Distortion-Aware Multilevel Coded Spreading Symbol Optimization for Future Navigation Signals

2.2.1 Legacy Scenario: Non-Constant-Envelope Signal With the BPSK-R Data Waveform

One possible approach is to optimize the MCS of the pilot exclusively, leaving the pulse shape of the data waveform unaltered, with the exception of a power adjustment. This scenario reflects a legacy configuration in which the data wave-form remains a BPSK-R(10). For this simple setup, the two-component complex composite signal x(t) power shares are defined as follows:

$\begin{array}{c}{\eta }_l=\frac{1}{N_{\mathrm{sc}}}\hspace{0.17em}\sum _{n=1}^{N_{\mathrm{sc}}}{w}_{ln}^2\in \left[0,1\right]\hspace{0.17em}\forall l\in L\end{array}$7

if no IM products are used. Consequently, the signal generation power shares must attain the following:

$\begin{array}{c}{\eta }_1+{\eta }_2=1\end{array}$8

To generate a composite signal with an MCS pilot and a BPSK-R data waveform, the weights of the latter waveform are set to the following:

$\begin{array}{c}{w}_{2\hspace{0.17em}n}=\sqrt{1-{\eta }_1}\forall n\in N\end{array}$9

with N = {1,..., N_sc} such that x (t) as defined in Equation (3) has an average power P. The selection of these weights results in a non-constant envelope for the signal, except for the cases where $\left|w_{11}\right|=\left|w_{12}\right|=\cdots =\left|w_{1\hspace{0.17em}{N}_{s\hspace{0.17em}c}}\right|$. For a defined pilot signal generation power share ƞ₁ ≤ 1, the set of possible solutions for the legacy scenario are as follows:

$\begin{array}{c}{W}_1\hspace{0.17em}\left({\eta }_1\right)=\{w_1\in R^{N_{s\hspace{0.17em}c}\times 1}|\sqrt{w_{1\hspace{0.17em}1}^2+w_{1\hspace{0.17em}2}^2+\cdots +w_{1\hspace{0.17em}{N}_{s\hspace{0.17em}c}}^2}\le \sqrt{{\eta }_1\hspace{0.17em}{N}_{s\hspace{0.17em}c}}\}\end{array}$10

This set describes an N_sc-dimensional sphere with radius $\sqrt{{\eta }_1{N}_{s\hspace{0.17em}c}}$, with the center at the origin.

2.2.2 Optimistic Scenario: Constant-Envelope Signal With the MCS Data Waveform

Another approach is to optimize the MCS of the pilot and choose an MCS for the data waveform such that the composite signal always has a constant envelope. This case can be considered as constant-envelope multiplexing (CEM), where the data waveform is a secondary user signal, but also acts as the IM component for the non-constant-envelope pilot waveform. In this optimistic scenario, the constellation points must reside on a circle with radius P and the center in the origin of the complex plane. Thus, we have the following:

$\begin{array}{c}1\stackrel{!}{=}{w}_{1\hspace{0.17em}n}^2+w_{2\hspace{0.17em}n}^2\hspace{0.17em}\forall n\in N\end{array}$11a

$\begin{array}{c}\Rightarrow w_{2\hspace{0.17em}n}=\pm \sqrt{1-w_{1\hspace{0.17em}n}^2}\hspace{0.17em}\forall n\in N\end{array}$11b

$\begin{array}{c}\Rightarrow -1\le w_{1\hspace{0.17em}n}\le +1\hspace{0.17em}\forall n\in N\end{array}$11c

which restricts the solution space of possible MCS weights. Choosing a weight w_1n for the pilot waveform for the n-th subchip essentially determines the amplitude of the weight w_2n of the data waveform, but leaves choices for the polarity. For simplicity, the weight w_2n of the data waveform in each n-th subchip is fixed to $\sqrt{1-w_{1\hspace{0.17em}n}^2}$, setting the polarity to always be +1. The set of possible constant-envelope solutions for the optimistic scenario is the intersection of W₁(ƞ₁) with an N_sc-dimensional cube with an edge length of 2 and the center at the origin:

$\begin{array}{c}{W}_2\hspace{0.17em}\left({\eta }_1\right)=W_1\hspace{0.17em}\left({\eta }_1\right)\cap {\left[-1,+1\right]}^{N_{s\hspace{0.17em}c}\times 1}\end{array}$12

