Shannon channel capacity theorem and its limitations

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.