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CAUSALITY AND SPECTAL FACTORIZATION

Mathematics sometimes seems a mundane subject, like when it does the ``accounting'' for an engineer. Other times it brings unexpected amazing new concepts into our lives. This is the case with the study of causality and spectral factorization. There are many little-known, amazing, fundamental ideas here I would like to tell you about. We won't get to the bottom of any of them but it's fun and useful to see what they are and how to use them.

Start with an example. Consider a mechanical object. We can strain it and watch it stress or we can stress it and watch it strain. We feel knowledge of the present and past stress history is all we need to determine the present value of strain. Likewise, the converse, history of strain should tell us the stress. We could say there is a filter that takes us from stress to strain; likewise another filter takes us from strain to stress. What we have here is a pair of filters that are mutually inverse under convolution. In the Fourier domain, one is literally the inverse of the other. What is remarkable is that in the time domain, both are causal. They both vanish before zero lag $\tau=0$.

Not all causal filters have a causal inverse. The best known name for one that does is ``minimum-phase filter.'' Unfortunately, this name is not suggestive of the fundamental property of interest, ``causal with a causal (convolutional) inverse.'' I could call it CwCI. An example of a causal filter without a causal inverse is the unit delay operator -- with Z-transforms, the operator Z itself. If you delay something, you can't get it back without seeing into the future, which you are not allowed to do. Mathematically, 1/Z cannot be expressed as a polynomial (actually, a convergent infinite series) in positive powers of Z.

Physics books don't tell us where to expect to find transfer functions that are CwCI. I think I know why they don't. Any causal filter has a ``sharp edge'' at zero time lag where it switches from nonresponsiveness to responsiveness. The sharp edge might cause the spectrum to be large at infinite frequency. If so, the inverse filter is small at infinite frequency. Either way, one of the two filters is unmanageable with Fourier transform theory which (you might have noticed in the mathematical fine print) requires signals (and spectra) to have finite energy which means the function must get real small in that immense space on the t-axis and the $\omega$ axis. It is impossible for a function to be small and its inverse be small. These imponderables get more managable in the world of Time Series Analysis (discretized time axis). The spectral factorization concept Interesting questions arise when we are given a spectrum and find ourselves asking how to find a filter that has that spectrum. Is the answer unique? We'll see not. Is there always an answer that is causal? Almost always, yes. Is there always an answer that is causal with a causal inverse (CwCI)? Almost always, yes.

Let us have an example. Consider a filter like the familiar time derivative (1,-1) except let us downweight the -1 a tiny bit, say $(1,-\rho)$ where $0<<\rho<1$.Now the filter $(1,-\rho)$has a spectrum $(1-\rho Z)(1-\rho/Z)$ with autocorrelation coefficients $(-\rho, 1+\rho^2,-\rho)$ that look a lot like a second derivative, but it is a tiny bit bigger in the middle. Two different waveforms, $(1,-\rho)$ and its time reverse both have the same autocorrelation. Spectral factorization could give us both $(1,-\rho)$ and $(\rho,-1)$but we always want the one that is CwCI. The bad one is weaker on its first pulse. Its inverse is not causal. Below are two expressions for the filter inverse to $(\rho,-1)$,the first divergent (filter coefficients at infinite lag are infinitely strong), the second convergent but noncausal.
\begin{eqnarray} {1\over \rho -Z} &=& { 1\over\rho}\ ( 1 +Z/\rho +Z^2/\rho^2+ \c... ...er \rho -Z} &=& {-1\over\rho}\ ( 1 + \rho/Z + \rho^2/Z^2 + \cdots)\end{eqnarray} (14)
(15)
(Please multiply each equation by $\rho -Z$ and see it reduce to 1=1).

So we start with a power spectrum and we should find a CwCI filter with that energy spectrum. If you input to the filter an infinite sequence of random numbers (white noise) you should output something with the original power spectrum.

We easily inverse Fourier transform the square root of the power spectrum getting a symmetrical time function, but we need a function that vanishes before $\tau=0$.On the other hand, if we already had a causal filter with the correct spectrum we could manufacture many others. To do so all we need is a family of delay operators to convolve with. A pure delay filter does not change the spectrum of anything. Same for frequency-dependent delay operators. Here is an example of a frequency-dependent delay operator: First convolve with (1,2) and then deconvolve with (2,1). Both these have the same amplitude spectrum so their ratio has a unit amplitude (and nontrivial phase). If you multiply (1+2Z)/(2+Z) by its Fourier conjugate (replace Z by 1/Z) the resulting spectrum is 1 for all $\omega$.

Anything whose nature is delay is death to CwCI. The CwCI has its energy as close as possible to $\tau=0$.More formally, my first book, FGDP, proves that the CwCI filter has for all time $\tau$ more energy between t=0 and $t=\tau$than any other filter with the same spectrum.

Spectra can be factorized by an amazingly wide variety of techniques, each of which gives you a different insight into this strange beast. They can be factorized by factoring polynomials, by inserting power series into other power series, by solving least squares problems, by taking logarithms and exponentials in the Fourier domain. I've coded most of them and still find them all somewhat mysterious.

Theorems in Fourier analysis can be interpreted physically in two different ways, one as given, the other with time and frequency reversed. For example, convolution in one domain amounts to multiplication in the other. If we were to express the CwCI concept with reversed domains, instead of saying the ``energy comes as quick as possible after $\tau=0$'' we would say ``the frequency function is as close to $\omega=0$ as possible.'' In other words, it is minimally wiggly with time. Most applications of spectral factorization begin with a spectrum, a real, positive function of frequency. I once achieved minor fame by starting with a real, positive function of space, a total magnetic field $\sqrt{H_x^2 +H_z^2}$ measured along the x-axis and I reconstructed the magnetic field components Hx and Hz that were minimally wiggly in space.