What is AWGN channel?

When a message $\mathbf{x}$ is sent to a noisy channel, the output message $\mathbf{y}$ is not the original one. If the channel is AWGN channel,the relation between input and output of the channel is $$\mathbf{y} = \mathbf{x} + \mathbf{n}$$ where $\mathbf{n} \sim N(0,\sigma^2)$ is the Gaussian noise.

The name Additive White Gaussian Noise (AWGN) comes from 3 aspect:

• The noise is added to the original message directly;
• In frequency domain, the noise has uniform power across the frequency band which is an analogy to the color white.
• In time domain, the noise is consistent to Gaussian noise.

Why we use AWGN channel?

AWGN channel model is an important model in communication system. The reason as follows:

• There are many kinds of noise in a communication system. Based on Central Limit Theorem, the summation of many random processes tends to be a Gaussian random variable.
• AWGN channel can not describe some complex situration very well, but it is simple enough and the result can give the researcher basic knowledge.

How to simulate in AWGN channel?

The behavior of AWGN channel relies on signal-to-noise ratio (SNR) which is defined as the ratio of signal power to the noise power, often expressed in decibels as
$$\mathrm{SNR} = 10\log_{10}(\frac{P_{signal}}{P_{noise}}), $$ where $P$ is the average power.

However $\mathrm{SNR}$ usually related to bandwidth, if we do not want to take bandwidth into consideration, we usually use the notation $E_b/N_0$. $E_b$ is the energy per information bit. $N_0$ is the noise power spectral density. It is a normalized signal-to-noise ratio (SNR) measure, also known as the “SNR per bit”.

In order to get $E_b$, we should know the relationship between $E_b$ and $E_s$ (the energy per modulated bit or the energy per symbol). $$E_s = R_c R_m E_b$$ where $R_c$ is the code rate, $R_m = \log_2(M)$ is the number of information bits that one symbol can express. For example, for BPSK, $R_m=\log_2(2)=1$, for QPSK, $R_m=\log_2(4)=2$. And $E_s$ is usually normalized as 1.

For AWGN channel, the noise power spectral density (N0) is given by, $$N_0 = 2 \sigma^2$$

Therefore $E_b/N_0$ can be represented as $$\frac{E_b}{N_0} = \frac{E_s}{R_c R_m N_0} = \frac{1}{R_c R_m 2 \sigma^2}$$ If we write in dB with BPSK, $$\frac{E_b}{N_0} \mathrm{(dB)}= 10\log_{10}\frac{1}{2 R_c \sigma^2}\mathrm{(dB)}$$ then we can get the variance of the noise by

$$\sigma^2= 1/(2 R_c 10^{\frac{E_b}{N_0}/10})$$

For same transmitted power, the code bit energy is less than the uncoded bit energy $$E_c = k/n* E_b$$