---
title: "Shannon channel capacity theorem and its implication"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:shannon-capacity-implication"
source: https://www.bctengineer.com/q/semester_6/communication_systems/shannon-capacity-implication
site: BCT Engineering
---

# Shannon channel capacity theorem and its implication

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2071 Magh (old course), 8 marks
- Video: [Channel Capacity by Shannon-Hartley | Basics, Proof & Maximum Bandwidth Condition](https://www.youtube.com/watch?v=3ekWsXeZ8TM) — 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/shannon-capacity-implication
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

State the Shannon Channel Capacity Theorem. Discuss the implication of this theory in communication system.

## Short answer

Shannon's theorem: capacity C = B log2(1+S/N) bits/s; if rate ≤ C, error-free transmission is possible with coding. Implications: sets an upper bound on data rate, shows bandwidth-power trade-off, noise limits rate not accuracy, and establishes a benchmark for code efficiency.

## Answer

#### Statement
**Shannon's channel capacity theorem (Shannon–Hartley law):** for a channel of bandwidth $B$ Hz with additive white Gaussian noise of power $N = N_0 B$ and average signal power $S$, the channel capacity is

$$
C = B \log_2\!\left(1 + \frac{S}{N}\right)\ \text{bits/s}
$$

If the information rate $R \le C$, there is a coding scheme that sends the information with an arbitrarily small probability of error. If $R > C$, the error probability cannot be made small, whatever coding is used.

#### Implications in communication systems
1. **Upper bound on data rate.** No modem or code can beat $C$. Real designs are judged by how close they get, e.g. LDPC/turbo codes come within about 1 dB.
2. **Bandwidth–power exchange.** The same $C$ can come from a large $B$ with low SNR, or a small $B$ with high SNR.
   - Spread spectrum and CDMA work at very low SNR by using wide bandwidth.
   - High-order QAM (e.g. 256-QAM in cable modems) uses high SNR to pack more bits into a narrow band.
3. **Error-free transmission over noisy channels is possible.** Noise limits the *rate*, not the *accuracy*. This motivates channel coding (error-correcting codes).
4. **Diminishing returns from power.** $C$ grows only logarithmically with $S/N$. At high SNR, doubling the power adds only about 1 bit/s/Hz.
5. **Limited gain from bandwidth.** As $B \to \infty$, $C \to 1.44\, S/N_0$, so capacity stays finite because noise grows with bandwidth.
6. **Shannon limit.** The minimum $E_b/N_0$ for reliable communication is $\ln 2 = -1.6$ dB. This is the benchmark for power-efficient systems such as deep-space links.
7. **Design guide.** It tells designers the best spectral efficiency $C/B$ for a given SNR, e.g. 3.46 b/s/Hz at 10 dB.

| Bandwidth-limited system | Power-limited system |
|---|---|
| High SNR, use M-ary QAM/PSK | Low SNR, use wide band + coding |
| e.g. telephone modems, microwave links | e.g. satellite, deep-space links |

**Example:** a telephone channel with $B = 3.1$ kHz and SNR = 30 dB has $C = 3100 \log_2(1001) \approx 30.9$ kb/s. This is close to the speed of the V.34 modem (33.6 kb/s), whose real SNR is slightly higher.

## Check yourself

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

1. What is the Shannon channel capacity of an AWGN channel?
   A. C = 2B log2 M
   B. C = B (S/N)
   C. C = B log2(1 + S/N)
   D. C = B log2(S/N)
   Answer: C. Shannon-Hartley gives C = B log2(1 + S/N) bits/s; 2B log2 M is Nyquist's noiseless result. (Key point: Shannon-Hartley)

2. What does the theorem say if the information rate R is at most C?
   A. A coding scheme exists with arbitrarily small error probability
   B. Error probability is always large
   C. Errors cannot be corrected
   D. Bandwidth must increase
   Answer: A. For R <= C reliable communication is possible with suitable coding; for R > C the error probability cannot be made small. (Key point: Meaning of C)

3. Why do spread spectrum systems work at very low SNR?
   A. They use fewer symbols
   B. They use higher power
   C. They avoid noise entirely
   D. They use wide bandwidth to keep capacity
   Answer: D. Capacity can be kept by trading SNR for bandwidth: large B with low SNR gives the same C as small B with high SNR. (Key point: Bandwidth-power trade)

4. As B tends to infinity, capacity becomes:
   A. About 1.44 S/N0, finite
   B. Infinite
   C. Equal to S/N
   D. Zero
   Answer: A. Noise power grows with bandwidth, so C saturates at 1.44 S/N0; the Shannon limit Eb/N0 is -1.6 dB. (Key point: Infinite bandwidth)

## Related in Communication Systems

- [Shannon channel capacity theorem and its limitations](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-capacity-limitations) — Communication Systems, chapter 6, asked 2×
- [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×
- [Digital communication system; Shannon-Fano for six symbols](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adcs-block-shannon-fano-six) — Communication Systems, chapter 6
- [Digital communication system elements; Shannon-Fano coding](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adcs-block-shannon-fano) — Communication Systems, chapter 6
- [Nyquist channel capacity theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-channel-capacity) — Communication Systems, chapter 6
- [Shannon-Hartley theorem; energy/power of e^(−2t)u(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-energy-e2t) — Communication Systems, chapter 1

## Related in other subjects

- [Shannon-Hartley channel capacity theorem](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ashannon-hartley-theorem) — Data Communication, chapter 1, asked 3×
- [State Nyquist and Shannon capacity formulae](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Anyquist-shannon-capacity-formula) — Data Communication, chapter 1, asked 2×
- [Entropy, information and channel capacity](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ainformation-entropy-capacity) — Data Communication, chapter 8
- [Channel capacity for 6 kHz, 6 dB](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acapacity-6khz-6db) — Data Communication, chapter 1