Shufflebrain

Have you ever looked directly through the teeth of a comb? If possible, try it in soft candle light. Observe the halos, the diffraction of light at the slits. If you haven't done so, and you can't find a candle or a comb, try this instead. Hold the tips of your thumb and index finger up to your eye and bring them together until they nearly touch. You should be able to see the halos overlap and occlude the slit just a tiny bit before your finger and thumb actually touch. Those halos are physical analogs of Fourier and Laplace transforms. In principle, the edges of your fingers do to the light wave squeezing through the slit what the Fourier analyst does with numerical values: execute a transformation from perceptual to transform space. What is transformed? The answer: the image the light waves would have carried to your eyes if the halos hadn't mutually transformed each other.

If the transform exists, it can be shown the transform space containing it necessarily exists. I say this not to propound a principle but to give the reader an impressionistic awareness of transform space. We have to intuit the ideal domain much as we would surmise that a sea is deep because a gigantic whale suddenly burst upon its surface. At the same time, our own finger tips give us a sense of reality.

Now I will put forth a principle. Although we cannot literally visualize the interior of a transform space, we can grasp the logical interplay of transformed entities. And sometimes, with the correct choice of a specific transform space, we can greatly simplify the meaning of an otherwise arcane idea.

Let me demonstrate the latter point by introducing a process called convolution, whose ramifications and underlying theorem (convolution theorem) we will soon be calling upon.

Convolution refers to the superimposition of independent sets, planes or magnitudes. Consider two initially separated set of dots, set A and set B, in perceptual space.[2] In the figure, A's dots line up on the horizontal axis at intervals of 5 units, say inches. Those of B are 2 inches apart and run obliquely up from left to right.

If we convolute A and B, we create a two-dimensional lattice. Using an asterisk to indicate the convolution operation, we can define the lattice as A*B, which we would read. "A convoluted on B." But what do we mean by convolute? And just how does convolution produce a lattice? The answer is rather complicated in perceptual space. But it's simple in a specific transform space, called reciprocal space (1/space).

We create 1/space from Fourier transforms. The Fourier transform of a line of dots is a grating; that is, a series of uniform lines. In 1/space the transform of A, let's call it T(A), is a grating made up of vertical lines whose spacings are the reciprocal of 5 inches--1/5 or 0.2 inches. The transform of B, T(B), is an array of oblique lines running downward from left to right, with 1/2 or 0.5 inch spacings (the reciprocal of 2 inches). We can superimpose the planes in T(A) and T(B). Because the transforms are lines, we can see directly that the superimposition of the two sets creates a grid, something we could not have envisaged with dots. The abstract operation corresponding to our superimposition is the same thing as uniting height and width to produce the area of a rectangle. This latter operation is multiplication. (A 2 inch vertical line and a 5 inch horizontal line produce a 2 by 5 rectangle.) And we can define the grid in transform space simply as T(A) x T(B).

Now let's take stock. First of all, perceptual and 1/space are reciprocal transformations of each other. Second, A and T(A) are transforms of each other, as are B and T(B). Therefore, the arcane and mysterious A*B--the lattice produced by convoluting A and B in perceptual space--is simply the reciprocal of T(A) x T(B). In other words, the asterisk in perceptual space is the equivalent of the multiplication sign in 1/space. We can't just up and call convolution multiplication. But we can see for ourselves that multiplication is the transform of convolution. Now multiplication is so much a part of our every day lives that we hardly think of it as an act of pure reason. But that's precisely what it is. And in opting for transform space, we made a small sacrifice in terms of intuition for a substantial gain in what we could avail to pure reason: simple arithmetic.

The convolution theorem, which we are about to employ, is the mathematical proof that indeed convolution is what we said it is--an arithmetic operation on transformation.[3] The critical lesson in our exercise is that while transform space is as remote as it ever was, it is far from incomprehensible. Fourier series have their corresponding Fourier transforms. And when attributes of waves do become incomprehensible in perceptual space, the appropriate transformation can put those attributes with the reach of reason.

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