Waves and Fourier sums

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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 , 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 as a numerical value. Instead we thought of it as a unit delay operator''. Now we think of the product numerically. For all possible numerical values of , each value is determined from the product of the two numbers and . 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.

Subsections
 Waves and Fourier sums

Next: FT as an invertible Up: Waves and Fourier sums Previous: Waves and Fourier sums

2013-01-06