AWGN EXIT

**The Additive White Gaussian Noise (AWGN) Channel** is the most popular channel in the field of communications and it is
considered as a good model for many applications as satellite and deep space communications. However, **AWGN** is a bit more complicated
in analysis than **BEC** and it is a more general channel. The channel is named this way because of its characterstics:

•* Additive:* Noise is added to symbols.

•* White:* Uniform power over all frequency bands.

•* Gaussian:* As it models many random processes and due to the central limit theorem (summation of many
random processes result in a normal distribution $N(0,\sigma^2)$ with 0 mean and $\sigma^2$ variance)

**The EXIT Chart** analysis is divided into two parts as stated previously:

•* The variable nodes decoder EXIT curve is given as an approximate expression:* $$I_{E}^{[V]} \approx J \left(\sqrt{(v-1)\,J^{-1} \left(I_{A}^{[V]} \right)^{2}+8R\,\dfrac{E_b}{N_o}} \right) \: (Regular)$$ $$I_{E}^{[V]} \approx \sum\limits_{i=1}^{v_{max}} \lambda_i \cdot J \left(\sqrt{(i-1)\,J^{-1} \left(I_{A}^{[V]} \right)^{2}+8R\,\dfrac{E_b}{N_o}} \right) \: (Irregular)$$
where $v$ in case of regular is the variable nodes degree (number of 1's in each column of

•* The check nodes decoder EXIT curve is given as an exact expression:* $$I_{E}^{[C]} \approx 1-J \left(\sqrt{c-1}\,J^{-1} \left(1-I_{A}^{[C]} \right) \right) \: (Regular)$$ $$I_{E}^{[C]} \approx 1-\sum\limits_{j=1}^{c_{max}} \rho_j \cdot J \left(\sqrt{j-1}\,J^{-1} \left(1-I_{A}^{[C]} \right) \right) \: (Irregular)$$
where $c$ in case of regular is the check nodes degree (number of 1's in each row of

Check nodes

Note: The J-Function is a monotonically increasing function that maps $\sigma$ to mutual information I as shown below, and the inverse J-Function can do the inverse operation.