Implications and limitations of Shannon-Hartley theorem

Shannon–Hartley theorem: the capacity of a channel of bandwidth $B$ Hz, disturbed by additive white Gaussian noise, with average signal power $S$ and noise power $N = N_0B$, is

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

If the information rate $R \le C$, there exists a coding scheme giving an arbitrarily small error probability; if $R > C$, errors cannot be made small.

Implications

1. Upper bound on rate: it gives the maximum error-free rate of any channel, a benchmark for real systems. Example: telephone line, $B = 3.1$ kHz, $S/N = 30$ dB (1000):

$$C = 3100\log_2(1001) = 30\ 898\ \text{bits/s} \approx 30.9\ \text{kbps}$$

This is why voice-band modems stopped near 33.6 kbps. 2. Bandwidth–SNR trade-off: the same capacity can be obtained with large bandwidth and low SNR, or small bandwidth and high SNR. To keep 30.9 kbps in half the bandwidth (1550 Hz) we need $S/N = 2^{30898/1550} - 1 \approx 1.0\times10^6$ (60 dB), much more power. Wideband systems (spread spectrum, FM, UWB) use bandwidth to save power. 3. Noise does not forbid error-free communication; it only limits the rate. Good channel coding (turbo, LDPC) approaches the limit. 4. Bandwidth limit: as $B \to \infty$, $C \to 1.44\,S/N_0$, and reliable transmission needs $E_b/N_0 \ge -1.6$ dB (Shannon limit). 5. Bandwidth efficiency plane: $C/B = \log_2(1 + S/N)$ separates possible from impossible regions for choosing modulation (e.g. 64-QAM needs high SNR).

Limitations

1. It assumes AWGN only; it does not directly apply to impulse noise, interference, fading and multipath channels.
2. It is an existence theorem: it does not tell how to build the code or modulator that reaches capacity.
3. Reaching capacity needs very long code words, so infinite delay and complexity; practical systems operate below $C$.
4. It assumes a Gaussian-distributed signal, average power constraint and a linear, band-limited channel; practical signals use finite constellations (e.g. 16-QAM), which reach less.
5. Increasing bandwidth gives only a limited gain (capacity saturates at $1.44\,S/N_0$), and increasing SNR gives only a logarithmic gain: doubling $S/N$ at high SNR adds only about $B$ bits/s.

Example of the logarithmic limitation: with $B = 3.1$ kHz, raising $S/N$ from 1000 to 2000 increases $C$ from 30.9 kbps to only about 34.0 kbps.