As discussed in §10.2, 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

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 [88]

where denotes the Dirac ``delta function'' or continuous
impulse function (§C.4.3). Thus, the allpass condition becomes

which implies

(C.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 (C.15) 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.