Convolution

Convolution - Convolution In this chapter and the next it...

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Unformatted text preview: Convolution In this chapter and the next it will seem that this physics course is actually just mathematics and defini- tions of words than have little to do with the real world. If you bear with me, we shall emerge on the other side to do some physics. Introduction Convolution is sometimes called Faltung which is German for folding , and is also described by terms such as running mean , cross-correlation function , smoothing , and so on. The convolution of two functions f(t) and g(t) is: h + t / X X f + u / g + t u / u Often we shall write this as: h + t / f + t / g + t / The above form is strictly notational. Do not use it, for example, with Mathematica which will interpret the asterisk as multiplication. Convolution turns out to be amazingly useful for a number of tasks. Here is one of them. We have a noisy Bessel function signal: 50 100 150 200 250-0.4-0.2 0.2 0.4 0.6 A noisy Bessel function We will convolve this signal with a Gaussian "kernel": convolution.nb 1-20-10 10 20 0.2 0.4 0.6 0.8 1 A Gaussian convolution kernel The result of the convolution smooths out the noise in the original signal: 50 100 150 200 250-0.2 0.2 0.4 Convolution of the signal with the kernel You will notice that in the above example, the signal and the kernel are both discrete time series, not continuous functions. In this case, the convolution is a sum instead of an integral: h i j 0 m f j g i j Here is an example. Choose f and g to be: f f ,f 1 , f 2 g g , g 1 Then: h j 0 m f j g j The f j terms are non-zero only for j between 0 and 2, so m is equal to 2. Also, g j is non-zero only for j equal to zero. So: h f g convolution.nb 2 Similarly: h 1 j 0 m f j g 1 j f g 1 f 1 g h 2 j 0 m f j g 2 j f 1 g 1 f 2 g h 3 j 0 m f j g 3 j f 2 g 1 Thus: h f g f g , f g 1 f 1 g , f 1 g 1 f 2 g , f 2 g 1 Now we form the convolution: h ' g f Then: + h ' / j 0 m g j f j where now, since...
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This note was uploaded on 09/11/2008 for the course ECEN 314 taught by Professor Halverson during the Spring '08 term at Texas A&M.

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Convolution - Convolution In this chapter and the next it...

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