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.

Shannon Limit and Reliable Decoding in BPSK

In BPSK communication, reliable decoding is possible even when noise power exceeds signal power. This is explained by the Shannon limit, whose theoretical minimum is approximately -1.59 dB in terms of E_b/N_0.

BPSK Constellation

BPSK uses two constellation points:

+√Eb   and   -√Eb
  

The distance between them is:

d = 2√Eb
  

Larger constellation distance improves noise immunity.

Why Decoding Works Below 0 dB

At negative E_b/N_0, constellation points overlap heavily due to noise, making individual symbols unreliable. However, modern error-correcting codes decode entire codewords rather than isolated bits.

• Redundancy is added during encoding
• The decoder analyzes long bit sequences
• Statistical patterns help recover the message

Shannon Capacity

C = B log2(1 + SNR)
  

Shannon showed that reliable communication becomes theoretically possible whenever:

Eb/N0 > -1.59 dB
  

Below this limit, no coding scheme can achieve reliable decoding.

The Physics of Signal and Noise

Understanding how Power, Energy, and Bandwidth interact through the definitions of S and N.

1. Signal Power (S = E_bC)

In physics, Power is defined as Energy divided by Time. In digital communications, we translate this relationship into bits and capacity.

S (Signal Power) Measured in Watts (Joules per second).

E_b (Energy per bit) The Joules assigned to a single bit.

C (Capacity) The number of bits sent per second.

If every bit requires a certain amount of energy (E_b), and you are sending a specific number of bits every second (C), the total power consumed is the product of the two:

S = Eb × C

Note: While engineers often use R (actual data rate), Shannon’s derivation uses C (theoretical maximum rate) to find the absolute limit of the channel.

2. Noise Power (N = N₀B)

This equation describes how noise is distributed across a frequency range. It assumes White Noise, where noise energy is spread evenly across the spectrum.

N = N0 × B

• N₀ (Noise Power Spectral Density): This represents the "thickness" of the noise, measured in Watts per Hertz (W/Hz).
• B (Bandwidth): The range of frequencies (Hertz) the signal occupies.

3. Combining for the Shannon Formula

By substituting these physical definitions into the Shannon Capacity equation, we can shift the focus from hardware-specific values (Watts and Hertz) to a universal measure of Energy Efficiency:

C = B log₂(1 + (Eb × C) / (N0 × B))

This substitution is what allows us to solve for the Shannon Limit (-1.59 dB), which dictates the minimum energy required to transmit one bit of information regardless of the bandwidth or power used.

Summary

Individual BPSK symbols may appear corrupted by noise, but the complete coded sequence still contains enough structure for accurate recovery. The power comes from:

Modulation + Error-Correcting Coding
  

Further Reading

1. Shannon’s Theorem vs. Shannon-Hartley Theorem: Full Comparison

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