2.3.1 Nonlinear Distortion

Typical processes modeled in a nonlinear fashion include digital distortion, quantization, and signal amplification. In this study, only signal amplification by an HPA is modeled. Other nonlinear phenomena such as digital distortion and quantization are beyond the scope of this work. Because there is no public information available on the HPA characteristics of any GNSS satellite, a generic, memoryless, phase-compensated Saleh model (Beck et al., 2022), resembling the characteristics of the HPA used by Rebeyrol et al. (2006), is applied. As stated by Rapp (1991), a memoryless amplification following the NSGU and FGUU in the passband can be expressed in baseband notation using input amplitude to output amplitude (AM/ AM) and input amplitude to output phase (AM/PM) characteristics. In this study, the AM/AM and AM/PM characteristics are given as follows:

$\begin{array}{cc}g\hspace{0.17em}(x\hspace{0.17em}\left(t\right))& =g_A\hspace{0.17em}(\left|x\hspace{0.17em}\left(t\right)\right|)\hspace{0.17em}{e}^{j\hspace{0.17em}\left(g_{\varphi }\hspace{0.17em}(\left|x\hspace{0.17em}\left(t\right)\right|)+\mathrm{atan2}\hspace{0.17em}(\mathrm{Im}\hspace{0.17em}\left\{x\hspace{0.17em}\left(t\right)\right\},\mathrm{Re}\hspace{0.17em}\left\{x\hspace{0.17em}\left(t\right)\right\})\right)}\end{array}$13a

$\begin{array}{c}{g}_A\hspace{0.17em}(\left|x\hspace{0.17em}\left(t\right)\right|)=\frac{p_1\hspace{0.17em}\left|x\hspace{0.17em}\left(t\right)\right|}{1+p_2{\left|x\hspace{0.17em}\left(t\right)\right|}^2}\end{array}$13b

$\begin{array}{c}{g}_{\varphi }\hspace{0.17em}(\left|x\hspace{0.17em}\left(t\right)\right|)=\frac{p_3{\left|x\hspace{0.17em}\left(t\right)\right|}^2}{1+p_4{\left|x\hspace{0.17em}\left(t\right)\right|}^2}+p_5\end{array}$13c

These characteristics are defined by the five parameters p₁ = 94.897 (small-signal gain), p₂ = 50.402 W⁻² (= 1 / $x_2^0$, where x₀ denotes the input saturation amplitude), p₃ = 24.715 W⁻², p₄ = 94.406 W⁻², and p₅ = −0.127. These parameters were derived by solving for the Saleh characteristic such that the HPA model has a small-signal gain of 39.5 dB and such that a constant-envelope input signal x(t) with a power of 10.0 dBm (= 0.01 W) has an output power of 46.0 dBm (≈ 39.8 W) with an input power backoff (IBO) of 3.0 dB and an output power backoff (OBO) of 0.5 dB. Figure 1 shows AM/AM and AM/PM functions for the HPA model used. With an average input power of 10 dBm and the resulting IBO and OBO, the HPA operates in the upper end of the transition region between the linear and saturation region. In this operating region, amplitude compression is already significant, resulting in correlation losses of non-constant-envelope inputs at the receiver due to mismatched replicas, as shown by Enneking et al. (2022).

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Figure 1

AM/AM and AM/PM characteristics of the memoryless HPA model

2.3.2 Linear Distortion

Filtering and path loss are typical processes that are modeled linearly. For the IMUX filter h_I (t), which models the band-limited behavior of digital-to-analog converter (DAC) results in a slightly non-constant input to the subsequent HPA, a phase-compensated sixth-order Butterworth characteristic with a two-sided 3-dB bandwidth of 204.60 MHz is assumed, inspired by the work presented by Rebeyrol (2007) in the context of GAL signal selection and optimization. We model the IMUX filter as identical for each payload characteristic, given the absence of publicly available data. However, a notable discrepancy in the behavior of DACs between satellite payloads due to aging and environmental effects is likely and may merit further investigation for fine-tuned distortion-aware signal designs, which is beyond the scope of this work. The OMUX filters $h_O^{\left(k\right)}\hspace{0.17em}\left(t\right)$ are modeled based on linear payload characterizations of nine GPS IIF satellites in the L5 band with NORAD IDs 36585, 37753, 38833, 39166, 39533, 39741, 40105, 40294, and 40534 (Thölert et al., 2019). Figure 2 shows the absolute step responses, where u(t) denotes the unit step function. Accounting for free space loss and transmitter and receiver antenna gains, we assume that γ_PL = −170.0 dB for the path loss constant. The receiver front-end characteristic $h_R^{\left(k\right)}\hspace{0.17em}\left(t\right)$ is implemented as a brick-wall filter with a two-sided bandwidth of 40.92 MHz. Consequently, the number of CPRCs is K = 9.

