---
title: "Information rate and coding efficiency: BCD vs Huffman"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:bcd-source-info-rate-huffman"
source: https://www.bctengineer.com/q/semester_6/communication_systems/bcd-source-info-rate-huffman
site: BCT Engineering
---

# Information rate and coding efficiency: BCD vs Huffman

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2072 Chaitra (CS II), 5 marks
- Video: [Huffman Coding (Basics, Algorithm, Procedure & Example) Explained in Digital Communication](https://www.youtube.com/watch?v=HmBH30NrM7c) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/bcd-source-info-rate-huffman
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

If a source emits symbols Xi = {A, B, C, D, E, F} in the BCD format with probabilities P(Xi) = {0.3, 0.1, 0.02, 0.15, 0.4, 0.03} at a rate Rs = 14.4 kbaud, find the following:
i) Information rate
ii) Coding efficiency both with BCD and Huffman coded signal

## Short answer

Entropy H = 2.0572 bits/symbol. Information rate = 14400 × 2.0572 ≈ 29.62 kbps. BCD uses 4 bits per symbol, efficiency = 2.0572/4 = 51.43%. Huffman coding yields average length 2.10 bits/symbol, efficiency = 2.0572/2.10 = 97.96%.

## Answer

**Given:** six symbols A–F with $P = \{0.3, 0.1, 0.02, 0.15, 0.4, 0.03\}$ (sum = 1); symbol rate $r_s = 14.4$ kbaud = 14 400 symbols/s; BCD format = 4 bits per symbol.

**Entropy:**

| Symbol | $p_i$ | $\log_2(1/p_i)$ | $p_i\log_2(1/p_i)$ |
|---|---|---|---|
| A | 0.3 | 1.7370 | 0.5211 |
| B | 0.1 | 3.3219 | 0.3322 |
| C | 0.02 | 5.6439 | 0.1129 |
| D | 0.15 | 2.7370 | 0.4105 |
| E | 0.4 | 1.3219 | 0.5288 |
| F | 0.03 | 5.0589 | 0.1518 |
| **Total** | 1 | | **2.0572** |

$$
H = 2.0572\ \text{bits/symbol}
$$

#### i) Information rate

$$
R = r_s H = 14\ 400 \times 2.0572 = 29\ 624.29\ \text{bits/s} \approx 29.62\ \text{kbps}
$$

#### ii) Coding efficiency

**BCD coding:** each symbol uses 4 bits, $\bar L = 4$.

$$
\eta_{BCD} = \frac{2.0572}{4} = 51.43\%
$$

(The line bit rate would be $14\ 400 \times 4 = 57.6$ kbps.)

**Huffman coding:**

**Reduction (sum of two lowest moved up as high as possible):**

| Stage | Probabilities (descending) | Combined |
|---|---|---|
| 1 | 0.4, 0.3, 0.15, 0.1, 0.03, 0.02 | 0.03 + 0.02 = 0.05 |
| 2 | 0.4, 0.3, 0.15, 0.1, 0.05 | 0.1 + 0.05 = 0.15 |
| 3 | 0.4, 0.3, 0.15, 0.15 | 0.15 + 0.15 = 0.3 |
| 4 | 0.4, 0.3, 0.3 | 0.3 + 0.3 = 0.6 |
| 5 | 0.6, 0.4 | 0.6 + 0.4 = 1 |

| Symbol | $p_i$ | Huffman code | $l_i$ |
|---|---|---|---|
| E | 0.4 | 1 | 1 |
| A | 0.3 | 01 | 2 |
| D | 0.15 | 001 | 3 |
| B | 0.1 | 0000 | 4 |
| F | 0.03 | 00010 | 5 |
| C | 0.02 | 00011 | 5 |

$$
\begin{aligned}
\bar L_H &= 0.4(1)+0.3(2)+0.15(3)+0.1(4)+0.03(5)+0.02(5) \\
&= 0.4+0.6+0.45+0.4+0.15+0.1 = 2.10\ \text{bits/symbol} \\
\eta_H &= \frac{2.0572}{2.10} = 97.96\%
\end{aligned}
$$

Bit rate with Huffman coding: $14\ 400 \times 2.1 = 30.24$ kbps.

**Answer:** $R \approx 29.62$ kbps; efficiency with BCD = **51.43 %**, with Huffman = **97.96 %**.

## Check yourself

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

1. How is the information rate computed?
   A. R = rs x H
   B. R = rs x L
   C. R = H / rs
   D. R = rs x log2(6)
   Answer: A. Information rate = symbol rate times entropy: 14 400 x 2.0572 = 29.62 kbps. (Key point: Information rate formula)

2. BCD uses 4 bits per symbol. With H = 2.0572, what is its coding efficiency?
   A. 51.43%
   B. 68.6%
   C. 97.96%
   D. 25.7%
   Answer: A. 2.0572/4 = 51.43%. (Key point: BCD efficiency)

3. In the Huffman reduction, which two probabilities are merged first for this source?
   A. 0.4 and 0.3
   B. 0.03 and 0.02
   C. 0.15 and 0.1
   D. 0.1 and 0.03
   Answer: B. Huffman always merges the two lowest probabilities: 0.03 + 0.02 = 0.05. (Key point: Huffman reduction step)

4. The Huffman average length is 2.10 bits. What is the efficiency?
   A. 97.96%
   B. 51.43%
   C. 85%
   D. 100%
   Answer: A. 2.0572/2.10 = 97.96%. (Key point: Huffman result)

## Related in Communication Systems

- [Importance of source coding; Huffman for eight symbols at 1000 symbols/s](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-coding-huffman-eight-1000) — Communication Systems, chapter 6
- [Importance of source coding; Shannon-Fano vs BCD (four symbols)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-coding-sf-bcd-four) — Communication Systems, chapter 6
- [Source vs channel coding; Huffman for five symbols](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-channel-huffman-five) — Communication Systems, chapter 6
- [Source coding; Huffman for five symbols (0.3, 0.25, 0.2, 0.15, 0.1)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-coding-huffman-five-s0) — Communication Systems, chapter 6
- [Importance of source coding; Shannon-Fano vs fixed length (seven symbols)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-coding-sf-fixed-seven) — Communication Systems, chapter 6
- [Importance of source coding; fixed vs Shannon-Fano (five symbols)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asource-coding-fixed-sf-five) — Communication Systems, chapter 6

## Related in other subjects

- [Huffman code for symbols A to E](https://www.bctengineer.com/question.md?key=semester_8%2Fmultimedia_system%3Ahuffman-numerical-abcde) — Multimedia System, chapter 5
- [Huffman code for symbols A0 to A4](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahuffman-a0-a4) — Data Communication, chapter 8
- [Huffman code for symbols A to E (0.4, 0.3, ...)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahuffman-table-abcde) — Data Communication, chapter 8
- [Huffman code for five symbols (0.2, 0.35, ...)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahuffman-five-symbols-02-035) — Data Communication, chapter 8