2. Signal model

At the beginning, pseudo random noise code modulated BPSK is adopted for the GPS C/A signal. With the development of other GNSS systems, such as BDS, Galileo and Glonass, the frequency spectrum for satellite navigation becomes increasingly more crowded. Therefore, BOC, AltBOC and similar signal modulation methods are adopted to degrade the cross correlation between signals. The expression of navigation signal based on BPSK and BOC are

(1)\begin{equation}\begin{aligned} {S_{\textrm{BPSK}}}(t ) & = AD(t )C(t )\exp ({j2\pi {f_0}t + {\phi_0}} )\\ {S_{\textrm{BOC}}}(t ) & = AD(t )SC(t )C(t )\exp ({j2\pi {f_0}t + {\phi_0}} )\end{aligned}\end{equation}

where A is the amplitude, $D(t )$is the navigation message, $C(t )$ is the code, ${f_0}$ is the central frequency, ${\phi _0}$ is the initial carrier phase and $\textrm{SC}(t )$ is the subcarrier of BOC.

Although acquisition method of these kinds of signals is similar, the acquisition calculation complexity of these signals is amplified by the high dynamic of LEO satellites. The proposed signal in this paper is designed to degrade the calculation complexity.

The proposed signal is a combination of CSS signal and traditional signal modulation methods. A CSS signal is used as the signal marker. Traditional signal modulation methods, such as BPSK and BOC, are used as the main body to maintain the performance of multi-access and carrier frequency tracking.

The expression of CSS is (Qian et al., Reference Qian, Ma and Liang2019)

(2)\begin{equation}{S_{\textrm{CSS}}}(t )= \left\{ {\begin{array}{@{}ll} {\exp \left( {j2{\rm \pi} {f_0}t \pm j2{\rm \pi} \left( {\dfrac{{\mu t - B}}{2}} \right)t} \right)}& {0 \le t < {T_s}}\\ 0& {\textrm{otherwise}} \end{array}} \right.\end{equation}

where ${f_0}$ is the central frequency, $\mu$ is the chirp rate and satisfies $\mu = B/{T_s}$, ${T_s}$ is the symbol length, B is the transmission bandwidth, the sign of $+$ makes a positive-chirp signal and $-$ makes a negative-chirp signal.

The multi-access signal of CSS is limited. Different CSS signals are constructed based on changing the chirp rate (Vangelista, Reference Vangelista2017). The performance of orthogonality degrades with the change of chirp rate. It is not enough to allocate one CSS signal for every satellite and maintain the orthogonality performance. Therefore, some work has to be done to design the signal for all the satellites.

The characteristic of the proposed signal is shown as follows.

1. (1) The signal comprises the CSS marker and main body of a traditional signal. The effective CSS signals are chosen for the markers based on the principle of orthogonality.

2. (2) The satellite signals will be separated into several groups. The same CSS marker is applied to all the satellites in one group and different groups adopt different CSS marker. The signals in one group should avoid collision based on other multi-access methods such as SDMA (Space Division Multi-Access) or others.

The structure of the proposed signal is shown in Figure 1 and the expression is shown as

(3)\begin{equation}s(t )= \left\{ {\begin{array}{@{}ll} {{S_{\textrm{CSS,gn}}}({t - k{T_p}} ) {e^{j2\pi {f_r}t}}}& \quad {0 \le t - k{T_p} < {T_c},k \in \textrm{N}}\\ {{S_{\textrm{Main,}}}_{\textrm{sn}}({t - k{T_p}} ) {e^{j2\pi {f_r}t}}}& \quad {{T_c} \le t - k{T_p} < {T_p},k \in \textrm{N}} \end{array}} \right.\end{equation}

where the subscript $\textrm{gn}$ is the index of group, $\textrm{sn}$ is the index of satellite, ${f_r}$ is the radio frequency, ${T_c}$ is the time length of the CSS marker and ${T_p}$ is the time interval between CSS markers.

Figure 1. Structure of the proposed signal

Considering that the earliest and most typical navigation signal is based on BPSK (IS-GPS-800, 2023), the main part of the proposed signal is supposed as BPSK in the following. The performance of the proposed signal is analysed and compared with the traditional signal.

