# Integrated Communication and Measurement System with BOC-Assisted OFDM
## 2 The Transmitting and Receiving System for the Composite Signal
### Signal Model
Meanwhile, to avoid interference in the demodulation of the OFDM signal caused by the peak of the main lobe of the BOC signal, the OFDM subcarrier signal corresponding to the subcarrier of the main lobe of the BOC signal is set to zero, namely, the OFDM signal does not exist at this subcarrier. Thus, orthogonality between the main lobe of the BOC signal and the OFDM signal is ensured. Furthermore, the power of the sidelobes of the BOC signal is sufficiently small to be equivalent to noise. Figure 2 shows the spectrum diagram of the fundamental frequency BOC (4, 1) signal, from which the sidelobe power is observed to be 15 dB lower than the main lobe power. Therefore, the interference with the OFDM signal is negligible. Fourth, the IFFT is applied after power control between \(F_{BOC}(u)\) and \(S_{k}^{\prime}\). Equation (5) is derived by combining Equations (3) and (4):  
\[ \begin{split} IFFT\big{[}\alpha\cdot S_{k}(m)+\beta\cdot F_{BOC} (u)\big{]}&=\frac{1}{M}\sum_{m=0}^{M-1}\left(\alpha\cdot S_{k}(m)+ \beta\cdot F_{BOC}(u)\right)\cdot e^{\beta 2\pi bu/M}\\ &=\frac{1}{M}\sum_{m=0}^{M-1}\left(\alpha\cdot S_{k}(m)+\beta \cdot\sum_{u=0}^{U}S_{b}(u)\cdot e^{-\beta 2\pi bu/U}\right)\cdot e^{\beta 2\pi bu/M}\\ &=\alpha\cdot\frac{1}{M}\sum_{m=0}^{M-1}S_{k}(m)\cdot e^{2\pi bu /M}+\beta\cdot\frac{1}{M}\sum_{m=0}^{M-1}\sum_{u=0}^{U-1}S_{b}(u)\cdot e^{- \beta 2\pi bu/U}\cdot e^{\beta 2\pi km/M}\end{split}  \tag{5}\]  
Figure 2: Spectrum diagram of the fundamental-frequency BOC (4, 1) signal (the square-wave subcarrier frequency is 4.096 MHz; the code frequency is 1.023 MHz). Finally, the serial data bits are obtained through parallel to serial (P/S) conversion, and carrier modulation is performed. Here, the function of carrier modulation is to shift the signal from the fundamental frequency to a high frequency. In conclusion, the composite signal is expressed as Equation (6):  
\[ \begin{split} S_{\text{OFDM\_BOC}}&=(IFFT[\alpha\cdot S _{k}(m)+\beta\cdot F_{\text{BOC}}])\cdot f_{c}(t)+n(t)\\ &=\left(\alpha\cdot\frac{1}{M}\sum_{m=0}^{M-1}S_{k}(m)\cdot e^{ |2\pi km/M}+\beta\cdot\frac{1}{M}\sum_{m=0}^{M-1}\sum_{u=0}^{U-1}S_{b}(u)\cdot e ^{-j2\pi bu/U}\cdot e^{j2\pi km/M}\right)\cdot e^{2\pi f_{c}t}+n(t)\\ &\quad\quad\text{where $\alpha$ and $\beta$ are the power ratios of the OFDM signal and the BOC signal, respectively, satisfying $\alpha+\beta=1$; $f_{c}(t)=e^{2\pi f_{c}t}$ is the carrier signal; $f_{c}$ is the carrier frequency; and $n(t)$ is additive white Gaussian noise.}\end{split}  \tag{6}\]  
According to the frequency-domain sampling theorem [26], when the number of frequency-domain sampling points is greater than or equal to the length of the signal sequence, the original signal sequence can be recovered from the frequency-domain sampling. Thus, the following equation is obtained:  
\[IFFT[F_{\text{BOC}}(u)]=\frac{1}{M}\sum_{m=0}^{M-1}\sum_{u=0}^{U-1}S_{b}(u) \cdot e^{-j2\pi bu/U}\cdot e^{j2\pi km/M}=S_{b}(u)=S_{BOC}(t) \tag{7}\]  
Hence, by combining Equations (6), (7), and (3), one finds that the composite signal is equivalent to Equation (8):  
\[ \begin{split} S_{\text{OFDM\_BOC}}&=(IFFT[\alpha \cdot S_{k}(m)+\beta\cdot F_{\text{BOC}}])\cdot f_{c}(t)+n(t)\\ &=(\alpha\cdot S_{OFDM}(t)+\beta\cdot S_{BOC}(t))\cdot e^{j2\pi f _{c}t}+n(t)\\ &=\left(\alpha\cdot S_{OFDM}(t)\cdot e^{|2\pi f_{c}t}+\beta\cdot S _{BOC}(t)\cdot e^{j2\pi f_{c}t}\right)+n(t)\\ &=\alpha\cdot S_{OFDM}^{\prime}(t)+\beta\cdot S_{BOC}^{\prime}(t) +n(t)\end{split}  \tag{8}\]  
where \(S_{OFDM}^{\prime}\) and \(S_{BOC}^{\prime}\) are the high-frequency OFDM signal and BOC signal after carrier modulation, respectively. The spectrum diagram of the composite signal is shown in Figure 3. The number of OFDM subcarriers must be \(2^{i}\). Therefore, the number of subcarriers is selected to be \(2^{4}~{}=~{}16\).