Ranging Performance Evaluation for Higher-Order Scalable Interplex

2.1 Overview

In satellite navigation, it is common practice to generate one composite complex navigation signal generated with N signal components (also referred to as spreading waveforms in the context of CDMA). The n-th spreading waveform is given as follows:

$\begin{array}{cc}{x}_n\left(t\right)=\sum _{m=-\infty }^{\infty }{c}_n\left(m{T}_{c,n}\right)p_n\left(t-m{T}_{c,n}\right)& n\in \left\{1,2,\dots ,N\right\}\end{array}$1

This waveform is generated with a dedicated binary ranging code (and possibly also data) sequence c_n(mT_c,n) ∈ {+1, −1}, pulse shape p_n(t – mT_c,n), and spreading code chip duration T_c,n = 1/f_c,n. All signal components x_n(t) are modulated by the input multiplexer (IMUX), which is represented by the complex function $M\{.\}$, to a composite complex signal:

$x\left(t\right)=M\left\{\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right)\right)\right\}$2

onto the same carrier frequency f₀ = ω₀/2π. In most cases, the IMUX adds IM terms to avoid large HPA distortion while operating the HPA close to its saturation point. The multiplexed signal x(t) is then fed to the transmit chain $T\{.\}$ consisting of the nonlinear satellite HPA, OMUX, and antenna, resulting in the amplified and radiated complex signal:

$y\left(t\right)=T\left\{x\left(t\right)\right\}$3

The resulting transmit passband signal y(t) is subsequently affected by the propagation channel $\mathbb{P}\{.\}$, resulting in the signal $\mathbb{P}\left\{y\left(t\right)\right\}$. The received passband signal is down-converted in the receiver to baseband, and the resulting signal is denoted by $D\left\{\mathbb{P}\left\{y\left(t\right)\right\}\right\}$. The receiver front-end filtering is modeled with a linear filter h_RX(t). Thermal receiver noise is modeled as a complex additive white Gaussian noise (AWGN) process n(t), whose real and imaginary parts are independent processes each with two-sided power spectral density (PSD) N₀/2. Thus, the analog complex baseband received signal is as follows:

$r\left(t\right)=h_{\text{RX}}\left(t\right)\ast D\left\{\mathbb{P}\left\{y\left(t\right)\right\}\right\}+n\left(t\right)\stackrel{△}{=}s\left(t\right)+n\left(t\right)$4

where * denotes the convolution operator. The received signal can then be represented as the sum of s(t), which denotes the noise-free, but distorted satellite signal on the ground, and n(t), which is the complex baseband AWGN. Figure 1 provides an overview of the navigation signal formation for the introduced system model. The following subsections elaborate on the models and assumptions applied to each building block within the simulation pipeline.

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FIGURE 1

System model simulating the navigation signal from signal generation, payload distortion, channel propagation, receiver front-end filtering, and additive noise

2.2 Signal Components

The large increase in the use of GNSSs in mass-market devices over the last decade has further fostered the idea of providing additional Galileo signals in the E1 band. These added signals or services might be specifically tailored to low-complexity receivers (Wallner et al., 2020, 2021), as the Galileo E1 open service has been shown to induce high complexity in GNSS receivers, demanding a significantly higher bandwidth and memory than GPS owing to its modulation, long spreading sequences, and used memory spreading range codes that cannot be generated by shift registers.

Hence, in the following, we consider four Galileo services to investigate the potential of IM term scaling when attempting to add an additional signal component to the already existing three Galileo E1 services. The current Galileo E1 signal is comprised of three services, with one service being binary offset carrier (BOC)-modulated and two being composite binary offset carrier (CBOC)-modulated and multiplexed to one constant-envelope signal (Rebeyrol et al., 2006). In a previous study, the scalable interplex with an added Galileo E1 signal was modeled as a four-channel interplex with the three previously mentioned components and a signal candidate (Beck et al., 2022b).

However, this approach leads to approximation errors that can be avoided by modeling the Galileo E1 signal as a result of a five-channel interplex, as proposed by Vergara et al. (2013). Hence, in this study, the first five signal components are modeled such that they represent the three existing Galileo services in the E1 band. The additional spreading waveform is a signal candidate tailored to low-end receivers (Enneking et al., 2022) and was investigated by Beck et al. (2022b) as a potential signal candidate for receivers with a relatively narrow reception bandwidth. The naming conventions and properties of the investigated signal components are described as follows:

1. BOC_cos-R(15, 2.5) is the first signal component. This component is BOC-modulated and cosine-phased and has a rectangular (R) pulse shape. The sub-carrier rate is f_sc,1 = 15 × 1.023 MHz, and the keying or chipping rate is f_chip,1 = 2.5 × 1.023 MHz. This BOC-modulation can be described via p₁(t) = h_BOC(t, f_sc,1, f_chip,1, cos(.)) with the following expression:

   $h_{\text{BOC}}\left(t,f_{sc},f_c,g(.)\right)=\{\begin{array}{cc}\text{sng}\left(g\left(2\pi f_{sc}t\right)\right)& 0\le t<1/f_c\\ 0& \text{else}\end{array}$5

   where sgn(.) denotes the sign function (Meurer & Antreich, 2017). This signal component is called E1A and is the signal of the Galileo PRS.