3. Correlation functions in acquisition of the proposed signal

The acquisition of navigation signal is the 3-D search of satellite index, time delay and Doppler. For MEO satellites, the maximum signal Doppler of receivers is approximately ±5 KHz. Considering that the relative velocity between the LEO satellite and the receiver can be as large as 7 Km/s, the signal Doppler can be as large as approximately ±37 KHz when the radio frequency is 1575⋅42 MHz.

For the traditional GPS C/A signal, the code rate is 1⋅023 Mcps (IS-GPS-800, 2023), and the signal main-lobe bandwidth ${B_{\textrm{ml}}}$ is two times the code rate. A typical sampling rate in acquisition is two times the code rate and a typical coherent integration time is 1 ms. The signal parameters above are adopted in the following analysis.

In the acquisition of the proposed signal, the local CSS signal correlates with the received signal. Therefore, the receiver obtains the auto-correlation of CSS when the current signal is the CSS marker and the cross-correlation of CSS and BPSK when the current signal is the main body. This part analyses the correlation performance.

3.1 Auto-correlation function

The ACF (auto-correlation function) of CSS is shown as follows (Kadlimatti and Fam, Reference Kadlimatti and Fam2016):

(4)\begin{equation}\begin{aligned} {R_{\textrm{CSS}}} & = \int_{{T_a}}^{{T_b}} {{e^{\left( {j2{\rm \pi} ({{f_0} + {f_e}} )({t - \tau } )+ j2{\rm \pi} \left( {\dfrac{{\mu ({t - \tau } )- B}}{2}} \right)({t - \tau } )} \right)}}{{\left( {{e^{j2{\rm \pi} {f_0}t + j2{\rm \pi} \left( {\dfrac{{\mu t}}{2} - B} \right)t}}} \right)}^\ast }dt} \\ & = {e^{ - j\phi }}\sin c({({{f_e} - \mu \tau } )({{T_b} - {T_a}} )} )({{T_b} - {T_a}} )\end{aligned}\end{equation}

where ${f_e}$ is the frequency error, $[{{T_a},{T_b}} ]$ is the integral range, and it satisfies $[{{T_a},{T_b}} ]\subset [{0,T} ]$,$\sin c(x )$ satisfies $\sin c(x )= \sin ({{\rm \pi} x} )/{\rm \pi} x$ and ‘$^\ast$’ is the conjugate operator in this equation.

When the signal parameters are set as the parameters above, the ACF of CSS under different signal delay and Doppler is shown in Figure 2.

Figure 2. ACF of CSS. (a) ACF under versus time delay and Doppler, (b) ACF when the Doppler is 0 Hz

The width of the correlation peak of CSS is $2/{B_{\textrm{ml}}}$, which is half the width of BPSK. Additionally, the time delay corresponding to the correlation peak changes with the Doppler.

Therefore, to lower the acquisition power loss, the correlation procedure of the proposed signal should be optimised based on the characteristic of CSS. First, the time delay search interval should be lower than BPSK to reduce the correlation power loss. This can be realised by increasing the sampling rate to two times of BPSK. Second, the signal Doppler should be reverified to obtain the actual time delay and signal Doppler. When searching the actual signal Doppler, the time delay of the local signal should be compensated to get the correlation peak. Suppose that the Doppler to be searched is ${d_k}$, the signal should be delayed with ${d_k}/\mu$. Additionally, the signal can be down sampled by two times to realise the normal sampling rate of BPSK.

When the signal Doppler is 37 KHz, the power loss of the maximum correlation peak is −0⋅16 dB. Therefore, it is not necessary to implement signal Doppler search when acquiring the CSS marker.

