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Begin with a causal filter response *c*_{t} 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
| |
(3) |

Next, we 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 power-series definition of the exponential
function:

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(4) |

Next, use this series expansion to rewrite
equation (3).
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(5) |

Each term in the infinite series corresponds to a causal response,
so the sum, *b*_{t}, 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:** Claerbout: Factorizing cross spectra
** Previous:** Level-phase functions
Stanford Exploration Project

7/5/1998