# Spectrum Modeling of Out-of-Band Intermodulation for Dual-Band RF Amplifiers in OFDM Modulation
## 2 Spectrum Model
### The Power Spectrum Density of Amplified Signals  
By using (13) and (16), \(P_{y_{3}}(f)\) can be derived as:  
\[P_{\tilde{y}_{3}}(f) =\mathcal{F}\{R_{\tilde{y}_{3}}(\tau)\} \tag{18}\] \[=\frac{9}{16}a_{3}^{2}\mathcal{F}\left\{R_{\tilde{y}_{2}}(\tau)(N _{1}B_{1})^{2}+2R_{\tilde{y}_{2}}(\tau)R_{\tilde{y}_{1}}^{2}(\tau)\right\}\]where \(\mathscr{F}\{\cdot\}\) is defined as the Fourier transform of \(\{\cdot\}\). Based on ([14], Equations (4), (25) and (49)), \(\mathscr{F}\{R_{\tilde{s}_{2}}(\tau)\}\) and \(\mathscr{F}\{R_{\tilde{s}_{2}}(\tau)R_{\tilde{s}_{1}}^{2}(\tau)\}\) can be described respectively as:  
\[\mathscr{F}\{R_{\tilde{s}_{2}}(\tau)\}=\left\{ \begin{array}{ll}\frac{N_{2}}{ 2},&\left|f\right|\leq B_{2}\\ 0,&\left|f\right|>B_{2}\end{array} \right. \tag{19}\]  
and  
\[\mathscr{F}\{R_{\tilde{s}_{2}}(\tau)R_{\tilde{s}_{1}}^{2}(\tau)\}=\left\{ \begin{array}{ll}\frac{N_{2}^{2}N_{2}}{8}\left(-f^{2}+4B_{1}B_{2}-B_{2}^{2} \right),&\left|f\right|\leq B_{2}\\ \frac{N_{2}^{2}N_{2}}{8}\left(4B_{1}B_{2}-2B_{2}\left|f\right|\right),&B_{2}< \left|f\right|\leq 2B_{1}-B_{2}\\ \frac{N_{2}^{2}N_{2}}{16}\left(2B_{1}+B_{2}-\left|f\right|\right)^{2},&2B_{1}- B_{2}<\left|f\right|\leq 2B_{1}+B_{2}\\ 0,&\left|f\right|\geq 2B_{1}+B_{2}\end{array} \right. \tag{20}\]  
By substituting (19) and (20) into (18), \(P_{\tilde{y}_{3}}(f)\) can be derived as:  
\[ \begin{array}{ll}P_{\tilde{y}_{3}}(f)=\\ \left\{\begin{array}{ll}\frac{1}{B_{2}}P_{o1}^{2}P_{o2}10^{-\frac{-ip_{3}}{5} }+\frac{1}{2B_{1}^{2}B_{2}}&\left|f\right|\leq B_{2}\\ \cdot P_{o1}^{2}P_{o2}10^{-\frac{-ip_{3}}{5}}\left(-f^{2}+4B_{1}B_{2}-B_{2}^{2} \right),&\left|f\right|\leq B_{2}\\ \frac{1}{2B_{1}^{2}B_{2}}P_{o1}^{2}P_{o2}10^{-\frac{-ip_{3}}{5}}&B_{2}<\left|f \right|\leq 2B_{1}-B_{2}\\ \cdot\left(4B_{1}B_{2}-2B_{2}\left|f\right|\right),&\\ \frac{1}{4B_{1}^{2}B_{2}}P_{o1}^{2}P_{o2}10^{-\frac{-ip_{3}}{5}}&2B_{1}-B_{2}< \left|f\right|\leq 2B_{1}+B_{2}\\ \cdot\left(2B_{1}+B_{2}-\left|f\right|\right)^{2},&\\ 0,&\left|f\right|>2B_{1}+B_{2}\end{array}\right.\end{array}  \tag{21}\]  
where \(P_{o1}\) is the linear part of the output power for \(\tilde{s}_{1}(t)\) and \(P_{o2}\) is the linear part of the output power for \(\tilde{s}_{2}(t)\)[11; 12], which are  
\[P_{o1}=\frac{a_{1}^{2}N_{1}B_{2}}{2} \tag{22}\]  
and  
\[P_{o2}=\frac{a_{1}^{2}N_{2}B_{2}}{2} \tag{23}\]  
Equation (21) describes the PSD of the first left part of the out-of-band intermodulation. Similarly, the right part of the out-of-band intermodulation \(P_{\tilde{y}_{4}}(f)\) can be calculated as:  
\[ \begin{array}{ll}P_{\tilde{y}_{4}}(f)=\\ \left\{\begin{array}{ll}\frac{1}{B_{1}}P_{o1}P_{o2}^{2}10^{-\frac{-ip_{3}}{5} }&\left|f\right|\leq 2B_{2}-B_{1}\\ +\frac{1}{2B_{1}^{2}B_{2}}P_{o1}P_{o2}^{2}10^{-\frac{-ip_{3}}{5}}\left(-f^{2} +4B_{1}B_{2}-B_{2}^{2}\right),&\left|f\right|\leq 2B_{2}-B_{1}\\ \frac{1}{B_{1}}P_{o1}P_{o2}^{2}10^{-\frac{-ip_{3}}{5}},&\\ +\frac{1}{4B_{1}B_{2}^{2}}P_{o1}P_{o2}^{2}10^{-\frac{-ip_{3}}{5}}\cdot\left[-f ^{2}+\left(2B_{1}-4B_{2}\right)\right.&2B_{2}-B_{1}<\left|f\right|\leq B_{1}\\ \cdot\left|f\right|-B_{1}^{2}+4B_{2}^{2}+4B_{1}B_{2}]\\ \frac{1}{4B_{1}B_{2}^{2}}P_{o1}P_{o2}^{2}10^{-\frac{-ip_{3}}{5}}\left(B_{1}+2B _{2}-\left|f\right|\right)^{2},&B_{1}<\left|f\right|\leq B_{1}+2B_{2}\\ 0,&\left|f\right|>B_{1}+2B_{2}\end{array}\right.\end{array}  \tag{24}\]  
Based on (12), we now could decompose \(P_{y}(f)\) into:  
\[P_{y}(f)=P_{y_{pushmd}}(f)+P_{y_{out-of-bm}}(f) \tag{25}\]where \(P_{y_{pushand}}(f)\) is attributed to linear amplification, in-band IM and CM, while \(P_{y_{out-of-hmd}}(f)\) is attributed to out-of-band only. \(P_{y_{pashand}}(f)\) can be written as ([14], Equation (54)), and the observations related to \(P_{y_{pashand}}(f)\) can be found in [14]. So as to obtain \(P_{y_{out-of-hmd}}(f)\), we need to shift \(P_{y_{3}}(f)\) and \(P_{y_{4}}(f)\) to carrier frequencies \(2f_{c_{1}}-f_{c_{2}}\) and \(2f_{c_{2}}-f_{c_{1}}\), respectively.