3.2 Cross-correlation function

When the current signal is not the CSS marker, the correlation result should not contain any correlation peak to avoid false alarm. Therefore, this part analyses the cross-correlation function (CCF) of CSS and BPSK. The expression of the cross correlation of local CSS and the received BPSK is shown as (Xie, Reference Xie2017)

(5)\begin{equation}{R_{\textrm{CCF}}}({\tau ,{f_d}} )= \frac{1}{{{T_s}}}\int_{ - \infty }^\infty {({{S_{\textrm{BPSK}}}({t - \tau } ){e^{j({2{\rm \pi} {f_d}({t - \tau } )+ {\theta_e}} )}}} ){S_{\textrm{CSS}}}(t )dt}\end{equation}

The cross correlation depends heavily on the pseudo codes. Therefore, it is analysed based on simulation. The GPS CA codes are adopted for the analysis. The CCF based on different CA codes is analysed and Figure 3 shows an.

Figure 3. CCF of CSS and BPSK under different time delay and Doppler

The CCF based on different CA code is basically the same. There is no correlation peak in CCF. The maximum correlation peak in the normalised CCF result is 0⋅0658, which is $- \textrm{23} \ {\cdot}\ \textrm{63}\,\textrm{dB}$. It is just 0⋅3 dB higher than the side peak of the CA code, which is $- \textrm{23} \ {\cdot}\ \textrm{94}\,\textrm{dB}$. The SNR (Signal to Noise Ratio) of the correlation result is usually above 15 dB to realise acceptable detection performance. This means that the CCF peak is far below the noise. Therefore, the CCF peak can be ignored when analysing the false alarm in the following.

4. Calculation complexity

The acquisition calculation complexity determines the resources that the receiver needs. For example, if the signal processing equipment is a CPU (Central Processing Unit), when the calculation complexity increases, the receiver will be unable to realise real-time acquisition of the signals with the same CPU. Therefore, calculation complexity is an important index of the signal performance. This part analyses the calculation complexity of the proposed signal.

Suppose that the receiver adopts a time parallel correlation algorithm for signal correlation. The following symbols are adopted in the paper. Here, ${N_s}$ is the number of the signal samples, ${N_{\textrm{FFT}}}$ is the number of FFT points, ${N_f}$ is the number of frequency search interval, ${N_{\textrm{grpn}}}$ is the number of satellites in one group and ${N_T}$ is the rate of ${T_p}$ to ${T_c}$. When the sampling rate of CSS is two times BPSK, ${N_s}$ and ${N_{\textrm{FFT}}}$ satisfy${N_{\textrm{FFT}}} = 2 \times \textrm{next}2\textrm{pow}({{N_s}} )$, ${N_{s,}}_{\textrm{CSS}} = 2{N_{s,}}_{\textrm{BPSK}}$ and ${N_{\textrm{FFT,CSS}}} = 2{N_{\textrm{FFT,BPSK}}}$.

The calculation complexity of CSS correlation is obtained based on the calculation complexity of the time parallel correlation algorithm (Zhu and Fan, Reference Zhu and Fan2015). It is combined with two times ${N_{\textrm{FFT,CSS}}}$ point complex FFT, one ${N_{\textrm{FFT,CSS}}}$ point complex multiplication and one ${N_{\textrm{FFT,CSS}}}$ point complex IFFT. Therefore, the total calculation complexity is

(6)\begin{align}\begin{aligned} \textrm{Multiplication}:{O_{M,}}_{\textrm{CSS}} & = {O_{M,\textrm{FFT,CSS}}} \times 2 + {O_{M\textrm{,IFFT,CSS}}} + {O_{M,\textrm{cor,CSS}}}\\ & = 2{N_{\textrm{FFT,CSS}}}{\log _2}({{N_{\textrm{FFT,CSS}}}} )\times 3 + 4{N_{\textrm{FFT,CSS}}} \end{aligned}\end{align}

(7)\begin{align}\begin{aligned} \textrm{Addition}:{O_{A,}}_{\textrm{CSS}} & = {O_{A,\textrm{FFT,CSS}}} \times 2 + {O_{A,\textrm{IFFT,CSS}}} + {O_{A,\textrm{cor,CSS}}}\\ & = 3{N_{\textrm{FFT,CSS}}}{\log _2}({{N_{\textrm{FFT,CSS}}}} )\times 3 + 2{N_{\textrm{FFT,CSS}}} \end{aligned}\end{align}

where O is used as the expression of calculation complexity.

