6.7.6 Spectral Efficiency

The spectral efficiency, η, of a modulation scheme is defined as the ratio of the baseband bit rate, R_b, to the RF bandwidth, B_RF, required to transmit the modulated waveform:

η = \frac{R_{b}}{B_{R F}}

(6.57)

Expressed in terms of symbol rate, R_s, and the number of modulation states, M, this relationship can be generalized as:

η = \frac{R_{s} \log_{2} M}{B_{R F}} = \frac{\log_{2} M}{B_{R F} T_{s}}

(6.58)

or an ideal Nyquist system employing zero roll-off pulse shaping (α = 0), the minimum RF bandwidth equals the symbol rate (BRF = Rs), so that B_RFT_s=1. In this limiting case:

η = \log_{2} M

(6.59)

Consequently, we have the following spectral efficiencies:

BPSK 1 bps/Hz.
QPSK 2 bps/Hz.
8-PSK 3 bps/Hz.

In practical systems B_RFT_s=(1+ α), where α is the roll-off factor of the filters in the detector circuit—typically α =0.2–0.35 in modern systems. Assuming a value of α =0.3, practical spectral efficiencies are:

BPSK 0.77 bps/Hz.
QPSK 1.54 bps/Hz.
8-PSK 2.31 bps/Hz.

These values demonstrate the fundamental trade-off between modulation order, bandwidth, and robustness: higher-order constellations increase throughput per unit bandwidth but require higher signal-to-noise ratios (SNRs) and more precise demodulation.