Waves and Fourier sums |

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

Look what happens to the coefficients when we multiply polynomials.

Identifying coefficients of successive powers of , we get

In matrix form this looks like

The following equation, called the ``convolution equation,'' carries the spirit of the group shown in (3)

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.

Waves and Fourier sums |

2013-01-06