# Download Fourier Transformation for Pedestrians (Fourier Series) by Tilman Butz PDF

By Tilman Butz

Covers Fourier transformation and Fourier series with a selected emphasis on window functions.

Written for college kids and practitioners who care for Fourier transformation.

Including many illustrations and easy-to-solve routines

Presents severe technological know-how in an a laugh way

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Extra info for Fourier Transformation for Pedestrians (Fourier Series)

Example text

For a > 1 the time axis will be stretched and, hence, the frequency axis will be compressed. For a < 1 the opposite is true. 27): +T /2a Cknew a = T +T /2 f (at)e −iωk t −T /2a a dt = T −T /2 1 f (t )e−iωk t /a dt a with t = at = Ckold with ωknew = ωkold . a Please note that we also have to stretch or compress the interval limits because of the requirement of periodicity. Here, we have tacitly assumed a > 0. For a < 0, we would only reverse the time axis and, hence, also the frequency axis. For the special case a = −1 we have: f (t) ↔ {Ck , ωk }, f (−t) ↔ {Ck ; −ωk }.

Alternatively to the polar representation, we can also represent the real and imaginary parts separately (cf. Fig. 7). Please note that |F (ω)| is no Lorentzian! If you want to “stick” to this property, you better represent the square of the magnitude: |F (ω)|2 = 3 The Residue Theorem is part of the theory of functions of complex variables. 1 Continuous Fourier Transformation 41 Fig. 6. Unilateral exponential function, magnitude of the Fourier transform and phase (imaginary part/real part) Fig.

K T This function is the expression of the partial sums of the unit step. In Fig. 13 we show some approximations. 14 shows the 49th partial sum. As we can see, we’re already getting pretty close to the unit step, but there are overshoots and undershoots near the discontinuity. Electro-technical engineers know this phenomenon Fig. 13. 4 Gibbs’ Phenomenon 29 Fig. 14. Partial sum expression of unit step for N = 49 when using ﬁlters with very steep ﬂanks: the signal “rings”. We could be led to believe that the amplitude of these overshoots and undershoots will get smaller and smaller, provided only we make N big enough.