Kotel’nikov’s optimum receiver: BPSK in white Gaussian noise

Vladimir Kotel’nikov’s 1947 thesis, The Theory of Optimum Noise Immunity, asked which receiver makes the fewest errors when known signals are buried in white Gaussian noise. For two antipodal signals the answer is a correlator (or matched filter) followed by a sign decision, with bit-error rate \(Q(\sqrt{2E_b/N_0})\) (ber_bpsk_awgn(), q_function()). A Monte Carlo simulation of bpsk() through noise reproduces the curve.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.communications import ber_bpsk_awgn, bpsk, q_function
from engineeringkit.communications.visualizers import plot_ber_curve

Monte Carlo against theory

rng = np.random.default_rng(1947)
ebn0_db = np.arange(0, 9)
simulated = []
for x_db in ebn0_db:
    ebn0 = 10 ** (x_db / 10)
    bits = rng.integers(0, 2, 400_000)
    received = bpsk(bits) + rng.normal(0, np.sqrt(1 / (2 * ebn0)), bits.size)
    simulated.append(np.mean((received > 0) != bits))
theory = ber_bpsk_awgn(10 ** (ebn0_db / 10))
for x, s, th in zip(ebn0_db, simulated, theory, strict=True):
    print(f"Eb/N0 = {x} dB: simulated {s:.2e}, theory {th:.2e}")
assert np.allclose(simulated[:6], theory[:6], rtol=0.1)
assert np.allclose(theory, q_function(np.sqrt(2 * 10 ** (ebn0_db / 10))))

fine = np.linspace(0, 10, 200)
ax = plot_ber_curve(fine, ber_bpsk_awgn(10 ** (fine / 10)), label="Q(sqrt(2 Eb/N0))")
ax.semilogy(ebn0_db, simulated, "o", label="simulation")
ax.legend()
ax.set_title("Coherent BPSK in AWGN")

plt.show()

Eb/N0 = 0 dB: simulated 7.85e-02, theory 7.86e-02
Eb/N0 = 1 dB: simulated 5.61e-02, theory 5.63e-02
Eb/N0 = 2 dB: simulated 3.74e-02, theory 3.75e-02
Eb/N0 = 3 dB: simulated 2.30e-02, theory 2.29e-02
Eb/N0 = 4 dB: simulated 1.24e-02, theory 1.25e-02
Eb/N0 = 5 dB: simulated 6.01e-03, theory 5.95e-03
Eb/N0 = 6 dB: simulated 2.41e-03, theory 2.39e-03
Eb/N0 = 7 dB: simulated 7.45e-04, theory 7.73e-04
Eb/N0 = 8 dB: simulated 1.72e-04, theory 1.91e-04