# Time Coding OTDM MIMO System Based on Singular Value Decomposition for 5G Applications
## 1 Introduction
This technique is promising for the future high performance and secure 5G systems when compared to the conventional OFDM system. The proposed technique increases the diversity gain resulting in a decrease in the fading effect of the multipath channel, thus enhancing the bit error rate (BER) performance and throughput of the system. The proposed technique also provides a secure data transmission to the desired user as his data is sent on the basis functions extracted from his own channel impulse response that cannot be decoded by other users, thus the desired user can only receive his data correctly. The remainder of the paper is organized as follows. Section 2 is devoted to a detailed description of the proposed system. System evaluation and simulation results are presented in Section 3. This is followed by the main conclusions showing  
where \(m=1\), \(2\),..., \(\frac{\mathrm{N}}{2}\)  
After modulation with M subcarriers and passing through a serial-to-parallel (S/P) converter, each of the modulated complex value symbols is carried by a channel-based orthogonal transform basis. The mapping method is basically done by a direct multiplication operation between each data symbol and the orthogonal basis vector. These orthogonal basis vectors are extracted from the SVD decomposition of the Toeplitz matrix of the channel impulse response \(Hb_{ij}\) between the \(i\)th receiving antenna and the \(j\)th transmitting antenna, where \(i=1,2\) and \(j=1,2\). This can be represented by  
\[Hb_{ij}=\left[ \begin{array}{ccccc}h_{ij}[0]&0&\cdots&\cdots&0\\ \vdots&h_{ij}[0]&0&\ddots&\vdots\\ \vdots&\vdots&h_{ij}[0]&\ddots&\vdots\\ h_{ij}[L-1]&\vdots&\vdots&\ddots&0\\ 0&h_{ij}[L-1]&\vdots&\ddots&h_{ij}[0]\\ \vdots&0&h_{ij}[L-1]&\ddots&\vdots\\ \vdots&\vdots&0&\ddots&\vdots\\ 0&0&\cdots&\ldots&h_{ij}[L-1]\\ \end{array} \right] \tag{3}\]  
then,  
\[Hb_{j}\ =\ \sum\
olimits_{i=1}^{2}Hb_{ij} \tag{4}\]  
where \(j=1\), \(2\) and \(Hb_{j}\in\mathrm{C}^{[(\frac{\mathrm{N}}{2}+\mathrm{L}-1)\times\frac{\mathrm{N }}{2}]}\). The receiver and the transmitter can extract the basic functions by decomposing the Toeplitz matrices using the SVD. Therefore, the Toeplitz matrices can be expressed in terms of the three new matrices as  
\[Hb_{j}=\ U_{n}E_{n}V_{n}^{H} \tag{5}\]  
where \(n=1,2\) and \(U_{n}\) and \(V_{n}^{H}\) are orthogonal matrices that are \(\in\mathrm{C}^{[(\frac{\mathrm{N}}{2}+\mathrm{L})\times\frac{\mathrm{N}}{2}]}\) and \(\in\mathrm{C}^{[\frac{\mathrm{N}}{2}\times\frac{\mathrm{N}}{2}]}\), respectively, and \(E_{n}\) is a diagonal matrix with real entires \(\in\mathrm{C}^{[\frac{\mathrm{N}}{2}\times\frac{\mathrm{N}}{2}]}\). The transformation matrix \(V_{n}\) is given by  
\[V_{n}=v_{n}(0)\ v_{n}(1)\ldots v_{n}(\frac{\mathrm{N}}{2}-1) \tag{6}\]  
where \(v(i)\) is the orthogonal basis vector of length \((\frac{\mathrm{N}}{2}-1)\). Figure 1: Space-time coded orthogonal transform division multiplexing (STC OTDM) transmitter technique. The transmitter maps each symbol to its corresponding basis functions by simple multiplication. This leads to two blocks of samples \(X_{1}\) and \(X_{2}\) referred to two OTDM symbols. This process can mathematically be expressed as  
\[X_{n}=\sum_{i=0}^{\text{N-1}}s_{n}[i]v_{n}[i]=V_{n}S_{n}\ldots\ldots\ldots\in \text{C}^{[\frac{N}{2}\times\text{l}]} \tag{7}\]  
where \(n=1,2\) and \(V_{n}\) is the Hermitian of \(V_{n}^{H}\). The transmitted data blocks \(X_{n}\) of length \(\frac{\text{N}}{2}\) can be represented by simple multiplication of Toeplitz matrices \(Hb_{j}\in\text{C}^{[(\frac{N}{2}+\text{L-1})\times\frac{N}{2}]}\) with the transmitted data blocks. The transmitted data blocks pass through a parallel-to-serial (P/S) converter before adding the guard period (L). Then, Alamouti space-time coding is applied to \(X_{1}\) and \(X_{2}\) in the frequency domain to generate the following code word matrix:  
\[X=\left[ \begin{array}{cc}X_{1}&X_{2}\\ -X_{2}^{*}&X_{1}^{*}\end{array}