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Figure 2

Step responses of linear payload characterizations of selected GPS IIF satellites

4.1 Problem Statement

The bi-objective OP is defined as follows:

$\underset{w_1\in {ℝ}^{N_{\text{sc}}}}{\min }\alpha J_1+(1-\alpha )J_2\left(\text{objective}\right)$21a

$\text{s.t.}{\eta }_1\le {\eta }_1^{max}\text{(maximum pilot power share)}$21b

${\varrho }_{11}\le {\varrho }_1^{\text{max}}\left(\text{maximum pilot LSPMPAR}\right)$21c

${\mathrm{SSC}}_{(1,q)}\le {\mathrm{SSC}}^{\text{max}}\forall q\in 𝒬\left(\text{maximum SSC}\right)$21d

$\left|w_{1n}\right|\ge w_1^{\text{min}}\forall n\in 𝒩\left(\text{minimum pilot weight amplitude}\right)$21e

$\left|w_{2n}\right|\ge w_2^{\text{min}}\forall n\in 𝒩\left(\text{minimum data weight amplitude}\right)$21f

$J_1\le J_1^{\text{max}}\left(\text{maximum differential range bias}\right)$21g

$J_2\le J_2^{\text{max}}\left(\text{maximum mean tracking jitter}\right)$21h

$\left|w_{1}n\right|\le w_1^{\text{max}}\forall n\in 𝒩\left(\text{maximum weight amplitude}\right)$21i

This problem minimizes the SD of the range biases and the mean of the TOA estimation error SDs of the pilot ranging signal over all K distortion realizations with the objective weight α ∈ [0,1] and the variable w₁ (i.e., the pilot weight vector). Setting α = 1 results in a pure differential bias optimization. Setting α = 0 results in a pure mean tracking jitter optimization. This comprehensive OP is subject to a number of constraints, the purpose of which is to ensure not only applicability, but also favorable pilot signal properties. The first constraint in Equation (21b) provides an upper bound for the power share of the pilot waveform ƞ₁ and acts only as an average power constraint for the pilot waveform. Equation (21c) constrains the largest side peak to main peak amplitude ratio (LSPMPAR), which is relevant in the context of robustness to multipath propagation. Considering noncoherent early-late processing (NELP), the LSPMPAR ${\varrho }_{11}=max\hspace{0.17em}\left\{\left|{\upsilon }_{1\hspace{0.17em}{n}^{\prime }}\right|/{\upsilon }_{{10}^{\prime }}|n^{\prime }\in P\right\}$ is defined as the ratio of the secondary stable-tracking side-peak with the largest amplitude to the amplitude of the main peak in the ACF of the pilot MCS, which is reflected by the NELP peak index set $P:=\left\{n^{\prime }\right|({\upsilon }_{1\hspace{0.17em}\left(n^{\prime }-1\right)}<{\upsilon }_{1\hspace{0.17em}{n}^{\prime }}>{\upsilon }_{1\hspace{0.17em}\left(n^{\prime }+1\right)}\vee {\upsilon }_{1\hspace{0.17em}\left(n^{\prime }-1\right)}>{\upsilon }_{1\hspace{0.17em}\left(n^{\prime }\right)}<{\upsilon }_{1\hspace{0.17em}\left(n^{\prime }+1\right)}\}$. Details regarding the derivation of the ACF of an MCS with any given weight vector are presented in Appendix A. The constraint in Equation (21d) uses the spectral separation coefficient (SSC) measure to reflect the requirements with respect to MAI in already occupied bands, such as the lower and upper L-band, to yield an optimized MCS waveform that does not adversely affect existing services. This study uses the following expression:

