Summary

Chapter 13 Summary: Equalization

Key Points

• 1.

  ISI arises from frequency-selective channels. When the channel impulse response spans multiple symbol periods, each received sample contains contributions from neighbouring symbols. The eye diagram visualises this degradation: ISI closes the eye, reducing noise margin and making symbol-by-symbol detection unreliable.

• 2.

  Linear equalizers trade ISI suppression for noise enhancement. The zero-forcing (ZF) equalizer inverts the channel ($W = 1/H$) and eliminates ISI but amplifies noise at spectral nulls. The MMSE equalizer ($W = H^*/(|H|^2 + N_0/E_s)$) balances ISI and noise, converging to ZF at high SNR and to the matched filter at low SNR.

• 3.

  The DFE cancels postcursor ISI without noise penalty. By feeding back past hard decisions through a second filter, the DFE avoids the noise enhancement that limits linear equalizers. Its Achilles heel is error propagation: one wrong decision can trigger a burst of subsequent errors.

• 4.

  MLSE via the Viterbi algorithm is the optimal receiver for ISI channels with AWGN, minimising sequence error probability. The ISI channel maps to a trellis with $M^L$ states, and the Viterbi algorithm finds the minimum-distance path in $O(K M^{L+1})$ time. Complexity grows exponentially with channel memory $L$.

• 5.

  Adaptive equalization enables operation on unknown, time-varying channels. The LMS algorithm provides $O(N_f)$ per-symbol adaptation via stochastic gradient descent, while RLS achieves faster convergence at $O(N_f^2)$ cost. Practical systems use a training preamble for initial convergence followed by decision-directed tracking.

• 6.

  Equalizer choice depends on channel and complexity constraints. OFDM avoids time-domain equalization entirely by converting the frequency-selective channel into flat subchannels. For single-carrier systems, the choice ranges from simple linear MMSE (low complexity) through DFE (moderate gains) to MLSE (optimal but exponential complexity).