Allpass Filter Sections

The allpass filter passes all frequencies with equal gain. This is in contrast with a lowpass filter, which passes only low frequencies, a highpass which passes high-frequencies, and a bandpass filter which passes an interval of frequencies. An allpass filter may have any phase response. The only requirement is that its amplitude response be constant. Normally, this constant is $\left\vert H(e^{j\omega T})\right\vert=1$.

From a physical modeling point of view, a unity-gain allpass filter models a lossless system in the sense that it preserves signal energy. Specifically, if $x(n)$ denotes the input to an allpass filter $H(z)$, and if $y(n)$ denotes its output, then we have

$$\sum_{n=-\infty}^\infty \left\vert x(n)\right\vert^2 = \sum_{n=-\infty}^\infty \left\vert y(n)\right\vert^2. \protect$$ (11.9)

This equation says that the total energy out equals the total energy in. No energy was created or destroyed by the filter. All an allpass filter can do is delay the sinusoidal components of a signal by differing amounts.

Appendix D proves that Eq. (10.9) holds if and only if

$$\left\vert H(e^{j\omega T})\right\vert = 1, \quad \forall \omega.$$

That is, a filter $H(z)$ is lossless if and only if it is an allpass filter having a gain of $1$ at every frequency $\omega$.