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We have already discussed
(page )
allpass filters, i.e.,
filters with constant unit spectra.
They can be written as
.In the frequency domain, P(Z)
can be expressed as , where
is real and is called the ``phase shift."
Clearly, for all real .It is an easy matter to make a filter
with any desired phase shiftwe
merely Fourier transform
into the time domain.
If is arbitrary,
the resulting time function is likely to be twosided.
Since we are interested in physical processes that are causal,
we may wonder what class of functions
corresponds to onesided time functions.
The answer is that the group delay
of a causal allpass filter must be positive.
Proof that for a causal allpass filter
is found in FGDP;
there is no need to reproduce the algebra here.
The proof begins from equation ()
and uses the imaginary part of the logarithm to get phase.
Differentiation with respect to yields a form
that is recognizable as a spectrum and hence is always positive.
A singlepole, singlezero allpass filter
passes all frequency components with constant gain
and a phase shift that can be adjusted
by the placement of the 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 farther away causes the delay to be
spread over more frequencies.
Complicated phase shifts or group delays can be built up
by cascading singlepole filters.
The above reasoning for a singlepole, singlezero allpass filter
also applies to many roots,
because the phase of each will add, and
the sum of will be greater than zero.
The Fourier dual to the positive group delay of a causal allpass filter
is that the instantaneous frequency of a certain class of analytic
signals must be positive.
This class of analytic signals is made up of
all those with a constant envelope function,
as might be approximated by field data
after the process of automatic gain control.
EXERCISES:

Let x_{t} be some real signal.
Let y_{t} =x_{t+3}
be another real signal.
Sketch the phase as a function of frequency
of the crossspectrum X(1/Z)Y(Z) as
would a computer that put all arctangents in the principal
quadrants .Label the axis scales.

Sketch the amplitude, phase, and group delay of the allpass filter
,where and
is small.
Label important parameters on the curve.

Show that the coefficients of an allpass,
phaseshifting filter made by cascading
(Z_{0}  Z) with
are real.

A continuous signal is the impulse
response of a continuoustime, allpass filter.
Describe the function in both time and
frequency domains.
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 allpass filter is positive?

A graph of the group delay 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.)
Next: PHASE OF A MINIMUMPHASE
Up: PHASE DELAY AND GROUP
Previous: Observation of dispersive waves
Stanford Exploration Project
10/21/1998