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Consider two real valued vectors $x$ and $y$. Suppose $x$ is $m$ dimensional and $y$ is $n$ dimensional with $n \ge m$. What is good notation for the function which returns an $n-m+1$-dimensional vector whose $i$th element is $$f(x,y)[i] = \sum_{j=0}^{m-1} x[j] y[i+j] \;?$$

Should it be

$$f(x,y) = x \star y \;?$$

  • Are you simply asking for people's opinion? If so, this is not a good question for this site. Check the Help Center. – robjohn Oct 30 '14 at 12:23
  • @robjohn I would like to know what is the standard mathematical notation, if there is one. Anything with references to sources would of course be even better. –  Oct 30 '14 at 12:24
  • Are you thinking of convolution? This is not convolution. – robjohn Oct 30 '14 at 12:29
  • @robjohn It's essentially the same as convolution if you reverse $x$ I think. –  Oct 30 '14 at 12:29
  • In the formula above, you can pull the $x[i]$ out front since it is independent of the summation. Convolution would be $$x\ast y[i]=\sum_{j=0}^ix[j]y[i-j]$$ – robjohn Oct 30 '14 at 14:52
  • @robjohn Oh sorry that was a silly typo by me. Thanks for pointing it out. –  Oct 30 '14 at 19:54
  • As long as you define what you mean by each symbol, it really doesn't matter what symbols you use. Using commonly used symbols reduces confusion. Is '$\star$' used in this way elsewhere? Is any other symbol? – robjohn Oct 31 '14 at 05:16
  • @robjohn This is my problem. I don't know the math literature well enough to find examples where discrete cross-correlation has been discussed before. My guess was from http://mathworld.wolfram.com/Cross-Correlation.html but that is not the discrete case. –  Oct 31 '14 at 07:48

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