By Tilman Butz

Intended to serve an "amusing textbook," this e-book belongs to an extraordinary style. it truly 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 services. Many illustrations and easy-to-solve workouts make the publication specially available, and its funny variety will upload to the excitement of studying from it.

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**Example text**

The ﬁnite time resolution ensures that there also is a signal at negative times, whereas it was 0 before convolution, ii. The maximum is not at t = 0 any more, iii. What can’t be seen straight away, yet is easy to grasp, is the following: the centre of gravity of the exponential function, which was at t = τ , doesn’t get shifted at all upon convolution. An even function won’t shift the centre of gravity! Have a go and check it out! It’s easy to remember the shape of the curve in Fig. 16. Start out with the exponential function with a “90◦ -vertical cliﬀ”, and then dump “gravel” Fig.

The convolution is commutative, distributive and associative. This means: commutative : f (t) ⊗ g(t) = g(t) ⊗ f (t). Here, we have to take into account the sign! Proof (Convolution, commutative). Substituting the integration variables: +∞ +∞ f (ξ)g(t − ξ)dξ = f (t) ⊗ g(t) = −∞ g(ξ )f (t − ξ )dξ −∞ with ξ = t − ξ . ). Associative : f (t) ⊗ (g(t) ⊗ h(t)) = (f (t) ⊗ g(t)) ⊗ h(t) (the convolution sequence doesn’t matter; proof: double integral with interchange of integration sequence). 7 (Convolution of a “rectangular function” with another “rectangular function”).

13). Please note the following: the interval, where f (t) ⊗ g(t) is unequal to 0, now is twice as big: 2T ! ), then also f (t) ⊗ g(t) would be symmetrical around 0. In this case we would have convoluted f (t) with itself. Now to a more useful example: let’s take a pulse that looks like a “unilateral” exponential function (Fig. 14 left): f (t) = e−t/τ for t ≥ 0 0 else . 3 Convolution, Cross Correlation, Autocorrelation, Parseval’s Theorem 49 h(t) T ✻ ✲ − T2 T 2 3T 2 t Fig. 13. Convolution h(t) = f (t) ⊗ g(t) Fig.