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As we learned from equation (19),
a minimumphase function is determined completely from its spectrum.
Thus its phase is determinable from its spectrum.
Likewise, we will see that, except for a scale,
the spectrum is determinable from the phase.
So far we have not discussed the fact that
spectral factorization implicitly uses Hilbert transformation.
Somehow we simply generated a phase.
To see how the phase arose,
recall equation (18) and (19):
 
(20) 
Where did come from?
We took U_{k} + i 0 to the time domain, obtaining u_{t}.
Then we multiplied u_{t} by a realvalued step function of time.
This multiplication in the time domain
is what created the phase,
because multiplication in the time domain
implies a convolution
in the frequency domain.
Recall that the Fourier transform of a realvalued step function
arises with Hilbert transform.
Multiplying in time with a step means that,
in frequency, U_{k} has been convolved
with phaseshift filter).
So U_{k} is unchanged and a phase, , has been generated.
This explanation will be somewhat clearer
if you review the Ztransform approach discussed
at the beginning of the chapter,
because there we can see both the frequency domain
and the time domain in one expression.
To illustrate different classes of discontinuity,
pulse, step, and slope,
Figure 13 shows another Hilberttransform pair.
hilb2
Figure 13
A Hilberttransform pair.
EXERCISES:

What is the meaning of minimumphase waveform if the
roles of the time and frequency domains
are interchanged?

Show how to do the inverse Hilbert transform:
given , find u.
What is the interpretation of the fact that we cannot get u_{0}?

Consider a model of a portion of the earth where x is the north
coordinate, +z represents altitude above the earth,
and magnetic bodies are distributed in the earth, creating
no component of
magnetic field in the eastwest direction.
We can show that the magnetic field h above the earth is represented by
Here F(k) is some spatial frequency spectrum.
 (a)
 By using Fourier transforms,
how do you compute h_{x}(x, 0) from
h_{z} (x, 0) and vice versa?
 (b)
 Given h_{z}(x, 0), how do you compute h_{z}(x, z)?
 (c)
 Notice that, at z = 0,
and that F(k) (k + k) is a onesided function of k.
With a total field magnetometer we observe that
What can you say about obtaining F(k) from this?
 (d)
 How unique are h_{x}(x) and h_{z}(x) if is given?

Test this idea:
write code to factor X(Z) into X(Z)=A(Z)B(Z),
where B(Z) is minimumphase and A(Z) is maximumphase.
Maximumphase means that Z^{N} A(1/Z) is minimumphase.
First compute .Then remove a linear trend in the phase of to get N.
Then split U
with its trend removed
into causal and anticausal parts
U(Z) = C^{}(1/Z) + C^{+}(Z).
Finally, form
and
.
Next: A BUTTERWORTHFILTER COOKBOOK
Up: SPECTRAL FACTORIZATION
Previous: Pathological examples
Stanford Exploration Project
10/21/1998