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Kolmogorov review

Kolmogorov 1939 spectral factorization provides a highly efficient Fourier method for calculating a minimum phase time domain function with a given power spectrum. The Kolmogorov spectral factorization method is commonly referred to as the Hilbert transform method within the signal processing community [e.g. Robinson and Treitel (1980)], although they are equivalent.

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)
Since is real and even, its time domain representation is also real and even. We can therefore isolate its causal part, C(Z), and its anti-causal part, :
 (5)
Once we have C(Z), we can easily obtain B(Z) through
 (6)
To verify that B(Z) of this form does indeed satisfy equation (), consider
 (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.

Next: Multi-dimensional factorization Up: Model of stochastic oscillations Previous: Model of stochastic oscillations
Stanford Exploration Project
5/27/2001