Chapter Summary

Chapter Summary

Key Points

• 1.

  Set partitioning is the geometric core of TCM. An Ungerboeck partition of a $2^{m+1}$-point constellation is a chain of successive binary splits in which each split is chosen to maximize the intra-subset minimum squared Euclidean distance $\Delta_i^2$. For 8-PSK: $\Delta_0^2 \approx 0.586 \to \Delta_1^2 = 2 \to \Delta_2^2 = 4 \to \Delta_3^2 = \infty$; for 16-QAM: $4 \to 8 \to 16 \to 32$ (clean doubling from the 2D lattice structure). Every level of splitting at least doubles the intra-subset MSED, and this geometric amplification is the source of TCM's coding gain.

• 2.

  Mapping by set partitioning separates coded from uncoded bits. A TCM scheme uses a rate-$\tilde{m}/(\tilde{m}+1)$ convolutional code to select a level-$(\tilde{m}+1)$ coset, then uses $m - \tilde{m}$ uncoded bits to select a point inside the coset. The coset-selecting bits are the ones that need coding protection; the uncoded bits ride the large intra-subset MSED for free. The result is coding gain without any bandwidth expansion: the constellation size doubles from $2^m$ to $2^{m+1}$ at fixed baud rate and fixed bit rate, absorbing the code rate.

• 3.

  Free Euclidean distance obeys a two-term lower bound. For any Ungerboeck TCM, $d_{\rm free}^2 \geq \min\{\Delta_{\tilde{m}+1}^2,\; d_{\rm free}^{(H)} \cdot \Delta_1^2\}$ — the first term bounds single-symbol parallel-transition errors, the second bounds longer error events via the Hamming distance of the convolutional code times the level-1 MSED. A good design balances the two terms; the celebrated 4-state 8-PSK TCM has them both at $d_{\rm free}^2 = 4$, giving $\gamma_c^{\rm dB} = 3.0$ dB over uncoded QPSK.

• 4.

  The three Ungerboeck design rules pick the label assignment. (R1) parallel transitions carry deepest-level coset points (same level-$(\tilde{m}+1)$ coset); (R2) transitions from or into a single state carry next-level coset points (same level-$\tilde{m}$ coset); (R3) uniform constellation usage. These rules jointly ensure $d_{\rm free}^2 \geq \Delta_{\tilde{m}+1}^2$ and are the direct outputs of the computer search that produced Ungerboeck's 1982 Tables I–III.

• 5.

  Viterbi + subset decoding is the ML decoder. For each received sample and each level-$(\tilde{m}+1)$ coset, the subset decoder finds the nearest in-coset point and records its squared Euclidean distance. Viterbi then runs over the state-pair trellis using those coset-level branch metrics. The per-symbol complexity is $\mathcal{O}(N_s \cdot 2^{\tilde{m}})$ ACS operations plus $\mathcal{O}(2^{m+1})$ distance evaluations. Traceback depth $\sim 5\nu$ to $7\nu$ keeps the bit-error floor tight.

• 6.

  Asymptotic coding gain is $\gamma_c = d_{\rm free}^2 / d_{\rm uncoded}^2$. At high SNR, the BER curve shifts horizontally by $10 \log_{10}(d_{\rm free}^2 / d_{\rm uncoded}^2)$ dB relative to the uncoded baseline at the same spectral efficiency. For 2D TCM this gain is 3–6 dB (the upper end reached by 64–256 state trellises); larger gains require higher-dimensional constellations as in V.34's 4D TCM. Shaping gain ($\leq 1.53$ dB, DShaping Gain (Preview)) is a separate quantity, added via constellation-shaping techniques in Chapter 4.

• 7.

  V.32/V.34 modems were TCM made flesh. V.32 (1984, 9.6 kbit/s) used 8-state 32-CROSS TCM for 4 dB gain over uncoded 16-QAM. V.34 (1994, 33.6 kbit/s) combined 16-state 4D TCM with shell-mapping shaping to reach within 1 dB of Shannon capacity on a typical telephone line. The V.34 modem was, for a decade, the most sophisticated commercial communications system in the world — and Ungerboeck's 1982 paper was the single most influential theoretical input to its design.

• 8.

  BICM supplanted TCM in wireless; Ungerboeck's partition idea lives on in probabilistic shaping. The interleaver in BICM converts a code's Hamming distance into diversity order, which beats TCM on fading channels (Caire–Taricco–Biglieri 1998). TCM remains optimal on AWGN-dominant links (satellite, optical), and the set-partitioning idea is alive in modern probabilistic amplitude shaping (PAS) for DVB-S2X and high-capacity optical transponders — the partition now carries the shaping gain instead of the coding gain, but the geometric insight is the same.