2. BOC_sin-R(1, 1) is the second signal component. The sub-carrier rate is f_sc,2 = 1 × 1.023 MHz, and the chipping rate is f_chip,2 = 1 × 1.023 MHz. The power ratio between the second and fourth component is 10 : 1, and the resulting pulse shape is p₂(t) = h_BOC(t, f_sc,2, f_c,2, sin(.)).

3. BOC_sin-R(1, 1) is the third signal component. The sub-carrier rate is f_sc,3 = 1 × 1.023 MHz, and the chipping rate is f_c,3 = 1 × 1.023 MHz. The power ratio between the third and fifth component is 10 : 1, and the resulting pulse shape is p₃(t) = h_BOC(t, f_sc,3,f_chip,3, sin(.)).

4. BOC_sin-R(6, 1) is the fourth signal component and is BOC-modulated. The sub-carrier rate is f_sc,4 = 6 × 1.023 MHz, and the chipping rate is f_c,4 = 1 × 1.023 MHz. The resulting pulse shape is p₄(t) = h_BOC(t, f_sc4, f_c,4, sin(.)). The second and fourth components have the same ranging code and are modulated in-phase to each to other such that $\sqrt{10/11}{p}_2\left(t\right)+\sqrt{1/11}{p}_4\left(t\right)$ forms a CBOC_sin-R(6, 1, 1/11, +), which corresponds to the Galileo E1B service.

5. BOC_sin-R(6, 1) is the fifth signal component. The sub-carrier rate is f_sc,5 = 6 × 1.023 MHz, and the chipping rate is f_c,5 = 1 × 1.023 MHz. The resulting pulse shape is p₅(t) = h_BOC(t, f_sc,5, f_c,5, sin(.)). The third and fifth components have the same ranging code and are modulated in anti-phase such that $\sqrt{10/11}{p}_3\left(t\right)-\sqrt{1/11}{p}_5\left(t\right)$ forms a CBOC_sin-R(6, 1, 1/11,−), which corresponds to the Galileo E1C service.

6. BPSK-RC(1, 0.22) is the sixth signal component and is binary phase-shift keying (BPSK)-modulated with a keying rate of f_c,6 = 1/T_c,6 = 1 × 1.023 MHz. The pulse is shaped with a raised-cosine (RC) with a roll-off factor of ν = 0.22. The pulse shape is p₆(t) = h_RC(t/T_chip,6, 0.22) with the following expression:

   $h_{RC}\left(t,\upsilon \right)={\left(1-\frac{\upsilon }{4}\right)}^{-\frac{1}{2}}\frac{\cos \left(\pi \upsilon t\right)}{1-{\left(2\upsilon t\right)}^2}\frac{\sin \left(\pi t\right)}{\pi t}$6

   as used in the Universal Mobile Telecommunications System (Proakis, 2001). This waveform is an example of a design tailored to receivers with a relatively low sampling rate and, consequently, a small receiver filtering bandwidth. The motivation for deploying such a signal would be to minimize filtering losses along the signal propagation path in order to maximize the signal-to-noise ratio (SNR) for users (Beck et al., 2022a). It can be argued that this signal is not necessarily an optimal tracking signal, owing to the fact that the auto-correlation function (ACF) is relatively smooth, which results in a larger TOA estimation error SD than a BPSK-R(1) waveform, such as the GPS L1 C/A signal. Moreover, the waveform itself is not a constant-envelope signal, which introduces additional complexities to the multiplexing process. Hence, there is a scarcity of robust multiplexing techniques capable of handling this type of signal as an input. For these reasons, the BPSK-RC(1, 0.22) was deemed unsuitable for the first generation of Galileo signals (Ávila Rodríguez, 2008). Nevertheless, a BPSK-RC(1, 0.22) signal could prove to be an effective acquisition signal, especially for the majority of GNSS receivers currently available in the low-end mass-market segment. These receivers are typically constrained in their power consumption and, therefore, rarely perform tracking for extended periods of time. Instead, these receivers primarily perform duty-cycled acquisition in order to determine a PNT solution (Enneking et al., 2022).

