ISI and RRC basic concepts

Question

I've been studying a kind of "introduction" for a few days now to get to talking about equalizers and ISI, and I would like to know if I have understood the things correctly or if I got confused:

For example, in PAM modulation we know that if we choose to use a rectangular pulse as the channel input (and if we do not filter well on the receive side) then that pulse on the receive side will "spread out" at the channel output interfering with adjacent pulses (the pulse tails overlap, creating so-called inter-symbol interference).

How to eliminate ISI? 3 ways:

1) Use "$\text{sinc}(x)$" pulses: by doing so, consecutive pulses will not have ISI because each will be sampled exactly at the instant when the tails of the other two (the one before and after) pass through zero.

• This type of pulse extends to infinity (in the time domain), so we have maximum band conservation -> good
• The ISI = 0 as just explained -> good
• This type of pulse (extending to infinity) is not realizable physically -> bad
• If a small timing error appears, the ISI appears -> bad

2) RRC filter: I apply a rectangular window to the $\text{sinc}(x)$ pulse and by varying the roll-off factor I adjust the trade-of between bandwidth and ISI:

• $\alpha$ high:
  • higher truncation, pulse tails decreasing more rapidly, lower ISI -> good
  • more bandwidth will be occupied because the change is more abrupt -> bad
• $\alpha$ small:
  • less truncation, we return to the $\text{sinc}(x)$ pulse of case 1) -> bad

3) Equalizer: a whole separate chapter I won't go into here.

If I have made mistakes please correct me, and if you want to clarify/deepen my statements please do so!

I thank you in advance.

Answer 1

The understanding may require some further explanation or clarification:

Sinc pulses and similar “Nyquist Pulses” do not eliminate ISI but allow us to constrain bandwidth in transmission without introducing ISI. (This is a slight distinction but as first introduced by the OP it may come across as if the purpose of such pulses is to somehow eliminate ISI that was previously there).

An RRC pulse is a “Root Raised Cosine Pulse” and on its own does introduce ISI. An RRC shaped waveform needs to be passed through a second RRC filter to then result in the “Raised Cosine” pulse shape which has no ISI. The second filter would be the matched filter in the receiver.