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What is difference between convolution and correlation?In simple words or in a nut shell?

As far as i am able to study is that Convolution is a mathematical operation used to express the relation between input and output of an LTI system. It relates input, output and impulse response of an LTI system while Correlation is a measure of similarity between two signals

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  • $\begingroup$ You got it very well, in a nutshell! That's exactly what they are. There are some further connection between them, but what you said is exactly to the point! $\endgroup$ – GKH Feb 23 at 8:19
  • $\begingroup$ In a system in which place value is valued,( like the decimal number system ), convolution is the actual way of multiplication to be used to obtain the product of two entities. $\endgroup$ – abhilash Feb 23 at 12:06
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Convolution:

$$ y(t) = h(t) \circledast x(t) = \int\limits_{-\infty}^{\infty} h(u) \, x(t-u) \ \mathrm{d}u $$

Cross Correlation:

$$ R_{xy}(t) = \int\limits_{-\infty}^{\infty} y(u) \, x(t+u) \ \mathrm{d}u $$

The difference between the two is effectively the sign on $u$ in $x(t-u)$ in the integral. That correlation is like convolution but with one of the signals flipped left to right is essentially the basis of the Matched Filter.

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  • $\begingroup$ missed conjugate in the second equation $\endgroup$ – abhilash Feb 23 at 10:48
  • $\begingroup$ @robert Do you mean,in convolution,we always have sign with u,that is opposite sign to the sign on u in correlation? $\endgroup$ – engr Feb 23 at 10:55
  • $\begingroup$ @robert Do you mean ,we will always have - sign,with u in convolution and positive sign with u in correlation?or vice versa also possible? $\endgroup$ – engr Feb 23 at 11:05
  • $\begingroup$ @robert Actually convolution is the flipped one, not the correlation. Correlation is the multiplication of the conjugated signal with a delayed( or advanced ) version of the second signal $\endgroup$ – abhilash Feb 23 at 11:35
  • $\begingroup$ @engr if we are able to find a function $x(t)$ where $x(t) = {u^*}(-t) $ then \begin{equation} x(t) \ast v(t) = u(t) \otimes v(t) \end{equation} $\endgroup$ – abhilash Feb 23 at 11:38

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