During the signal reverification, ${N_f}$ signal Doppler search and three code phase search are necessary to control the power loss. Therefore, the calculation complexity for satellite index search and Doppler acquisition is composed of $({{N_{\textrm{grpn}}} \times {N_f} \times 3} )$ times ${N_{s,\textrm{BPSK}}}$ point complex multiplication. It is shown as

(8)\begin{align}\begin{aligned} \textrm{Multiplication}:{O_{M,DpAq}} & = {O_{M,\textrm{commul}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3\\ & = 4{N_{s,\textrm{BPSK}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3 \end{aligned}\end{align}

(9)\begin{align}\begin{aligned} \textrm{Addition}:{O_{A,DpAq}} & = {O_{A,\textrm{cor}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3\\ & = 2{N_{s,\textrm{BPSK}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3 \end{aligned}\end{align}

where ${O_{M,\textrm{commul}}}$ is the calculation complexity of ${N_{s\textrm{,BPSK}}}$ point complex multiplication.

Considering that the location of the correct CSS marker is random during the signal length ${T_p}$, the average number of CSS correlation is ${N_T}/2$, which means that the average calculation complexity of the proposed signal for one satellite is

(10)\begin{align}\textrm{Multiplication}:{O_{M,\textrm{PS}}} = \left( \begin{array}{l} ({2{N_{\textrm{FFT,CSS}}}{{\log }_2}({{N_{\textrm{FFT,CSS}}}} )\times 3 + 4{N_{\textrm{FFT,CSS}}}} )\times {N_T}/2\\ + 4{N_{s,\textrm{BPSK}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3 \end{array} \right)/{N_{\textrm{grpn}}}\end{align}

(11)\begin{align}\textrm{Addition}:{O_{A,\textrm{PS}}} = \left( \begin{array}{l} ({3{N_{\textrm{FFT,CSS}}}{{\log }_2}({{N_{\textrm{FFT,CSS}}}} )\times 3 + 2{N_{\textrm{FFT,CSS}}}} )\times {N_T}/2\\ + 2{N_{s,\textrm{BPSK}}} \times {N_{\textrm{grpn}}} \times {N_f} \times 3 \end{array} \right)/{N_{\textrm{grpn}}}\end{align}

The acquisition calculation complexity of the BPSK signal comprises $({{N_f} + 1} )$ times${N_{\textrm{FFT,BPSK}}}$ point FFT, one ${N_{s,\textrm{BPSK}}}$ point complex multiplication, ${N_f}$ times ${N_{\textrm{FFT,BPSK}}}$ point complex multiplication and one ${N_{\textrm{FFT,BPSK}}}$ point IFFT. It is shown as

(12)\begin{align}\textrm{Multiplication}:{O_{M,\textrm{BPSK}}} = \left( \begin{array}{l} 2({{N_f}\textrm{ + }2} ){N_{\textrm{FFT,BPSK}}}{\log_2}({{N_{\textrm{FFT,BPSK}}}} )\\ \textrm{ + }4{N_f}{N_{\textrm{FFT,BPSK}}} + 4{N_{s,\textrm{BPSK}}} \end{array} \right)\end{align}

(13)\begin{align}\textrm{Addition}:{O_{A,\textrm{BPSK}}} = \left( \begin{array}{l} 3({{N_f} + 2} ){N_{\textrm{FFT,BPSK}}}{\log_2}({{N_{\textrm{FFT,BPSK}}}} )\\ + 4{N_f}{N_{\textrm{FFT,BPSK}}} + 2{N_{s,\textrm{BPSK}}} \end{array} \right)\end{align}

For calculators, the resource needed for multiplication is less than that of addition. Therefore, the calculation complexity of multiplication is mainly considered and it is shown in Figure 4.

Figure 4. Calculation complexity versus ${N_T}$

From the above analysis, the following conclusions are obtained.

1. (1) The average calculation complexity of the proposed signal increases with the increment of ${N_T}$ and the decrement of ${N_{\textrm{grpn}}}$.

