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# FOURIER TRANSFORM

We first examine the two ways to visualize polynomial multiplication. The two ways lead us to the most basic principle of Fourier analysis that

 A product in the Fourier domain is a convolution in the physical domain

Look what happens to the coefficients when we multiply polynomials.
 (1) (2)
Identifying coefficients of successive powers of Z, we get
 (3)
In matrix form this looks like
 (4)
The following equation, called the convolution equation,'' carries the spirit of the group shown in (3)
 (5)

The second way to visualize polynomial multiplication is simpler. Above we did not think of Z as a numerical value. Instead we thought of it as a unit delay operator''. Now we think of the product X(Z) B(Z) = Y(Z) numerically. For all possible numerical values of Z, each value Y is determined from the product of the two numbers X and B. Instead of considering all possible numerical values we limit ourselves to all values of unit magnitude for all real values of .This is Fourier analysis, a topic we consider next.

Next: FT as an invertible Up: Waves and Fourier sums Previous: Waves and Fourier sums
Stanford Exploration Project
12/26/2000