# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 2 Matrix Theory in Modern Wireless Communication Systems
### SVD for Modeling MIMO Channels  
The channel capacity, defined as the maximum data rate at which data can be sent with arbitrarily low probability of error, was reported for the Gaussian channel by [PERSON] in his seminal paper [1]. Currently, achieving higher and higher data rates with tolerable error rates and fixed transmitter power is one of the most demanding targets for the current wireless communication systems. For example, the evolution from LTE to 4 G LTE-Advanced communication standard requires an increase in the downlink data rate from 150 Mbps to 1 Gbps and an uplink data rate increase from 75 Mbps to 500 Mbps [2]. Theoretically, there are many ways to improve the channel capacity, like increasing the signal power, reducing the noise power, exploitation of space/time/frequency diversity, multiplexing, etc. However, due to the power limitation and unexpected properties of the receiver noise, diversity and multiplexing have been two popular design strategies on which researchers focused extensively during the last decade. Among these techniques, using multiple transmit and multiple receive antennas has been advocated as one of the most efficient methods to achieve the spatial multiplexing gain. MIMO signaling is defined as the communication system with multiple antennas at both the transmitter and receiver, and it can be modeled as a system with multiple inputs and multiple outputs. A MIMO communication system with \(N_{t}\) transmit and \(N_{r}\) receive antennas is shown in Figure 1. This system can be depicted by the following matrix representation:\[\left[ \begin{array}{c}y_{1}\\ \vdots\\ y_{N_{r}}\end{array} \right]=\left[ \begin{array}{ccc}h_{11}&\cdots&h_{1N_{t}}\\ \vdots&\ddots&\vdots\\ h_{N_{t}1}&\cdots&h_{N_{t}N_{t}}\end{array} \right]\left[ \begin{array}{c}x_{1 }\\ \vdots\\ x_{N_{t}}\end{array} \right]+\left[ \begin{array}{c}\
u_{1}\\ \vdots\\ \
u_{N_{r}}\end{array} \right], \tag{1}\]  
or simply by \(\mathbf{y}=\mathbf{Hx}+\boldsymbol{\
u}\), where \(\mathbf{x}\) represents the \(N_{t}\times 1\) dimensional vector of transmitted data symbols, \(\mathbf{y}\) denotes the \(N_{r}\times 1\) dimensional vector of received data symbols, \(\mathbf{H}=[h_{ij}]\) stands for the transmit (propagation channel) matrix with the entry \(h_{ij}\) denoting the gain from transmitter \(j\) to receiver \(i\), and \(\boldsymbol{\
u}\) is assumed to represent the white Gaussian noise vector. From (1), one can infer that  
\[y_{i}=\sum_{j=1}^{N_{t}}h_{ij}x_{j}+\
u_{i},\;i=1,2,\cdots,N_{r}.\]  
In other words, the output symbol \(y_{i}\) is a linear combination of the input symbols, which makes it difficult to recover the transmitted data symbols. This unfavorable phenomenon is referred to as the intersymbol interference (ISI) and results in signal aliasing. However, it is shown next that the MIMO channel can be decomposed into independent signal paths by the help of SVD. This feature enables to design efficient decoding schemes while achieving the spatial multiplexing gain to improve the channel capacity. The SVD representation of channel matrix \(\mathbf{H}\) takes the form:  
\[\mathbf{H}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{H}, \tag{2}\]  
where \(\mathbf{U}\in\mathbb{C}^{N_{r}\times N_{r}}\) and \(\mathbf{V}\in\mathbb{C}^{N_{t}\times N_{t}}\) are unitary matrices, \(\boldsymbol{\Sigma}\in\mathbb{R}^{N_{r}\times N_{t}}\) is a diagonal matrix with \(r\) nonzero diagonal elements, where \(r\leq\min(N_{r},N_{t})\). The parallel decomposition of the MIMO channel into a series of parallel single-input single-output (SISO) channels can be obtained by pre-multiplying the transmitted signal with \(\mathbf{V}\) and post-multiplying the received signal with \(\mathbf{U}^{H}\), as shown in Figure 2.  
In the new coordinate system, the relationship between the input and output of channel takes the form:  
\[