---
title: "Short note: Convolutional encoding and decoding"
subject: "Wireless Communication"
chapter: 6
key: "semester_7/wireless_communication:convolutional-coding"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/convolutional-coding
site: BCT Engineering
---

# Short note: Convolutional encoding and decoding

Wireless Communication · Chapter 6: Speech and Channel Coding

- Asked in: 2074 Bhadra, 5 marks
- Video: [Convolutional Codes (Basics, Designing & Parameters) Explained in Digital Communication](https://www.youtube.com/watch?v=hPNfiFmx5Dk) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/convolutional-coding
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-6

## Question

Write a short note on convolutional encoding and decoding.

## Short answer

Convolutional coding uses shift-register memory; the example (2,1,3) encoder with generators 111 and 101 gives output 11 10 00 01 01 11 for input 1011. Decoding uses the Viterbi algorithm on a trellis with Hamming branch metrics and survivor paths, correcting single errors.

## Answer

**Convolutional coding** is a forward error-correction method in which each group of *k* input bits produces *n* output bits that depend on the current input and the previous (K − 1) inputs stored in a shift register. It is described by (n, k, K), with code rate r = k/n and constraint length K.

**Encoding** – example (2, 1, 3) encoder, rate 1/2, generators g₁ = 111, g₂ = 101:

```text
 input ->[ m ]->[ s1 ]->[ s2 ]
          |       |       |
          +--(⊕)--+--(⊕)--+---> v1 = m ⊕ s1 ⊕ s2
          |               |
          +------(⊕)------+---> v2 = m ⊕ s2
          output: v1 v2 (alternately)
```

Input 1011 followed by two flushing zeros gives output **11 10 00 01 01 11**. The encoder can be shown as a state diagram, a tree, or a **trellis** (states = 2ᴷ⁻¹ = 4).

**Decoding** – the **Viterbi algorithm** (maximum likelihood) is standard:
1. For each received bit pair, compute branch metrics (Hamming distance) on the trellis.
2. At each state, add-compare-select: keep only the survivor path with the smallest total metric.
3. At the end, trace back from state 00 to get the decoded bits.

Example: received 11 10 **1**0 01 01 11 (one error) is decoded back to 1011, as the best path has metric 1. Soft-decision decoding gives about 2 dB extra gain. Sequential decoding is an alternative for long K.

Used in GSM (rate 1/2, K = 5), IS-95 (K = 9) and satellite links.

## Check yourself

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

1. In a (2,1,3) encoder with g1 = 111 and g2 = 101, what is v2?
   A. m xor s1 xor s2
   B. m xor s1
   C. m xor s2
   D. s1 xor s2
   Answer: C. g2 = 101 taps the input m and the last register s2. (Key point: Encoder output)

2. For the (2,1,3) code, what are the rate and number of states?
   A. Rate 2, 3 states
   B. Rate 1/2, 4 states
   C. Rate 1/3, 8 states
   D. Rate 1/2, 8 states
   Answer: B. r = k/n = 1/2 and states = 2^(K-1) = 4. (Key point: Code rate and states)

3. Which method is the standard maximum-likelihood decoder?
   A. Viterbi algorithm
   B. Zero forcing
   C. RAKE
   D. LMS
   Answer: A. The Viterbi algorithm finds the minimum-metric trellis path. (Key point: Decoding method)

4. How much extra gain does soft-decision decoding give approximately?
   A. 0 dB
   B. About 20 dB
   C. About 2 dB
   D. About 10 dB
   Answer: C. Soft decision gives about 2 dB over hard decision. (Key point: Soft decision gain)

5. For received 11 10 10 01 01 11 (one error), what is the best path metric and decoded data?
   A. Metric 0, data 1011
   B. Metric 1, data 1011
   C. Metric 2, data 1101
   D. Metric 1, data 1100
   Answer: B. The best path differs from the received sequence in one bit and decodes to 1011. (Key point: Error correction example)

## Related in Wireless Communication

- [CDMA encoding and decoding](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acdma-encoding-decoding) — Wireless Communication, chapter 7, asked 2×

## Related in other subjects

- [Convolutional encoding of 10011 with state and trellis](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aconvolutional-encoder-10011) — Communication Systems, chapter 8
- [Weighted Huffman encoding and decoding](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahuffman-jasta-lai) — Data Communication, chapter 8
- [Trellis decoding of 1/2 rate conv. code (3-bit errors)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconv-trellis-3bit) — Data Communication, chapter 8, asked 2×
- [Trellis decoding of 1/2 rate conv. code (2-bit errors)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconv-trellis-2bit) — Data Communication, chapter 8, asked 2×