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
![$ N$](http://www.dsprelated.com/josimages_new/mdft/img35.png)
discrete
frequencies
![$ \omega_k$](http://www.dsprelated.com/josimages_new/mdft/img677.png)
uniformly spaced between 0 and the
sampling
rate
![$ \omega_s \isdeftext 2\pi f_s$](http://www.dsprelated.com/josimages_new/mdft/img678.png)
. All that remains is to understand
the purpose and function of the summation over
![$ n$](http://www.dsprelated.com/josimages_new/mdft/img80.png)
of the pointwise
product of
![$ x(n)$](http://www.dsprelated.com/josimages_new/mdft/img45.png)
times each
complex sinusoid. We will learn that
this can be interpreted as an
inner product operation which
computes the
coefficient of projection of the
signal ![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
onto
the complex
sinusoid
![$ \cos(\omega_k t_n) + j \sin(\omega_k t_n)$](http://www.dsprelated.com/josimages_new/mdft/img679.png)
. As
such,
![$ X(\omega_k)$](http://www.dsprelated.com/josimages_new/mdft/img680.png)
, the DFT at frequency
![$ \omega_k$](http://www.dsprelated.com/josimages_new/mdft/img677.png)
, is a measure of
the amplitude and phase of the complex sinusoid which is present in
the input signal
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
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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