In the following investigation, the Galileo E1 signal will be generated as a six-channel interplex, but the TOA estimation error SD will be evaluated by using four replicas that correspond to the services currently transmitted by Galileo, including the added component. These replicas are as follows:

${\overline{x}}_1\left(t\right)=x_1\left(t\right)$(7a)

${\overline{x}}_2\left(t\right)=\sqrt{\frac{10}{11}}{x}_2\left(t\right)+\sqrt{\frac{1}{11}}{x}_4\left(t\right)$(7b)

${\overline{x}}_3\left(t\right)=\sqrt{\frac{10}{11}}{x}_3\left(t\right)-\sqrt{\frac{1}{11}}{x}_5\left(t\right)$(7c)

${\overline{x}}_4\left(t\right)=x_6\left(t\right)$(7d)

which will be used to evaluate the ACFs and CCFs. Figure 2 illustrates the PSDs of the signal replicas.

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FIGURE 2

PSDs of the signal replicas

2.3 Multiplexing

This paper focuses on scalable interplex, which is a modified version of the N-channel interplex, as described by Butman and Timor (1972). The N-channel interplex is a phase modulation (PM) system resulting in the following composite signal:

$x\left(t\right)=M\left\{x_1\left(t\right),x_2\left(t\right),\cdots ,x_N\left(t\right)\right\}=\sqrt{2P^{\left(N\right)}}\sin \left({\omega }_{0}t+\Theta \left(t\right)\right)$8

where P^(N) denotes the total mean power, ω₀ denotes the angular carrier frequency, and Θ(t) denotes the PM term, which requires N spreading waveforms and N modulation angles as input. The PM term for the N-channel interplex is as follows:

$\begin{array}{cc}\Theta \left(t\right)& =\left({\theta }_1+\sum _{n=2}^N{\theta }_n{x}_n\left(t\right)\right)x_1\left(t\right)\\ & ={\theta }_1{x}_1\left(t\right)+{\theta }_2{x}_2\left(t\right)x_1\left(t\right)+{\theta }_3{x}_3\left(t\right)x_1\left(t\right)+\cdots +{\theta }_n{x}_n\left(t\right)x_1\left(t\right)\end{array}$9

with θ₁,…, θ_N denoting the modulating angles, which essentially determine the power distribution of the input waveforms and the implicitly generated IM terms when the input waveforms are power-normalized. Furthermore, the value range of the modulation angles is relevant to the correct generation of the composite CBOC_sin-R(6, 1, 1/11, +) and CBOC_sin-R(6, 1, 1/11,−) waveforms, as π < θ_n < 2π causes a polarity flip of the respective signal component n with respect to the primary component if and only if θ₁ = π/2. The N-channel interplex reduces the power dissipated in the IM terms already in a two-component setup compared with QPSK. When introducing the N-channel interplex with a sine carrier, it is helpful to set the modulation angle θ₁ = π/2 of the primary waveform x 1(t), as it collapses the number of IM terms N_IM = 2^N–1 – N (Vergara et al., 2013). The CEM of the Galileo E1B, E1C, and PRS signals on the E1 carrier, as described by Rebeyrol et al. (2006), can be represented as a five-channel interplex of the respective five bipolar sub-components with a target power ratio of 2:1 between PRS and E1B and 1:1 between E1B and E1C (Vergara & Antreich, 2013). Thus, the use of eleven IM terms is necessary for the Galileo E1 multiplex when the five-channel interplex analogy is applied. To minimize the influence of approximation errors when inspecting the Galileo E1 signal with an added non-binary component, a six-channel interplex is used to derive the scalable interplex formulation, as, for instance, reported by Ortega et al. (2020). The six-channel interplex can be given as follows:

$x\left(t\right)=\sqrt{2P^{\left(6\right)}}\sin \left(\begin{array}{c}{\omega }_{0}t+{\theta }_1{x}_1\left(t\right)+{\theta }_2{x}_2\left(t\right)x_1\left(t\right)+{\theta }_3{x}_3\left(t\right)x_1\left(t\right)\\ +{\theta }_4{x}_4\left(t\right)x_1\left(t\right)+{\theta }_5{x}_5\left(t\right)x_1\left(t\right)+{\theta }_6{x}_6\left(t\right)x_1\left(t\right)\end{array}\right)$10

which is referred to in the latter as the standard six-channel interplex.

