Sum orthogonality

The same concept with somewhat simpler notation is in FIT.htm

The basis for evaluating sums is the relation

In the event that , this becomes

This is explicitly zero for all m not equal to a multiple of N or 0. For m = to a multiple of N, the sum is exactly N so that explicitly

Fitting the data

I want to fit the data between -T/2 and T/2 as a set of 2M+1 coefficients such that

Equation 1

In general this means that I want to minimize

For a discrete set of N mid-point data points

Minimize

Setting the derivative with respect to c_m equal to zero yields

Interchanging the second sum and noting that the second term is the complex conjugate of the first

Which in general is a set of M coupled linear equations to be solved for 2M+1 complex values of c_i. A wonderful thing happens over this region when t_k is substituted into the equation.

The sums over k in the second term are orthogonal. This means

Equation 2 Coddet

Equation 1 is extendable to all time. This makes d(t) explicitly periodic over all time. In this case