CW Interference

This post shows the theory of CW interference and Matlab source codes for validating the theory.
- The received satellite signal can be represented as
$$s(t)=A_sC(t)D(t)cos(2\pi(f_c+f_d)t+\Phi_0)$$

1. Determining $$(I_s+jQ_s)(m)$$ after tracking phase:
Suppose that the local code $$C(t-\tau)$$ and local carrier frequency is $$f_c$$
$$(I_s+jQ_s)(m)=\int_{mT_d}^{(m+1)T_d}s(t)e^{j2\pi f_{lc}}C(t-\tau)dt$$
$$=\int_{\frac{-T_d}2}^{\frac{Td}2}s(t)e^{-j*2\pi f_{lc}}C(t-\tau)dt$$
$$=\frac12\int_{\frac{-T_d}2}^{\frac{Td}2}A_sC(t)C(t-\tau)D(m)[e^{-j2\pi (f_c+f_d)t}+e^{-j2\pi (f_c+f_d)t}]e^{-j2\pi f_{lc}}e^{j\Phi_m}$$
$$=\frac12A_sD(m)\int_0^{T_d}C(t)C(t-\tau)[e^{-j2\pi \Delta f_c t}+e^{-j2\pi \Sigma f_c t}]dt$$
C(t) is a CA code which is repeated every 1ms. Therefore, C(t) can be expressed as a series of harmonics with the fundamental frequency - $$f_0=1/Tc$$
$$C(t)=\sum_nC_ne^{j2\pi nf_0t}$$
$$C(t-\tau)=e^{j2\pi\tau}\sum_nC_ne^{j2\pi nf_0t}$$
$$C(t)C(t-\tau)=\sum_iC_ie^{j2\pi if_0t}\sum_pC_pe^{j2\pi pf_0t}$$
$$=\sum_i\sum_kC_iC_ke^{j2\pi (i+k)f_0t}$$
$$=\sum_kR_ke^{j2\pi kf-0t}$$
If we denote $$R_k$$ as:
$$R_k=\sum_i\sum_jC_iC_j$$ where i+j=k
$$\frac12A_sD(m)\int_{\frac{-T_d}2}^{\frac{Td}2}\sum_kR_ke^{j2\pi kf_0t}[e^{-j2\pi \Delta f_c t}+e^{j2\pi \Sigma f_c t}]dt$$
$$=\frac12A_sD(m)\sum_k\int_{\frac{-T_d}2}^{\frac{Td}2}R_ke^{j2\pi kf_0t}[e^{-j2\pi \Delta f_c t}+e^{j2\pi \Sigma f_c t}]dt$$
It's clear that
$$|\int_{\frac{-T_d}2}^{\frac{T_d}2}e^{j2\pi ft}dt|=|sinc(fT_d))|<|\frac1{\pi fT_d}|\approx0$$
when  $$f>\frac1{T_d}$$
And $$|kf_0+\Sigma f_c|>|-F_{max}(CA)+\Sigma f_c|>>\frac1{T_d}$$
$$|kf_0-\Delta f_c|<\frac1{T_d}$$ if and only if k=0
$$(I+jQ)(m)=\frac12A_sD(m)R_0sinc(\Delta f_cT_d)$$
2. Determining $$(I_i+jQ_i)(m)$$ of interference
$$(I_i+jQ_i)(m)=\frac{A_i}2e^{j\Phi_i}\int_{-\frac{T_d}2}^{\frac{T_d}2}C(t)[e^{j2\pi f_it}+e^{-j2\pi f_it}]e^{-j2\pi f_{LC}t}dt$$
$$=\frac{A_i}2e^{j\Phi_i}\int_{-\frac{T_d}2}^{\frac{T_d}2}C(t)[e^{j2\pi \Delta f_it}+e^{j2\pi \Sigma f_it}]dt$$
$$=\frac{A_i}2e^{j\Phi_i}\int_{-\frac{T_d}2}^{\frac{T_d}2}\sum_nC_ne^{-j2\pi\frac{n}{Tc}t}[e^{j2\pi \Delta f_it}+e^{j2\pi \Sigma f_it}]dt$$
$$=\frac{A_i}2T_de^{j\Phi_i}C_nsinc((\Delta f_i-\frac{n}{Tc})T_d)$$
where $$|\Delta f_i-\frac{n}{T_c}|<T_d$$