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
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:
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.