# 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. 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 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 Feb 23 '20 at 11:35
• @engr if we are able to find a function $x(t)$ where $x(t) = {u^*}(-t)$ then $$x(t) \ast v(t) = u(t) \otimes v(t)$$ Feb 23 '20 at 11:38