Lossless Analog Filters
As discussed in §B.2, the an allpass filter can be defined
as any filter that preserves signal energy for every input
signal . In the continuoustime case, this means
where
denotes the output signal, and
denotes the
L2 norm of
. Using the
Rayleigh energy theorem
(
Parseval's theorem) for
Fourier transforms [
87],
energy preservation can be expressed in the
frequency domain by
where
and
denote the Fourier transforms of
and
, respectively,
and frequencydomain L2
norms are defined by
If
denotes the
impulse response of the
allpass
filter, then its
transfer function
is given by the
Laplace transform of
,
and we have the requirement
Since this equality must hold for every input signal
, it must be
true in particular for complex
sinusoidal inputs of the form
, in which case [
87]
where denotes the Dirac ``delta function'' or continuous
impulse function (§E.4.3). Thus, the allpass condition becomes
which implies

(E.15) 
Suppose
is a rational analog filter, so that
where
and
are polynomials in
:
(We have normalized so that is monic () without
loss of generality.) Equation (E.15) implies
If
, then the allpass condition reduces to
,
which implies
where
is any real phase constant. In other words,
can be any unitmodulus
complex number. If
, then the
filter is allpass provided
Since this must hold for all
, there are only two solutions:
 and , in which case
for all .

and , i.e.,
Case (1) is trivially allpass, while case (2) is the one discussed above
in the introduction to this section.
By analytic continuation, we have
If
is real, then
, and we can write
To have
, every
pole at
in
must be canceled
by a zero at
in
, which is a zero at
in
.
Thus, we have derived the simplified ``allpass rule'' for real analog
filters.
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