Given an input X(Z) and an output Y(Z), we know that the spectra of the two are the same, i.e., .The existence of an infinitude of all-pass filters tells us that an infinitude of wavelets can have the same spectrum. Wave propagation without absorption is modeled by all-pass filters. All-pass filters yield a waveform distortion that can be corrected by methods discussed in chapter .
The simplest example of an all-pass filter is the delay operator itself. Its phase as a function of is simply .
A less trivial example of phase distortion can be constructed from a single root Z_{r}, where Z_{r} is an arbitrary complex number. The ratio of any complex number to its complex conjugate, say (x+iy)/(x-iy), is of unit magnitude, because, taking and ,the ratio is .Thus, given a minimum-phase filter ,we can take its conjugate and make an all-pass filter P(Z) from the ratio .A simple case is
(45) | ||
(46) |
(47) |
disper
Figure 13 Examples of causal all-pass filters with real poles and zeros. These have high frequencies at the beginning and low frequencies at the end. |
The denominator of equation (47) tells us that we have a pole at Z_{r}. Let this location be .The numerator vanishes at
(48) |
The all-pass filter (47) outputs a complex-valued signal, however. To see real outputs, we must handle the negative frequencies in the same way as the positive ones. The filter (47) should be multiplied by another like itself but with replaced by ; i.e., with Z_{r} replaced by .The result of this procedure is shown in Figure 14.
A general form for an all-pass filter is ,where A(Z) is an arbitrary minimum-phase filter. That this form is valid can be verified by checking that .