${\mathrm{SSC}}_{(1,q)}=\frac{{\int }_{-B/2}^{+B/2}{G}_1\left(f\right)G_q\left(f\right)df}{{\int }_{-\infty }^{+\infty }{G}_1\left(f\right)df{\int }_{-\infty }^{+\infty }{G}_q\left(f\right)df}$22

as, for instance, defined by Betz (2015) for a two-sided front-end bandwidth B with units of 1/Hz, where G₁ (f) denotes the power spectral density (PSD) of the MCS pilot waveform and G_q (f) denotes the PSD of another waveform using an index q ∈ Q for indicating waveform pairs for which the SSC shall not be exceeded. Details regarding the derivation of the PSD of an MCS are presented in Appendix B. Equations (21e) and (21f) aim to restrict solutions with subchip weights close to 0 to avoid receiver dynamic issues. Equations (21g) and (21h) limit the SD of the range biases and the mean of the TOA estimation error SDs. These constraints are, to some extent, also reflected in the objective, but they are incorporated in the constraint set to ensure suitable solutions for a pure differential bias or tracking error jitter optimization. The constraint in Equation (21i) is only relevant for the constant-envelope approach in the optimistic scenario and limits the value space of each weight w_1n, such that a real-valued w_2n exists for every n ∈ N.

4.2 Problem Reformulation

The OP is a nonlinear and nonconvex programming problem. One method for handling the constraints is the introduction of penalty functions with corresponding penalty coefficients (Nocedal & Wright, 2006). The problem can then be stated as follows:

$\begin{array}{c}\underset{w\in {ℝ}^{N_{\text{sc}}}}{\min }\tilde{\alpha }\widetilde{J_1}+(1-\tilde{\alpha })\widetilde{J_2}+{\mu }_1{\left({\eta }_1-{\eta }_1^{\max }\right)}_{+}^2+{\mu }_2{\left({\varrho }_{11}-{\varrho }_1^{\text{max}}\right)}_{+}^2\\ +{\mu }_3\sum _{q=1}^Q{\left(SS{C}_{(1,q)}-SS{C}^{\text{max}}\right)}_{+}^2+{\mu }_4\sum _{n=1}^{N_{\text{sc}}}{\left(w_1^{\text{min}}-\left(w_{1n}\right)\right)}_{+}^2+{\mu }_5\sum _{n=1}^{N_{\text{sc}}}{\left(w_2^{\text{min}}-\left(w_{2n}\right)\right)}_{+}^2\\ +{\mu }_6{\left(J_1-J_1^{\text{max}}\right)}_{+}^2+{\mu }_7{\left(J_2-J_2^{\text{max}}\right)}_{+}^2\end{array}$23

where ${\left[z\right]}_{+}^2={\left(max\hspace{0.17em}\left\{0,z\right\}\right)}^2$ denotes an l²-merit function, ${\stackrel{\sim }{J}}_1$ is the normalized objective J₁, ${\stackrel{\sim }{J}}_2$ represents the normalized objective J₂, and µ₁,..., µ₇ ∈ ℝ₊ denotes the real-valued non-negative penalty function coefficients. Normalizing the objective functions into the domain of [0,1] was imperative for deriving meaningful trade-off solutions in the numerical implementation. This objective was accomplished by systematically adjusting the parameter α. In the absence of adequate scaling, the magnitudes of the objective values can become imbalanced, resulting in biased solutions and skewing of the intended trade-off. In ensuring a balanced optimization process, it was necessary to normalize the objectives prior to the application of the bi-objective optimization framework. A thorough examination of the normalization process in the context of multi-objective optimization has been reported by Miettinen (1998). The value space constraint may be implemented in such a way that a solver must not leave $W_1\hspace{0.17em}\left({\eta }_1\right)$ or $W_2\hspace{0.17em}\left({\eta }_1\right)$, respectively. One approach for finding sufficient local minima is an initial brute-force search on the convex hull of $W_1\hspace{0.17em}\left({\eta }_1\right)$ or $W_2\hspace{0.17em}\left({\eta }_1\right)$ to find suitable start vectors for a subsequent search with the downhill simplex method.

