Multidimensional autoregression

Basic blind deconvolution

Here are the basic definitions of blind deconvolution: If a model $m_t$ (with FT $M$ ) is made of random numbers and convolved with a ``source waveform'' (having FT) $F^{-1}$ it creates data $D$ . From data $D$ you find the model $M$ by $M=FD$ . Trouble is, you typically do not know $F$ and need to estimate (guess) it hence the word ``blind.''

Suppose we have many observations or many channels of $D$ so we label them $D_j$ . We can define a model $M_j$ as

$$M_j = \frac{D_j}{ \sqrt{\sum_j D^\ast D}}$$ (27)

so blind deconvolution removes the average spectrum.

Sometimes we have only a single signal $D$ but it is quite long. Because the signal is long, the magnitude of its Fourier transform is rough, so we smooth it over frequency, and denote it thus:

$$M = \frac{D}{ \sqrt{\ll D^\ast D\gg}}$$ (28)

Smoothing the spectrum makes the time function shorter. Indeed, the amount of smoothing may be chosen by the amount of shortness wanted.

These preliminary models are the most primative forms of deconvolved data. They deal only with the amplitude spectrum. Most deconvolutions involve also the phase. The generally chosen phase is one with a causal filter. A casual filter $f_t$ (vanishes before $t=0$ ) with FT $F$ is chosen so that $M=FD$ is white. Finding this filter is a serious undertaking, normally done in a one-dimensional space. Here, taking advantage of the helix, we do it in space of any number of dimensions.

For reasons explained later, this is equivalent to minimizing the energy output of a filter beginning with a one, $(1,f_1,f_2,f_3,\cdots)$ . The inverse of this filter $1/F$ is often called ``the impulse response'', or ``the source waveform''. Whether it actually is a source waveform depends on the physical setup as well as some mathematical assumptions we will learn.

Multidimensional autoregression