Candidate Design of New Service Signals in the NavIC L1 Frequency Band

SYSTEM MODEL

NavIC transmits interoperable open civilian service signals over the L1 frequency band via second-generation NavIC satellites. These multi-level signals have a constant envelope modulation (Upadhyay et al., 2024; Upadhyay & Bhadouria, 2021; Upadhyay et al., 2020) and meet the PSD criteria of MBOC, i.e., MBOC(6,1,1/11), which is essential for interoperability (Hein et al., 2006).

We represent a multi-level signal (s_SBOC(t)) as follows:

$s_{SBOC}\left(t\right)=s_{SBOC}^p\left(t\right)+j{s}_{SBOC}^d\left(t\right)$1

where $s_{SBOC}^d\left(t\right)$ and $s_{SBOC}^p\left(t\right)$ are the data and pilot components of s_SBOC(t). We represent the data and pilot components of the multi-level signal as follows:

$s_{SBOC}^p\left(t\right)=\sqrt{\alpha }{s}_{BOC(1,1)}^p\left(t\right)-\sqrt{\beta }{s}_{BOC(6,1)}^p\left(t\right)$2

$s_{SBOC}^d\left(t\right)=\sqrt{\gamma }{s}_{BOC(1,1)}^d\left(t\right)+\sqrt{\eta }{s}_{BOC(6,1)}^d\left(t\right)$3

where α, β, γ, and η are the coefficients of the BOC(1,1) and BOC(6,1) components, which satisfy the following relationships:

$\alpha +\beta +\gamma +\eta =1;\eta =\frac{\alpha \beta }{\gamma };\alpha +\gamma =\frac{10}{11};\beta +\eta =\frac{1}{11}$4

η is a parameter that depends on α, β, and γ and controls the envelope of the composite signal. As shown in Equations (1)–(4), the envelope of the multi-level signal has a constant modulus, i.e., |s_SBOC(t)| = 1, and satisfies the MBOC PSD criteria. Time-domain plots and correlation properties of the data and pilot components of the multi-level signal are displayed in Figure 1, showing that both the data and pilot components are multi-level and have a non-smooth main lobe of the auto-correlation curve.

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FIGURE 1

Characteristics of multi-level data and pilot signals; time-domain plots of multilevel data and pilot signals (left); autocorrelation plots of multi-level data and pilot signals (right)

Multi-level modulation components make it difficult to include additional signals while maintaining the envelope of the composite signal. Although research on constant-envelope multiplexing of multi-level signals is limited, recent studies have presented an MCEMIC technique to add signal power deviation in order to maximize multiplexing efficiency. The MCEMIC approach has higher multiplexing effectiveness than CEMIC for bipolar signals over a single frequency while preserving signal power (Bhadouria et al., 2022).

Here, we extend the MCEMIC technique for multiplexing multi-level signals. Furthermore, to satisfy the power ratio given in Equation (4) for the multi-level signal to maintain the MBOC PSD, we fuse constraints related to the power ratio in the optimization framework of the MCEMIC technique to support backward compatibility for the already planned multi-level signal. Figure 2 shows the overall block-level implementation of the onboard MCEMIC technique based on a satellite navigation payload.

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FIGURE 2

Block diagram of a typical satellite navigation payload with the MCEMIC technique

Figure 2 shows a transmitter in which four signals are multiplexed via MCEMIC to create a constant-envelope composite signal (s¹(t),s²(t),s³(t), and s⁴(t)). The composite signal (s(t)) is up-converted by an RF oscillator and amplified once (HPA). Generating a composite signal (s_RF(t)) with a constant envelope maximizes HPA efficiency. A bandpass filter (BPF) limits out-of-band emission and transmits in-band signals without distortion. We propose adding NSSs to the NavIC L1 multi-level signal. NSSs must be multiplexed with the multi-level signal. The satellite navigation payload must multiplex multi-level data (multi-level-D), multi-level pilot (multi-level-P), NSS-data (NSS-D), and NSS-pilot (NSS-P) components. The multiplexed composite signal (s(t)) shown in Figure 1 is represented as follows:

$s\left(t\right)={\sum }_{i=1}^4{s}^i\left(t\right)+IM\left(t\right)$5

where:

$s^i\left(t\right)={\sum }_{n=-\infty }^{\infty }{c}_n^i{p}^i\left(t-\frac{n}{R_c^i}\right)$6

Here, $c_n^i$ is the product of the spreading code and the data of the i^th signal, and $R_c^i$ is the code rate of the spreading code of the i^th signal. IM(t) is the superposition of multiple intermodulation terms inserted to ensure that the envelope of s(t) has a constant modulus. The pulsed signal pⁱ(t) has a non-zero amplitude for $t\in \left[0,1/R_c^i\right)$ and can be represented as follows:

