The Mathematics
A modulated symbol is:
s(t) = I·cos(2πfct) − Q·sin(2πfct)
where (I, Q) is the constellation point. Through an AWGN channel:
(I', Q') = (I + nI, Q + nQ) where nI, nQ ~ N(0, σ²)
and σ² = 1/(2·SNR_linear).
Bit error rate
The probability of decoding incorrectly depends on the minimum distance between points and noise level:
- QPSK at 20 dB SNR: BER ≈ 10⁻⁵ (one error per 100,000 bits).
- 64-QAM at 20 dB SNR: BER ≈ 10⁻³ — two orders of magnitude worse.
This is the fundamental trade-off: spectral efficiency vs robustness. Wi-Fi and LTE switch between modulation orders in real time — high-order QAM when signal is strong, falling back to QPSK when conditions degrade.
Further reading
- QAM (Wikipedia)
- Constellation diagram
- Proakis, Digital Communications