--- title: "Shannon channel capacity theorem and its limitations" subject: "Communication Systems" chapter: 6 key: "semester_6/communication_systems:shannon-capacity-limitations" source: https://www.bctengineer.com/q/semester_6/communication_systems/shannon-capacity-limitations site: BCT Engineering --- # Shannon channel capacity theorem and its limitations Communication Systems · Chapter 6: Baseband Digital Data Transmission - Asked in: 2072 Chaitra (CS II), 2+3 marks; 2071 Chaitra (CS II), 1+3 marks (asked 2 times) - 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-limitations - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6 ## Question Explain Shannon channel capacity theorem. Write down theoretical limitations of this theorem. ## Short answer Channel capacity C = B log2(1+S/N) bps. Limitations: only a limit, not a construction method; needs long codes causing delay; as B→∞, C→1.44 S/N0; ultimate Eb/N0 ≥ −1.6 dB; assumes AWGN only, not fading or interference; capacity grows logarithmically with SNR. ## Answer #### Shannon channel capacity theorem For a channel of bandwidth $B$ Hz with additive white Gaussian noise, the maximum rate of error-free information transfer is $$ C = B \log_2\!\left(1 + \frac{S}{N}\right)\ \text{b/s} $$ where $S/N$ is the signal-to-noise power ratio, with $N = N_0 B$. If the information rate $R \le C$, suitable coding can make the error probability as small as desired. If $R > C$, it cannot. Example: a telephone channel with $B = 3.1$ kHz and $S/N = 30$ dB (1000) has $C = 3100 \log_2 1001 \approx 30.9$ kb/s. #### Theoretical limitations - **It gives a limit, not a method.** It proves good codes exist but does not show how to build them. Reaching $C$ needs very long code words, so delay and complexity become very large. - **Infinite bandwidth gives finite capacity:** as $B \to \infty$, $C \to 1.44\, S/N_0$, because noise power grows with bandwidth. - **Shannon limit:** reliable transmission needs $E_b/N_0 \ge \ln 2$ (−1.6 dB), even with unlimited bandwidth. - **Only for AWGN.** It assumes Gaussian, white, additive noise and a linear, band-limited channel. Fading, impulse noise, interference and non-linearity are not covered. - It assumes Gaussian-distributed signals and average power limits. Real systems with fixed constellations fall short of $C$. - Capacity grows only logarithmically with SNR, so raising power gives diminishing returns. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A telephone channel has B = 3.1 kHz and S/N = 30 dB. What is the approximate capacity? A. About 3.1 kb/s B. About 30.9 kb/s C. About 93 kb/s D. About 310 kb/s Answer: B. S/N = 1000, so C = 3100 log2(1001) is about 30.9 kb/s. (Key point: Channel capacity) 2. What does the theorem NOT tell us? A. That a rate above C cannot be error-free B. How to construct the codes that reach capacity C. That capacity depends on SNR D. That noise limits capacity Answer: B. It proves good codes exist but gives no method; approaching C needs very long code words, with large delay and complexity. (Key point: Constructive limitation) 3. Why does infinite bandwidth not give infinite capacity? A. Because signal power falls with B B. Because noise power N0 B grows with B, so C tends to 1.44 S/N0 C. Because the sampling rate is limited D. Because the theorem fails above 1 GHz Answer: B. More bandwidth admits proportionally more noise, so capacity saturates at 1.44 S/N0. (Key point: Infinite bandwidth) 4. Which is an assumption of the Shannon capacity formula? A. Fading channel B. Additive white Gaussian noise on a linear band-limited channel C. Impulse noise D. Non-linear channel Answer: B. The theorem covers AWGN only; fading, impulse noise, interference and non-linearity are outside it. 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