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Fourier Transformation for Pedestrians by Tilman Butz

By Tilman Butz

Intended to serve an "amusing textbook," this booklet belongs to an extraordinary style. it really is written for all scholars and practitioners who care for Fourier transformation. Fourier sequence in addition to non-stop and discrete Fourier transformation are coated, and specific emphasis is put on window features. Many illustrations and easy-to-solve routines make the e-book in particular obtainable, and its funny variety will upload to the excitement of studying from it.

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Tackling Infinity ∞ Derive the result for the infinite series k=1 1/k 4 using Parseval’s theorem. Hint: Instead of the triangular function try a parabola! 8. Smoothly Given is the function f (t) = [1−(2t)2 ]2 for −1/2 ≤ t ≤ 1/2 with periodic continuation. 63) and argue how the Fourier coefficients Ck must depend on k. Check it by calculating the Ck directly. 1 Continuous Fourier Transformation Preliminary remark : Contrary to Chap. 1, here we won’t limit things to periodic f (t). The integration interval is the entire real axis (−∞, +∞).

G(t0 ) = 2π δ(t0 − t) δ(t0 + t) + 2 2 (cf. ). Finally we get: h(t0 ) = e− 2 σ 1 2 2 t0 δ(t0 − t) δ(t0 + t) + . 2 2 54 2 Continuous Fourier Transformation Now the only thing left is to Fourier-transform h(t0 ). The integration over the δ-function actually is fun: ∼ +∞ e− 2 σ 1 f (t) ≡ H(ω0 ) = 2 2 t0 −∞ = e− 2 σ 1 2 2 t δ(t0 − t) δ(t0 + t) −iω0 t0 + dt0 e 2 2 cos ω0 t. Now, this was more work than we’d originally thought it would be. But look at what we’ve gained in insight! This means: the convolution of a Gaussian distribution in the frequency domain results in exponential “damping” of the cosine term, where the damping happens to be the Fourier transform of the frequency distribution.

12) +∞ f (t )δ(t − t )dt = f (t) . 9). For f (t) = 1 we get: FT(δ(t)) = 1. 13) The impulse, therefore, requires all frequencies with unity amplitude for its Fourier representation (“white” spectrum). Conversely: FT(1) = 2πδ(ω). 14) The constant 1 can be represented by a single spectral component, viz. ω = 0. No others occur. As we have integrated from −∞ to +∞, naturally an ω = 0 will also result in infinity for intensity. We realise the dual character of the forward and inverse transformations: a very slowly varying function f (t) will have a very high spectral density for very small frequencies; the spectral density will go down quickly and rapidly approaches 0.

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