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Begin with a causal filter response ct and its associated C(Z).
The Z-transform C(Z) is evaluated,
giving a complex value for each real .This complex value is exponentiated to get another value, say
Next, we inverse transform back to bt.
We will prove the amazing fact that bt 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 power-series definition of the exponential
function:
Next, use this series expansion to rewrite
equation ().
Each term in the infinite series corresponds to a causal response,
so the sum, bt, is causal.
The factorials in the denominators
assure us that the power series always converges,
i.e., it is finite for any finite x.
The inverse wavelet to B(Z) is also causal
because it is e-C(Z).
(Unfortunately the words ``minimum phase'' distract people
from the equivalent property of genuine interest,
that the causal wavelet has a causal inverse
so we can use feedback filters.)
Next: SUMMARY AND COMPUTATION
Up: Rickett, et al.: STANFORD
Previous: Level-phase functions
Stanford Exploration Project
7/5/1998