---
title: "MLSE equalization"
subject: "Wireless Communication"
chapter: 5
key: "semester_7/wireless_communication:mlse-equalization"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/mlse-equalization
site: BCT Engineering
---

# MLSE equalization

Wireless Communication · Chapter 5: Equalization and Diversity

- Asked in: 2071 Magh, 4 marks
- Video: [GEL7114 Module 4.1 Maximum Likelihood Sequence Estimation (MLSE)](https://www.youtube.com/watch?v=tL8lTuQR3dY) — Leslie Rusch
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/mlse-equalization
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-5

## Question

Explain with block diagram the concept of Maximum Likelihood Sequence Estimation equalization.

## Short answer

MLSE equalization processes the received signal through a matched filter and channel estimator. It forms a trellis of M^L states based on ISI memory and uses the Viterbi algorithm to select the sequence minimizing accumulated squared error. This provides optimum detection but complexity grows exponentially with channel memory length.

## Answer

**Maximum Likelihood Sequence Estimation (MLSE) equalization** is a non-linear equalizer that decides on the most probable **sequence** of transmitted symbols, rather than one symbol at a time. It is optimum for channels with ISI and white Gaussian noise.

```text
 r(t) +---------+ y(k) +----------------+  a^(k)
 ---->| Matched |----->| MLSE: Viterbi  |------->
      | filter  |      | trellis search |  data
      +---------+      +----------------+
           ^                   ^
           |   +-----------+   |
           +---| Channel   |---+
               | estimator |  (CIR h0..hL from
               +-----------+   training sequence)
```

**Working**

1. The **matched filter** maximises SNR and its output is sampled once per symbol to give y(k).
2. The **channel estimator** uses the known training sequence to estimate the discrete channel impulse response h₀ … h_L.
3. Because ISI comes from the last *L* symbols, the channel behaves like a finite-state machine with Mᴸ states, drawn as a **trellis**.
4. For every path in the trellis the expected output ŷ(k) = Σ hⱼ·a(k−j) is formed and the metric |y(k) − ŷ(k)|² is accumulated.
5. The **Viterbi algorithm** keeps only one survivor (lowest metric) per state at each step, so the search is efficient.
6. The surviving path with the least total metric is the maximum-likelihood sequence and is output.

**Merits:** minimum sequence error probability; handles deep spectral nulls better than linear equalizers.
**Limitation:** complexity grows exponentially with channel memory (Mᴸ states). GSM receivers use a Viterbi (MLSE) equalizer.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does an MLSE equalizer decide?
   A. The most probable sequence of transmitted symbols
   B. Each symbol independently using a threshold
   C. The inverse of the channel response
   D. The noise variance at each sample
   Answer: A. MLSE picks the most probable whole sequence rather than deciding one symbol at a time. (Key point: Sequence decision)

2. For an M-ary alphabet and ISI spanning the last L symbols, how many trellis states does MLSE have?
   A. M x L
   B. L^M
   C. M^L
   D. 2L
   Answer: C. The channel acts as a finite-state machine with M^L states, which is why complexity grows exponentially with memory. (Key point: Number of trellis states)

3. Why does the MLSE receiver need a channel estimator?
   A. To amplify the received signal
   B. To estimate the impulse response h0..hL from the training sequence so expected outputs can be computed
   C. To generate the PN code
   D. To sample the matched filter faster
   Answer: B. The estimated CIR is used to form the expected output for each trellis path, which is compared with the received sample. (Key point: Role of channel estimator)

4. How does the Viterbi algorithm keep MLSE search efficient?
   A. It tests every possible sequence fully
   B. It keeps only one survivor with the lowest metric per state at each step
   C. It ignores the metric and uses the last symbol
   D. It discards all paths except the first
   Answer: B. Keeping one lowest-metric survivor per state avoids exhaustive search over all sequences. (Key point: Viterbi survivor)

## Related in Wireless Communication

- [Interleaving and MLSE](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Ainterleaving-mlse) — Wireless Communication, chapter 5
- [MLSE and time diversity](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Amlse-time-diversity) — Wireless Communication, chapter 5
- [Equalization fundamentals and RAKE receiver](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aequalization-fundamentals-rake) — Wireless Communication, chapter 5
- [Direct sequence spread spectrum (DSSS)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adsss-block-diagram) — Wireless Communication, chapter 4

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