Decon in the log domain with variable gain

MINIMUM PHASE EXTENSION

A minimum phase wavelet can be made from any causal wavelet by taking it to Fourier space, and exponentiating. The proof is straightforward: Let $U(Z)= 1 + u_1 Z + u_2 Z^2 +\cdots$ be the $Z$ transform ( $Z=e^{i\omega}$ ) of any causal function $u_t$ . Consider $e^{U(Z)}$ . Although we would always do this calculation in the Fourier domain, the easy proof is in the time domain. The power series for an exponential $e^U= 1 + U + U^2/2! + U^3/3! +\cdots$ has no powers of $1/Z$ (because U has no such powers), and it always converges because of the powerful influence of the denominator factorials. Likewise $e^{-U}$ , the inverse of $e^U$ , always converges and is causal. Thus both the filter and its inverse are causal. This is the essense of minimum phase.

We seek to find two functions, one strictly causal the other strictly anticausal.

$$U^+ = u_1 Z + u_2 Z^2+ \cdots$$ (5)

$$U^- = u_{-1}/Z + u_{-2}/Z^2+ \cdots$$ (6)

Notice $U$ , $U^2$ , etc do not contain $Z^0$ . Thus the coefficient of $Z^0$ in $e^U=1+U+U^2/2!+\cdots$ is unity. Thus $a_0=b_0=1$ .

$$e^{U^+} \ =\ A = 1 + a_1 Z + a_2 Z^2 + \cdots$$ (7)

$$e^{U^-} \ =\ B = 1 + b_1 /Z + b_2 /Z^2 + \cdots$$ (8)

Define $U = U^- + U^+$ . The decon filter is $AB=e^U$ and the source waveform is its inverse $e^{-U}$ .

Consider $U(\omega)=\ln AB$ the log spectrum of the filter. We will be adjusting the various $u_t$ , all of them but not $u_0$ which is the average of the log spectrum. The other $u_t$ cannot change the average; they merely cause the log spectrum to oscillate.

Decon in the log domain with variable gain