# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 2 Matrix Theory in Modern Wireless Communication Systems
### Matrix Representation of OFDM  
Orthogonal frequency-division multiplexing (OFDM) is another core technology employed by modern wireless communication systems. It is the basic signaling technology in the recently proposed long-term evolution (LTE) wireless communication standard and has been already adopted as the basic signaling scheme in many communication standards such as Wireless Local Area Network (WLAN), Wireless Metropolitan Area Network (WMAN), Wireless Regional Area Network (WRAN), Wireless Personal Area Network (WPAN), Digital Audio Broadcasting (DAB), and Digital Video Broadcasting (DVB). OFDM represents a multicarrier system which divides the whole spectrum into several narrow frequency bands, and the data is transmitted in a parallel manner along all these narrow frequency bands. The basic signaling mechanism of OFDM [4] is depicted graphically in Figure 3.  
To simplify the exposition, we have represented all the operations between channel input vector \(\mathbf{x}\) and output vector \(\mathbf{y}\) as a system with transfer matrix \(\mathbf{H}\), such that \(\mathbf{y}=\mathbf{H}\mathbf{x}+\boldsymbol{\
u}\), where \(\boldsymbol{\
u}\) stands for the additive noise vector. OFDM implements an inverse discrete Fourier transform (IDFT) at the transmitter side and a discrete Fourier transform (DFT) at the receiver side. However, in practice, the fast Fourier transform (FFT) algorithm is widely used to compute DFT due to its fast implementation and computational efficiency. This is the reason why the processing blocks are marked in Figure 3 with FFT and IFFT instead of DFT and IDFT. Given \(x[n]\), \(n=0,1,\cdots,N-1\), the \(N\)-point DFT of \(x[n]\) is defined as  
\[X[k]=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi\text{rad}}{N}},\;k=0,1,\cdots,N-1.\]  
The sequence \(x[n]\) can be recovered back by means of the IDFT as  
\[x[n]=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X[k]e^{j\frac{2\pi\text{rad}}{N}},\;n=0,1,\cdots,N-1.\]  
It is easy to show that the DFT operation can be represented as a matrix multiplication  
\[\mathbf{X}=\mathbf{W}\mathbf{x},\]  
where \(\mathbf{X}=[X[0],X[1],\cdots,X[N-1]]^{T}\), \(\mathbf{x}=[x[0],x[1],\cdots,x[N-1]]^{T}\), and the DFT matrix \(\mathbf{W}\) is a unitary matrix expressed as  
\[\mathbf{W}=\frac{1}{\sqrt{N}}\left[ \begin{array}{ccccc}1&1&1&\cdots&1\\ 1&w&w^{2}&\cdots&w^{N-1}\\ 1&w^{2}&w^{4}&\cdots&w^{2(N-1)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&w^{N-1}&w^{2(N-1)}&\cdots&w^{(N-1)^{2}}\end{array} \right] \tag{5}\]  
Figure 3: Orthogonal frequency-division multiplexing (OFDM). with \(w=e^{-j\frac{2\pi}{N}}\). Additionally, the IDFT operation can be expressed as  
\[\mathbf{x}=\mathbf{W}^{-1}\mathbf{X}=\mathbf{W}^{H}\mathbf{X},\]  
since \(\mathbf{W}^{-1}=\mathbf{W}^{H}\) is due to the fact that \(\mathbf{W}\) is a unitary matrix. Matrix Analysis of OFDM  
Consider a discrete channel \(\mathbf{H}\) with coefficients \(h[n]\), \(n=0,1,\cdots,M\). The cyclic prefix [4] is implemented to generate the new sequence \(\hat{x}[n]\), \(n=-M,-M+1,\cdots,N-1\), where \(\hat{x}[n]=x[n\bmod N]\). Specifically,  
\[\hat{x}[-M],\cdots,\hat{x}[-1],\hat{x}[0],\cdots,\hat{x}[N-1]=x[N-M],\cdots,x[ N-1],x[0],\cdots,x[N-1]. \tag{6}\]  
This yields the output \(y[n]=\hat{x}[n]*h[n]+\
u[n]\). Introducing the simplified notations \(\hat{x}[n]\) as \(\hat{x}_{n}\), \(h[n]\) as \(h_{n}\), etc., it turns out that the relation between the input and output of the channel can be expressed as  
\[\left[ \begin{array}{c}y_{N-1}\\ y_{N-2}\\ \vdots\\ y_{0}\end{array} \right]=\left[ \begin{array}{cccccc}h_{0}&h_{1}&\cdots&h_{M }&0&\cdots&0\\ 0&h_{0}&\cdots&h_{M-1}&h_{M}&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&h_{0}&\cdots&h_{M-1}&h_{M}\end{array} \right]\left[ \begin{array}{c} \tilde{x}_{N-1}\\ \vdots\\ \tilde{x}_{0}\\ \tilde{x}_{-1}\\ \vdots\\ \tilde{x}_{-M}\end{array} \right]+\left[ \begin{array}{c}v_{N-1}\\ v_{N-2}\\ \vdots\\ v_{0}\end{array} \right]. \tag{7}\]  
Plugging (6) into (7) leads to  
\[\left[ \begin{array}{c}y_{N-1}\\ y_{N-2}\\ \vdots\\ y_{0}\end{array}