$p^i\left(t\right)={\sum }_{k=0}^{M^i-1}{a}_k^i{\psi }_{T_s}^i\left(t-k{T}_s^i\right)$7

where Mⁱ is the modulation index of the i^th signal and $T_s^i\left\{1/\left(M^i{R}_c^i\right)\right\}$ is the sub-chip interval of the i^th signal. aⁱ defines the shape vector of length Mⁱ for the i^th signal, and $a_k^i$ is the k^th element. ${\psi }_{T_s}^i\left(t\right)$ is a unit rectangular pulse of duration $T_s^i$.

Considering sⁱ(t) of the form $s^i\left(t\right)={\sum }_{n=-\infty }^{\infty }\sqrt{P^i}{c}_n^i{p}^i\left(t-\frac{n}{R_c^i}\right)e^{j\left(2\pi \left(f^i-f_0\right)t+{\varphi }^i\right)}$ in

Equation (5), we obtain the following:

$s\left(t\right)={\sum }_{i=1}^4{\sum }_{n=-\infty }^{\infty }\sqrt{P^i}{c}_n^i{p}^i\left(t-\frac{n}{R_c^i}\right)e^{j\left(2\pi \left(f^i-f_0\right)t+{\varphi }^i\right)}+IM\left(t\right)$8

where Pⁱ is the power of the i^th signal; fⁱ and φⁱ are the frequency and initial phase of the i^th signal, respectively; and f₀ is the common center frequency for all signals. We impose the condition fⁱ = f₀; i ∈ [1,4] when multiplexing signals over a single frequency. This condition may not hold when multiplexing signals over multiple frequencies.

We define the general form of the MCEMIC method to obtain the formulation of IM(t) such that s(t) = Ae^jϕ, where A is an arbitrary constant taken as A = 1 with no loss of generality and ϕ is an arbitrary phase. Based on Equation (5), IM(t) can be represented as follows:

$IM\left(t\right)=e^{j\varphi }-{\sum }_{i=1}^4{\sum }_{n=-\infty }^{\infty }\sqrt{P^i}{c}_n^i{p}^i\left(t-\frac{n}{R_c^i}\right)e^{j\left(2\pi \left(f^i-f_0\right)t+{\varphi }^i\right)}$9

The solution of Equation (9) is nontrivial; multiple values of the design parameters Pⁱ and ϕⁱ satisfy Equation (9) simultaneously. However, with the assumption that all of the onboard signals of the form shown in Equation (7) are synchronized and periodic, the span of aⁱ and $e^{j\left(2\pi \left(f^i-f_0\right)t\right)}$ becomes finite. Hence, Equation (9) can be solved by posing it in an optimization framework with Pⁱ and ϕⁱ as the design (optimization) parameters for a given aⁱ and fⁱ. For a uni-level signal, aⁱ is bipolar, i.e., $a_k^i=\pm 1;k\in \left[0,M^i-1\right]$, and for multi-level cases, aⁱ takes Kⁱ(Kⁱ ≤ Mⁱ) values such that $0<a_k^i\le 1;k\in \left[0,M^i-1\right]$. In the case of single-frequency transmission, $e^{j2\pi \left(f^i-f_0\right)t}\triangleq 1$; whereas for multi-frequency cases, $e^{j2\text{À}\left(f^i-f_0\right)t}\triangleq e^{j2\text{À}\left(f^i-f_0\right)n{T}_s}$, 0 ≤ n ≤ L provided that L {= 1/(fⁱ – f₀)T_s} is an integer, where $T_s\left\{=1/LCM\left(R_c^i\right)\right\}$ is the sampling duration. Here, LCM(·) is defined as the least common multiple. The intermodulation term (IM(t)) is also a multi-level chip sequence that is orthogonal to the signal sⁱ(t) (Yao et al., 2017) and satisfies the following:

$E\left(IM\left(t\right)s^{i^{*}}\left(t\right)\right)=c_i^{H}P\lambda =0$10

where c_i is an $F\times 1\hspace{1em}\left(F\le L\prod _{i=1}^4{K}^i\right)$ vector defined as $c_i=c_n^i{a}^i{e}^{j\left(2\pi \left(f^i-f_0\right)n{T}_s\right)}$. P is a positive definite diagonal matrix, and λ is the N_λ × 1 realization of IM(t). Furthermore, it is shown that $c_i^H\lambda =0$; the basis of λ, i.e., $\overline{C}$, is obtained from Gram-Schmidt orthonormal expansion of the matrix C = [c₁, c₂,..., c₄] (Leon et al., 2012), which results in an augmented matrix $C_0=[C,\overline{C}]$.