4.3 Trade-Offs Along Pareto Curves

As a result of the optimization of the bi-objective OP with a varying objective weight α, a swarm of solutions can be found that form a Pareto curve defined by a finite set of solutions, each found for an individual objective weight α. Each solution along the Pareto curve represents a potential optimal candidate, contingent upon the specific trade-off criteria. One straightforward approach for selecting a signal candidate is to inspect the performance enhancement relative to a chosen performance, which serves as a reference point. We use the following two performance change parameters:

$x_{1,m}=J_{1,m}^{\ast }/J_1^{\text{ref}}-1\forall m\in ℳ$24a

$x_{2,m}=J_{2,m}^{\ast }/J_2^{\text{ref}}-1\forall m\in ℳ$24b

to assess the change in joint ranging performance of the m-th optimal solution point $\left(J_{1,m}^{\ast },J_{2,m}^{\ast }\right)$ on the Pareto curve with respect to the reference performance point $\left(J_1^{\text{ref}},J_2^{\text{ref}}\right)$, where each Pareto curve consists of M optimal solutions defining the index set M = {1,...,M}. If $X_{1,m}<0$ or $X_{2,m}<0$, the differential range bias or tracking jitter is reduced, respectively. In this study, the reference point $\left(J_1^{\text{ref}},J_2^{\text{ref}}\right)$ is characterized by the transmission of two BPSK-R(10) waveforms in I and Q, representing the current status quo of GPS L5. Using the two performance change parameters $X_{1,m}$ and $X_{2,m}$ defined by Equations (24a) and (24b), a multitude of trade-off strategies might be used to achieve a meaningful trade-off, including the following strategies:

$\arg \text{min}{X}_{1,m}\forall m\in \left\{m\in ℳ|X_{2,m}\le 0\}\right(\text{Trade-off 1})$25a

$\arg \text{min}{X}_{2,m}\forall m\in \left\{m\in ℳ|X_{1,m}\le 0\}\right(\text{Trade-off 2})$25b

$\arg \text{min}{X}_{1,m}+X_{2,m}\forall m\in ℳ\left(\text{Trade-off 3}\right)$25c

$\arg \text{min}{X}_{1,m}{X}_{2,m}\forall m\in ℳ\left(\text{Trade-off 4}\right)$25d

Trade-off 1 in Equation (25a) aims to minimize the SD of the range biases while not increasing the mean of the TOA estimation error SDs. In contrast, trade-off 2 in Equation (25b) aims to minimize the mean of TOA estimation error SDs while not increasing the SD of the range biases. The goal of the latter rationales, expressed by Equations (25c) and (25d), is to minimize the sum and product of the change parameters, respectively.

5.1 Simulation Parameters

This study examines the optimization of an MCS for a pilot waveform in two distinct scenarios. The first scenario entails a reshaping of the pilot waveform’s chip shape, while the data waveform’s chip shape remains unaltered. The second scenario involves a reshaping of the pilot and data waveform’s chip shape, with the objective of optimizing the ranging performance of the pilot while maintaining a constant envelope for the complex signal. The selection of these scenarios is intended to illustrate the range of potential performance improvements that can be achieved. The respective simulation parameters of the defined scenarios are provided in Table 1, where f_b= 1/T_b = 1.023 MHz denotes the base frequency relevant to GNSS. In both scenarios, the noise density is set to N₀ = −204 dBW/Hz, resulting in a carrier-to-noise density ratio (C/N₀) in the range of 42 to 43 dB-Hz for the composite signal. The early-late correlator spacing is set to △ = T_b / 20 ≈ 48.88 ns. The primary range codes for pilot and data components of currently deployed GPS L5 payloads were used for the range codes in this simulation. Furthermore, ${\eta }_1^{max}=0.5$, ${\varrho }_1^{max}=0.5$, SSC^max =−71.86 dB/Hz (i.e., self-SSC of BPSK-R(10)), $w_1^{min}=0.1$, $w_2^{min}=0.1$, $c_0\hspace{0.17em}{J}_1^{max}=10.0m$, and $c_0\hspace{0.17em}{J}_2^{max}=10.0m$. The SSCs relevant to the SSC constraint are the pilot MCS self-SSC, the data/pilot MCS SSC, and the SSC of the pilot MCS and BPSK-R(10). The penalty function coefficients were selected in a manner that ensured that the magnitude of the corresponding penalty function was comparable to that of the objective function. The penalty function coefficients were set to µ₃ = 1, µ₁ = 10², µ₆ = µ₇ = 10³, and µ₂ = µ₄ = µ₅ = 10⁴.

