Nyquist’s signalling rate: 2B pulses per second

Harry Nyquist’s “Certain topics in telegraph transmission theory” (1928) showed that a channel of bandwidth \(B\) can carry at most \(2B\) independent pulses per second without intersymbol interference (nyquist_symbol_rate()). Sinc-shaped pulses spaced \(1/2B\) apart are each zero at all the other sampling instants, so every pulse can be read without seeing its neighbours. Sending faster makes the pulses blur together.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.communications import bpsk, nyquist_symbol_rate

Sinc pulses at the Nyquist rate

B = 1000.0
rate = float(nyquist_symbol_rate(B))
rng = np.random.default_rng(1928)
symbols = bpsk(rng.integers(0, 2, 24))
t = np.linspace(-0.002, 0.014, 4001)


def waveform(symbol_rate):
    T = 1 / symbol_rate
    return sum(a * np.sinc(2 * B * (t - k * T)) for k, a in enumerate(symbols)), np.arange(symbols.size) * T


fig, axes = plt.subplots(2, 1, figsize=(8, 5), sharex=True)
for ax, r in zip(axes, [rate, 1.5 * rate], strict=True):
    x, instants = waveform(r)
    samples = np.interp(instants, t, x)
    isi = np.max(np.abs(samples - symbols)[: int(0.012 * r)])
    print(f"{r:6.0f} baud through B = {B:.0f} Hz: worst sampling error {isi:.3f}")
    ax.plot(t * 1e3, x)
    ax.plot(instants * 1e3, samples, "ko", ms=4)
    ax.set(xlim=(-1, 12), title=f"{r:.0f} baud ({r / rate:.1f} x Nyquist)")
axes[-1].set_xlabel("time [ms]")
fig.tight_layout()
x, instants = waveform(rate)
assert np.allclose(np.interp(instants[:20], t, x), symbols[:20], atol=5e-3)

plt.show()

2000 baud through B = 1000 Hz: worst sampling error 0.000
3000 baud through B = 1000 Hz: worst sampling error 1.793