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Begin with a causal response c_{t} and its associated C(Z).
The Ztransform C(Z) could be evaluated,
giving a complex value for each real .This complex value could be exponentiated to get another value, say
 
(15) 
Next, we could inverse transform back to b_{t}.
We will prove the amazing fact that b_{t} must be causal too.
First notice that if C(Z) has no negative powers of Z,
then C(Z)^{2} does not either.
Likewise for the third power or any positive integer power,
or sum of positive integer powers.
Now recall the basic powerseries definition of the exponential
function:
 
(16) 
Including equation (15) gives the
exponential of a causal:
 
(17) 
Each term in the infinite series corresponds to a causal response,
so the sum, b_{t}, is causal.
(If you have forgotten the series for the exponential function,
then recall that the solution to dy/dx=y
is the definition of the exponential function y(x)=e^{x},
and that the power series satisfies the differential equation
term by term, so it must be the exponential too.
The factorials in the denominators
assure us that the power series always converges,
i.e., it is finite for any finite x.)
Putting one polynomial into another or one infinite series
into another is an onerous task, even if it does lead to a wavelet
that is exactly causal.
In practice we do operations that are conceptually the same,
but for speed we do them with discrete Fourier transforms.
The disadvantage is periodicity,
i.e., negative times are represented computationally like
negative frequencies.
Negative times are the last half of the elements of a vector,
so there can be some blurring of late times into negative ones.
eZ
Figure 9
Exponentials.

 
Figure 9 gives examples
of equation (17) for C=Z and C=4Z.
Unfortunately, I do not have an analytic calculation to confirm the
validity of these examples.
1
Next: Finding a causal wavelet
Up: SPECTRAL FACTORIZATION
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Stanford Exploration Project
10/21/1998