5.2 Restriction to a Symmetric MCS

To apply a heuristic approach for finding suitable start vectors by evaluating the nonconvex OP, a dimensionality reduction is key for handling the complexity involved. A trivial way to reduce the dimension of the OP from N_sc to N_sc /2 is to consider only MCSs of a special structure. This work considers waveforms with even N_sc and that are, with respect to the pulse center, axially symmetric (A-sym) or point symmetric (P-sym) MCSs. Consequently, only MCSs with the following weight relationships for the pilot waveform were considered in this investigation:

$A-sym:w_{ln}=+w_{l(N_{\text{sc}}-n+1)}\forall (l,n)\in ℒ\times 𝒩$26a

$P-sym:w_{ln}=-w_{l(N_{\text{sc}}-n+1)}\forall (l,n)\in ℒ\times 𝒩$26b

The symmetry constraint is inspired by the structure of a majority of navigation signals currently broadcast by GNSS, but it is also imposed to effectively reduce the computational complexity of the OP. However, the introduction of the symmetry assumption requires a case distinction for A-sym and P-sym MCS. Furthermore, distortion-aware nonsymmetric MCSs, which may perform better than symmetric MCSs, are excluded from the set of solutions and are therefore beyond the scope of this work.

5.3 Pareto Curves

The Pareto curves for the bi-objective OP illustrate the set of optimal solutions for varying objective weights. The objective weight enables the formulation of a trade-off between the two opposing objectives of differential range bias and mean tracking jitter minimization. Figure 3 illustrates the performance of the found solutions of the optimization, given the chosen simulation parameters with a varying objective weight α ∈ [0,1] for each scenario and symmetry assumption. In addition to the Pareto curves, which are the result of a distortion-aware optimization, Figure 3 also depicts the current reference operating point, which employs a BPSK-R(10) pilot and data component with c₀J₁= 6.59 cm and c₀J₂= 1.26 m. In addition, the performance of BOC(10,5) (c₀J₁ = 8.20 cm, c₀J₂= 0.84 m) and BOC(10,10) (c₀J₁ = 7.71 cm, c₀J₂ = 0.72 m) signals is shown, which exhibit a lower mean tracking error jitter but experience an increased differential bias compared with the BPSK-R(10) signal. Furthermore, the performance of a Gabor bandwidth-optimized (GBO) MCS is also presented for a fair comparison. The optimization strategy for deriving the GBO MCS uses as an input the PSD of the pulse basis functions, whose exact shape are dependent on T_c and N_sc, and a target two-sided front-end bandwidth. Details on finding a GBO MCS for those parameters are briefly provided in Appendix C based on the work of Zhang et al. (2011a). The performance of the GBO MCS is shown in Figure 3 for a fixed T_c= T_b/10 and N_sc = 10, but varying target front-end bandwidths. In this study, the GBO MCS with minimal mean tracking jitter is optimized for a front-end bandwidth of 27.621 MHz (c₀J₁ = 8.54 cm, c₀J₂ = 0.59 m). GBO MCSs that are optimized for a front-end bandwidth of 23.529 MHz (c₀J₁= 8.26 cm, c₀J₂ = 0.62 m) or higher violate the LSPMPAR constraint ${\varrho }_1^{max}=0.5$. Based on the solutions along the Pareto curves with respect to the GPS L5 performance and the performance of the GBO MCS, multiple observations can be made that are specific to this example:

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Figure 3

Pareto curves for N_sc = 10

1. Distortion-aware optimized A-sym MCSs can achieve a simultaneous reduction in differential range bias or mean tracking jitter with respect to the reference operating point.

2. Distortion-aware optimized P-sym MCSs effectively reduce mean tracking jitter, but systematically increase the differential range bias with respect to the reference operating point.

3. In the majority of instances, the optimistic scenario demonstrates superior performance in comparison to its legacy counterpart for the same symmetry constraint imposed on the MCS.

