# Transmitter Diversity Gain Technique Aided Irregular Channel Coding for Mobile Video Transmission
## 4 Proposed System Setup
### Iterative Source Channel Decoding Setup:  
The input alphabet is referred to the set of elementary input signals and the individual signals as an input symbols. Consequently, the output alphabet is a set of regions in channel output space and the individual signal as an output symbols. The channel output symbol is generated by the following expression, provided that the bits are independent of each other [17; 44]. \[P[\hat{y}_{\tau}\mid y_{\tau}]=\prod_{k=1}^{K}P[\hat{y}_{\tau}(k)\mid y_{\tau} (k)] \tag{20}\]  
where \(\hat{y}_{\tau}\) is the received information on the transmission of \(y_{\tau}\). For a specific bit \(y_{\tau}(\lambda)\), the output extrinsic information \(P\left[\hat{y}_{\tau}^{[ext]}\mid y_{\tau}^{[ext]}\right]\) can be achieved as follow. \[P\left[\hat{y}_{\tau}^{[ext]}\mid y_{\tau}^{[ext]}\right]=\prod_{k=1,k\
eq \lambda}^{K}P[\hat{y}_{\tau}(k)\mid y_{\tau}(k)] \tag{21}\]  
Eventually, the channel output information is combined with the A-priori knowledge of that corresponding symbol in order to achieve the resultant Log-Likelihood Ratio (LLR) extrinsic value, as expressed below. \[ \begin{array}{l}\text{LLR}[y_{\tau}(\lambda)]=\\ \log\left(\frac{\sum_{y_{[ext]}}p\left[y_{\tau}^{[ext]}|y_{\tau}(\lambda)=+1 \right]\cdot P\left[\hat{y}_{\tau}^{[ext]}|y_{\tau}^{[ext]}\right]}{\sum_{y_{ \tau}^{[ext]}}p\left[y_{\tau}^{[ext]}|y_{\tau}(\lambda)=-1\right]\cdot P\left[ \hat{y}_{\tau}^{[ext]}|y_{\tau}^{[ext]}\right]}\right)\end{array}  \tag{22}\]  
Mutual information or A-priori knowledge is a measure of information that one random variable has about the others [45]. Entropy becomes the self-information of a random variable. Mutual information is a more general quantity, called a relative entropy, a measure of the distance between two probability distributions. Symbol-wise mutual \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Error Protection Scheme**} & \multirow{2}{*}{**Order of Concatenation**} & \multicolumn{4}{c}{**Code Rate**} \\ \cline{3-5}  & & **Inner Code** & **Outer Code** & **Overall** \\ \hline Regular Error Protection & \multirow{2}{*}{[A-B-C]} & \multirow{2}{*}{[A-B-C]} & \multirow{2}{*}{URC rate-1} & \multirow{2}{*}{IRCC Uni-code rate-\(3/4\)} & \multirow{2}{*}{\(3/4\)} \\ (REP) & & & & & \\ \hline Irregular Error Protection & \multirow{2}{*}{[A-B-C]} & \multirow{2}{*}{[A-B-C]} & \multirow{2}{*}{URC rate-1} & \multirow{2}{*}{IRCC Multi-code rate-\(3/4\)} & \multirow{2}{*}{\(3/4\)} \\ Scheme-1 (IREP-1) & & & & & \\ \hline Irregular Error Protection & \multirow{2}{*}{[B-A-C]} & \multirow{2}{*}{[B-A-C]} & \multirow{2}{*}{URC rate-1} & \multirow{2}{*}{IRCC Multi-code rate-\(3/4\)} & \multirow{2}{*}{\(3/4\)} \\ Scheme-2 (IREP-2) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Error Protection Schemes and their Constituent Codes. information between symbol \(S_{i}\) and received Rayleigh fading channel output \(\hat{y}_{\tau}\) is given by the following equation [21]:  
\[I(S_{i};\hat{y}_{\tau})=\frac{1}{L}\cdot\sum_{n=1}^{L}\int_{-\infty}^{+\infty}p( \hat{y}_{\tau}\mid S_{i}=a_{n})\times ld\frac{p(\hat{y}_{\tau}\mid S_{i}=a_{n}) }{p(\hat{y}_{\tau})}dz \tag{23}\]  
where \(a_{n}\) represents the current signaling element. The conditional Probability Density Function \((PDF)\) of Equation (23) is given as:  
\[p(\hat{y}_{\tau}\mid S_{i}=a_{n})=\frac{1}{\sqrt{2\pi\sigma}}\cdot\exp\left[ \frac{\left[-(\hat{y}_{\tau}-an)^{2}\right]}{2\sigma^{2}}\right] \tag{24}\]  
where  
\[p(\hat{y}_{\tau})=\frac{1}{L}\sum_{n=1}^{L}p(\hat{y}_{\tau}\mid S_{i}=a_{n}) \tag{25}\]  
At the receiver side in the iterative decoding paradigm, the constellation signal is iteratively decoded by exchanging the mutual information between the inner and outer decoder, which iteratively collaborate in order to give the highest possible output to the source decoder.