## The DFT

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*:

*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.

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Sinusoid Problems