The main idea for yielding a scalable interplex expession starting with Equation (10) is the application of trigonometric identities, as shown in Appendix A, such that the reformulated expression is a sum of products of the sine and cosine of the modulation angles and a product of the signal components. Each additive term that contains only one single signal component x_n(t) is a useful signal term. All other terms that contain a product of signal components are designated as IM terms, constituting the complex IM component (i.e., complex sum of all IM terms). A generalized expression for yielding this additive expression for any N-channel interplex was developed by Vergara et al. (2013). We note that Ortega Espluga et al. (2020) published a written formulation of this expression for the six-channel interplex when a cosine carrier is selected. By using a sine carrier and setting θ₁ = π/2, waveforms from x₂(t) to x_N(t) are I-broadcast, whereas the primary component x₁(t) is Q-transmitted. Setting the parameters as such, the scalable six-channel interplex can be expressed as follows:

$x\left(t\right)=\sqrt{2P^{\left(6\right)}}\sin \left({\omega }_{0}t\right)I\left(t\right)+\sqrt{2P^{\left(6\right)}}\cos \left({\omega }_{0}t\right)Q\left(t\right)$11

with:

$\begin{array}{ccc}I\left(t\right)=& -\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_2\left(t\right)& \left(\text{Signal} \text{term} 1\right)\\ & +\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_2\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_{11}& \left(\text{IM} \text{term} 11\right)\\ & +\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_2\left(t\right)x_4\left(t\right)x_5\left(t\right){\kappa }_{13}& \left(\text{IM} \text{term} 13\right)\\ & +\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_2\left(t\right)x_4\left(t\right)x_6\left(t\right){\kappa }_{12}& \left(\text{IM} \text{term} 12\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_2\left(t\right)x_3\left(t\right)x_4\left(t\right){\kappa }_{16}& \left(\text{IM} \text{term} 16\right)\\ & -\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_2\left(t\right)x_3\left(t\right)x_4\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_1& \left(\text{IM} \text{term} 1\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_2\left(t\right)x_3\left(t\right)x_5\left(t\right){\kappa }_{15}& \left(\text{IM} \text{term} 15\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_2\left(t\right)x_3\left(t\right)x_6\left(t\right){\kappa }_{14}& \left(\text{IM} \text{term} 14\right)\\ & -\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_3\left(t\right)& \left(\text{Signal} \text{term} 3\right)\\ & +\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_3\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_8& \left(\text{IM} \text{term} 8\right)\\ & +\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_3\left(t\right)x_4\left(t\right)x_5\left(t\right){\kappa }_{10}& \left(\text{IM} \text{term} 10\right)\\ & +\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_3\left(t\right)x_4\left(t\right)x_6\left(t\right){\kappa }_9& \left(\text{IM} \text{term} 9\right)\\ & -\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_4\left(t\right)& \left(\text{Signal} \text{term} 4\right)\\ & +\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_4\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_7& \left(\text{IM} \text{term} 7\right)\\ & -\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_5\left(t\right)& \left(\text{Signal} \text{term} 5\right)\\ & -\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_6\left(t\right)& \left(\text{Signal} \text{term} 6\right)\end{array}$12

and:

$\begin{array}{ccc}Q\left(t\right)=& +\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)& \left(\text{Signal} \text{term} 1\right)\\ & \cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_{17}& \left(\text{IM} \text{term} 17\right)\\ & -\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_4\left(t\right)x_5\left(t\right){\kappa }_{19}& \left(\text{IM} \text{term} 19\right)\\ & -\cos \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_4\left(t\right)x_6\left(t\right){\kappa }_{18}& \left(\text{IM} \text{term} 18\right)\\ & -\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_3\left(t\right)x_4\left(t\right){\kappa }_{22}& \left(\text{IM} \text{term} 22\right)\\ & +\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_3\left(t\right)x_4\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_2& \left(\text{IM} \text{term} 2\right)\\ & -\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_3\left(t\right)x_5\left(t\right){\kappa }_{21}& \left(\text{IM} \text{term} 21\right)\\ & -\cos \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_3\left(t\right)x_6\left(t\right){\kappa }_{20}& \left(\text{IM} \text{term} 20\right)\\ & -\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_3\left(t\right){\kappa }_{26}& \left(\text{IM} \text{term} 26\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_3\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_4& \left(\text{IM} \text{term} 4\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_3\left(t\right)x_4\left(t\right)x_5\left(t\right){\kappa }_6& \left(\text{IM} \text{term} 6\right)\\ & +\sin \left({\theta }_2\right)\sin \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_3\left(t\right)x_4\left(t\right)x_6\left(t\right){\kappa }_5& \left(\text{IM} \text{term} 5\right)\\ & -\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\cos \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_4\left(t\right){\kappa }_{25}& \left(\text{IM} \text{term} 25\right)\\ & +\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\sin \left({\theta }_4\right)\sin \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_4\left(t\right)x_5\left(t\right)x_6\left(t\right){\kappa }_3& \left(\text{IM} \text{term} 3\right)\\ & -\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\sin \left({\theta }_5\right)\cos \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_5\left(t\right){\kappa }_{24}& \left(\text{IM} \text{term} 24\right)\\ & -\sin \left({\theta }_2\right)\cos \left({\theta }_3\right)\cos \left({\theta }_4\right)\cos \left({\theta }_5\right)\sin \left({\theta }_6\right)x_1\left(t\right)x_2\left(t\right)x_6\left(t\right){\kappa }_{23}& \left(\text{IM} \text{term} 23\right)\end{array}$13