2. (2) The average calculation complexity of the proposed signal for one satellite is basically lower than that of BPSK. When ${N_{\textrm{grpn}}}$ is 4 and ${N_T}$ is 10, the average calculation complexity of the proposed signal is just 26 % of that of BPSK.

Therefore, the proposed signal is able to decrease the calculation complexity of the receiver compared to the traditional signal.

5. Mean acquisition time

Mean acquisition time (MAT) shows the time to successfully acquire the signal for the receiver. This part compares the MAT of the receiver when the other parameters are constraint as the same.

5.1 Expression of the MAT

Park et al. (Reference Park, Choi, Lee and Kim2002) analysed the expression of the MAT based on the statistical model. The expression is

(14)\begin{equation}\begin{aligned} {T_{M,\textrm{BPSK}}} & = ({{{\bar{N}}_C} - 1} ){T_D}({1 + {k_p}({1 - {{({1 - {P_f}} )}^{{M_p}}}} )} )\left( {\dfrac{{2 - {P_D}}}{{{P_D}}}} \right)\\ & + \dfrac{{{T_D}}}{{{P_D}}}({1 + {k_p}({1 - {{({1 - {P_f}} )}^{{M_p} - 1}}} )} )\end{aligned}\end{equation}

where ${\bar{N}_C}$ is the number of search blocks, ${T_D}$ is the dwell time for single search block, ${P_D}$ is the probability of detection for single detection, ${T_f}$ is the penalty time for false alarm, ${k_p}$ is the penalty coefficient (${T_f} = {k_p}{T_D}$), ${M_p}$ is the number of parallel search cell and ${P_f}$ is the probability of false alarm.

The penalty time of BPSK is basically ${T_D}$, because the receiver is able to carry out another correlation to verify the acquisition result during the dwell time.

The MAT of the CSS marker is comparable to the expression above. For the proposed signal, the number of search blocks is the number of the possible positions of the CSS marker in one signal period, which is ${N_T}$. The number of search cells in each search block is two times that of BPSK. The false alarm penalty time is equal to the time of signal Doppler acquisition. The MAT of the proposed signal contains the time of signal Doppler acquisition based on the MAT of CSS marker. Therefore, the MAT of the proposed signal is

(15)\begin{equation}{T_{M,\textrm{proposed}}} = \left( \begin{array}{l} ({{N_T} - 1} ){T_D}({1 + {k_{p,\textrm{pro}}}({1 - {{({1 - {P_f}} )}^{{M_{p,\textrm{CSS}}}}}} )} )\left( {\dfrac{{2 - {P_{D,\textrm{CSS}}}}}{{{P_{D,\textrm{CSS}}}}}} \right)\\ + \dfrac{{{T_D}}}{{{P_{D,\textrm{CSS}}}}}({1 + {k_{p,\textrm{pro}}}({1 - {{({1 - {P_f}} )}^{{M_{p,\textrm{CSS}}} - 1}}} )} )+ {k_{p,\textrm{pro}}}{T_D} \end{array} \right)\end{equation}

The value of ${k_{p,\textrm{pro}}}$ can be expressed as ${k_{p,\textrm{pro}}} = \lceil{{N_{\textrm{grpn}}} \times {N_f} \times 3/{N_{\textrm{mp}}}} \rceil$. The penalty time is the dwell time to deal with the calculation during the Doppler acquisition. In software receivers, the processing time is determined by the calculation complexity. Considering that the maximum calculation complexity is the CSS correlation, the processing procedures that are of lower calculation complexity are also able to be solved in the dwell time ${T_D}$. Therefore, the maximum number of parallel correlations satisfies

(16)\begin{equation}{N_{\textrm{mp}}} = \lfloor{({{O_{M,\textrm{FFT,CSS}}} + {O_{M,\textrm{IFFT,CSS}}} + {O_{M,\textrm{cor,CSS}}}} )/{O_{M,\textrm{cor}}}} \rfloor\end{equation}

and $\lfloor x \rfloor$ and $\lceil x \rceil$ are separately the floor integer and the ceiling integer. The ${O_{M,\textrm{FFT,CSS}}}$ of the local signal is ignored because it can be restored in advance instead of calculating in real time.

