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Annex B - Summing Complex Exponential Series.

A geometric progression of form:

equation

Can be summed from N0 to N1 inclusive as follows:

equation

So..

equation

Subtracting the second equation from the first yields:

equation

or..

equation

In particular, if N0=0, N1=N-1, this reduces to:

equation

If a is of form:

equation

then we have:

equation

If we look at the pure real (fraction) in the above expression, we can observe that the numerator is zero for any integral value of x. The denominator is zero for any x which is an integral multiple of N. In the cases where x is an integral multiple of N, both the numerator and denominator are zero, but application of L'Hopitals rule tells us that the fraction has magnitude N. The fraction is zero for all other integral x.

So, to recap, as far as integral values of x are concerned, the sum is zero unless x is also multiple of N (including zero), in which case the magnitude of the sum is N. In fact, by considering distinct cases of N odd and N even, it is easy to show than the sum is always +N if x is a multiple of N. (This is not altogether surprising if we recall that original form of the expression as a sum of complex exponentials.)

In Annex A, we were presented with an expression of form:

equation

This can be related to our result by associating..

equation

Now both m and n range over 0..N-1, so..

equation

Therefore, we need not consider any other integral multiples of N other than x=0, (n=m).

Conclusion:

equation

equation

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