---
title: "Adaptive equalization algorithms (any two)"
subject: "Wireless Communication"
chapter: 5
key: "semester_7/wireless_communication:adaptive-equalization-algorithms"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/adaptive-equalization-algorithms
site: BCT Engineering
---

# Adaptive equalization algorithms (any two)

Wireless Communication · Chapter 5: Equalization and Diversity

- Asked in: 2073 Magh, 4 marks
- Video: [Adaptive Equalization](https://www.youtube.com/watch?v=ktIYXkPDx0U) — J. Jayalakshmi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/adaptive-equalization-algorithms
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-5

## Question

Explain briefly adaptive equalization algorithms (any two).

## Short answer

LMS updates taps as w(k+1)=w(k)+μ e(k)x(k) to minimize mean square error; it is simple but converges slowly when input eigenvalue spread is large. RLS recursively minimizes weighted squared error using an inverse correlation matrix, converging much faster but with O(N²) complexity and possible numerical instability.

## Answer

An **adaptive equalization algorithm** updates the tap weights w of the equalizer so that the error e(k) = d(k) − d̂(k) between the desired and actual output becomes small. Two common ones:

#### 1. Least Mean Square (LMS) algorithm

- Minimises the mean square error E[e²(k)] using a stochastic gradient (steepest descent) step.
- Update equations:

```text
 d^(k)  = wᵀ(k)·x(k)          (filter output)
 e(k)   = d(k) - d^(k)        (error)
 w(k+1) = w(k) + μ·e(k)·x(k)  (weight update)
```

- μ is the step size: larger μ gives faster convergence but more misadjustment; stability needs 0 < μ < 2/λ_max.
- Needs only about 2N+1 operations per iteration (N = taps), so it is simple and robust.
- Disadvantage: slow convergence when the eigenvalue spread of the input correlation matrix is large.

#### 2. Recursive Least Squares (RLS) algorithm

- Minimises the weighted sum of past squared errors Σ λⁿ⁻ⁱ e²(i), where λ (0 < λ ≤ 1) is a forgetting factor.
- Recursively updates the inverse correlation matrix P(n) and a gain vector k(n):

```text
 k(n)   = P(n-1)x(n) / [λ + xᵀ(n)P(n-1)x(n)]
 e(n)   = d(n) - wᵀ(n-1)x(n)
 w(n)   = w(n-1) + k(n)·e(n)
 P(n)   = [P(n-1) - k(n)xᵀ(n)P(n-1)] / λ
```

- Converges about an order of magnitude faster than LMS and is insensitive to eigenvalue spread.
- Needs about 2.5N² + 4.5N operations per iteration and can be numerically unstable.

| Point | LMS | RLS |
|---|---|---|
| Criterion | Mean square error | Weighted least squares |
| Convergence | Slow | Fast |
| Complexity | ~2N+1 | ~2.5N²+4.5N |
| Stability | Good | Can be unstable |

(Zero forcing is a third simple algorithm that makes the combined channel–equalizer response have zero ISI.)

## Check yourself

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

1. What does LMS minimise and how?
   A. Weighted sum of past errors by matrix inversion
   B. Peak distortion exactly
   C. Mean square error by a stochastic gradient step
   D. Bit error rate directly
   Answer: C. LMS uses steepest descent on E[e^2]. (Key point: LMS criterion)

2. Compared with LMS, RLS...
   A. converges faster and is insensitive to eigenvalue spread but is costlier
   B. converges slower and cheaper
   C. has the same complexity
   D. is always numerically stable
   Answer: A. RLS needs about 2.5N^2 + 4.5N operations and can be numerically unstable. (Key point: RLS advantages)

3. What is lambda in RLS?
   A. Step size
   B. Forgetting factor weighting past errors, 0 < lambda <= 1
   C. Number of taps
   D. Noise power
   Answer: B. It weights older squared errors less. (Key point: Forgetting factor)

## Related in Wireless Communication

- [Interleaving and adaptive equalization of ISI](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Ainterleaving-adaptive-equalizer) — Wireless Communication, chapter 5
- [Short note: Adaptive equalization](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aadaptive-equalization-note) — Wireless Communication, chapter 5

## Related in other subjects

- [Adaptive and non-adaptive routing algorithms](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aadaptive-nonadaptive-routing) — Telecommunications, chapter 8, asked 3×