### 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 continuous-time 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 frequency-domain 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 functionE.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 unit-modulus complex number. If , then the filter is allpass provided

Since this must hold for all , there are only two solutions:
1. and , in which case for all .
2. 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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