More General Allpass Filters

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More General Allpass Filters

We have so far seen two types of allpass filters:

The series combination of feedback and feedforward comb-filters is allpass when their delay lines are the same length and their feedback and feedforward coefficents are the same. An example is shown in Fig.1.24.
Any delay element in an allpass filter can be replaced by an allpass filter to obtain a new (typically higher order) allpass filter. The special case of nested first-order allpass filters yielded the lattice digital filter structure of Fig.1.26.

We now discuss allpass filters more generally in the SISO case. (See Appendix D of [460] for the MIMO case.)

Definition: A linear, time-invariant filter is said to be lossless if it preserves signal energy for every input signal. That is, if the input signal is , and the output signal is $y(n) = (h\ast x)(n)$ , we must have

$\displaystyle \sum_{n=-\infty}^{\infty} \left\vert y(n)\right\vert^2 = \sum_{n=-\infty}^{\infty} \left\vert x(n)\right\vert^2.$

In terms of the

signal norm $\left\Vert\,\,\cdot\,\,\right\Vert _2$ , this can be expressed more succinctly as

$\displaystyle \left\Vert\,y\,\right\Vert _2^2 = \left\Vert\,x\,\right\Vert _2^2.$

Notice that only stable filters can be lossless since, otherwise, $\left\Vert\,y\,\right\Vert$ is generally infinite, even when $\left\Vert\,x\,\right\Vert$ is finite. We further assume all filters are causal^2.11 for simplicity. It is straightforward to show the following:

It can be shown [460, Appendix D] that stable, linear, time-invariant (

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Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Allpass Filters
          More General Allpass Filters