit-modulus gain
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:
for some fixed integer
, where
is again some constant phase,
constrained to be 0
or
in the real-filter case.
We consider only causal filters here, so
.
- Every finite-order, single-input, single-output (SISO),
lossless IIR filter (recursive allpass filter) can be written as
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:
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Download JFB.pdf
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