---
title: "Interleaving and adaptive equalization of ISI"
subject: "Wireless Communication"
chapter: 5
key: "semester_7/wireless_communication:interleaving-adaptive-equalizer"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/interleaving-adaptive-equalizer
site: BCT Engineering
---

# Interleaving and adaptive equalization of ISI

Wireless Communication · Chapter 5: Equalization and Diversity

- Asked in: 2081 Baisakh, 2+6 marks
- Video: [Wireless Technology | Adaptive Equalizer](https://www.youtube.com/watch?v=04ltaflxFB8) — RANJI RAJ
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/interleaving-adaptive-equalizer
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-5

## Question

Define interleaving. Explain how equalization offsets the ISI introduced by the multipath time-dispersive channel using an example of an adaptive equalizer.

## Short answer

Interleaving rearranges coded bits to spread fade bursts into isolated errors. An adaptive linear equalizer acts as an inverse channel filter, using a training sequence to compute the error and update tap weights via LMS; it then tracks the channel with decision-directed adaptation, periodically retraining to handle time variations.

## Answer

#### Interleaving
**Interleaving** is the process of rearranging coded bits in time before transmission so that burst errors from deep fades are spread into random, isolated errors after de-interleaving, where the channel decoder can correct them.

#### How equalization offsets ISI
In a multipath channel, delayed copies of a symbol overlap with later symbols, causing **inter-symbol interference (ISI)**. If the channel impulse response is f(t), the received signal is y(t) = x(t) ⊛ f(t) + n(t). An **equalizer** h_eq(t) is chosen so that the combined response is an impulse:
```text
f(t) ⊛ h_eq(t) = δ(t)
F(f) · H_eq(f) = 1   →  H_eq(f) = 1 / F(f)
```
So the equalizer is an inverse filter of the channel. Since the mobile channel changes with time and position, the equalizer must be **adaptive**.

#### Example: adaptive linear (transversal) equalizer
```text
y_k ─┬─[T]─┬─[T]─┬─ ... ─[T]─┐
     │     │     │           │
   (x)w₀  (x)w₁ (x)w₂ ...  (x)w_N
     └─────┴────(Σ)──────────┘
                 │ d̂_k
                 ├──────►[Decision]──► x̂_k
                 v              │
     x_k ───►(+)(−)◄────────────┘ (tracking)
     (training)  │ e_k
                 v
          [LMS: w ← w + α e_k y_k]
```
**Operation**
1. **Training mode:** the transmitter sends a known training sequence (e.g. 26-bit midamble in GSM). The receiver compares the equalizer output d̂_k with the known symbol x_k to form the error e_k = x_k − d̂_k.
2. **Weight update:** an adaptive algorithm (e.g. LMS) adjusts the taps to reduce the mean square error:
```text
w_(k+1) = w_k + α e_k y_k
```
   The taps converge near the optimum ŵ = R⁻¹p, so the filter undoes the channel's ISI.
3. **Tracking mode:** during user data, the receiver's own decisions x̂_k replace the training symbols, so the equalizer keeps following slow channel changes.
4. **Retraining:** training is repeated periodically (in every TDMA burst), because over a long time the channel can change too much for decision-directed tracking to follow reliably.

**Example:** a two-path channel with ISI from one earlier symbol (f = [1, 0.5]) needs an inverse filter approximately [1, −0.5, 0.25]; the adaptive equalizer finds these taps automatically from the training sequence.

## Check yourself

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

1. What does interleaving do to a burst of errors caused by a deep fade?
   A. It removes them by adding extra parity bits
   B. It spreads them into isolated single errors after de-interleaving
   C. It lowers the transmit power needed during the fade
   D. It converts them into longer bursts the code can detect
   Answer: B. After de-interleaving the burst bits fall into different code words, so each has few errors that the decoder can correct. (Key point: Purpose of interleaving)

2. What condition defines the equalizer response relative to the channel F(f)?
   A. H_eq(f) = F(f)
   B. H_eq(f) = 0 at all frequencies
   C. H_eq(f) = 1/F(f), so the combined response is an impulse
   D. H_eq(f) = F(f)^2
   Answer: C. The equalizer is the inverse filter of the channel so f(t) convolved with h_eq(t) is delta(t), giving no ISI. (Key point: Equalizer as inverse filter)

3. Why must the equalizer in a mobile channel be adaptive?
   A. The transmit power changes with time
   B. The modulation changes every symbol
   C. The noise is always zero
   D. The channel changes with time and position
   Answer: D. The mobile channel varies as the user moves, so fixed taps would not stay matched. (Key point: Why adaptive)

4. Which tap update is used by the LMS adaptive equalizer?
   A. w(k+1) = w(k) + alpha * e(k) * y(k)
   B. w(k+1) = w(k) - e(k)
   C. w(k+1) = alpha * w(k) / e(k)
   D. w(k+1) = y(k) / e(k)
   Answer: A. LMS moves the weights in proportion to the error times the input, reducing mean square error. (Key point: LMS weight update)

## Related in Wireless Communication

- [Training and tracking modes of adaptive equalizer](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aadaptive-equalizer-training-tracking) — Wireless Communication, chapter 5
- [ISI, RAKE and linear transversal equalizer](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aisi-rake-linear-transversal-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
- [Adaptive equalization algorithms (any two)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aadaptive-equalization-algorithms) — Wireless Communication, chapter 5
- [Parameters of mobile multipath channel](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Amobile-multipath-channel-parameters) — Wireless Communication, chapter 3
- [Path loss and multipath channel parameters](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Apath-loss-multipath-parameters) — Wireless Communication, chapter 3

## Related in other subjects

- [Geometric properties from channel offsets](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aopen-channel-cross-section-from-table-properties) — Hydraulics, chapter 6
- [Interleaving; SSTF, SCAN, C-SCAN order](https://www.bctengineer.com/question.md?key=semester_6%2Foperating_system%3Ainterleaving-winchester-disk-scheduling) — Operating System, chapter 6
- [TDM frame, time slot and 24-channel system](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atdm-frame-24-channel) — Telecommunications, chapter 3
- [Nyquist criteria for zero ISI; duobinary difficulties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-zero-isi-duobinary-difficulties) — Communication Systems, chapter 6