Quadrature modulation: QPSK doubles the rate at the same error rate

Charles Cahn’s 1960 analysis of combined phase and amplitude modulation formalised what quadrature carriers make possible. Cosine and sine carriers at the same frequency are orthogonal, so each can carry an independent BPSK stream. Gray-labelled QPSK (qpsk()) sends two bits per symbol in the bandwidth of one. Its bit-error rate per unit \(E_b/N_0\) equals that of BPSK (ber_qpsk_awgn()).

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.communications import ber_qpsk_awgn, qpsk
from engineeringkit.communications.visualizers import plot_constellation

A noisy QPSK constellation

rng = np.random.default_rng(1960)
ebn0 = 10 ** (6 / 10)
bits = rng.integers(0, 2, 400_000)
symbols = qpsk(bits)
es_n0 = 2 * ebn0
noise = (rng.normal(size=symbols.size) + 1j * rng.normal(size=symbols.size)) * np.sqrt(1 / (2 * es_n0))
received = symbols + noise
decided = np.column_stack((received.real > 0, received.imag > 0)).astype(int).reshape(-1)
ber = np.mean(decided != bits)
print(f"Eb/N0 = 6 dB: simulated QPSK BER {ber:.2e}, theory {float(ber_qpsk_awgn(ebn0)):.2e}")
assert np.isclose(ber, ber_qpsk_awgn(ebn0), rtol=0.1)

fig, axes = plt.subplots(1, 2, figsize=(9, 4))
plot_constellation(received[:3000], ax=axes[0], label="received")
plot_constellation(qpsk([0, 0, 0, 1, 1, 0, 1, 1]), ax=axes[0], label="ideal")
axes[0].set_title("QPSK at Eb/N0 = 6 dB")

Eb/N0 = 6 dB: simulated QPSK BER 2.41e-03, theory 2.39e-03

Text(0.5, 1.0, 'QPSK at Eb/N0 = 6 dB')

Two bits per symbol in one bandwidth

t = np.linspace(0, 4, 800, endpoint=False)
i_rail, q_rail = np.repeat(symbols[:4].real, 200), np.repeat(symbols[:4].imag, 200)
passband = i_rail * np.cos(2 * np.pi * 3 * t) - q_rail * np.sin(2 * np.pi * 3 * t)
recovered_i = 2 * np.mean((passband * np.cos(2 * np.pi * 3 * t)).reshape(4, -1), axis=1)
assert np.allclose(recovered_i, symbols[:4].real)
axes[1].plot(t, passband)
axes[1].set(xlabel="time [symbols]", title="Quadrature carriers share one band")
fig.tight_layout()

plt.show()