It is important to note that the cardinality of the vector c_i, i.e., F, is critical for maximizing the multiplexing efficiency. The signal type determines the cardinality of cⁱ ; for example, if c_i ; i ∈ [1,4] makes the orthonormal basis and strict equality is held, i.e., $F=L{\prod }_{i=1}^4{K}^i$.

Bipolar single-frequency multiplexing has less cardinality than multi-level multi-frequency multiplexing. Signal multiplexing for multi-level signals requires care, as this process is more complicated. When enumerating signal value combinations, we must consider the fact that multi-level data and pilot signals are not independent. The next segment discusses the challenges that arise in multi-level multiplexing design.

In matrix notation, Equation (5) is written as follows:

${\sum }_{i=1}^4{s}^i\left(t\right)\triangleq C{w}_s;\lambda \triangleq \overline{C}{w}_{im}$11

Hence, $s\left(t\right)\triangleq C_{0}w$, where $w={\left[w_s^T,w_{im}^T\right]}^T$ and $w_s={\left[\sqrt{P^1},\dots ,\sqrt{P^4}\right]}^T$.

The MCEMIC-based cost function for reducing the power of intermodulation terms and minimizing the deviation in the power of the desired signals is formulated as follows:

$C\left(W\right)=\frac{1}{2}\Vert \Lambda w-b{\Vert }^2$12

where $b={\left[e^{j{\varphi }^i},Z\right]}^T$. Z is a zero vector with length N_λ, and ϕⁱ is a vector containing the initialization phases of w_s weights. Λ is an L × L diagonal matrix that preserves the power of the desired signals to support backward compatibility, given by the following:

$\Lambda =\left[\begin{array}{cc}{\Lambda }_E& 0\\ 0& I\end{array}\right]$13

where L is the length of the vector w, which is defined by L = 4 + N_λ. Λ_E is a 4 × 4 power-sharing preservation matrix denoted by the following:

${\Lambda }_E=\left[\begin{array}{cccc}\frac{{\epsilon }_1}{\sqrt{P_1}}& 0& 0& 0\\ 0& \frac{{\epsilon }_2}{\sqrt{P_2}}& 0& 0\\ & & \ddots & \\ 0& 0& 0& \frac{{\epsilon }_4}{\sqrt{P_4}}\end{array}\right]$14

where ε₁ is the penalty factor to minimize the deviation of power-sharing of the i^th desired signal.

The optimization is formulated as follows:

$\arg \underset{w}{\min }\left(C\right(w\left)\right)$15

such that:

$\left|C_{0}w\right|=1$16

However, we add another constraint for multi-level signals in Equations (15) and (16), which we require in order to meet the MBOC PSD criteria for interoperability in the L1 frequency band. According to Equation (4), this constraint is given as follows:

$\frac{\alpha +\gamma }{\beta +\eta }=10$17

Considering w_s(1) and w_s(2) as the weights of the $s_{SBOC}^p\left(t\right)$ and $s_{SBOC}^d\left(t\right)$ signals, respectively, we can use Equation (1) to obtain the following:

$\frac{\alpha +\beta }{\gamma +\eta }=\frac{w_s\left(1\right)}{w_s\left(2\right)}=\sqrt{\frac{64}{46}}={\zeta }_{SBOC}$18

Hence, we extend the MCEMIC method for multi-level and multi-frequency signals with an additional multi-level-based PSD constraint. Using the Lagrange method (Pierre, 1986), the cost function for optimization is represented as follows:

$ℒ\left(w\right)=C\left(w\right)+\frac{{\lambda }_1}{2}{s}^{H}Gs+{\lambda }_2{w}^{H}Dw$19

where D is derived based on Equation (18):

$D=\left[\begin{array}{cc}{D}_E& 0\\ 0& 0\end{array}\right]$20

D_E is a 2 × 2 matrix, given by the following:

$D_E=\left[\begin{array}{cc}1& -{\zeta }_{SBOC}\\ -{\zeta }_{SBOC}& {\zeta }_{SBOC}^2\end{array}\right]$21

The second term in Equation (19) arises from the constant-modulus constraint given in Equation (16) (Yao et al., 2017). Here, $G\left\{=\mathrm{diag}\left(g_1,g_2,,g_{N_{\lambda }}\right)\right\}$ is a diagonal matrix whose entries are given by the following:

$g_k=\{\begin{array}{cc}1,& k\in G^{+}\\ -1,& k\in G^{-}\end{array}$22

where $G^{+}$ represents the L / 2 higher values of the magnitude-wise sorted values present in s and, similarly, $G^{-}$ represents the L / 2 lower values in s.

