26. The Channel Capacity with a Peak Power Limitation
Average vs. Peak Power
Section 25 assumed average power was limited to \(P\). Many systems also have peak power constraints — the instantaneous signal amplitude cannot exceed \(A\).
This changes the optimal input distribution and the capacity formula.
The Constraint
Under this constraint, the signal cannot be Gaussian (Gaussian has infinite support). The maximum entropy distribution on \([-A, A]\) with given variance is more complex.
Capacity Formula
For additive white Gaussian noise with peak amplitude constraint \(A\) and noise variance \(\sigma^2\):
(approximate form; exact requires numerical optimization).
More precisely, capacity is found by optimizing over input distributions on \([-A, A]\):
Comparison: Peak vs. Average
| Constraint | Optimal Input | Capacity Trend |
|---|---|---|
| Average power \(P\) | Gaussian | \(\sim W \log(P/N)\) |
| Peak amplitude \(A\) | Non-Gaussian (often uniform-like) | \(\sim W \log(A/\sigma)\) |
| Both | Truncated/constrained Gaussian | Complex optimization |
For the same numerical power (\(P = A^2\)), peak-limited capacity is lower because the Gaussian input is forbidden.
Optical Communication
Fiber optic channels are fundamentally peak-power limited:
- Laser intensity cannot be negative
- High peak intensity causes nonlinear effects in the fiber
- The "linear regime" requires peak power constraints
Capacity is lower than the Shannon–Hartley formula would suggest, and modern research focuses on nonlinear channel capacity.
Modern Wireless Systems
Cellular and WiFi have both constraints:
- Average: limited by regulations and battery
- Peak: limited by amplifier linearity (PAPR — Peak-to-Average Power Ratio)
High PAPR signals (like OFDM) are problematic. Techniques to reduce PAPR include:
- Clipping (deliberate distortion)
- Tone reservation
- Active constellation extension
These are all attempts to stay near the Shannon limit while respecting peak constraints.
End of Part IV. Next: Part V — The Rate for a Continuous Source