SNR Limits: Shannon-Hartley

The Capacity Formula

Claude Shannon proved that the capacity $C$ of a channel is limited by two primary physical factors: the Bandwidth (frequency range) and the Signal-to-Noise Ratio (SNR).

Where:

• C: Channel Capacity (bits per second).
• B: Bandwidth (Hz).
• S/N: Signal-to-Noise Power Ratio (linear, not dB).
• N: Gaussian Noise power.

Converting dB to Linear

In the field, we measure SNR in decibels (dB). To use the Shannon formula, we must convert the log-scale dB back to the linear power ratio:

Example: An SNR of 30dB corresponds to a linear ratio of 1000. Under these conditions, a 20MHz Wi-Fi channel has a theoretical limit of roughly $20 \times \log_2(1001) \approx 200 \text{ Mbps}$.

Spectral Efficiency (bits/Hz)

A key metric derived from the Shannon-Hartley theorem is Spectral Efficiency, defined as the ratio of the channel capacity to the bandwidth:

This value tells us how many bits per second we can push through a single Hertz of spectrum. In a 5G environment using 256-QAM modulation, spectral efficiency can reach up to 8 bits/Hz. However, as SNR drops (e.g., as you move further from a cell tower), the modulation scheme must "downshift" to lower orders like 16-QAM or QPSK to remain within the physical limits of the channel.

The Normalized SNR: $E_b/N_0$

While SNR is useful for general engineering, communication theorists often use Energy per Bit to Noise Power Spectral Density ($E_b/N_0$). This normalization allows us to compare different modulation and coding schemes regardless of the bandwidth used.

Where $R$ is the actual data rate. Shannon proved that there exists an ultimate physical floor for $E_b/N_0$ of approximately -1.59 dB. Below this point, no amount of advanced mathematics or error correction can ever reconstruct the original message—the noise becomes the message.

Approaching the Limit: Turbo Codes and LDPC

For decades after Shannon published his paper in 1948, engineering systems were far from the theoretical capacity. It wasn't until the invention of Turbo Codes in the 1990s and Low-Density Parity-Check (LDPC) codes (now used in 5G and DVB-S2) that we began to close the Shannon Gap.

These Forward Error Correction (FEC) algorithms use iterative mathematical processes to "guess" the original data even when the signal is severely mangled by noise, allowing us to operate within a fraction of a decibel of the Shannon Limit.

Increasing Capacity: MIMO and MCS

How do we bypass these limits in modern hardware?

• MIMO: Multiple-Input Multiple-Output uses spatial multiplexing to create multiple "virtual channels," effectively multiplying the capacity by the number of spatial streams.
• High-Order Modulation (QAM): Encoding more bits per symbol allows us to pack more information into the same bandwidth, but requires a significantly cleaner SNR (lower noise floor).
• Spectral Reuse: Utilizing higher frequencies (mmWave) where larger bandwidth chunks are available.

The Future: Terahertz and Beyond

As we reach the peak of SNR efficiency in the Gigahertz range, the only path forward for 6G and beyond is moving into higher frequency bands (Terahertz), where vast amounts of Bandwidth (B) are still available, even if the SNR remains challenging at those ranges.