Relationship between the (continuous)
Fourier transform and the discrete Fourier transform.
Left column: A continuous function (top) and its Fourier transform (bottom).
Center-left column: If the function is periodically repeated, its Fourier transform becomes zero except at discrete points.
Center-right column: Conversely, if the function is discretized (multiplied by a
Dirac comb), its Fourier transform becomes periodic.
Right column: If a function is both discrete and periodic, then so is its Fourier transform. The situation in the right column is mathematically identical to the discrete Fourier transform.