The gradient of C(w) is given by the following:

${\nabla }_{w}C\left(w\right)={\kappa }_1\left({\Lambda }^T\Lambda w-{\Lambda }^{T}b\right)+{\kappa }_2\left(C_0^{H}Gs\right)+{\kappa }_{3}Dw$23

The gradient-descent-based iterative update is given as follows:

$w_{i+1}=w_i-\mu \left({\nabla }_{w}C\left(w\right)\right)$24

where μ is a parameter that controls the step size, κ₁ is a weight factor for the receiver transparency error, κ₂ is the constant-envelope error weight factor, and κ₃ is the PSD error weight factor in the proposed optimization framework. It is critical to observe that the magnitude of the constant-modulus error term is greater than that of the other errors. Hence, it is intuitive to maintain higher values for κ₁ and κ₃ compared with κ₂ . Accordingly, we set μ = 1e⁻⁵, κ₁ = 10, κ₂ = 1, and κ₃ = 10 in the analysis.

It is essential to note that the proposed algorithm is an iterative-type method. Hence, careful attention is needed when selecting the initial values of the optimization parameter. Not all values provide equal multiplexing efficiency. Moreover, a solution with higher efficiency may not have practical relevance for the overall system performance. For example, the high-efficiency multiplexing solution may reduce the power-sharing of one or more signals below the acquisition/demodulation threshold. Hence, incorporating the system engineering aspects into the signal design requires a practical solution. The following section describes the criteria for selecting the initial power-sharing of the desired signal. Moreover, it also considers the requirements to minimize inter- and intra-system interference by selecting the optimum modulation and frequency of the proposed NSS in the NavIC L1 frequency band.

SYSTEM ENGINEERING ASPECTS FOR SIGNAL DESIGN

This section explains details of the system engineering aspects that are essential for NSS design. We explore the link design to ensure the availability of the required minimum received isotropic power (RIP) level of the NSS.

RESULTS AND DISCUSSION

Comparison of MCEMIC and CEMIC Methods for Multi-Level Signals

To compare the performance of MCEMIC with that of the existing CEMIC method (Yao et al., 2017), we simulated multiplexing of the four signals depicted in Figure 3. It can be observed that both signal 1 and signal 2 have multiple levels. In addition, the initial power distribution among signals 1, 2, 3, and 4 is 20%, 30%, 20%, and 30%, respectively. Furthermore, signal 1 and signal 2 are maintained in the in-phase channel, whereas signal 3 and signal 4 are maintained in the quadrature channel.

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FIGURE 3

Time-domain plots of four signals

We investigate both single- and multi-frequency cases. As shown in Figure 4, the proposed MCEMIC methods have a higher multiplexing efficiency than the existing CEMIC method (Yao et al., 2017). The first four bars represent the power distribution for the desired signals, while the fifth bar represents the total power of the intermodulation signals. For case scenarios involving a single frequency, MCEMIC is 1.5% more efficient than CEMIC. Moreover, MCEMIC achieves a 0.8% improvement in multiplexing efficiency for multi-frequency cases.

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FIGURE 4

Comparison of MCEMIC and CEMIC methods for multi-level signal multiplexing: single-frequency multiplexing (left) and multi-frequency multiplexing (right)

MCEMIC-BASED SIGNAL DESIGN AND PERFORMANCE ANALYSIS

This section discusses the results of the proposed NSS in the NavIC L1 frequency band with the existing NavIC multi-level signal. The phasing of the multi-level data and multi-level pilot components has not been changed to maintain backward compatibility, and the power-sharing ratio between the multi-level data and multi-level pilot components was not modified to meet the MBOC PSD. We analyze two options in light of L1 congestion. The first option is multiplexing multi-level signals and the NSS over a single frequency, where the NSS and multi-level signals have the same center frequency. Here, we use split-spectrum approaches to explore frequency diversity. The second option is multiplexing multi-level signals and the NSS over multiple frequencies. In this case, a signal frequency offset isolates the PSDs. Based on interference analysis, we explore two modulation methods for the NSS. We suggest cosine phasing BOC, i.e., BOCc(15,2.5), for the single-frequency case and BPSK(5) for the multi-frequency case with a 15.345-MHz NSS offset.

