Additive White Gaussian Noise (AWGN)

AWGN's PSD and its iDFT

AWGN_PSD.png

AWGN_PSD_iDFT.png

In words, each noise sample in a sequence is uncorrelated with every other noise sample in the same sequence. Both positive and negative value can be acquired with equal probablity, therefore, mean value of a white noise is zero.

AWGN's instant value distribution

The probability distribution of the noise samples is Gaussian with a zero mean, i.e., in time domain, the samples can acquire both positive and negative values and in addition, the values close to zero have a higher chance of occurrence while the values far away from zero are less likely to appear. This is shown in Figure below. As a result, the time domain average of a large number of noise samples is equal to zero.

AWGN_Gaussian.png

AWGN's Power

As with all densities, the value $N_0$ is the amount of noise power $P_w$ per unit bandwidth $B$ .
$\begin{equation} N_0 = \frac{P_w}{B} \end{equation}$
For the case of real sampling, we can plug $B=F_S/2$ in the above equation and the noise power in a sampled bandlimited system is given as
$\begin{equation} P_w = N_0\cdot B =N_0\cdot \frac{F_S}{2} \end{equation}$
Thus, the noise power is directly proportional to the system bandwidth at the sampling stage.