## 2 Theoretical Derivation of Short-Code Systems  
This paper takes the commonly used BPSK-DSSS system as an example to discuss the anti-single-frequency interference performance of the system. The principal diagram of the DSSS system receiver is shown in Figure 1.  
Assuming that the receiver has achieved accurate carrier synchronization and code synchronization of the received signal, the Doppler frequency shift and transmission delay are not considered in this paper. The received signal through the front-end band-pass filter is as follows:  
\[r(t)=s(t)+n(t)+j(t) \tag{1}\]  
where \(s(t)\) is a useful DSSS signal; \(n(t)\) is a white Gaussian noise with mean bilateral power spectral density \(N_{0}/2\); \(j(t)\) is a single-frequency interference; and \(s(t)\) and \(j(t)\) are modeled as follows:  
\[ \begin{array}{l}s(t)=\sqrt{2\mathcal{P}}d(t)c(t)\cos(2\pi f_{0}t+\varphi_{0} )\\ j(t)=\sqrt{2\mathcal{I}}\cos(2\pi f_{j}t+\varphi_{j})\end{array}  \tag{2}\]  
Figure 1: Receiver model of DSSS system. In (2), P and I denote the input signal power and interference power, respectively; \(d(t)\) is the information sequence to be decoded, with a value of \(\pm 1\) and a bit duration of \(T_{b}\); and \(c(t)\) is the pseudo-random \(\pm\) 1 sequence, with a cycle period of \(N\) and a chip duration of \(T_{c}\). For the DSSS system using short-period PN sequence, \(T_{c}=T_{b}/N,f_{0}\) is the carrier frequency of the received signal; \(\varphi_{0}\) is the random phase; and \(f_{j}\) and \(\varphi_{j}\) are the frequency and phase of the interference, respectively. The power spectral density of \(c(t)\) is given by  
\[S_{c}(f)=\frac{1}{N^{2}}\delta(f)+\frac{N+1}{N^{2}}\sum_{ \begin{array}{c}n=- \infty\\ n\
eq 0\end{array} }^{\infty}\sin c^{2}(\pi T_{c}f)\delta\bigg{(}f-\frac{n}{NT_{c }}\bigg{)} \tag{3}\]  
After despreading and demodulating, the signal output by the low-pass filter can be written as  
\[s^{\prime}(t)=\left[\int_{t_{0}}^{t_{0}+T_{b}}s(t)2\cos(2\pi f_{0}t+\varphi_{ 0})c(t)dt\right]_{LPF}=\pm\sqrt{2P}T_{b} \tag{4}\]  
After the single-frequency interference is processed,  
\[ \begin{array}{l}\dot{r}(t)=\left[\int_{t_{0}}^{t_{0}+T_{b}}j(t)2\cos(2\pi f _{0}t+\varphi_{0})c(t)dt\right]_{LPF}\\ =\sqrt{2I}\int_{t_{0}}^{t_{0}+T_{b}}\cos(2\pi\Delta ft+\Delta\varphi)c(t)dt \end{array}  \tag{5}\]  
In the above, \(\Delta f=f_{j}-f_{0}\) is the frequency offset of the interference relative to the signal carrier, and \(\Delta\varphi=\varphi_{j}-\varphi_{0}\) is the phase deviation. Therefore, the decision statistic of the \(m\)-th data bit sent to the decision maker can be written as  
\[A(m)=A_{d}(m)+A_{N}(m)+A_{j}(m) \tag{6}\]  
Here, \(A_{d}(m)\) is the response to the signal with a value of \(\sqrt{2P}T_{b}\), positive or negative, depending on the data bit; \(A_{N}(m)\) is a zero-mean Gaussian random variable with a variance of \(N_{0}T_{b}\), which is the response of the correlation receiver to additive white Gaussian noise; and \(A_{j}(m)\) is the receiver's response to interference, which can be written as  
\[A_{j}(m)=\int_{t_{0}+mT_{b}}^{t_{0}+(m+1)T_{b}}\cos(2\pi\Delta ft+\Delta\varphi )c(t)dt \tag{7}\]  
According to the periodicity of the spread spectrum sequences, the above equation can be rewritten as  
\[A_{j}(m)=\int_{t_{0}}^{t_{0}+T_{b}}\sqrt{2I}\cos(2\pi\Delta ft+\Delta\varphi+ m\Delta\theta)c(t)dt \tag{8}\]  
where \(\Delta\theta=2\pi\Delta fT_{b}\). For any given \(\Delta f\) and \(\Delta\varphi\), the above formula can be expanded into  
\[ \begin{array}{l}A_{j}(m)=\sqrt{2I}\cos(m\Delta\theta+\Delta\varphi)\int_{t_ {0}}^{t_{0}+T_{b}}\cos(2\pi\Delta ft)c(t)dt\\ -\sqrt{2I}\sin(m\Delta\theta+\Delta\varphi)\int_{t_{0}}^{t_{0}+T_{b}}\sin(2\pi \Delta ft)c(t)dt\end{array}  \tag{9}\]  
Defining \(C1=\int_{t_{0}}^{t_{0}+T_{b}}\cos(2\pi\Delta ft)c(t)dt\), and \(C2=\int_{t_{0}}^{t_{0}+T_{b}}\sin(2\pi\Delta ft)c(t)dt\), the above formula can be simplified as  
\[A_{j}(m)=C\cos(m\Delta\theta+\alpha+\Delta\varphi) \tag{10}\]  
where \(C=\sqrt{C1^{2}+C2^{2}}\), and \(\alpha=\arctan(C2/C1)\).