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 inner product operation which
computes the coefficient of projection of the signal onto
the complex sinusoid
. As
such,
, the DFT at frequency , is a measure of
the amplitude and phase of the complex sinusoid which is present in
the input signal at that frequency. This is the basic function of
all linear transform summations (in discrete time) and integrals (in
continuous time) and their kernels.