In this section we consider filters with
constant unit spectra, that is,
.In other words, in the frequency domain *B*(*Z*)
takes the form where
is real and is called the *phase shift*.
Clearly for all real .It is an easy matter to construct a filter
with any desired phase shift;
one merely Fourier transforms
into the time domain.
If is arbitrary,
the resulting time function is likely to
be two-sided. Since we are interested in physical processes
which are causal, we may wonder what class of functions
corresponds to one-sided time functions.
The easiest way to proceed is to begin with a simple case
of a single-pole, single-zero all-pass filter.
Then more elaborate all-pass filters can be made up
by cascading these simple filters. Consider the filter

(35) |

Note that this is a simple case of functions
of the form , where *A*(*Z*) is a
polynomial of degree *N* or less. Now observe that
the spectrum of the filter *p*_{t} is indeed a
frequency-independent constant. The spectrum is

(36) |

Multiply top and bottom on the left by *Z*. We now have

(37) |

It is easy to show that
for the general form .If *Z _{0}* is chosen outside the unit circle,
then the denominator of (35) can be
expanded in positive powers of

2-17
The pole of the all-phase filter
lies outside the unit circle
and the zero is inside.
They lie on the same radius line.
Figure 17 |

From Section 2.2 (on minimum phase) we see
that the numerator of *P* is not minimum phase
and its phase is augmented by as goes from 0 to .Thus the average delay
is positive.
Not only is the average positive but, in fact,
the group delay turns out to be positive in every
frequency. To see this, first note that

(38) |

The phase of the all-pass filter (or any complex number) may be written as

(39) |

(40) |

Using (38) the group delay is now found to be

(41) |

The numerator of (41) is a positive real number (since ), and the denominator is of the form , which is a spectrum and also positive. Thus we have shown that the group delay of this causal all-pass filter is always positive.

Now if we take a filter and follow it will an all-pass filter, the phases add and the group delay of the composite filter must necessarily be greater than the group delay of original filter. By the same reasoning the minimum-phase filter must have less group delay than any other filter with the same spectrum.

In summary, a single-pole, single-zero all-pass filter
passes all frequency components with constant gain
and a phase shift which may be adjusted by the
placement of a pole. Taking *Z _{0}* near the unit circle
causes most of the phase shift to be concentrated
near the frequency where the pole is located.
Taking the pole further away causes the delay to be
spread over more frequencies. Complicated phase shifts
or group delays may be built up by cascading several
single-pole filters.

- An example of an all-pass filter is the time function . Calculate a few lags of its autocorrelation by summing some infinite series.
- Sketch the amplitude, phase, and group delay of the all-pass filter where and is small. Indicate important parameters on the curve.
- Show that the coefficients of an all-pass, phase-shifting filter made by cascading with are real.
- A continuous time function is the impulse
response of a continuous-time, all-pass filter.
Describe the function in both time domain and
frequency domain. Interchange the words
*time*and*frequency*in your description of the function. What is a physical example of such a function? What happens to the statement: ``The group delay of an all-pass filter is positive."? - A graph of the group delay in equation (41) shows to be positive for all . What is the area under in the range .(HINT: This is a trick question you can solve in your head.)

10/30/1997