# Analytical Framework of CP-Free Multiuser OFDM System for Coordinated Multi-Point at mmWave
## 2 System Model
### Signal Model
#### 2.3.1 Without HPA  
In this subsection, Algorithm 1 is introduced to highlight various steps for estimating the system-performance-related parameters without HPA implementation. In the case of a system without HPA, the signal obtained at the user \(k\) can be represented as  
\[\bar{Y}_{k}=H_{k}X_{0}+n_{01}+\bar{H}_{k}\bar{X}_{0}+n_{02} \tag{16}\]  
where \(n_{01}\) and \(n_{02}\) are the AWGN noises associated with transmitted signals from first and second BS. Equation (16) for user \(k\) can be written in an elaborated form as  
\[\bar{Y}_{k} =H_{k}(X_{1}+X_{2}+X_{3}+X_{4})+\bar{H}_{k}(\bar{X}_{1}+\bar{X}_{ 2}+\bar{X}_{3}+\bar{X}_{4})+n_{01}+n_{02}\] \[=H_{k}(X_{k})+\bar{H}_{k}(\bar{X}_{k})+H_{k\
eq k}\big{(}\sum(X_{ \bar{k}})\big{)}+\bar{H}_{k\
eq k}\big{(}\sum(\bar{X}_{\bar{k}})\big{)}+n_{01k}\] \[=\underbrace{H_{k}(X_{k})+\bar{H}_{k}(\bar{X}_{k})}_{\text{user}} +\underbrace{H_{k\
eq k}\big{(}\sum(X_{\bar{k}})\big{)}+\bar{H}_{k\
eq k}\big{(} \sum(\bar{X}_{\bar{k}})\big{)}}_{\text{interference}}+n_{01k} \tag{17}\]  
where the first term is user \(k\)'s own signal, the second term is the interference signal, the third is the effective noise (\(n_{01k}=n_{01}+n_{02}\)) with noise variance \(\sigma^{2}_{01k}\), \(\sum(. )\) is indicative of summation of variables \(k\), and \(\bar{k}=1,2,3,4\). The received signal-to-noise ratio for user \(k\) (\(SNR_{k}\)) can be written as  
\[SNR_{k}=\frac{||H_{k}(X_{k})+\bar{H}_{k}(\bar{X}_{k})||^{2}}{N_{k}\sigma^{2}_{0 1k}} \tag{18}\]  
As the signal model presented in Equation (17) corresponds to the undesired signal with the sum of noise and interference signal, the received signal-to-interference-noise ratio for the user \(k\) (\(SINR_{k}\)) can be written as  
\[SINR_{k}=\frac{||H_{k}(X_{k})+\bar{H}_{k}(\bar{X}_{k})||^{2}}{||H_{\bar{k} \
eq k}(\sum(\bar{X}_{\bar{k}}))+\bar{H}_{\bar{k}\
eq k}(\sum(\bar{X}_{\bar{k}} ))||^{2}+N_{\bar{k}}\sigma^{2}_{01k}} \tag{19}\]  
The ergodic achievable rate \(R_{k}\) for user \(k\) is a logarithmic function of \(SINR_{k}\) and it can be written as \[R_{k} =\mathbb{E}\{\log_{2}(1+SINR_{k})\}\] \[=\mathbb{E}\Bigg{\{}\log_{2}\Bigg{(}1+\frac{||H_{k}(X_{k})+\bar{H}_{ k}(X_{k})||^{2}}{||H_{\bar{k}\
eq k}(\Sigma(X_{\bar{k}}))+\bar{H}_{\bar{k}\
eq k}( \Sigma(\bar{X}_{\bar{k}}))||^{2}+N_{k}\sigma_{01k}^{2}}\Bigg{)}\Bigg{\}} \tag{20}\]  
Considering identical transmission power from each of the two BSs ranging from 500 W to 5000 W and keeping typically assumed noise variance \(\sigma_{01k}^{2}\) for each receiving end of user \(k\) is of \(-90\) dBm, which corresponds to 10-12 W, the received SINR and ergodic achievable rate \(R_{k}\) can be estimated. As the interference term presented in Equation (17) is negligible in comparison with the user's desirable signal, the signal received at the four users can be written with user interference terms neglected as  
\[Y_{1} =(H_{1}W_{1}+\bar{H}_{1}\bar{W}_{1})\hat{X}_{1}+n_{1} \tag{21a}\] \[Y_{2} =(H_{2}W_{2}+\bar{H}_{2}\bar{W}_{2})\hat{X}_{2}+n_{2}\] (21b) \[\bar{Y}_{k} =H_{k}(X_{1}+X_{2}+X_{3}+X_{4})+\bar{H}_{k}(\bar{X}_{1}+\bar{X}_{ 2}+\bar{X}_{3}+\bar{X}_{4})+n_{01}+n_{02}\]0 (21c) \[\bar{Y}_{k} =H_{k}(X_{1}+X_{2}+X_{3}+X_{4})+\bar{H}_{k}(\bar{X}_{1}+\bar{X}_{ 2}+\bar{X}_{3}+\bar{X}_{4})+n_{01}+n_{02}\]1  
where \(n_{1}\), \(n_{2}\), \(n_{3}\), and \(n_{4}\) have been substituted into Equations (21a)-(21d) instead of \(n_{01k}\) from Equation (17). ```
Initialization: Signal-to-Noise ratio (SNR) = \(-10\) to \(10\). Inputs: \(X_{0}\), \(\bar{X}_{0}\), \(H_{0}\), \(\bar{H}_{0}\), \(n_{01}\), and \(n_{02}\).