For a length
complex sequence
,
, the
*discrete Fourier transform* (DFT) is defined by

We are now in a position to have a full understanding of the transform *kernel*:

The kernel consists of samples of a complex sinusoid at discrete frequencies uniformly spaced between 0 and the sampling rate . All that remains is to understand the purpose and function of the summation over of the pointwise product of times each complex sinusoid. We will learn that this can be interpreted as an

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