Relationship between spectral efficiency and PAR

Question

Low PAR means less fluctiations in the magnitude of a signal and hence analog signal processing (e.g., power amplifier) is tremendously reduced. For example, constant envelope modulation schemes like BPSK, GMSK can use a switching amplifier which are >90% efficient. The downside is low spectral efficiency.

On the other side, high-order QAM or OFDM have high spectral efficiencies but the PAR becomes very high (up to 12dB for OFDM) and hence power amplifiers only reach 10-20% effiency.

Specieal cases and tricks aside (like Crest Factor Reduction, Digital Predistortion, ...) and leaving SNR out of the picture, is there a first order relationship between spectral efficiency and PAR? Any theoretical bounds?

If not, are there at least published plots that plot spectral efficiency vs PAR for various standards/modulation formats? (I did not find any)

Answer 1

I'm with BlackMath here – a PAPR vs SE graph makes little sense, since you would be denying the differen BER at fixed SNRs you'd get.

Let me give you an example:

256-PSK has a PAPR of 1 – the "best" you can get. It's way lower than the PAPR of 256-QAM.
Both have the same spectral efficiency (which doesn't depend on the constellation shape at all, but only on the number of constellation points).

You would, however, never use 256-PSK, though, because it's terribly low on distance between constellation points.

Since you could define a PSK with any number > 1 of constellation points, and any PSK has PAPR of 1 ("doesn't get any better"), that graph would be mighty boring.

Aside from PSK, there's, for spectral efficiency-sensitive applications, only QAM (and things that are pretty much QAM, e.g. Amplitude shift-PSK). So, that graph, if only considering constellations that "make sense but aren't PSK" would in the end only compare different power-of-2 QAMs. There's little surprise in that – you simply get the PAPR of these QAM constellations – which is a bit ugly to calculate analytically, but holds little surprise: the higher the number of bits, the higher your PAPR in a QAM.

Instead of hoping for a constellation, I'd simply use e.g. Python to give me grids of $2^n$ points centered on 0+0j, and simply calculate PAPR on these over n. That's less than 50 lines of code :)