Shannon Limit Explained: Negative SNR, Eb/No and Channel Capacity

Understanding Negative SNR and the Shannon Limit

An explanation of Signal-to-Noise Ratio (SNR), its behavior in decibels, and how Shannon's theorem defines the ultimate communication limit.

Signal-to-Noise Ratio in Shannon’s Equation

In Shannon's equation, the Signal-to-Noise Ratio (SNR) is defined as the signal power divided by the noise power:

SNR = S / N

Since both signal power and noise power are physical quantities, neither can be negative. Therefore, the SNR itself is always a positive number.

However, engineers often express SNR in decibels:

SNR(dB)

When SNR = 1, the logarithmic value becomes:

SNR(dB) = 0

When the noise power exceeds the signal power (SNR < 1), the decibel representation becomes negative.

Behavior of Shannon's Capacity Equation

Shannon’s channel capacity formula is:

C = B log₂(1 + SNR)

For SNR = 0:

log₂(1 + SNR) = 0

When SNR becomes smaller (including negative values in dB), the expression approaches zero but never becomes negative.

This means that reliable decoding of information is still possible even when the SNR expressed in dB is negative.

Spread Spectrum and Negative SNR

Negative SNR values in dB occur frequently in spread spectrum communication systems. Spread spectrum is a coding technique that spreads the signal energy across a wider bandwidth.

Although the raw SNR may appear extremely low, the coding structure allows the receiver to correctly decode the information.

However, there exists a fundamental limit defined by Shannon known as the Shannon Limit.

SNR Per Bit (Eb/No)

The most important parameter in digital communications is the signal-to-noise ratio per bit, represented as:

Eb / No

Starting from Shannon's capacity equation:

C = B log₂(1 + SNR)

Rewriting:

C/B = log₂(1 + SNR)

Using the relationship:

SNR = S / (NoB)

And defining signal power as:

S = EbC

We obtain:

z = SNR = (Eb / No) × (C / B)

Deriving the Shannon Limit

As bandwidth becomes extremely large while the data rate remains finite:

B → ∞
C/B → 0
z → 0

Using the limit:

lim (1 + z)^(1/z) = e

We obtain the minimum required energy per bit:

Eb / No = 1 / log₂(e) ≈ 0.693

Expressed in decibels:

Eb / No (dB) ≈ -1.6 dB

This value represents the Shannon Limit.

What the Shannon Limit Means

• Reliable decoding is possible down to Eb/No ≈ −1.6 dB.
• Below this value, error rates increase exponentially.
• No coding scheme can outperform this theoretical limit.

Example: Spread Spectrum Processing Gain

In a spread spectrum system with a bandwidth-to-data-rate ratio of:

B / C = 100

The resulting SNR becomes:

SNR = (Eb / No) × (C / B)

In decibels:

SNR(dB) = Eb/No(dB) − 20 dB

If the system operates near the Shannon limit:

SNR ≈ -1.6 dB − 20 dB = -21.6 dB

This demonstrates how extremely low SNR values can still allow reliable communication when advanced coding techniques or spread spectrum methods are used.

Shannon's theorem proves that reliable communication is theoretically possible close to this limit, although it does not specify how to construct the optimal coding schemes needed to achieve it.