- 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
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