---
title: "Message vs information; 5 symbols per 10 μs"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:message-info-five-symbols-10us"
source: https://www.bctengineer.com/q/semester_6/communication_systems/message-info-five-symbols-10us
site: BCT Engineering
---

# Message vs information; 5 symbols per 10 μs

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2070 Chaitra (CS II), 2+3 marks
- Video: [Calculate Source Entropy and Information Rate | Communication system.](https://www.youtube.com/watch?v=pi1XA8j5dcY) — ELECTRICAL HUB
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/message-info-five-symbols-10us
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

Differentiate between message and information? A discrete source is emitting one of 5 possible symbols per 10 microsec. The probabilities are 1/2, 1/4, 1/8, 1/16 and 1/16. Find (a) Symbol rate (b) Source entropy (c) Information rate

## Short answer

Message is the actual symbol; information is the uncertainty removed, quantified as –log2(p) bits. Symbol rate = 10^5 symbols/s. Entropy = 1.875 bits/symbol. Information rate = 187.5 kbps.

## Answer

#### Message vs information

| Message | Information |
|---|---|
| The actual symbol or sequence produced by the source | Measure of uncertainty removed by receiving the message |
| Has physical form and meaning | Depends only on probability, not meaning |
| Counted in symbols | Measured in bits: $I = \log_2(1/p)$ |
| A certain message is still a message | A certain message ($p = 1$) carries 0 bits |

#### Numerical

One of 5 symbols every $10\ \mu$s with $P = \{\tfrac12, \tfrac14, \tfrac18, \tfrac1{16}, \tfrac1{16}\}$.

**(a) Symbol rate**

$$
r = \frac{1}{10\times10^{-6}} = 10^5\ \text{symbols/s}
$$

**(b) Source entropy**

| Symbol | $p_i$ | $\log_2(1/p_i)$ | $p_i\log_2(1/p_i)$ |
|---|---|---|---|
| x1 | 0.5 | 1.0000 | 0.5000 |
| x2 | 0.25 | 2.0000 | 0.5000 |
| x3 | 0.125 | 3.0000 | 0.3750 |
| x4 | 0.0625 | 4.0000 | 0.2500 |
| x5 | 0.0625 | 4.0000 | 0.2500 |
| **Total** | 1 | | **1.8750** |

$$
\begin{aligned}
H &= \tfrac12(1) + \tfrac14(2) + \tfrac18(3) + \tfrac1{16}(4) + \tfrac1{16}(4) \\
&= 0.5 + 0.5 + 0.375 + 0.25 + 0.25 = 1.875\ \text{bits/symbol}
\end{aligned}
$$

**(c) Information rate**

$$
R = rH = 10^5 \times 1.875 = 1.875\times10^5\ \text{bits/s} = 187.5\ \text{kbps}
$$

**Answer:** $r = 10^5$ symbols/s, $H = 1.875$ bits/symbol, $R = 187.5$ kbps.

## Check yourself

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

1. What is true about information compared with a message?
   A. Information depends on probability, not meaning; a certain message carries 0 bits
   B. Information depends on the meaning of the message
   C. A certain message carries 1 bit
   D. Information is counted in symbols
   Answer: A. I = log2(1/p), so p = 1 gives 0 bits. (Key point: Message vs information)

2. For P = {1/2, 1/4, 1/8, 1/16, 1/16}, what is H?
   A. 1.5 bits
   B. 1.875 bits
   C. 2.0 bits
   D. 2.32 bits
   Answer: B. 0.5 + 0.5 + 0.375 + 0.25 + 0.25 = 1.875 bits/symbol. (Key point: Entropy)

3. With r = 10^5 symbols/s, what is the information rate?
   A. 187.5 kbps
   B. 100 kbps
   C. 232 kbps
   D. 18.75 kbps
   Answer: A. R = rH = 10^5 x 1.875 = 187.5 kbps. (Key point: Information rate)

## Related in Communication Systems

- [Symbol rate, entropy and information rate (6 symbols per 10 μs)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asymbol-rate-six-symbols-10us) — Communication Systems, chapter 6
- [Message vs information; entropy](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amessage-vs-information-entropy) — Communication Systems, chapter 6
- [Entropy and information rate of 6-level PCM signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aentropy-info-rate-5khz-six-levels) — Communication Systems, chapter 6
- [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

## Related in other subjects

- [Entropy and information rate, 5 outcomes](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aentropy-info-rate-5-outcomes) — Data Communication, chapter 8