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, $Z=e^{i \omega \Delta t}$ is the unit delay operator, and functions can be evaluated either in the frequency domain as functions of $\omega$, 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  

$${\bar B}(1/Z) B(Z)=S(Z).$$ (3)

Since S(Z) is a power spectrum, it is non-negative by definition for all $\omega$; 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,

$$U(\omega)=\ln \left[ S(\omega) \right].$$ (4)

Since $U(\omega)$ 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, ${\bar C}(1/Z)$:

$$U(Z)={\bar C}(1/Z) +C(Z).$$ (5)

Once we have C(Z), we can easily obtain B(Z) through

$$B(\omega)=e^{C(\omega)}.$$ (6)

To verify that B(Z) of this form does indeed satisfy equation (), consider

$$\begin{eqnarray} {\bar B}(1/Z) B(Z) & = & e^{{\bar C}(1/Z)} e^{C(Z)} \\ & = & e^{{\bar C}(1/Z) +C(Z)} \\ & = & e^{U(Z)} \\ & = & S(Z).\end{eqnarray}$$ (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.