# Transmitter Diversity Gain Technique Aided Irregular Channel Coding for Mobile Video Transmission
## 3 Preliminaries & System Design Criteria
\[\mathbf{V}_{t}^{1}=\frac{x_{t}^{1}\cdot\mathbf{V}_{t-1}^{1}+x_{t}^{2}\cdot \mathbf{V}_{t-1}^{2*}}{\sqrt{|\mathbf{V}_{t-1}^{1}|^{2}|+|\mathbf{V}_{t-1}^{2 }|^{2}}} \tag{3}\]  
\[\mathbf{V}_{t}^{2}=\frac{x_{t}^{1}\cdot\mathbf{V}_{t-1}^{2}-x_{t}^{2}\cdot \mathbf{V}_{t-1}^{2*}}{\sqrt{|\mathbf{V}_{t-1}^{1}|^{2}|+|\mathbf{V}_{t-1}^{2 }|^{2}}} \tag{4}\]  
Differential encoded symbols \(\mathbf{V}_{t}^{1}\) and \(\mathbf{V}_{t}^{2}\) are then spread into \(\bar{\mathbf{C}_{1}}\) and \(\bar{\mathbf{C}_{1}}\) using STS encoder. The consecutive differential encoded symbols at both transmitting antennas can be represented while using the following mapping [10,11]. \[y_{t}^{1}=\frac{1}{\sqrt{2}}(\bar{\mathbf{C}_{1}}\cdot\mathbf{V}_{t}^{1}+\bar {\mathbf{C}_{2}}\cdot\mathbf{V}_{t}^{2}) \tag{5}\]  
\[y_{t}^{2}=\frac{1}{\sqrt{2}}(\bar{\mathbf{C}_{1}}\cdot\mathbf{V}_{t}^{1}-\bar {\mathbf{C}_{2}}\cdot\mathbf{V}_{t}^{2}) \tag{6}\]  
In the case of N transmitting antenna, for the transmission purpose the average transmission power of each antenna \(E_{s}=1/N\). The digital signal (\(r_{t}^{j}\)) at receiving antenna \(j\) after demodulation where \(C_{t}^{j}\) represents the codeword and \(\eta_{t}^{j}\) represents the path gain from the transmission antenna to the receiving antenna is given by the following equation [5]. \[r_{t}^{j}=\sum_{i=1}^{N}\alpha_{i,j}C_{t}^{i}\sqrt{E}+\eta_{t}^{j} \tag{7}\]  
The input and output relationships of Equation (7) are called the fading channel model, where \(\alpha\) represents the channel path gain. The design criteria have been briefly studied for space time codes in [2], which guarantee maximum coding and diversity gain to improve the mutual information b/w the input and output of the system as a design criterion. For example, the \(C_{1}\) codeword is selected at the transmission side from the codebook for transmission and, at the receiver, an error occurred if the decoder mistakenly estimates \(C_{2}\). \[\mathbf{C}^{2}=\left( \begin{array}{cccc}\mathbf{C}_{1,1}^{2}&\mathbf{C}_{1,2}^ {2}&\cdots&\mathbf{C}_{1,N}^{2}\\ \mathbf{C}_{2,1}^{2}&\mathbf{C}_{2,2}^{2}&\cdots&\mathbf{C}_{2,N}^{2}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{C}_{T,1}^{2}&\mathbf{C}_{T,2}^{2}&\cdots&\mathbf{C}_{T,N}^{2}\\ \end{array} \right) \tag{8}\]  
If the codebook contains L number of codewords, then the probability of error using the union bound is given as:  
\[P\Big{(}\text{ error }\mid\mathbf{C}^{1}\text{ is sent }\Big{)}\leq\sum_{i=2}^{I}P \Big{(}\mathbf{C}^{1}\rightarrow\mathbf{C}^{i}\Big{)} \tag{9}\]  
For an error matrix \(D(C_{1},C_{2})=C_{2}-C_{1}\), the pairwise error probability to be defined [2] in terms of non negative real number eigenvalues \(\lambda_{n}\geq 0\) of matrix \(\mathbf{A}(C_{1},C_{2})=D(C_{1},C_{2})_{H}\cdot D(C_{1},C_{2})=\left(C_{2}-C _{1}\right)_{H}\cdot\left(C_{2}-C_{1}\right)\), where \(H\) represents the conjugate transpose of the variable. \[P(\mathbf{C}\rightarrow\text{error})\leq\left(\prod_{n=1}^{r}\lambda_{n} \right)^{-M}\frac{E_{s}}{4N_{0}}^{-rM} \tag{10}\]  
or  
\[P\Big{(}\mathbf{C}^{1}\rightarrow\mathbf{C}^{2}\Big{)}\leq\frac{\left(\frac{ E_{s}}{4N_{0}}\right)^{-rM}}{\left(\prod_{n=1}^{r}\lambda_{n}\right)^{M}} \tag{11}\]  
Figure 1: Space Time Spreading (STS) scheme. Figure 2: Differential Space Time Spreading (DSTS) Scheme. or  
\[P\Big{(}\mathbf{C}^{1}\rightarrow\mathbf{C}^{2}\Big{)}\leq\frac{4^{rM}}{(\prod_{n =1}^{r}\lambda_{n})^{M}\gamma^{rM}} \tag{12}\]  
The diversity gain of the space time code is the rank of matrix \(A(C_{1},C_{2})\) or the rank of the difference matrix \(D(C_{1},C_{2})\) multiplied with the number of receiving antennas, which is the power of SNR in the denominator of Equation (12), i.e., rM, while the product of the nonzero eigenvalue matrix \(A(C_{1},C_{2})\) corresponds to the coding gain. Consequently, a design criterion for a full diversity scheme is possible for M transmitting antennas and N receiving antennas if the matrix \(A(C_{i},C_{i})\) is a full rank for all possible codewords and it is called a rank criterion.