# Spectrum Modeling of Out-of-Band Intermodulation for Dual-Band RF Amplifiers in OFDM Modulation
## 2 Spectrum Model
### Calculating the Correlation for the Dual-Band OFDM Output Signals
\[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\] \[+[a_{2}\ddot{s}_{2}(t)+\frac{3}{4}a_{3}\dot{s}_{2}^{3}(t)+\frac{3} {2}a_{3}\ddot{s}_{2}\ddot{s}_{1}^{2}]cos(2\pi f_{c_{2}}t)\] \[+\frac{3}{4}a_{3}\dot{s}_{1}^{2}(t)\dot{s}_{2}(t)[cos(4\pi f_{c_{1 }}t+2\pi f_{c_{2}}t)+cos(4\pi f_{c_{1}}t-2\pi f_{c_{2}}t)]\] \[+\frac{3}{4}a_{3}\dot{s}_{2}^{2}(t)\dot{s}_{1}(t)[cos(2\pi f_{c_{1 }}t+4\pi f_{c_{2}}t)+cos(4\pi f_{c_{2}}t-2\pi f_{c_{1}}t)]\] \[+\frac{1}{4}a_{3}\dot{s}_{1}^{3}cos(6\pi f_{c_{1}}t)\] \[+\frac{1}{4}a_{3}\dot{s}_{2}^{3}cos(6\pi f_{c_{2}}t)\]  
To simplify the expression of (3), six new terms \(\dot{y}_{1}(t)\), \(\dot{y}_{2}(t)\), \(\dot{y}_{3}(t)\), \(\dot{y}_{4}(t)\), \(\dot{y}_{5}(t)\), and \(\dot{y}_{6}(t)\) are defined as:  
\[\dot{y}_{1}(t)=a_{1}\ddot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3 }{2}a_{3}\ddot{s}_{1}(t)\dot{s}_{2}^{2}(t) \tag{4}\]  
\[\ddot{y}_{2}(t)=a_{1}\ddot{s}_{2}(t)+\frac{3}{4}a_{3}\dot{s}_{2}^{3}(t)+\frac{ 3}{2}a_{3}\ddot{s}_{2}(t)\ddot{s}_{1}^{2}(t) \tag{5}\]  
\[\dot{y}_{3}(t)=\frac{3}{4}a_{3}\dot{s}_{1}^{2}(t)\dot{s}_{2}(t) \tag{6}\]  
\[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]0  
\[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]1  
\[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]2  
Using (4)-(9), (3) can be rewritten as:  
\[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]3 \[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]4 \[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]5 \[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]6 \[y(t) =[a_{1}\dot{s}_{1}(t)+\frac{3}{4}a_{3}\dot{s}_{1}^{3}(t)+\frac{3}{2} a_{3}\dot{s}_{1}\dot{s}_{2}^{2}]cos(2\pi f_{c_{1}}t) \tag{3}\]7  
When the frequency segmentation between the two carrier-frequency bands \(\Delta f=\left|f_{c_{2}}-f_{c_{1}}\right|\) is large, the last six terms in (10) centered at \((2f_{c_{1}}\pm f_{c_{2}})\), \((2f_{c_{2}}\pm f_{c_{1}})\), \(3f_{c_{1}}\) and \(3f_{c_{2}}\), respectively, are considered far away from the spectrum bands of interest. Therefore, after the band-pass filtering, the out-of-band spectral components in (10) are removed [14]. In this article, we now consider the situation that the frequency segmentation \(\Delta f=\left|f_{c_{2}}-f_{c_{1}}\right|\) is small. Therefore, the two terms in (10) centered at \(2f_{c_{2}}-f_{c_{1}}\) and \(2f_{c_{1}}-f_{c_{2}}\) are difficult to be filtered, thus remain in consideration. After the band-pass filtering, the last four terms in (10) centered at \((2f_{c_{1}}+f_{c_{2}})\), \((2f_{c_{2}}+f_{c_{1}})\), \(3f_{c_{1}}\) and \(3f_{c_{2}}\), respectively, are far away from the passband and are thus removed using band-pass filtering.