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.
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