# Analytical Framework of CP-Free Multiuser OFDM System for Coordinated Multi-Point at mmWave
## 2 System Model
### Signal Model
As CoMP-JT is concerned, the estimated channel matrices for four users with first BS can be written as \(H_{1}\), \(H_{2}\), \(H_{3}\), and \(H_{4}\), where each is of \(N_{K}\times N_{T}\) matrix in size. The joint-channel matrix \(H\) is expressed as follows:  
\[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]  
Four matrices \(\bar{H}_{1}\), \(\bar{H}_{2}\), \(\bar{H}_{3}\), and \(\bar{H}_{4}\) are considered and each is \((3\times N_{K})\times N_{T}\) in size for designing transmit precoding matrix; \(W_{1}\), \(W_{2}\), \(W_{3}\), and \(W_{4}\) is for four users and each is of \(N_{T}\times N_{K}\) in size as follows:  
\[\bar{H}_{1} =\left[H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H} \tag{9a}\] \[\bar{H}_{2} =\left[H_{1}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\] (9b) \[\bar{H}_{3} =\left[H_{1}^{H}\;H_{2}^{H}\;H_{4}^{H}\right]^{H}\] (9c) \[\bar{H}_{4} =\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\right]^{H} \tag{9d}\]  
where \((\cdot)^{H}/[\cdot]^{H}\) are indicative of conjugate transformation. On singular value decomposition (SVD) of \(\bar{H}_{1}\), \(\bar{H}_{2}\), \(\bar{H}_{3}\), and \(\bar{H}_{4}\), we obtain  
\[\bar{H}_{1} =U_{1}\bar{\wedge}_{1}[\bar{V_{1}}^{\;non-zero}\;\bar{V_{1}}^{\; zero}]^{H} \tag{10a}\] \[\bar{H}_{2} =U_{2}\bar{\wedge}_{2}[\bar{V_{2}}^{\;non-zero}\;\bar{V_{2}}^{\; zero}]^{H}\] (10b) \[\bar{H}_{3} =U_{3}\bar{\wedge}_{3}[\bar{V_{3}}^{\;non-zero}\;\bar{V_{3}}^{\; zero}]^{H}\] (10c) \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]0  
The matrices \(U_{1}\), \(U_{2}\), \(U_{3}\), and \(U_{4}\) are all \((3\times N_{K})\times N_{T}\) in size; the matrices \(\bar{\wedge}_{1}\), \(\bar{\wedge}_{2}\), \(\bar{\wedge}_{3}\), and \(\bar{\wedge}_{4}\) are all \(N_{T}\times N_{T}\) in size; and the matrices \(\bar{V}_{1}\), \(\bar{V}_{2}\), \(\bar{V}_{3}\), and \(\bar{V}_{4}\) are all \(N_{T}\times N_{T}\) in size. Their all nonzero matrices are all in \(N_{T}\times(N_{T}-\bar{L})\) in size and their all zero matrices are all in \(N_{T}\times\bar{L}\) in size. The zero matrices of \(\bar{V}_{1}\), \(\bar{V}_{2}\), \(\bar{V}_{3}\), and \(\bar{V}_{4}\) are assumed to have zero singular values and such matrices are considered to be the precoding matrices [29]:  
\[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]1 \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]2 (11b) \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]3 (11c) \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]4  
On multiplication of precoding matrix \(\widehat{X}_{k}\) with respective precoding matrix of individual user, four precoded matrices for four users are obtained, each with \(N_{T}\times\bar{L}\) in size as follows:  
\[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]5 \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]6 (12b) \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]7 (12c) \[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]8Summing all precoded matrix data for first BS (BS 1), we obtain  
\[PL_{NLOS}(d)=61.4+34\log_{10}(d)+X_{\sigma} \tag{7}\]9  
With an adaptation of the procedure presented in Equation (7) through Equation (10) in the case of the second BS, the computed four precoded matrices for four users will be \(\bar{W}_{1}\), \(\bar{W}_{2}\), \(\bar{W}_{3}\), and \(\bar{W}_{4}\) and we obtain four precoded matrices for four users, each with \(N_{T}\times\bar{L}\) in size as  
\[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]0 \[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]1 (14b) \[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]2 (14c) \[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]3  
Summing all the precoded matrix data for second BS (BS 2), we obtain  
\[H=\left[H_{1}^{H}\;H_{2}^{H}\;H_{3}^{H}\;H_{4}^{H}\right]^{H}\in\mathbb{C}^{4N _{K}\times N_{T}} \tag{8}\]4