A Guide on Spectral Methods Applied to Discrete Data in One Dimension

Figure 9

In (a), the function from (48) is sampled in the time range . The finite time interval corresponds to a rectangular weight function. The continuous spectrum (arrows in (b)) is then convoluted with the spectral representation of that rectangular window. The black line indicates the convolution step for one left spectral part. Note that only the Dirac impulse generates a value, because all other components remain zero. The dashed line in (b) indicates the limitation to a maximum frequency, which corresponds to a convolution in the time domain (a). Open symbols show the sampling (a) and the result of the DFT (b).

(a) Sampled signal with

(b) Spectrum and folding kernel