# Analytical Framework of CP-Free Multiuser OFDM System for Coordinated Multi-Point at mmWave
## 2 System Model
### Signal Model
#### 2.3.2 With HPA  
In this subsection, Algorithm 2 is introduced to highlight various steps for estimating system-performance-relayed parameters with HPA implementation. In each BS, the precoded data signals in each of the eight channels are passed through nonlinear HPA for the sake of improving power-related effectiveness as well as minimizing the nonlinear distortion. The nonlinear HPA amplifier has been considered in [29], where the complex envelope of the input signal exerted to the HPA in each channel is  
\[x_{k}(t)=\rho(t)e^{\left|\hat{\varphi}_{k}(t)\right|} \tag{25}\]  
Now, for the case of each channel, the complex envelope of the output signal can be exhibited by  
\[z_{k}(t)=A_{k}[\rho(t)]e^{\left|\rho(\varphi_{k}(t)+\varphi_{k}|\rho(t))\right|} \tag{26}\]  
where \(A_{k}[\rho]\) and \(\psi_{k}[\rho]\) denotes, respectively, the AM/AM and AM/PM conversion of the nonlinear amplifier. Generally, the operating point of the amplifier is denoted by \"back-off\" and the IBO and the output back-off (OBO) can be noted as  
\[IBO=20\log_{10}\left(\frac{A_{sat}}{\sigma_{IN}}\right)\] dB (27a) \[OBO=20\log_{10}\left(\frac{A_{o}}{\sigma_{OUT}}\right)\] dB (27b)  
where \(\sigma_{IN}^{2}\) denotes the mean power of the HPA input signal, \(\sigma_{OUT}^{2}\) denotes the mean power of the transmitted signal, \(A_{o}\) denotes the maximum output amplitude, and \(A_{sat}\) denotes the input amplitude corresponding to the maximum output amplitude. However,on eliminating nonlinear distortion in HPA, the output signal matrix from the first and second BS may be designated as \(\overrightarrow{X_{0}}\) and \(\overrightarrow{X}_{0}\). ```
Initialization: Signal-to-Noise ratio (SNR) = \(-10\) to \(10\) (as cited in Algorithm 1). Inputs: \(x_{k}(t)\). Outputs: IBO, OBO, and \(X_{k=1,2,3,4}^{\prime\prime\prime}\) while(different SNR)do  Step 1: Calculate \(z_{k}(t)\): HPA output signal from its input signal \(x_{k}(t)\) from Equation (26)  Step 2: Apply memoryless Saleh model for a travelling wave tube as \[A_{k}[\rho(t)]=A_{sat}^{2}\frac{\rho(t)}{\rho^{2}(t)+A_{sat}^{2}}\] \[\psi_{k}[\rho(t)]=\frac{\pi}{3}\frac{\rho^{2}(t)}{\rho^{2}(t)+A_{sat}^{2}}\quad rad\] where \(A_{sat}\) denotes the amplifier input saturation voltage, \(\rho(t)\) denotes the amplitude of the input signal applied to HPA. Step 3: Calculate \(Y_{k}^{\prime\prime}\): Received signal for user \(k\) with HPA implementation from Equation (28)  Step 4: Use \(H_{\textit{eqk},k=1,2,3,4}\): Equivalent channel for user \(k(=1,2,3,4)\) as considered in Step 4 of Algorithm 1  Step 5: Calculate \(\overrightarrow{\overline{Y}_{k}^{\prime\prime}}\): Received signal for user \(k\) with interfering signal neglected  Step 6: Obtain IBO: Input back-off from Equation (27a)  Step 7: Obtain OBO: Output back-off from Equation (27b)  Step 8: Obtain \(X_{k=1,2,3,4}^{\prime\prime\prime}\): Detected signal for user \(k(=1,2,3,4)\) using signal detection technique [31] from Equations (30a)-(30d)  end while
```  
**Algorithm 2**For user \(k\) with HPA implementation  
In the case of implementing HPA, the signal received at the user \(k\) can be written as  
\[Y_{k}^{\prime\prime}=H_{k}\overleftrightarrow{X_{0}}+n_{001}+\bar{H}_{k} \overrightarrow{X}_{0}+n_{002} \tag{28}\]  
where \(n_{001}\) and \(n_{002}\) are the AWGN noises associated with transmitted HPA-implemented signals from the first and second BS. In processing the signal model presented in Equation (28), the equivalent channel matrices [Equation (22a-d)] have been incorporated merely to obtain approximate results.