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It is to apply the function of a system on its input so as to get the output of the system. The system may be time-variant or time-invariant.
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The convolution in time domain is equivalent to the multiplication in the frequency domain. The multiplication in time domain is equivalent to the convolution in the frequency domain.
It's easy! When you have the response of a linear system to a definite input you can obtain the system response to every desired input, simply by the means of Convolution Integral. This is the reason why we use it. But if you want to discover it through mathematical way you can refer to the engineering mathematics books.
It is a math operation involving summation of all products of a signal at 1 point with all possible shifted versions of the other.
It may appear 'convoluted' or complicated in time domain hence its name yet its implications are straight fwd if we view in freq domain - spectrum multiplication !!
I think convolution operation is not restricted to only LTI system, but, can be applied to nonlinear time-variant system. We just use the LTI convolution operation because of the simple mathematical form. Also, everyday physical phenomena are of smooth feature, we can apply it to restricted parts of each physical phenomenon.
Some Firends say: " Convolution is used only for time invarient systems". This is not true! You can use a convolution for a time varient system, too. In that case the function under the integral will be time varient and the convolution formula is different.
convolution is applicable to both LTI and linear time variant systems.To understand convolution just imagine it as a sum of outputs for time shifted impulses as inputs.
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