# convolution vs correlation? [duplicate]

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

• 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! – GKH Feb 23 '20 at 8:19
• 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. – abhilash Feb 23 '20 at 12:06

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.

• missed conjugate in the second equation – abhilash Feb 23 '20 at 10:48
• @robert Do you mean,in convolution,we always have sign with u,that is opposite sign to the sign on u in correlation? – engr Feb 23 '20 at 10:55
• @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? – engr Feb 23 '20 at 11:05
• @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 – abhilash Feb 23 '20 at 11:35
• @engr if we are able to find a function $x(t)$ where $x(t) = {u^*}(-t)$ then $$x(t) \ast v(t) = u(t) \otimes v(t)$$ – abhilash Feb 23 '20 at 11:38