# Transmitter Diversity Gain Technique Aided Irregular Channel Coding for Mobile Video Transmission
## 3 Preliminaries & System Design Criteria
Contrary, the coding gain is the distance between two codes and it is equal to \(det(A(C_{1},C_{2}))\). The coding gain can be increased by maximizing the minimum determinate of matrix \(A(C_{1},C_{2})\), which is the determinant criterion [2]. Because the symbols in STBC are always orthogonal, the two orthogonal symbols transmitted from 2 Tx antennas at \(t_{1}\) and \(t_{2}\) can be represented with \(C_{1}= \begin{bmatrix}x_{1}&x_{2}\\ -x_{2}^{*}&x_{1}^{*}\end{bmatrix} \). The first Tx antenna transmit the original signal while the second transmit the conjugate version of the original signal. For instance, at the receiver, the following pair of symbols received: \(C_{2}= \begin{bmatrix}x_{1}^{{}^{\prime}}&x_{2}^{{}^{\prime}}\\ -x_{2}^{*}&x_{1}^{*}\end{bmatrix} \). The difference metric can be expressed as: \(D(C_{1},C_{2})= \begin{bmatrix}x_{1}^{{}^{\prime}}-x_{1}&x_{2}^{{}^{\prime}}-x_ {2}\\ x_{2}^{*}-x_{2}^{{}^{\prime}}&x_{1}^{*}-x_{1}^{*}\end{bmatrix} \). The determinant of the difference matrix \(det[C_{1},C_{2}]=[x_{1}^{{}^{\prime}}-x_{1}]^{2}+[x_{2}^{{}^{\prime}}-x_{2}]^{2}\) is equal to 2 if \(C_{1}\
eq C_{2}\), otherwise it is zero, which shows that the difference matrix is a full rank for Alamouti code and it satisfies the determinant criteria. Therefore, STBC has to be a good code, if (a) the rank of the difference matrix b/w any two codewords is a full rank and (b) if there exists a fast maximum likelihood decoding algorithm. The space time block codes from orthogonal designs guarantee maximum diversity and fast maximum likelihood designs because of the special structure of orthogonal designs [41]. SP modulation was designed for obtaining full diversity in the time-correlated Rayleigh fading channel by combining orthogonal designs with sphere packing inspired by space-time codes [8]. A recursive orthogonal design function can be constructed, as given below [8; 41]. Let \(G1(x)=x_{1}I_{1}\), where \(I_{1}\) is the identity matrix. \[G_{2^{k}}(x_{1},\cdots,x_{k+1})=\left[ \begin{array}{cc}G_{2^{k-1}}(x_{1}, \cdots,x_{k})&x_{k+1}I_{2^{k-1}}\\ -x_{k+1}^{*}I_{2^{k-1}}&G_{2^{k-1}}^{\mathcal{H}}(x_{1},\cdots,x_{k})\end{array} \right] \tag{13}\]  
with \(x_{1},x_{2},.....x_{k+1}\) are complex variables and \(x_{1}^{*},x_{2}^{*},.....x_{k+1}^{*}\) are their conjugate version. Similarly, \(\mathcal{H}\) denotes the transpose and conjugate of \(G_{2^{k-1}}\). A set of SP signals can be constructed from orthogonal design of \(G_{2^{k}}\), as given below. \[C=\sqrt{2^{k}/(k+1)}G_{2^{k}}(x_{1},x_{2},\cdots,x_{k+1}) \tag{14}\]  
Here, \(\sqrt{2^{k}/(k+1)}\) is the normalization factor for energy constraint and \(k+1/2^{k}\) is the symbol rate. For instance, the SP signals for two Tx antennas are given below by exploring the above equation. \[C=\sqrt{2^{k}/(k+1)}G_{2}(x_{1},x_{2})=\sqrt{2^{k}/(k+1)}\bigg{[} \begin{array} []{cc}G_{1}(x_{1})&x_{2}\\ -x_{2}^{*}&G_{1}^{\mathcal{H}}(x_{1})\end{array} \bigg{]}=\sqrt{2^{k}/(k+1)} \bigg{[}\frac{x_{1}\bigm{|}x_{2}}{-x_{2}^{*}\bigm{|}x_{1}^{*}}\bigg{]} \tag{15}\]  
Similarly, for four and eight Tx antennas, the SP signals can be obtained as:  
\[C=\sqrt{2^{k}/(k+1)}G_{4}(x_{1},x_{2},x_{3})=\sqrt{2^{k}/(k+1)}\bigg{[} \begin{array} []{cc}G_{2}(x_{1},x_{2})&x_{3}I_{2}\\ -x_{3}^{*}I_{2}&G_{2}^{\mathcal{H}}(x_{1},x_{2})\end{array} \bigg{]} \tag{16}\]\[=\sqrt{2^{k}/(k+1)}\left[ \begin{array}{cccc|cc}x_{1}&x_{2}&x_{3}&0\\ -x_{2}^{*}&x_{1}^{*}&0&x_{3}\\ \hline-x_{3}^{*}&0&x_{1}^{*}&-x_{2}\\ 0&-x_{3}^{*}&x_{2}^{*}&x_{1}\end{array} \right] \tag{17}\]  
\[C=\sqrt{2^{k}/(k+1)}\,G_{8}(x_{1},x_{2},x_{3},x_{4})=\sqrt{2^{k}/(k+1)}\left[ \begin{array}{cccc|cc}G_{4}(x_{1},x_{2},x_{3})&x_{4}I_{4}\\ -x_{4}^{*}I_{4}&G_{4}^{H}(x_{1},x_{2},x_{3})\end{array} \right] \tag{18}\]  
\[=\sqrt{2^{k}/(k+1)}\left[