# Why is N-point DFT approximated by the sinc function?

While looking into DFT leakage, I've came across the author saying that

"..., the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function."

Other than the fact that the fourier transform of the rectangle function is the sinc function, I have no idea why the statement above is true.

## 1 Answer

When a waveform of arbitrary length is cut into a length that can be fed to an N-point DFT (with N being finite), that is the same as multiplying that original waveform by a rectangular window of that given finite length. Multiplication in the time domain is the same as convolution in the frequency domain with the Fourier transform of that multiplying function, which happens to be a Sinc function for a rectangular window (or more precisely, for finite length real data, a conjugate mirrored Dirichlet (or periodic Sinc) function, which is pretty close to a regular Sinc, except near DC or Fs/2 or with very short DFTs. More details here ).

Another way to look at it is that any pure sinusoid of a frequency that is not exactly equal to one of the DFT basis vectors will thus not be orthogonal to any of the DFT basis vectors. Thus any non-zero magnitude of that sinusoid must end up partially represented in every element of a DFT result (since only orthogonal vector dot products will end up zero). To interpolate the magnitude of that original non-orthogonal pure sinusoid, one might use the Whittaker–Shannon interpolation formula which happens to use a Sinc kernel to weight the contributions from all the DFT result elements into which energy from the original sinusoid was spread.