The Bingham formula

During my time at the Politecnico di Torino I attended a Communication Systems course which focused on systems such as ADSL/VDSL, DVB-T, LTE and modulation techniques such as PSK, QAM, DMT/OFDM. That being said, the one formula we always used to compute the cardinality of the constellation used was the famous Bingham formula.

When I was looking at the different parameters of this formula I figured I would do a search on Google to see if there was more information to be found. However, a search for “Bingham formula” resulted in not a single useful link. If this is such a famous and used formula, why is there nothing on the searchable web about it? Not even an entry on Wikipedia? This is an honest question, if someone knows please tell me.

What does this formula look like you ask? Here it is:

Bingham Formula

m: the number of bits
1/10: used when solving for a BER of P(e) = 10^-7, 1/14 if you’re using a BER of 10^-10 (e.g. for DVB-T)
Prx: received power
Pn : noise power
Yc/Ys: non-ideality in relation to Shannon + recovery by coding gain
Ym: margin parameters taking other losses into account

When computing the achievable bitrate for a given band, you’d use Rb = Rs * m, where Rs is the symbol rate. Not going in to much detail here as I could spend another three paragraphs on things such as delay spread, guard time and cyclic prefix, just know that once you have solved the equation and have m, you know which QAM you can use. For example, if m = 3, you can use a 2^m = 2^3 = 8-QAM constellation, so that would mean 3 bits per symbol.

The difference with the Shannon formula is that you take into account some non-ideal properties, if you want the ideal bitrate, just take Yc/Ys = 1 and neglect 1/Ym.

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