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As far as i have studied and understood, convolution is the process by which we can get/determine output of LTI systems While reading one web link about convolution, i came across certain notation ,that i couldn't understand as shown highlighted in attached photo

1)Are these notations referring to delay/shift in input and impulse response?

2)Also it mentions that **Periodic or circular convolution is also called as fast convolution as shown highlighted in last line of 2nd photo attached. Is it idea correct?**

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1)Are these notations referring to delay/shift in input and impulse response?

Yes

2)Also it mentions that **Periodic or circular convolution is also called as fast convolution

That's a bit of a misrepresentation. For a non-trivial length of signal and/or impulse response the fastest way to implement convolution is to multiply in the frequency domain. That does indeed implement circular convolution. That's typically NOT what you actually want. If you want to leverage frequency domain multiplication for linear convolution, you need to add the right amount of zero padding and/or overlap handling to your algorithm. Read up on "Overlap Add" or "Overlap Save" algorithm.

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    $\begingroup$ What do you mean by "non-trivial length of signal and/or impulse response"?? $\endgroup$
    – engr
    Feb 19 '20 at 5:13
  • $\begingroup$ If signal or impulse response are only a few samples long, direct convolution is typically the most efficient way. Once you have hundreds or thousands of samples frequency domain convolution is much more efficient. $\endgroup$
    – Hilmar
    Feb 20 '20 at 0:30
  • $\begingroup$ In 2nd line of your answer,you wrote"multiply in the frequency domain",here you mean time domain multiplication or frequency domain convolution?you also referred to this phenomena in your comment,where you said it frequency domain convolution $\endgroup$
    – engr
    Feb 21 '20 at 10:50

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