4. In a setup that accounts for distortions, the performance of the GBO MCS is worse than that of P-sym MCSs in the low-tracking-jitter region owing to a model mismatch and is also unable to reduce differential biases.

In light of these observations, it can be concluded that distortion-aware optimized MCSs are capable of achieving enhanced ranging performance in terms of differential range bias and mean tracking jitter in realistic scenarios when compared with the current status quo, as well as when compared with more wideband waveforms such as the BOC(10,5) and BOC(10,10) and even GBO MCSs. The advantage of this distortion-aware navigation signal design methodology is that it produces a set of optimal solutions from which a decision-maker can select the most suitable option. For instances in which it is challenging to monitor differential range biases, a decision-maker may select an MCS that minimizes these biases while maintaining tracking jitter performance in relation to a reference operating point. Conversely, for scenarios in which differential range biases are readily assessable and distributed to each user, a decision-maker may select an MCS that solely minimizes mean tracking jitter, relying on robust differential range bias monitoring and propagation to users. The primary limitation of this approach is that the set of solutions is finite and can only be augmented through an iterative search process. In the context of this case study, we identified MCSs that demonstrated the capacity to either reduce the differential range bias by 6.4% while simultaneously increasing the mean tracking jitter relative to the reference operating point by 2.1% or to reduce the mean tracking jitter by 56.2% while concurrently increasing the differential range bias by 31.0% relative to the aforementioned reference operating point. A decision-maker must evaluate whether the performance improvements in terms of differential range bias or mean tracking error jitter reduction are sufficient to justify replacing rectangular waveforms with distortion-aware optimized MCS waveforms.

5.4 Power Losses of Two-Component Signals Along the Signal Propagation Chain

The spectral occupation of an MCS waveform’s PSD beyond the transmitter and receiver bandwidths is an inherent property of this type of waveform. Consequently, each MCS signal candidate will spill signal energy beyond the respective filter passbands, resulting in a loss of signal power. To illustrate the influence of the optimization weight α, the choice of signal generation scenario, and MCS symmetry on signal power loss, Figure 4 provides an illustration of the signal power after each step in signal formation. The HPA signal power loss is determined by the ratio of the IMUX-filtered and subsequently amplified complex HPA power output of a constant-envelope signal that possesses a mean signal power equivalent to that of the IMUX-filtered signal. As shown in Figure 4, it is evident that for all scenarios and MCS symmetries, the pure solution that exclusively minimizes the tracking jitter (i.e., α = 0) leads to greater signal power losses compared with the solution where a combination of the differential range bias and tracking jitter (in the shown example, α = 0.97) is minimized. A comparison of the legacy scenario with the optimistic signal generation scenario reveals that the former exhibits higher signal power at the receiver stage, yet experiences greater losses at the HPA due to the heightened non-constancy that arises from using the BPSK-R(10) as the data waveform. Conversely, in the optimistic scenario, although the solutions exhibit marginal losses induced by the HPA, they tend to experience a greater overall reduction in signal power.

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Figure 4

Cumulative signal power loss of two-component signal power for selected MCSs on the Pareto curve, with N_sc = 10

5.5 Characteristics of Selected Optimal MCSs

To further analyze elements of the Pareto curves shown above, this section presents the characteristics of selected distortion-aware optimized MCSs. In the event that a decision-maker chooses to implement an optimized MCS, a number of trade-off rationales, as outlined in Equations (25a)-(25d), could be used to select an MCS for each scenario. The choice of these trade-off rationales is intended to show a sufficient range of distortion-aware optimized MCSs, but these options may be suboptimal for specific use cases.

5.5.1 Time Domain

One domain meriting further investigation is the time domain of the optimized MCSs. Figures 5 and 6 illustrate the optimal MCSs in the time domain, which are elements of the Pareto curve selected according to the trade-off rationale for each scenario, where the notation MCS_p merely refers to use of the MCS for the pilot waveform. Correspondingly, the notation MCS_d, which will be used in the subsequent section, denotes the MCS of the data waveform. As illustrated in Figure 5, the MCS in the legacy scenario resulting from the first, second, and fourth trade-off rationales are A-sym. In contrast, the MCS resulting from the third trade-off rationale is P-sym. Although the maximum amplitude subchip weight constraint is inactive for the legacy scenario, no amplitude subchip weight greater than 1 is part of any solution. Figure 6 shows that the MCSs resulting from the first and second trade-off rationales are A-sym. In contrast, the MCSs resulting from the third and fourth trade-off rationales are P-sym.

