r"""
Nyquist's signalling rate: 2B pulses per second
===============================================

Harry Nyquist's "Certain topics in telegraph transmission theory" (1928)
showed that a channel of bandwidth :math:`B` can carry at most :math:`2B`
independent pulses per second without intersymbol interference
(:func:`~engineeringkit.communications.nyquist_symbol_rate`). Sinc-shaped
pulses spaced :math:`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()