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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 continuous-time case, this means
where
denotes the output signal, and
denotes the
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 frequency-domain
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
![$\displaystyle \left\vert H(j\omega)\right\vert = 1, \quad \forall\, \omega\in(-\infty,\infty). \protect$](img1978.png) |
(E.13) |
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.13) implies
If
, then the allpass condition reduces to
,
which implies
where
is any real phase constant. In other words,
can be any unit-modulus 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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