## AWGN Noise: Mean and Variance in Practical Systems

## 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 = P`_{signal} / P_{noise}

Where:

**P**is the average power of the signal._{signal}**P**is the power (variance) of the noise._{noise}

### 2. Noise Variance:

`P`_{noise} = σ^{2}

### 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:

`σ`^{2} = P_{signal} / SNR (linear)

## Example Calculation

If the average signal power `P`

is 1 (which is typical when the signal is normalized), and the SNR is 20 dB, then:_{signal}

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

The noise variance will be:

`σ`^{2} = 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:

`σ`^{2} = 1 / SNR (linear)