Gaussian Noise and AWGN

• 📘 Overview
• 📘 Mean, Variance
• 📘 SNR Definition, and Noise Variance Calculation, and SNR dB to Linear
• 📘 SNR dB to Linear
• 📘 MATLAB Code
• 📚 Further Reading

What is Gaussian Noise?

Gaussian noise is a random signal whose amplitude follows a Gaussian (normal) distribution.

p(x) = (1 / √(2πσ²)) e^{-(x-μ)² / (2σ²)}

Where:

• μ = mean
• σ² = variance

It is widely used in communication systems because many natural noise sources follow this distribution.

Difference Between Gaussian Noise and AWGN

Feature	Gaussian Noise	AWGN
Definition	Noise with Gaussian distribution	Gaussian + Additive + White
Additive	Not necessarily	Always additive
White (flat spectrum)	Not required	Yes
Usage	General noise model	Communication systems

AWGN Noise: Mean and Variance in Practical Systems

In practical communication systems, Additive White Gaussian Noise (AWGN) is modeled with a zero mean and variance based on the signal power and signal-to-noise ratio (SNR).

AWGN Mean

In most practical systems, the mean of the AWGN noise is set to zero. This is because AWGN is symmetric around zero, making it equally likely to increase or decrease the signal.

Why Zero Mean? A zero mean ensures that the noise doesn’t introduce a consistent bias to the signal.

Mean of AWGN = 0

AWGN Variance

The variance of AWGN is determined based on the signal power and the desired SNR.

1. SNR Definition:

SNR = Psignal / Pnoise

Where:

• P_signal is the average power of the signal.
• P_noise is the power (variance) of the noise.

2. Noise Variance:

Pnoise = σ2

3. Convert SNR from dB to Linear Scale:

If SNR is given in decibels (dB), convert it to a linear scale:

SNR (linear) = 10^(SNR (dB) / 10)

4. Variance Calculation:

Rearranging the SNR formula, we get:

σ2 = Psignal / SNR (linear)

Example Calculation

If the average signal power Psignal is 1 (which is typical when the signal is normalized), and the SNR is 20 dB, then:

SNR (linear) = 10^(20 / 10) = 100

The noise variance will be:

σ2 = 1 / 100 = 0.01

Practical Use

The signal power may not always be 1, so you'll need to calculate it or have an estimate. The SNR can vary based on the channel conditions or the design of the communication system. Practical systems often use SNR values between 0 dB (noisy channel) to 30 dB (clean channel).

In summary, the AWGN noise has a zero mean and its variance depends on the signal power and the desired SNR. For normalized signals, you can use the formula:

σ2 = 1 / SNR (linear)







Online Simulator

Choose Input Type: Sine Wave

Number of Samples:

AWGN Mean:

AWGN Variance:

SNR (dB):

Amplitude:

Frequency (Hz):

Theory: Sine Wave, Noise, Variance, and SNR

In our simulation, we generate a noisy sinusoidal signal:

x[n] = A * sin(2π f n / N) + w[n]
  

Where:

• A = amplitude of the sine wave
• f = frequency of the sine wave
• N = total number of samples
• w[n] = additive noise with zero mean and variance σ²

The variance of the noise determines how strong the noise is compared to the sine wave:

σ² = P_signal / (10^(SNR_dB / 10))
  

The Signal-to-Noise Ratio (SNR) in decibels is given by:

SNR_dB = 10 * log10(P_signal / σ²)
  

For a pure sinusoid of amplitude A, the average signal power is:

P_signal = A² / 2
  

Using these formulas, we can interconvert between variance and SNR, allowing us to control the noise strength and quality of the generated signal.

Other Simulations →

Further Reading

1.  
   AWGN Online Simulator
2. Gaussian random variable and its PDF in MATLAB
3. Generation of Gaussian Random Noise using Box-Mullar Transform
4. Difference between AWGN and Rayleigh Fading
5. Gaussian vs Uniform Distribution in MATLAB