Chapter Summary

Chapter Summary

Key Points

• 1.

  The ISI channel $y[n] = \sum_{k=0}^{L} h[k] x[n-k] + w[n]$ is an inference problem on the hidden symbol sequence $\{x[n]\}$. The maximum-likelihood sequence estimator (MLSE) minimizes $\|\mathbf{y} - \mathbf{h} \star \mathbf{x}\|^2$ over $\mathbf{x} \in \mathcal{A}^T$.

• 2.

  The Viterbi algorithm solves MLSE in $O(|\mathcal{A}|^{L} T)$ time by dynamic programming on the channel trellis with $|\mathcal{A}|^{L}$ states — exponential in channel memory, but linear in block length.

• 3.

  Linear equalizers are convex-quadratic and cheap: ZF inverts the channel ($W_{\text{ZF}}(f)=1/H(f)$) and enhances noise wherever $|H(f)|$ is small; MMSE trades residual ISI against noise ($W_{\text{MMSE}}(f)=H^*(f)/(|H(f)|^2 + N_0)$). The MMSE equalizer is the Wiener filter for the equalization problem.

• 4.

  The MMSE-DFE combines a forward filter with a symbol-by-symbol feedback filter that cancels post-cursor ISI. When past decisions are correct, the MMSE-DFE approaches the MFB (matched-filter bound) and closes most of the gap to MLSE. Error propagation is the cost of its feedback structure.

• 5.

  Frequency-selective single-carrier processing is the time-domain dual of OFDM: both diagonalize the channel, but in different bases. The per-subcarrier MMSE equalizer of OFDM is the diagonal form of the MMSE equalizer of this chapter.