Next: SUMMARY AND COMPUTATION Up: Claerbout: Factorizing cross spectra Previous: Level-phase functions

The exponential of a causal is causal.

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
 (3)
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:
 (4)
Next, use this series expansion to rewrite equation (3).
 (5)
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: Claerbout: Factorizing cross spectra Previous: Level-phase functions
Stanford Exploration Project
7/5/1998