Identically, when comparing the MAT of the traditional BPSK signal and that of the proposed signal, the calculation complexity is also considered. The maximum number of parallel search blocks in ${T_D}$ satisfies the following expression:

(17)\begin{equation}{N_{\textrm{mb}}} = \left\lfloor {\left( \begin{array}{l} {O_{M,\textrm{FFT,CSS}}} + {O_{M,\textrm{IFFT,CSS}}}\\ + {O_{M,\textrm{cor,CSS}}} - {O_{M,\textrm{FFT,BPSK}}} \end{array} \right)/({{O_{M,\textrm{IFFT,BPSK}}} + {O_{M,\textrm{cor,BPSK}}}} )} \right\rfloor\end{equation}

and the number of search blocks is ${\bar{N}_c} = \lceil{{N_f}/{N_{\textrm{mb}}}} \rceil$.

5.2 Results of MAT

The probability of detection has to be obtained to get the MAT. When using a square-law detector, the probability of detection is expressed as follows (Liu et al., Reference Liu, Kou, Huang, Wang and Yao2019):

(18)\begin{equation}\begin{aligned} \lambda & = \sum\limits_{i = 1}^{{N_{ncoh}}} {({m_I^2({i|{H_1}} )+ m_Q^2({i|{H_1}} )} )} = 2\sum\limits_{i = 0}^{{N_{ncoh}} - 1} {\textrm{SN}{\textrm{R}_i}} \\ & = 2{D_{\textrm{DS}}}\sum\limits_{i = 0}^{{N_{ncoh}} - 1} {D({{f_d},{f_a},{f_c},{T_c},i} )C{N_0} \ {\cdot}\ {T_{coh}}} \end{aligned}\end{equation}

where ${D_{\textrm{DS}}}$ is the front-end power loss of the receiver, $D({{f_d},{f_a},{f_c},{T_c},i} )$ is the correlation loss, $\textrm{C}{\textrm{N}_0}$ is the Carrier to Noise power spectral density Ratio (CNR), ${T_{\textrm{coh}}}$ is the coherent integration time length, ${N_{\textrm{ncoh}}}$ is the times of the non-coherent accumulation and $\textrm{SN}{\textrm{R}_i}$ is the signal to noise rate for the $ith$ non-coherent integration.

Suppose that ${D_{\textrm{DS}}}$ is identical for the proposed signal and BPSK. From the results of Section 3.1, it can be obtained that the maximum correlation loss of CSS is $- 1 \ {\cdot}\ 04\,\textrm{dB}$ when the Doppler is 37 KHz and the sampling rate is $2\,{B_{\textrm{ml}}}$, while the Doppler loss and the code phase loss are separately $- 0 \ {\cdot}\ 91\,\textrm{dB}$ and $- 1 \ {\cdot}\ 16\,\textrm{dB}$ for the traditional BPSK signal.

Suppose that ${P_f}$ is 1 × 10⁻⁶, then the probability of detection versus CNR and the MAT versus CNR is shown in Figures 5 and 6.

Figure 5. Probability of detection versus CNR

Figure 6. MAT of BPSK and CSS versus CNR

From the above analysis, the following conclusions are obtained.

1. (1) The probability of detection of the proposed signal is higher than the traditional BPSK for the same CNR because of the correlation loss advantage.

2. (2) The MAT of the proposed signal increases with the increment of ${N_{\textrm{grpn}}}$. This is because the increment of ${N_{\textrm{grpn}}}$ leads to the increment of false alarm penalty time. When ${N_{\textrm{grpn}}}$ is 5, the MAT of the proposed signal is higher than that of BPSK when the CNR is higher than 46 dBHz.

3. (3) The advantage of the proposed signal is greater when the CNR is lower than 45 dBHz. This is mainly determined by the advantage of probability of detection. When ${N_{\textrm{grpn}}}$ is 4 and the CNR is 45 dBHz, the MAT of the proposed signal is just 91⋅8 % of the MAT of BPSK.

Therefore, the proposed signal is able to decrease the mean acquisition time of the receiver under the constraint of calculation complexity.