The link architecture determines the NSS data-to-pilot signal power-sharing ratio, i.e. ζ_nss, and GNSS receivers use the pilot signal instead of the data signal for improved acquisition and tracking performance. The receiver uses a coherent phase-locked loop, as the pilot signal contains no data. This approach delivers a 6-dB relative advantage. In addition, the receiver uses a longer integration interval to acquire/track the pilot signal, leading to higher pilot signal power-sharing.

Single-Frequency and Multi-Level Multiplexing Analysis

The NSS-D and NSS-P signals are modulated with BOCc(15,2.5) for a single-frequency multiplexing option. Four signals must be multiplexed to create the constant-envelope composite signal. The multi-level data and pilot signals are multi-level, whereas the NSS-D and NSS-P signals are bipolar. To evaluate the performance of the proposed multiplexing method, we explore a total of four power-sharing combinations between existing multi-level signals and the NSS (multi-level: 30%, 40%, 50%, and 70%; NSS: 70%, 60%, 50%, and 30%). Because the NSS-D and NSS-P components can have four different value combinations in the initial phases, we evaluate sixteen possible combinations for the proposed scheme. Table 3 displays the findings for ζ_nss = 1/3. For ζ_nss = 1/2, details are provided in Appendix II of this manuscript. However, it is observed that the performance for ζ_nss = 1/2 is inferior when considering the overall system requirements. Hence, we consider ζ_nss = 1/3 for the analysis. We studied various power-sharing options for individual signals, evaluating all potential combinations of NSS-D and NSS-P phasing. Case 4 shows the most efficient multiplexing, but case 15 shows optimal performance. Case 15 has nearly comparable multi-level and NSS powers, at 41% each. Figure 5 displays the simulation constellation diagram and PSD for case 15. The composite signal has a constant envelope and incorporates multi-level modulation and BOCc(15,2.5).

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FIGURE 5

Constellation (left) and PSD (right) for single-frequency multiplexing with ζ_nss = 1/3

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TABLE 3

Power-Sharing Options (in %) for Various Phase Relationships of NSS-D and NSS-P Signals with ζ_nss = 1/3 for Multiplexing with Multi-Level Signals Over a Single Frequency

η denotes the multiplexing efficiency; I indicates the in-phase channel; Q denotes the quadrature channel.

It is important to observe that the number of constellation points is significantly lower than the expected value, i.e., 32 instead of 64. This difference is primarily due to the dependence between the multi-level data and pilot signals. Because these signals are not independent, we must consider this condition when enumerating the possible combinations of signals to generate a matrix C₀. In this case, any mismatch results in a high correlation loss of up to 3 dB, rendering the overall multiplexing technique ineffective. Furthermore, because the number of effective constellation points is reduced, the degrees of freedom for optimization are decreased, resulting in slightly lower multiplexing efficiency.

Multi-Frequency and Multi-Level Multiplexing Analysis

Based on the spectral compatibility analysis, we consider BPSK(5) modulation offset by 15.345 MHz from the 1575.42-MHz center frequency on the higher side as the desired option for multi-frequency multiplexing. Similar to single-frequency multiplexing, this case also considers two values of ζ_nss; i.e., ζ_nss = 1/3 and ζ_nss = 1/2. Table 4 shows the results for ζ_nss = 1/3. We observe that case 15 results in optimum performance based on the system requirements. Furthermore, the multi-frequency analysis for case 15 achieves approximately 1.7% higher efficiency than single-frequency multiplexing. The power-sharing ratio between multi-level data and multi-level pilot signals is preserved for the case of MCEMIC. In contrast, for the case of CEMIC, the power-sharing ratio changes by approximately 1%, thus violating the PSD requirement. Figure 6 shows the simulated constellation plot and the PSD of the proposed option for the multi-frequency multiplexing case. We used a sampling rate of 61.38 MHz to simulate this signal.

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FIGURE 6

Constellation (left) and PSD (right) for multi-frequency multiplexing with ζ_nss = 1/3

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TABLE 4

Power-Sharing Options (in %) for Various Phase Relationships of NSS-D and NSS-P Signals with ζ_nss = 1/3 for Multiplexing with Multi-Level Modulation Signals Over Multiple Frequencies

η denotes the multiplexing efficiency; I represents the in-phase channel; Q represents the quadrature channel.

It is important to observe that the number of constellation points is significantly less than the expected value, i.e., 64 against 256 possible constellation points. As already mentioned, the multi-level data and multi-level pilot signals are dependent. Moreover, the sampling of carrier results is performed in order pairs, which causes the sampled offset carrier vector to be correlated across time, resulting in a further reduction in the constellation points. Consequently, we must use additional caution when enumerating the possible combinations.