Chapter Summary

Chapter Summary

Key Points

• 1.

  The turbo principle. When a system decomposes into subsystems coupled through an interleaver, efficient SISO decoders for each subsystem can exchange extrinsic log-likelihood ratios iteratively. Only extrinsic information may cross the interleaver, because re-including a-priori or channel inputs causes positive feedback that breaks the iterative dynamics.

• 2.

  Turbo codes as loopy belief propagation. The parallel concatenated convolutional code has a factor graph with two trellis subgraphs joined at the information bits through the interleaver. BCJR on each component is exact sum-product on a tree; the full iterative decoder is loopy BP on the PCCC factor graph with flooding schedule and LLR-encoded messages.

• 3.

  EXIT charts. Under the symmetric-Gaussian model for extrinsic LLRs, a SISO block is summarised by a one-dimensional transfer function $T(I_A; \rho)$ mapping a-priori mutual information to extrinsic mutual information. The iteration $I_A^{(t+1)} = T_2(T_1(I_A^{(t)}))$ converges to $I_A = 1$ if and only if the EXIT tunnel between $T_1$ and $T_2^{-1}$ is open. The turbo cliff is the SNR at which the tunnel first closes.

• 4.

  Turbo equalization. Treating an ISI channel as an inner rate-1 convolutional encoder, a soft MMSE-PIC equalizer exchanges extrinsic LLRs with the outer decoder. Soft interference cancellation uses decoder feedback to subtract the mean of interfering symbols; the LMMSE filter is matched to the residual-interference variance. This architecture approaches matched-filter-bound performance after a handful of iterations.

• 5.

  Iterative MIMO detection. The same soft-cancellation / LMMSE template applies to coded MIMO with a per-layer SISO detector. Variance matching is essential: as decoder feedback sharpens, the per-stream interference variance shrinks and the LMMSE filter becomes closer to matched filtering, opening the EXIT tunnel.

• 6.

  Expectation propagation. EP is a principled generalisation of Gaussian message passing that matches moments of a true posterior marginal under an approximating Gaussian. For MIMO detection EP replaces the LMMSE-PIC Gaussian prior with a moment-matched Gaussian that incorporates the discrete constellation prior, yielding large gains for high-order QAM at moderate complexity.

• 7.

  EXIT-chart design methodology. Iterative-receiver codes are engineered by plotting the detector/equalizer EXIT curve against candidate decoder EXIT curves and selecting a code whose curve squeezes through with a narrow but open tunnel. This shifts the design axis from minimum distance to EXIT-curve matching and is the basis of 3GPP code selection for turbo and LDPC schemes.

• 8.

  Practical limits. EXIT analysis assumes infinite block length, i.i.d. symmetric-Gaussian messages, and negligible cycle effects. Finite-length blocks, short interleavers, and correlated messages introduce a gap of a few tenths of a dB between predicted and simulated thresholds. Error floors are dominated by low-weight codewords and cycle structure, which EXIT analysis cannot see.