# Relationship between spectral efficiency and PAR

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)

• you shouldn't be putting OFDM on the same level as QAM – QAM is a constellation, whereas OFDM is a multicarrier transmission scheme (which can use any linear modulation underneath, be it a PSK or QAM, for example). – Marcus Müller Jan 19 '19 at 2:30
• I don't think there is any intrinsic relationship between spectral efficiency and PAR. They are two different metrics measuring two different things. It's good to consider them separately to get the big picture of a system like OFDM. – BlackMath Jan 19 '19 at 2:36

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 :)