Here, a scalar IM steering coefficient κ_j ∈ [0, 1] is introduced for each IM term (Vergara et al., 2013). The introduction of the IM steering coefficients κ_j with $j\in J_{\text{IM}}=\left\{1,\dots ,26\right\}$ allows the scaling of each IM term and therefore the shaping of the IM component and the whole signal constellation. This approach permits scalable interplex to modify the composite complex signal x(t) considering an HPA realization, thereby reducing the power of the IM component while simultaneously increasing the power of the useful signal components. The first of the two extreme cases is ${\kappa }_j=1 \forall j\in J_{\text{IM}}$, which results in the scalable interplex being equal to the standard interplex when strictly bipolar spreading waveforms are used with x_n(t) ∈ {−1, +1}. In this case, the signal x(t) has a constant envelope. In the case of non-bipolar, power-normalized spreading waveforms, Equation (11) with Equations (12) and (13) is an approximation of the standard interplex (Equation (10)). In this study, the utilization of the BPSK-RC(1, 0.22) waveform as x₆(t), which is not a bipolar waveform, thus indicates that the scalable interplex employed is an approximation of the standard interplex even if ${\kappa }_j=1 \forall j\in J_{\text{IM}}$. The latter of the two extreme cases is the complete omission of IM terms, where ${\kappa }_j=0 \forall j\in J_{\text{IM}}$. This scenario results in severe HPA distortion and is not a desired configuration for any realistic HPA characteristic. In the latter, for the purpose of achieving a more compact notation, all IM steering coefficients are aggregated into a single vector, designated as follows:

$\kappa ={\left[{\kappa }_1 {\kappa }_2\cdots {\kappa }_j\cdots {\kappa }_{26}\right]}^T\in {\left[0,1\right]}^{26\times 1}$14

The scalable interplex formulation (Equation (11)) with Equations (12) and (13) further allows one to chose a power distribution among the user signals by solving for the modulation angles θ₁, …, θ_N. A key metric for CEM methods is the multiplexing efficiency:

${\eta }_{\text{MUX}}=1-\frac{P_{\text{IM}}}{P^{\left(N\right)}}$15

which assesses the power of unusable IM component P_IM over the average total signal power of the composite signal x(t):

$P^{\left(N\right)}=\sum _{n=1}^N{P}_n^{\left(N\right)}+P_{\text{IM}}$16

where $P_n^{\left(N\right)}$ denotes the average power of the n-th signal component x_n(t). A multiplexing method with η_MUX = 1 allocates all signal power P^(N) only to useful signal terms and therefore does not contain any IM terms. The N-channel interplex has the worst performance with respect to the multiplexing efficiency η_MUX if the modulation angles are chosen in such a way that the power of each signal component is equal, resulting in a uniform power distribution. This scenario causes the highest possible IM power (Butman & Timor, 1972) and therefore the lowest multiplexing efficiency η_MUX. Thus, an unevenness in the power distribution of the signal components is advantageous and is also realized with the Galileo E1 signals (Rebeyrol et al., 2006).

In the context of Galileo E1 with an additional component implemented as a six-channel interplex, similar to the work conducted by Ortega et al. (2020), it is simple to implicitly set the power distribution of the user signals and the IM power by choosing the following power ratios:

$\alpha =\frac{P_1^{\left(6\right)}}{P_2^{\left(6\right)}+P_4^{\left(6\right)}}=\frac{P_1^{\left(6\right)}}{P_3^{\left(6\right)}+P_5^{\left(6\right)}}$17

and:

$\beta =\frac{P_1^{\left(6\right)}}{P_6^{\left(6\right)}}$18

between the primary and other components. The fixed power ratio is given as follows:

$10=\frac{P_2^{\left(6\right)}}{P_4^{\left(6\right)}}=\frac{P_3^{\left(6\right)}}{P_5^{\left(6\right)}}$19

and is set to correctly construct the CBOC components given in the Galileo interface control document (ICD).

