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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

DSP_CS
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  • 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 08:19
  • 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

1 Answers1

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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.

robert bristow-johnson
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  • 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? – DSP_CS 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? – DSP_CS 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

    \begin{equation} x(t) \ast v(t) = u(t) \otimes v(t) \end{equation}

     – abhilash Feb 23 '20 at 11:38
  • @abhilash what you are trying to demonstrate in your last comment is that when x(t)=u(-t), convolution and correlation will be equal/same?? and your equation ??what idea is conveyed by your equation? is it also says that when x(t)=u(-t), convolution and correlation will be equal/same – DSP_CS Feb 23 '20 at 12:38
  • @abhilash what do you intend by * ,by which you raised u,when already you have changed(made negative)sign of time in u(t)?? – DSP_CS Feb 23 '20 at 12:44
  • The $$ is the complex conjugation operator. $(a + j b)^ = a - j b$. – TimWescott Feb 23 '20 at 16:29
  • @abhilash , you are correct. i was assuming real $x(t)$, $y(t)$, and $h(t)$ and real $t$ and $u$ in all cases. – robert bristow-johnson Feb 23 '20 at 18:24
  • "Do you mean,in convolution,we always have sign with $u$,that is opposite sign to the sign on $u$ in correlation?" yes @engr i do mean that. this is why the impulse response of a matched filter resembles the signal you're trying to detect, but it's flipped left-to-right. – robert bristow-johnson Feb 23 '20 at 18:27