Polarity preserving decon in N log N'' time

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## 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 -transforms. Given a time function its -transform is . When you identify and the -transform is clearly a Fourier series. An example of a causal function is . It is causal because for likewise, has no powers of .

We may exponentiate by a frequency domain method or a time domain method. Easiest is the frequency domain method. Write for all , then Fourier transform to time. More interesting is the time domain method. The polynomial U has no powers of . The power series for an exponential is . Inserting the polynomial for U into the power series for gives us a new polynomial (infinite series) that has no powers of . 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 be an amplitude spectrum with logarithm . The exponential is the inverse of the logarithm

 (1)

Both and are real symmetric functions of . In the time domain, corresponds to an autocorrelation. In the time domain, merely corresponds to a real symmetric function . Adding some phase function to will shift the time function , likely shifting each frequency differently.
 (2)

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

 Polarity preserving decon in N log N'' time

Next: Mostly causal decon Up: GETTING THE BEST OF Previous: GETTING THE BEST OF

2012-05-10