Fourier Transformation for Pedestrians by Tilman Butz

Author:Tilman Butz
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Fig. 4.3For (fictitious only) we quite clearly see the analogy with the -function

4.1.3 Definition of the Discrete Fourier Transformation

Now we want to determine the spectral content of the series using discrete Fourier transformation. For this purpose, we have to make the transition in the definition of the Fourier series:

(4.10)

with periodic in :

(4.11)

In the exponent we find , meaning that can be eliminated. The prefactor contains the sampling raster , so the prefactor becomes . During the transition from (4.10) to (4.11) we tacitly shifted the limits of the interval from to to to , something that was okay, as we integrate over an integer period and was assumed to be periodic in . The sum has to come to an end at , as this sampling point plus reaches the limit of the interval. Therefore we get, for the discrete Fourier transformation:

Definition 4.1

(Discrete Fourier transformation)

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