Owing to the non-discrete range of values inherent to a BPSK-RC(1, 0.22) waveform, a fair comparison between scalable interplex and a potent CEM method such as CEMIC would prove to be challenging. To illustrate this point, a case study could be conducted to compare the performance of scalable interplex with BPSK-RC(1, 0.22) and CEMIC with a discrete derivative of BPSK-RC(1, 0.22). For example, an MCS with 16, 32, or 64 sub-chips could be used as a discrete derivative of BPSK-RC(1, 0.22). This prompts the following question: With which replica would a receiver correlate such a signal? Assuming the implementation of a BPSK-RC(1, 0.22) waveform for low-complexity receivers would be advantageous, as employing an identical replica would inevitably result in a correlation loss due to replica mismatch for CEMIC, rendering this analysis potentially unfair. Consequently, a comparison of QCEM and CEM is beyond the scope of this paper.

2.4 Transmitter

A model of a GNSS satellite transmitter $T\{.\}$ may comprise an HPA, OMUX, and antenna model (Beck et al., 2022a). To solely investigate the interplay between the constellation shaping of the multiplexer when applying scalable interplex and the distortion introduced by the HPA, the effect of band limitations on the transmitter side was neglected in this paper. Hence, the OMUX and antenna were assumed to have an ideal all-pass characteristic without phase distortion such that the transmitter model effectively comprises only an HPA characteristic.

As there is no publicly available information on the detailed shape of GNSS satellite amplifier characteristics, a generic memory-less HPA model was used, which assumes a perfect HPA pre-distortion. The used HPA model is a Saleh model with AM/AM and AM/PM characteristics:

$A\left(\left|x\left(t\right)\right|\right)=\frac{a_A\left|x\left(t\right)\right|}{1+b_A{\left|x\left(t\right)\right|}^2}$(20a)

$\vartheta \left(x\left(t\right)\right)=\text{atan}2\left(\text{Im}\left\{x\left(t\right)\right\},\mathrm{Re}\left\{x\left(t\right)\right\}\right)$(20b)

where the two parameters a_A and b_A = 1/|x₀|² denote the small signal gain and the input amplitude |x₀| at which the maximum output amplitude A₀ = a_A/(2|x₀|) is achieved (Saleh, 1981). The formation of the transmit signal is as follows:

$y\left(t\right)=T\left\{x\left(t\right)\right\}=A\left(\left|x\left(t\right)\right|\right)e^{i\vartheta \left(x\left(t\right)\right)}$21

corresponding to the application of AM/AM and AM/PM mappings onto the multiplexed input x(t).

2.5 Channel and Receiver Model

This paper considers only path loss in the propagation channel such that the satellite signal on the ground can be expressed as $\mathbb{P}\left\{y\left(t\right)\right\}=\gamma y\left(t-{\tau }_0\right)$, where $\gamma \in ℝ$ denotes the channel coefficient, which models the signal path loss, and ${\tau }_0\in ℝ$ denotes the propagation delay. The receiver applies only frequency down-conversion and front-end filtering with a brick-wall filter h_RX(t) such that the received signal can be expressed as $r\left(t\right)=h_{\text{RX}}\left(t\right)\ast D\left\{\gamma y\left(t-{\tau }_0\right)\right\}+n\left(t\right)$. The effect of quantization lies beyond the scope of this paper.

3.1 Performance Measures

This subsection presents the five performance metrics assessed in this study. The first metric is the joint receiver power efficiency, which serves as the objective of the scalable interplex optimization process. The latter four metrics are the civil signal power recovery, scalable interplex IM power ratio, joint receiver power efficiency gain, and TOA estimation error SD, which are used to evaluate the viability of scalable interplex in the chosen scenario.

3.2 Configuration of Signal Power Distribution and Amplifier Operating Region

This study examines two distinct six-channel interplex operating states, which will henceforth be designated as power modes. The power modes are distinguishedby differing values of the power ratio β, which represents the power ratio of the first to the sixth signal component. The modulation angles are essentially determined by the defined power ratios a and β among the user signals and their polarity with respect to each other. The selected power modes result in varying levels of baseline multiplexing efficiency η_MUX, as detailed in Table 1. Power ratios between signal components of a similar scale can be found for GPS (Thölert et al., 2018) and Galileo signals (Rebeyrol et al., 2006). In power mode I, it is assumed that the additional signal has the same power as the E1B and E1C signals that are already broadcast. In power mode II, the additional signal is assumed to have a power level equal to 1/3 of that of the PRS signal or 2/3 of that of the E1B signal. The power ratio α = 2 between PRS and each already present open service signal is as in the Galileo ICD and achieved by the Galileo E1 signal multiplex described by Rebeyrol et al. (2006).

