---
title: "Duobinary error; entropy; capacity as B → ∞"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:duobinary-entropy-capacity-infinite-b"
source: https://www.bctengineer.com/q/semester_6/communication_systems/duobinary-entropy-capacity-infinite-b
site: BCT Engineering
---

# Duobinary error; entropy; capacity as B → ∞

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2081 Baisakh (CS II), 2+2+4 marks
- Video: [Channel Capacity by Shannon Hartley Explained | Equation, Proof, and Maximum Capacity Condition](https://www.youtube.com/watch?v=lQHjREaGJVc) — 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/duobinary-entropy-capacity-infinite-b
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

Mention how can we minimize error in duo-binary coding. Define information and entropy. Calculate the upper limit of the channel capacity as the bandwidth of the channel (B) tends to infinity.

## Short answer

Duobinary error propagation is minimized by precoding dk = bk ⊕ dk-1, enabling symbol-by-symbol detection. Information I = log2(1/p) bits; entropy H = -∑ p log2 p. As B → ∞, channel capacity C∞ = (S/N0) log2 e ≈ 1.44 S/N0 bits/s, limited by noise.

## Answer

#### Minimizing error in duo-binary coding

In duobinary signalling the received sample is $c_k = a_k + a_{k-1}$, so the decoder normally finds the present bit by subtracting the previous decision: $\hat a_k = c_k - \hat a_{k-1}$. One wrong decision then spoils all the following decisions. This is called **error propagation**. It is reduced or removed by:

- **Precoding:** before the duobinary filter, form $d_k = b_k \oplus d_{k-1}$ (modulo-2 sum). The receiver can then decide each bit from the present sample alone: if $|c_k| < 1$ (i.e. $c_k = 0$) then $\hat b_k = 1$; if $|c_k| > 1$ ($c_k = \pm 2$) then $\hat b_k = 0$. An error stays in one bit only.
- Using the correct decision thresholds ($\pm 1$) for the three-level signal and good receive filtering to keep noise low.

#### Information and entropy

- **Information:** the information carried by a message $m_k$ of probability $p_k$ is $I_k = \log_2 \frac{1}{p_k}$ bits. A less likely message carries more information; a certain message ($p_k = 1$) carries none.
- **Entropy:** the average information per symbol of a source with $M$ symbols:

$$
H = \sum_{k=1}^{M} p_k \log_2 \frac{1}{p_k}\ \text{bits/symbol}, \qquad 0 \le H \le \log_2 M
$$

#### Upper limit of channel capacity as $B \to \infty$

By the Shannon–Hartley theorem, with signal power $S$ and white noise of one-sided PSD $N_0$ (noise power $N = N_0 B$):

$$
C = B\log_2\left(1 + \frac{S}{N_0 B}\right)
$$

Multiply and divide by $S/N_0$ and put $x = \dfrac{S}{N_0 B}$:

$$
\begin{aligned}
C &= \frac{S}{N_0}\cdot\frac{N_0 B}{S}\log_2\left(1+\frac{S}{N_0B}\right) = \frac{S}{N_0}\cdot\frac{1}{x}\log_2(1+x) \\
&= \frac{S}{N_0}\log_2 (1+x)^{1/x}
\end{aligned}
$$

As $B \to \infty$, $x \to 0$ and $\lim_{x\to 0}(1+x)^{1/x} = e$. Therefore

$$
C_\infty = \lim_{B\to\infty} C = \frac{S}{N_0}\log_2 e = 1.44\,\frac{S}{N_0}\ \text{bits/s}
$$

So increasing bandwidth does not give unlimited capacity, because noise power also grows with $B$. Capacity saturates at $1.44\,S/N_0$.

Putting $S = E_b C$ at the limit gives $\dfrac{E_b}{N_0} = \ln 2 = 0.693$, i.e. **−1.6 dB** (the Shannon limit): no system can communicate error-free below this $E_b/N_0$.

**Answer:** $C_\infty = 1.44\,S/N_0$ bits/s.

## Check yourself

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

1. Why do errors propagate in duobinary decoding without precoding?
   A. Each decision uses the previous decision (a_k = c_k - a_{k-1})
   B. The filter is unstable
   C. Noise is correlated
   D. The sampling is late
   Answer: A. One wrong decision corrupts all following ones; precoding removes this. (Key point: Error propagation)

2. How does precoding help?
   A. d_k = b_k XOR d_{k-1}, so each bit is decided from the present sample alone
   B. It removes noise
   C. It increases the levels to five
   D. It doubles bandwidth
   Answer: A. After precoding, c_k = 0 means b=1 and c_k = +-2 means b=0, so errors stay in one bit. (Key point: Precoding)

3. In the B-to-infinity derivation, which limit is used?
   A. lim 1/x = 0
   B. lim (1+x)^x = e
   C. lim x log x = 0
   D. lim (1+x)^(1/x) = e as x -> 0
   Answer: D. With x = S/(N0 B), C = (S/N0) log2 (1+x)^(1/x) which tends to (S/N0) log2 e. (Key point: Limit step)

4. What is the upper limit of capacity as B tends to infinity?
   A. 1.44 S/N0
   B. 0.693 S/N0
   C. S/N0 squared
   D. Infinite
   Answer: A. C_inf = (S/N0) log2 e = 1.44 S/N0 because noise power grows with B. (Key point: Result)

## Related in Communication Systems

- [Information and entropy; capacity as bandwidth → ∞](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ainformation-entropy-capacity-infinite) — Communication Systems, chapter 6
- [Entropy and information rate; Shannon limit as bandwidth → ∞](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aentropy-info-rate-shannon-limit) — Communication Systems, chapter 6
- [Correlative coding; minimizing error in duobinary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acorrelative-coding-duobinary-error) — Communication Systems, chapter 6
- [Shannon-Hartley theorem; capacity limit 1.44S/N₀](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-1-44) — Communication Systems, chapter 6, asked 2×
- [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
- [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

## Related in other subjects

- [Entropy, information and channel capacity](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ainformation-entropy-capacity) — Data Communication, chapter 8