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Figure 5

Selected optimal MCSs along the Pareto curve for the legacy scenario

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Figure 6

Selected optimal MCSs along the Pareto curve for the optimistic scenario

5.5.2 Power Spectral Densities

As illustrated in Figures 7 and 8, the evaluation of the PSD of the selected MCSs indicates that the trade-offs that prioritize a reduction or disallow an increase in the differential range bias with respect to the reference operating point (i.e., trade-offs 1 and 2) result in the allocation of signal power to the center of the band. This power allocation is performed with the objective of preventing an accumulation of distortion at the band edges, where distortions are more prevalent. The main lobe of these MCSs is observed to have a width that is greater than that of the BPSK-R(10). The secondary side lobes exhibit a reduction in signal strength and (partially) extend beyond the target reception bandwidth of 40.92 MHz. Conversely, the trade-offs that prioritize the minimization of the mean tracking jitter over the differential range bias (i.e., trade-offs 3 and 4) allocate the signal power closer to the band edges of the payload and receiver front-end characteristics while accounting for distortions. The respective MCSs exhibit two primary main lobes, which contain the majority of the signal power. The incorporation of distortion into the signal design process enables more accurate model matching, thereby enabling distortion-aware MCSs to outperform purely GBO MCSs in realistic signal propagation settings.

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Figure 7

PSD of selected optimal MCSs along the Pareto curve for the legacy scenario

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Figure 8

PSD of selected optimal MCSs along the Pareto curve for the optimistic scenario

5.5.4 Auto-Correlation Functions

It is of interest to examine the shape of the ACFs of the selected MCSs, as illustrated in Figures 9 and 10. In the context of the presented case study, the application of trade-offs 1 and 2 results in the generation of MCSs with an ACF that exhibits a single main peak and no secondary side peaks. Employing these MCSs would allow for unambiguous tracking, similar to BPSK-R(10) signals. The application of trade-offs 3 and 4 results in the selection of MCSs with a main peak exhibiting a steeper slope than the BPSK-R(10) waveforms and, consequently, superior tracking performance. However, these MCSs also display secondary side peaks, necessitating the implementation of tracking methods capable of resolving code-phase ambiguity. The LSPMPAR constraint is relevant for the latter MCSs, which prevents the LSPMPAR from being larger than ${\varrho }_1^{max}=0.5$. It should be noted that the ACF of the GBO MCS may exhibit a main peak with a steeper slope, corresponding to a larger Gabor bandwidth, than that of the distortion-aware MCS. However, in the presence of distortions, the GBO MCS is susceptible to stronger correlation losses and stronger smoothing of the CCF, resulting in inferior performance compared with the distortion-aware MCS, as evidenced by Figure 3.

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Figure 9

ACF of selected optimal MCSs along the Pareto curve for the legacy scenario

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Figure 10

ACF of selected optimal MCSs along the Pareto curve for the optimistic scenario

5.5.5 Multipath Error Envelopes

An often-used metric for illustrating the range of errors induced by multipath propagation is the multipath ranging error envelope (MREE), which assumes a two-ray signal propagation with a line-of-sight (LOS) and a non-LOS (NLOS) path with a time delay, where the power of the NLOS path is attenuated by 3 dB compared with the LOS path. Figures 11 and 12 show the MREEs of the selected MCS for each scenario. The MREE were calculated by assuming a bandwidth of 40.92 MHz and an early-late spacing of △ = T_b / 20. The majority of the selected distortion-aware MCSs result in an earlier tapering and reduction of the MREE in comparison to the BPSK-R(10) MREE, indicating that they are more effective in suppressing errors induced by multipath propagation. The MREEs of low-tracking-jitter MCSs exhibit multiple humps that decrease in size rather than a single larger hump, which is a consequence of their steeper-sloped multi-peaked ACF. These findings suggest the potential for enhanced multipath robustness when utilizing distortion-aware MCSs.

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Figure 11

Multipath error envelopes for the legacy scenario

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Figure 12

Multipath error envelopes for the optimistic scenario