Three scenarios are investigated for which the HPA was driven in either its saturation, transition, or linear region. To facilitate a fair comparison of ranging performances for different HPA operation regions, the small signal gain a_A and maximum output amplitude A₀ were chosen such that an operating point with an input power of 0 dBW (= 1 W) would result in the same HPA output power of P_TX = 15.5 dBW (≈ 35.5 W). This scenario was achieved by setting the maximum output amplitude A₀ such that a certain OBO is attained and then solving for the small signal gain a_A. Table 2 shows the applied parameters for the three realistic HPA models and an additional idealized HPA model. The fourth HPA is a theoretical example of an HPA with idealized amplification (i.e., linear amplification with a_A and no amplitude compression). This HPA characteristic is used as a reference, as any realistic Saleh-HPA model imposes an implicit average power constraint. An overly severe down-scaling of IM terms can deteriorate the joint receiver efficiency metric as correlation losses occur, owing to a mismatch in the received signal and signal replica. Nevertheless, an idealized HPA model permits the complete omission of IM terms, as amplitude compression does not occur. Consequently, the results yielded by this model indicate the upper or lower limits that can be achieved, depending on the metric in question. The corresponding AM/AM characteristics for each HPA model and the common operation point are shown in the amplitude domain in Figure 3.

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FIGURE 3

AM/AM mappings of applied HPA models

3.3 Optimization of Scalable Interplex IM Steering Vectors

Vergara and Antreich (2013) presented the scalable five-channel interplex as a two-dimensional optimization problem (OP), with the variables being the IM steering coefficients for all I-IM terms and Q-IM terms, which reduces the computational complexity but scales IM terms with less granularity. To ascertain the full potential of scalable interplex, this paper employs a refined approach and aims to optimize the IM steering coefficient of each IM term individually. The refined scalable six-channel interplex OP is 26-dimensional, as there exist a total of 26 IM steering coefficients, as shown in Equations (12) and (13). The OP can then be expressed as follows:

$\begin{array}{cc}\underset{\kappa }{\max }& {\eta }_{\text{eff}}\left(\kappa \right)\\ \text{s.t}.& \kappa \le 1\\ & \kappa \ge 0\end{array}$31

where 1 and 0 denote 26 × 1 all-one and all-zero vectors, respectively. However, because of the inherent high dimensionality and nonlinearity of the problem, a solution was reached via four distinct steps. The approach for identifying suitable IM steering vectors was based on an initial dimensional reduction of the 26-dimensional problem, which could be solved through a meaningful approach in an acceptable runtime. It should be noted, however, that this dimensional reduction is likely to yield a local optimum rather than the global optimum of the OP. A detailed analysis for deriving the global optimum is beyond the scope of this paper. The initial two steps were undertaken with the objective of optimizing the IM steering coefficients, with a distinction made between those transmitted in I-phase (i.e., IM terms 1, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16) and those transmitted in Q-phase (i.e., IM terms 2, 3, 4, 5, 6, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26). In the initial stage of the process, a two-dimensional brute-force search is conducted, employing a method of iterative grid shrinking. The search grid is a uniformly spaced 21 × 21 grid comprising 441 test points within the domain [0, 1] × [0, 1]. The brute-force search was implemented as a greedy search, whereby the optimal result was passed to a subsequent brute-force search that shrunk the search grid by a factor of 0.5. This process was repeated three times before proceeding to the subsequent step. In the second step, the best result was further optimized by using the Nelder–Mead simplex method (NMM). In the third step, a further distinction is made between high-frequency and low-frequency IM coefficients. The IM terms were designated as high-frequency terms if they contained the product of five signal terms (i.e., IM terms 1–6) or as low-frequency terms if they contained the product of three signal terms (i.e., IM terms 7–26).With the distinction of I-phase or Q-phase at the outset, a four-dimensional problem was formulated and solved using the NMM. In the final step, all IM terms were considered independently, thus necessitating a 26-dimensional optimization based on the optimization result of the four-dimensional problem. This procedure was performed for each HPA model and power mode combination.

The resulting power distributions, fractions of IM power spent compared with the standard interplex, and joint receiver efficiency gains for each power mode and HPA combination are reported in Table 3. In all combinations with HPA 1–3, neither the full use nor the full omission of IM terms is found to be optimal. Depending on the HPA model and power mode, the IM power ratio ν_opt for the scalable interplex ranges between 11.94% and 22.81%. The scalable interplex method exhibits the inherent capability to allocate the power distribution among the signal terms in a manner that is solely determined by the modulation angles θ₁,…, θ_N, irrespective of the chosen IM steering vector κ (Vergara & Antreich, 2013). Consequently, the power reduction of the IM component by scalable interplex, as observed in the studied cases, results in a redistribution of over 77.19% of the IM power to the useful signal components, in comparison to the standard interplex approach. The power share P_IM,opt/P⁽⁶⁾ of the optimized IM component reaches a maximum in cases where saturation-region HPA 1 is employed, as this HPA model features the lowest OBO, necessitating a greater IM component power allocation compared with the other HPA models with larger OBOs. When the idealized HPA model 4 is applied, the almost full omission of IM terms was found to be optimal, as the maximization of η_eff is implicitly constrained by the use of the HPA model and therefore its maximum output power. Indeed, using HPA 4 corresponds to neglecting an AM/AM deformation constraint.

