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I am reviewing some bandlimited communications theory and something is bothering me -

My book says increasing the roll-off factor makes the received sequence more tolerant to timing recovery errors but it's very clear that the strongest peaks are not at the sampling instances. Thus it doesn't seem that the system is very tolerant to timing errors regardless of the roll-off factor.

It looks like to make the peaks equate to the sampling times you would need to half the transmission rate.

Am I wrong? What's going on?

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An easy way to see what's going on is by plotting the signal's eye diagram.

With low roll-off factors, the eye diagram will show you that deviating a small amount from the optimum sampling time will decrease the error margin. Note that the signal peaks are not the ideal sampling points. This is the eye diagram of a sinc pulse (zero roll-off):

enter image description here

With larger roll-off factors, the eye remains quite open for a larger period of time around the optimum sampling time. This is why it's said that these pulses are more tolerant of timing errors. This is the eye diagram when using pulses with roll-off equal to 1:

enter image description here

Mind you, this doesn't mean that you can be sloppy about timing synchronization. It's just that small errors in timing will result in larger variation in the sample values of low roll-off pulses compared to large roll-off pulses.

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  • $\begingroup$ Thanks, I still don't see how a higher roll off factor would make the system more tolerant to timing mismatches though. In the case of timing recovery algorithms like the Gardner it's implicitly assumed that symbols don't overlap causing removing symmetry - how is this dealt with? $\endgroup$ – FourierFlux Nov 11 '18 at 17:21
  • $\begingroup$ Gardner works fine as long as the roll-off factor is not too small. If you could be more precise about what is still unclear, I could try to edit my answer accordingly. $\endgroup$ – MBaz Nov 11 '18 at 19:59
  • $\begingroup$ I will try to be. I will try to post some graphs later. Basically in the case of a Raised Cosine modulation scheme the symbols interfere with each other except at the ideal sampling times. There is a dissymmetry in the received signal on each side of the ideal sampling time because of this. This seems like it would break the Gardner timing recovery loop since it seeks a position of equal strength on each side of the ideal sampling point. $\endgroup$ – FourierFlux Nov 11 '18 at 23:40
  • $\begingroup$ @FourierFlux OK -- but that is not what you originally asked. I suggest: 1) read Gardner's paper, where he explains that his algorithm is (somewhat) tolerant of "self-noise" (which is what you're concerned with). 2) read the many answers involving Gardner on this website. 3) Implement Gardner yourself (it takes maybe 50 lines of Matlab code total) and see for yourself how it behaves with different pulses. 4) Edit your question (or ask a new one) if necessary. $\endgroup$ – MBaz Nov 11 '18 at 23:47
  • $\begingroup$ I will make a new question in a bit, thanks. I am still confused though since it still seems to want to seek out a balance on either of the sampling time which doesn't exist in RRC pulse-shaped signals. I have a book which talks about using it for them though so I guess I am missing something. $\endgroup$ – FourierFlux Nov 13 '18 at 5:04

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