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When using multi-level Pulse Amplitude Modulation to transmit a message, how should the receiver handle noise?

For example, suppose there are 5 levels (0,1,2,3,or 4) and the following message is transmitted: {2,5,1,4,0,0,0,2,3,2,3,2,3}

On the receiving end you get:

{2,5,1,4,0,0,0,2,2,3,3,2,2}, which is clearly incorrect.

So the amplitudes that are a distance of 2 or more get received correctly, but amplitudes with a distance of 1 from each other are more difficult.

What are some techniques used to handle this kind of noise?

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    \$\begingroup\$ This is basically the question that led Shannon to invent the whole field of Information Theory. The seminal paper is here. But answering the question without giving a complete course in communications systems (which I'm not qualified to do) is beyond me. \$\endgroup\$
    – The Photon
    Mar 19 '20 at 23:03
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    \$\begingroup\$ Or if you just want a 1-word answer: Error correcting codes. \$\endgroup\$
    – The Photon
    Mar 19 '20 at 23:04
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When you say handle, you have to be more specific. You need to specify the tolerable error rate of your system (it can't be 0 since there is always noise present). Generally, in communications systems there is a trade-off between data rate and error rate. In your example, you increased the data rate by allowing more levels. However, as you noticed, you paid the price in increased error rate. Note that if you allowed only 3 levels (say 1,3 and 5) then there would always be at least a distance of 2 between successive transmissions which, as you also noticed, reduces the error rate. However, the cost is a reduced data rate. Aas already noted, error correcting codes are another approach to handling noise but the effect is the same: reduced error rate but also reduced data rate (since the coding requires extra bits).

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So you have PAM.

We don't know how precisely your levels are transmitted, or how precisely the receiver slices between levels, or where-in-time the decisions are made.

So, assume you want the PAM to settle to 1% accuracy at decision time, or 40dB.

Given a NEPER is 8.7dB, and 40/8.7 is about 4.5 tau, if your bit times are 45 us, you need a 10us time constant RC LPF on your datalink.

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