Confusions regarding convolution?

Question

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?**

Answer 1

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.