OFDM: many subcarriers from one inverse DFT

Robert Chang (1966) showed that subcarriers whose spectra overlap remain orthogonal if they are spaced by the symbol rate. Stephen Weinstein and Paul Ebert (1971) generated the whole set with a single inverse DFT (ofdm_modulate()) and prefixed each block with a copy of its tail. This cyclic prefix turns a dispersive channel’s linear convolution into a circular one, so after the receiver’s DFT (ofdm_demodulate()) every subcarrier sees one complex gain, removed by a single division. The idea now carries Wi-Fi, LTE and digital television.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.communications import ofdm_demodulate, ofdm_modulate, qpsk
from engineeringkit.communications.visualizers import plot_constellation

A multipath channel with and without the cyclic prefix

rng = np.random.default_rng(1971)
n_sc, n_blocks = 64, 40
symbols = qpsk(rng.integers(0, 2, 2 * n_sc * n_blocks))
channel = np.array([1.0, 0.0, 0.6 - 0.3j, 0.0, 0.0, 0.25j])
response = np.fft.fft(channel, n_sc)
fig, axes = plt.subplots(1, 3, figsize=(12, 3.8))
for ax, cp in zip(axes[:2], [0, 8], strict=True):
    tx = ofdm_modulate(symbols, n_sc, cp)
    rx = np.convolve(tx, channel)[: tx.size]
    equalised = (ofdm_demodulate(rx, n_sc, cp).reshape(-1, n_sc) / response).reshape(-1)
    error = np.sqrt(np.mean(np.abs(equalised - symbols) ** 2))
    print(f"cyclic prefix {cp}: RMS error after one-tap equalisation = {error:.2e}")
    plot_constellation(equalised, ax=ax)
    ax.set_title(f"cyclic prefix = {cp} samples")
assert error < 1e-12

cyclic prefix 0: RMS error after one-tap equalisation = 3.26e-01
cyclic prefix 8: RMS error after one-tap equalisation = 3.78e-16

The channel seen by each subcarrier

With the prefix, the demodulated pilot divided by the transmitted symbol is exactly the channel’s frequency response at that subcarrier.

estimate = (ofdm_demodulate(np.convolve(ofdm_modulate(symbols[:n_sc], n_sc, 8), channel)[: n_sc + 8], n_sc, 8)) / symbols[:n_sc]
assert np.allclose(estimate, response)
axes[2].plot(np.abs(response), label="|H(k)|")
axes[2].plot(np.abs(estimate), "k.", ms=3, label="pilot estimate")
axes[2].set(xlabel="subcarrier", ylabel="gain", title="Frequency-selective fading")
axes[2].legend()
fig.tight_layout()

plt.show()