Following Claerbout 1992,
we will describe the method briefly with *Z* transform notation.
In this notation, is the unit delay
operator, and functions can be evaluated either in the frequency
domain as functions of , or in the time domain as the
coefficients of the polynomial in *Z*.
Causal functions can, therefore, be written as polynomials with
non-negative powers of *Z*, whereas anti-causal functions contain
non-positive powers of *Z*.

The spectral factorization problem can be summarized as given a power
spectrum, *S*(*Z*), we must find a minimum phase function such that

(3) |

Since *S*(*Z*) is a power spectrum, it is non-negative by definition for
all ; however, the Kolmogorov process places the additional
requirement that *S*(*Z*) contains no zeros. If this is the case, then we
can safely take its logarithm,

(4) |

(5) |

(6) |

(7) | ||

(8) | ||

(9) | ||

(10) |

*B*(*Z*) will be causal since *C*(*Z*) was causal, and a power series
expansion proves that the exponential of a causal function is also
causal. It is also clear that 1/*B*(*Z*)=*e*^{-C(Z)} will also be causal
in the time domain. Therefore, *B*(*Z*) will be causal, and will have a
causal inverse. Hence *B*(*Z*) satisfies the definition of minimum phase
given above.

5/27/2001