Polarity preserving decon in ``N log N'' time

Spectral factorization

A causal function is one that vanishes at negative time. Too short a summary is to say the exponential of a causal is a causal. What is meant is if we take the Fourier transformation of a causal function, exponentiate it, and then inverse transform we will again have a causal function. This is the heart of spectral factorization, an obscure mathematical calculation addressing interesting practical applications.

Start with $Z$ -transforms. Given a time function $(1,u_1,u_2,u_3,\cdots)$ its $Z$ -transform is $U(Z) = 1 +u_1Z +u_2Z^2 +u_3Z^3+\cdots$ . When you identify $Z=e^{i\omega\Delta t}$ and $Z^5=e^{i\omega 5\Delta t}$ the $Z$ -transform is clearly a Fourier series. An example of a causal function is $u_\tau$ . It is causal because $u_\tau=0$ for $\tau<0$ likewise, $U(Z)$ has no powers of $1/Z$ .

We may exponentiate $U(Z)$ by a frequency domain method or a time domain method. Easiest is the frequency domain method. Write $e^{U(Z(\omega))}$ for all $\omega$ , then Fourier transform to time. More interesting is the time domain method. The polynomial U has no powers of $1/Z$ . The power series for an exponential is $e^U= 1 + U + U^2/2! + U^3/3! +\cdots$ . Inserting the polynomial for U into the power series for $e^U$ gives us a new polynomial (infinite series) that has no powers of $1/Z$ . Furthermore, this new polynomial always converges because of the powerful influence of the denominator factorials. Thus we have shown that the ``exponential of a causal is a causal''.

Let $\bar{S}(Z(\omega))$ be an amplitude spectrum $\bar S(\omega)>0$ with logarithm $\bar U= \log \bar S$ . The exponential is the inverse of the logarithm

$$\bar{S} = e^{\log \bar{S}} \ =\ e^{\bar U}$$ (1)

Both $\bar S$ and $\bar U$ are real symmetric functions of $\omega$ . In the time domain, $\vert\bar S\vert^2$ corresponds to an autocorrelation. In the time domain, $\bar U$ merely corresponds to a real symmetric function $\bar u_\tau$ . Adding some phase function $\Phi(\omega)$ to $\bar U$ will shift the time function $s_\tau$ , likely shifting each frequency differently.

$$S = e^{\log \bar{S}+i\Phi} \ =\ e^{\bar U+i\Phi} \ =\ e^U$$ (2)

Keeping $\bar U$ fixed keeps the spectrum $S^\ast S$ fixed. Let $u_\tau$ now correspond to the Fourier transform of $U(\omega)=\bar U+i\Phi$ . The time symmetric part of $u_\tau$ corresponds to $\bar U(\omega)$ while the antisymmetric part of $u_\tau$ corresponds to the newly added phase $\Phi(\omega)$ . How shall we choose $\Phi(\omega)$ ? Let us choose the antisymmetric part of $u_\tau$ instead, choose it to cancel the symmetric part of $u_\tau$ on the negative $\tau$ axis. In other words, let us choose $u_\tau$ to be causal. Recalling that ``exponentials of causals are causal'' we have thus created a causal $s_\tau$ . Hooray! Hooray because $s_\tau$ has the same spectrum $\bar S$ that we started with. We started with a spectrum $\bar S$ and constructed a causal wavelet $s_\tau$ with that spectrum. Good trick! This is called ``spectral factorization.'' Causal decon is simply taking your data $D$ and dividing by a causal source waveform $S$ .

Polarity preserving decon in ``N log N'' time