25. Channel Capacity with an Average Power Limitation

The Most Famous Formula in Information Theory

For a continuous channel with bandwidth \(W\) Hz, additive white Gaussian noise of power \(N\), and average signal power limited to \(P\):

\[ \boxed{C = W \log_2\left(1 + \frac{P}{N}\right)} \]

This is the Shannon–Hartley Law. It appears in every communications textbook, on Wikipedia, and in the design of every wireless and wired communication system.

Derivation

Step 1: Degrees of Freedom

A signal band-limited to \(W\) for duration \(T\) has \(N = 2WT\) degrees of freedom (samples).

Step 2: Power Per Sample

Total signal power \(P\) over time \(T\) with \(2WT\) samples:

\[ \text{Power per sample} = \frac{P}{2W} \]

Total noise power \(N = N_0 W\) over the same band, so noise per sample:

\[ \text{Noise per sample} = \frac{N}{2W} = \frac{N_0}{2} \]

Step 3: SNR Per Sample

\[ \text{SNR per sample} = \frac{P/N_0 W}{1/2} = \frac{P}{N} \]

Step 4: Capacity Per Sample

For a Gaussian channel with SNR = \(P/N\) per sample:

\[ C_{\text{per sample}} = \frac{1}{2}\log_2\left(1 + \frac{P}{N}\right) \]

Step 5: Total Capacity

Multiply by \(2W\) samples per second:

\[ C = 2W \cdot \frac{1}{2}\log_2\left(1 + \frac{P}{N}\right) = W \log_2\left(1 + \frac{P}{N}\right) \]

Understanding the Formula

Bandwidth \(W\)

More bandwidth → more degrees of freedom per second → linear increase in capacity.

Power \(P\)

More power → higher SNR → logarithmic increase in capacity. Doubling power adds 1 bit/Hz (or \(W\) bits/second total).

Noise \(N\)

Less noise → higher SNR → logarithmic increase. Reducing noise by factor of 2 adds 1 bit/Hz.

The Logarithm

The diminishing returns: to double capacity, you must quadruple \(P/N\) (increase SNR by ~6 dB).

Numerical Examples

System	Bandwidth	SNR (dB)	Capacity
Telephone voice	3.4 kHz	30 dB	~34 kbps
WiFi (802.11n)	40 MHz	20 dB	~266 Mbps
4G LTE	20 MHz	15 dB	~100 Mbps
Fiber optic	50 THz	20 dB	~50 Tbps (theoretical)
Deep space (X-band)	500 MHz	-5 dB	~100 Mbps

Spectral Efficiency

Define spectral efficiency:

\[ \eta = \frac{C}{W} = \log_2(1 + \text{SNR}) \quad \text{bits/sec/Hz} \]

Modern systems approach this limit:
- LTE: ~4–6 bits/sec/Hz
- 5G mmWave: ~8–10 bits/sec/Hz
- Theoretical maximum at 20 dB SNR: ~6.7 bits/sec/Hz

The Waterfall Curve

As SNR increases, spectral efficiency increases logarithmically:

graph LR
    subgraph "Spectral Efficiency vs SNR"
        direction TB
        A["SNR = -10 dB<br/>η ≈ 0.1"] -->
        B["SNR = 0 dB<br/>η = 1"] -->
        C["SNR = 10 dB<br/>η ≈ 3.5"] -->
        D["SNR = 20 dB<br/>η ≈ 6.7"] -->
        E["SNR = 30 dB<br/>η ≈ 10"]
    end

Practical Limits

No real system achieves the full Shannon limit due to:
- Non-Gaussian noise
- Imperfect channel estimation
- Finite block lengths (not \(N \to \infty\))
- Implementation complexity
- Peak (not average) power constraints

Modern codes get within 0.5–1.0 dB of the limit.

Historical Impact

Before Shannon, engineers didn't know if there was a fundamental limit. The Shannon–Hartley law:
- Set the theoretical ceiling for all communication systems
- Showed that bandwidth and power are fundamentally tradable
- Proved that error-free transmission is possible even at low SNR (just with low spectral efficiency)
- Justified investments in wider bandwidth (spread spectrum, mmWave, optical) over pure power increases

Next: §26 — The Channel Capacity with a Peak Power Limitation