# Analytical Framework of CP-Free Multiuser OFDM System for Coordinated Multi-Point at mmWave
## 2 System Model
### Signal Model  
A framework is considered where a BS is composed of \(N_{T}\) transmitting antennas and each of the \(K\) users is furnished with \(N_{K}\) receiving antennas under the scenario of multiuser downlink transmission. The digitally modulated complex symbol vector for the user \(k\) at the transmit side is \(x_{k}\) and it is spread through multiplying with the one-dimensional CHT matrix \(W_{k}\), which is addressed in [26]. The CHT matrix of order 8 for the users can be written as  
\[ \begin{array}{ccccccccc}W_{1}&=&[1&j&-1&-j&1&j&-1&-j]\\ W_{2}&=&[1&-j&-1&j&1&-j&-1&j]\\ W_{3}&=&[1&1&j&j&-1&-1&-j&-j]\\ W_{4}&=&[1&-1&j&-j&-1&1&-j&j]\end{array}  \tag{1}\]  
Further, the CHT signal for user \(k\) can be expressed as follows:  
\[X_{k}=W_{k}x_{k} \tag{2}\]  
where \(k=1,2,3,4\). In this study, we consider a special case of four users and proceed with our calculations accordingly. The data symbol vector \(X_{k}\) is transformed into matrix \(\bar{X}_{k}\) of a size \(\bar{N}\times L\), where \(L\) is the number of OFDM symbols to be produced and \(\bar{N}\) is the number of active subcarriers in each OFDM symbol. The \(l\)th data symbol vector \(\bar{X}_{k,l}[\bar{n}]\) of the matrix \(\bar{X}\) with \(\bar{n}=1,2,3,\ldots\ldots\ldots,\bar{N}\) and \(l=1,2,3,\ldots\ldots\ldots,L\) is oversampled through adding \(\bar{N}\) number of zeros in both ends. Considering \(\bar{N}\) as the number of null subcarriers and concatenating all the data samples of \(\bar{X}_{k,l}[\bar{n}]\), the input \(l\)th data symbol vector \(\bar{X}_{k,l}[\bar{n}]\) is fed into the OFDM modulation process as follows:  
\[\bar{X}_{k,l}= \begin{bmatrix}0_{\bar{N}\times 1}\\ \bar{X}_{k,\bar{n},\bar{l}}\\ 0_{\bar{N}\times 1}\end{bmatrix}  \tag{3}\]  
where \(0_{\bar{N}\times 1}\) is a zero vector of a size (\(\bar{N}\times 1\)). The signal vector \(\bar{X}_{k,l}\) contains \(N\) (\(=\bar{N}+2\times\bar{N}\)) number of samples and its \(n\)th (\(n=0,1,2,\ldots,N-1\)) sample is modulated onto one subcarrier. The time-domain \(l\)th OFDM-modulated complex signal for \(m\)th time sample index can be noted down as  
\[\bar{X}_{k,l}[m]=\frac{1}{N}\sum_{n=0}^{N-1}\bar{X}_{k,l}[n]e^{-\frac{2\pi m }{N}} \tag{4}\]  
As cyclic prefix is not preferred in this paper and considering all the time samples of \(\bar{X}_{k,l}[m]\), \(\bar{X}_{k,l}\) can be treated as the \(l\)th CP-free OFDM data symbol matrix. A root-raised cosine filter with roll-off factor \(\alpha\) spanning over \(\beta\) number of information-bearing samples of the \(l\)th OFDM symbol vector \(\bar{X}_{k,l}\) can be written as [32]:  
\[h_{src}(\bar{n})=\frac{ \begin{bmatrix}\frac{4\alpha}{\pi}\frac{\cos((1+\alpha )\pi\bar{n})}{\beta}\end{bmatrix} + \begin{bmatrix}(1-\alpha)\frac{\sin((1- \alpha)\pi\bar{n})}{\beta}\end{bmatrix} }{\sqrt{\beta} \begin{bmatrix}1- \begin{pmatrix}\frac{4\alpha\bar{n}}{\beta}\end{bmatrix} ^{2}\end{bmatrix}} \tag{5}\]  
where \(\bar{n}=0,1,2,3,\ldots\ldots\ldots,\beta\eta\), \(\eta\) denotes the number of samples per information-bearing sample of the \(l\)th OFDM symbol vector \(\bar{X}_{k,l}\), and the number of filter coefficients is \(\beta\eta+1\). With the application of the root-raised cosine filter \(h_{src}(\bar{n})\) on the data vector \(\bar{X}_{k,l}[m]\), we can get the filtered output as  
\[\overrightarrow{X}_{k,l}[\bar{m}]=\bar{X}_{k,l}[m]*h_{src}(\bar{n}) \tag{6}\]  
where \(\bar{m}=0,1,2,3,\ldots\ldots\ldots,\bar{N}+\beta\eta-1\) and \(\bar{N}=N\eta-(\eta-1)\). The symbol \(*\) indicates the linear convolution. The precoders are designed by utilizing a paradigm of path loss under NLOS environment by estimating the MIMO fading channel. At 28 GHz mmWave frequency, the model covering path loss is given by [33]  
\[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]  
where \(d\) denotes the distance in meter between the transmitter and user's end. \(X_{\sigma}\) denotes log-normal random variable composed of zero mean and standard deviation \(\sigma\).