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?**

https://dsp.stackexchange.com/questions/18130/understanding-discrete-time-convolution-in-lti-systems — jomegaA, Feb 18 '20 at 07:22
If you perform the convolution sum either delaying the impulse response of a system or delaying the input, you will end up with $y(n)$, which must be the same. — jomegaA, Feb 18 '20 at 07:25
Further what was demonstrated in your attachment is the "Commutative property" of convolution. — jomegaA, Feb 18 '20 at 07:47
https://en.wikipedia.org/wiki/Circular_convolution You may understand when it is periodic convolution with this article — jomegaA, Feb 18 '20 at 08:02

score 2 · Answer 1 · answered Feb 18 '20 at 18:47

2

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.

answered Feb 18 '20 at 18:47

Hilmar

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What do you mean by "non-trivial length of signal and/or impulse response"?? – DSP_CS Feb 19 '20 at 05:13
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. – Hilmar Feb 20 '20 at 00:30
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 – DSP_CS Feb 21 '20 at 10:50

Confusions regarding convolution?

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