# 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
\begin{split}\mathbf{\hat{y}}&=\mathbf{U}^{H}( \mathbf{Hx}+\boldsymbol{\
u})\\ &=\mathbf{U}^{H}(\mathbf{H}\mathbf{V}\hat{\mathbf{x}}+\boldsymbol{ \
u})\\ &=\mathbf{U}^{H}(\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{H} \mathbf{V}\hat{\mathbf{x}}+\boldsymbol{\
u})\\ &=\boldsymbol{\Sigma}\mathbf{\hat{x}}+\boldsymbol{\hat{v}}.\end{split}  \tag{3}\]  
Figure 1: Multiple-input multiple-output (MIMO) channel. Figure 2: Parallel decomposition. Assume \(\mathbf{\Sigma}=\text{diag}(\sigma_{1},\sigma_{2},\cdots,\sigma_{r},0,\cdots,0)\). Thus, the MIMO channel can be decomposed into \(r\) independent SISO channels. Specifically, the \(i\)th SISO channel has a gain \(\sigma_{i}\) and a noise factor \(\tilde{v}_{i}\), and assumes the representation  
\[\tilde{y}_{i}=\sigma_{i}\tilde{x}_{i}+\tilde{v}_{i}.\]  
This implies that all the efficient signaling and decoding strategies well investigated in the SISO context could be transferred mutatis-mutandis to the MIMO framework with a minimal effort thanks to SVD. Moreover, by assuming that the channel state information, or equivalently, the channel matrix \(\mathbf{H}\) is known at the receiver but not at the transmitter, [PERSON] [3] showed that the MIMO channel capacity can be expressed as  
\[C=\sum_{i=1}^{r}\log_{2}\left(1+\frac{\sigma_{i}^{2}\rho}{N_{t}}\right), \tag{4}\]  
where \(\rho\) is the transmitted signal power and the additive receiver noise is assumed to have unit power. This result is in perfect accordance with (3) since the MIMO capacity in (4) can be interpreted as the summation of independent SISO channel capacities. Under the assumption that the entries of \(\mathbf{H}\) follow a zero-mean spatially white (ZMSW) model, the law of large numbers implies that \(\lim\limits_{N_{t}\rightarrow\infty}\frac{1}{N_{t}}\mathbf{H}\mathbf{H}^{H}= \mathbf{I}_{N_{r}}\), and the capacity can be expressed asymptotically [4] in the presence of a large number of transmit/receive antennas as  
\[C=N_{r}\log_{2}(1+\rho).\]  
This implies that as \(N=\min\left(N_{r},N_{t}\right)\) grows large, the channel capacity increases linearly with \(N\)[5]. This fact as well as some other results for MIMO capacity analysis [6; 7] makes MIMO signaling superior to SISO signaling and it also explains the reason why currently MIMO techniques are so appealing in a broad range of applications such as wireless communications, radar, sonar, geo-physical (seismic) signal processing, wireless sensor networks, and antenna array processing. The proposed framework for determining the capacity of MIMO linear time-invariant channels could be further extended to determine the capacity of memoryless nonlinear MIMO channels subject to a power constraint [8]. Reference [8] re-formulates this problem within a functional framework and employs variational calculus techniques to determine the capacity-achieving input distribution of memoryless nonlinear MIMO channels. As a special case, the capacity-achieving input distribution for the linear MIMO channel is re-established in [8].