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Title: Convolution
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Series: Advent of Mathematical Symbols
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YouTube-Title: Convolution
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Bright video: https://youtu.be/55BaHGo-2jg
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Dark video: https://youtu.be/olYeagTr-JI
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: aoms21_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,757 –> 00:00:03,871 The mathematical symbol of today is the convolution.
2 00:00:04,043 –> 00:00:06,086 Denoted with such a star.
3 00:00:07,129 –> 00:00:11,800 In fact in mathematics you find different possibilities to define the convolution.
4 00:00:11,956 –> 00:00:14,449 Depending which problem you consider.
5 00:00:15,457 –> 00:00:19,554 Most importantly you find a continuous version and a discrete version.
6 00:00:20,586 –> 00:00:24,143 Now in this video i want to show you the ordinary continuous version.
7 00:00:24,257 –> 00:00:28,429 So we consider functions f and g defined on the real number line.
8 00:00:29,871 –> 00:00:35,148 Then for these two functions we define the convolution of f and g.
9 00:00:36,200 –> 00:00:42,142 In fact this convolution should give us a new function also defined on the real number line.
10 00:00:43,000 –> 00:00:48,515 More precisely “f star g” is again a function from R to R.
11 00:00:49,686 –> 00:00:54,988 and the idea to get this new function is to mix f and g together.
12 00:00:55,657 –> 00:01:00,642 Moreover this mixing works in two different directions inside the integral.
13 00:01:01,457 –> 00:01:04,631 Of course this is what we should explain in a formula.
14 00:01:05,829 –> 00:01:08,915 So lets look at “f star g” at a given point x.
15 00:01:09,929 –> 00:01:17,994 Then we define this value by an integral that goes through all numbers. So we start with - infinity and go to infinity.
16 00:01:19,229 –> 00:01:24,288 Of course inside we have the functions f and g and the variable name for the integration.
17 00:01:25,500 –> 00:01:28,685 and for this i want to use the greek letter Tau.
18 00:01:29,514 –> 00:01:33,912 This is very common, because in some sense the variable can stand for time.
19 00:01:34,957 –> 00:01:39,216 So here you see we have the first function f at the point Tau
20 00:01:39,416 –> 00:01:42,662 Times the function g at another point.
21 00:01:43,600 –> 00:01:50,112 and now i already told you g should go the other direction so what we need inside is (-Tau).
22 00:01:51,343 –> 00:01:55,654 Moreover we also shift the point here by our given point x.
23 00:01:56,643 –> 00:02:01,182 and that’s what we have. This product of the two functions is integrated.
24 00:02:01,914 –> 00:02:07,362 and if this integral exists for all points x in R we have the convolution.
25 00:02:08,457 –> 00:02:14,945 Indeed this convolution has some nice properties and therefore it is very important in a lot of applications.
26 00:02:15,814 –> 00:02:19,529 For this reason i hope that you now can remember the definition.
27 00:02:20,300 –> 00:02:23,722 Then have a nice day and see you next time. Bye!
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Quiz Content
Q1: What is the correct definition of the convolution $f \ast g$?
A1: $(f \ast g)(x) = \int_{-\infty}^\infty f(y) g(x - y) , dy $
A2: $(f \ast g)(x) = \int_{-\infty}^\infty f(y) g(y - x) , dy $
A3: $(f \ast g)(x) = \int_{-\infty}^\infty f(x) g(y - x) , dy $
A4: $(f \ast g)(x) = \int_{-\infty}^\infty f(x) g(y) , dy $
Q2: What is not correct for the convolution?
A1: $ f \ast g = g \ast f$
A2: $ f \ast (g \ast h) = (f \ast g) \ast h$
A3: $ f \ast (g + h) = f \ast g + f \ast h $
A4: There is a function $h$ such that $f \ast h = f$ for all functions $f$.
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Last update: 2024-11
