DVB-S: the error rate after the Viterbi decoder against Table 3

comnum.Dvbs implements the DVB-S transmitter of ETSI EN 300 421 V1.1.2 and a receiver aligned on the burst: the energy dispersal of the MPEG-2 transport stream (the PRBS \(1 + X^{14} + X^{15}\) reset every 8 packets), the outer code RS(204, 188), the convolutional interleaver of 12 branches (depths \(17 j\)), the inner code of constraint length 7 (\(X\) from \(171_8\), \(Y\) from \(133_8\)) punctured to 2/3, 3/4, 5/6 or 7/8 by Table 2, and the Gray QPSK of Figure 5.

Table 3 gives the \(E_b/N_0\) per useful bit (before the outer code) at which the BER after the Viterbi decoder is \(2 \cdot 10^{-4}\), the input of the outer decoder for a quasi-error-free output: 4.5, 5.0, 5.5, 6.0 and 6.4 dB from 1/2 to 7/8, a modem implementation margin of 0.8 dB included. An ideal modem should reach the BER at Table 3 minus 0.8 dB. The card dvbs_table3 measures it with \(10^6\) bits per point at 3.60, 4.04, 4.50, 5.00 and 5.38 dB, 0.10 to 0.22 dB below. This script draws the waterfalls with fewer bits.

The bits on I and Q. The punctured bits leave in time order, \(X\) before \(Y\) at every step, and alternate between I and Q: at rate 3/4, I carries \(X_1 Y_2\) and Q carries \(Y_1 X_3\), the rows of Table 2.

import matplotlib.pyplot as plt
import numpy as np

import comnum
from comnum.plot import BACKGROUND, BLUE, GREY, INK, PURPLE, RULE, RUST, TEAL

The BER after the Viterbi decoder: twelve bursts of 100 packets per point through the whole chain, the errors counted on the bytes entering the inner coder.

rng, noise = comnum.Rng(421), np.random.default_rng(421)
table3 = {"1/2": 4.5, "2/3": 5.0, "3/4": 5.5, "5/6": 6.0, "7/8": 6.4}
colors = [BLUE, RUST, TEAL, PURPLE, GREY]
fig, ax = plt.subplots()
for (rate, required), color in zip(table3.items(), colors, strict=True):
    link = comnum.Dvbs(rate)
    ebn0 = np.arange(required - 1.6, required - 0.39, 0.2)
    ber = []
    for e in ebn0:
        noise_var = 1 / comnum.db2lin(link.esn0_db(e))
        errors = bits = 0
        for _ in range(12):  # about 2.2e6 bits per point
            tx = link.outer(comnum.dvb_packets(100, rng))
            x = link.inner(tx)
            w = (noise.normal(size=len(x)) + 1j * noise.normal(size=len(x))) * np.sqrt(
                noise_var / 2
            )
            rx = link.inner_decode(x + w, noise_var)
            errors += int(np.unpackbits(tx ^ rx).sum())
            bits += 8 * len(tx)
        ber.append(errors / bits if errors else np.nan)
    ax.semilogy(
        ebn0, ber, marker="o", color=color, markeredgecolor=BACKGROUND, label=f"rate {rate}"
    )
    ax.plot(required - 0.8, 2e-4, marker="*", markersize=12, color=color, markeredgecolor=INK)
ax.axhline(2e-4, color=RULE, linewidth=0.8)
ax.annotate("Table 3 − 0.8 dB (★)", (3.0, 2.6e-4), color=INK, fontsize=8)
ax.set(xlabel="Eb/N0 per useful bit (dB)", ylabel="BER after the Viterbi decoder", ylim=(1e-6, 1e-2),
       title="DVB-S, AWGN, soft Viterbi decoding")  # fmt: skip
ax.legend(loc="lower left")
fig.tight_layout()

The Gray QPSK of Figure 5, label \(b_I b_Q\).

c = comnum.Dvbs("1/2").constellation
p = np.asarray(c.points)
fig, ax = plt.subplots(figsize=(4.5, 4.5))
ax.plot(p.real, p.imag, "o", color=BLUE, markeredgecolor=BACKGROUND)
for label, z in enumerate(p):
    ax.annotate(format(label, "02b"), (z.real, z.imag), textcoords="offset points", xytext=(0, 8),
                ha="center", color=INK)  # fmt: skip
ax.axhline(0, color=RULE, linewidth=0.6)
ax.axvline(0, color=RULE, linewidth=0.6)
ax.set(aspect="equal", xlim=(-1.2, 1.2), ylim=(-1.2, 1.2), xlabel="I", ylabel="Q",
       title="DVB-S QPSK (Figure 5)")  # fmt: skip
fig.tight_layout()