To summarize, in the full IM case, the available transmit power remains unused, which can be exploited by the scalable interplex methodology. In the IM omission case, the signal components are transmitted with the maximum available power but are more severely affected by HPA distortions such that this configuration is sub-optimal. Intermediate levels of IM power use were found to be optimal. The optimal IM steering coefficient vector κ_opt depends on the HPA model and the interplex power mode.

4.1 Simulation Parameters

The total path loss is set to −174 dB, which incorporates the transmitting antenna gain, implementation losses, atmospheric losses, free space loss, and receiving antenna gain. Based on the target average output power of 15.5 dBW for the HPA models, the reception power amounts to −158.5 dBW, which is a typical value of reception power in GNSSs according to Joseph (2010). The noise density was set to N₀ = −204 dBW/Hz, which corresponds to a receiver noise temperature at room temperature (Joseph, 2010). The modeled observation time window and correlator spacing were T₀ = 20 ms and Δ = 9.7752 ns = T_c,1/40 = T_c,2/100 = T_c,3/100 = T_c,4/100 = T_c,5/100 = T_c,6/100, respectively. Furthermore, the receiver was modeled ideally, by applying no quantization and correlating the filtered satellite contribution s(t) with the replicas ${\overline{x}}_i\left(t\right)$, as indicated in Equation (7). The receiver front-end has a brick-wall characteristic:

$H_{\text{RX}}\left(f,B\right)=ℱ\left\{h_{\text{RX}}\left(t,B\right)\right\}=\{\begin{array}{cc}1& \left|f\right|<B/2\\ 1/2& \left|f\right|=B/2\\ 0& \text{else}\end{array}$32

where B denotes the two-sided receiver bandwidth and $ℱ\{.\}$ denotes the Fourier transform. The TOA estimation error SD results are presented for varying front-end bandwidths B.

4.2 Analysis of Civil Signal Power Recovery

In this first assessment, the scenario of adding one new signal component is compared with the status quo, where the Galileo signal can be generated via a five-channel interplex. Adding a new component would not only diminish the signal power of the legacy signals, owing to the fraction of power allocated to the new signal, but would also result in an over-proportional increase in power spent on IM terms. When transitioning from a five-channel interplex with eleven IM terms to a six-channel interplex with 26 IM terms, the multiplexing efficiency alone would decrease from 86.49% to a range of 72.07%–75.68% when the standard interplex is applied as the CEM method for the investigated power modes (see Table 1). The civil legacy signal power would be decreased in a range from 1.249 dB to 1.761 dB, depending on the chosen power mode when the standard interplex is applied with the assumption of a constant transmit power, as reported by the gain $\tilde{G}$ in the third column of Table 4.

Applying the scalable six-channel interplex instead would result in consistent improvements, as less power is spent on IM terms and, therefore, the power share among useful signal components increases. Compared with the current status quo, scalable interplex with six components would still lead to a decrease in signal power given by G_opt, as shown in the fourth column of Table 4. However, the decrease induced by adding the new component would be less severe and could improve the situation; moreover, this decrease may even enable the transmission of a new component on current payloads (if a certain loss of signal power is deemed to be acceptable and an HPA lifetime assessment is positive). For the realistic HPA scenarios (1–3), 1.002–1.275 dB of civil legacy signal power could be recovered when a scalable interplex is adopted rather than the standard interplex. The power recovery achieved by scalable interplex is greater for power modes in which a more even power distribution is chosen (i.e., power mode I with α = β = 2), as the baseline multiplexing efficiency is worse than that of more uneven power distributions.

4.3 Analysis of Ranging Performance

This subsection presents a discussion on whether adopting a scalable interplex instead of a standard interplex scheme directly yields an increase in ranging performance for each signal component. The ranging performance is evaluated by using the TOA estimation error SD for each HPA and power mode for varying receiver bandwidths B. Scalable interplex enhances the SNR portion in Equation (27), but employing QCEM instead of CEM will distort the shape of the CCF, as nonlinear effects introduced by the HPA affect the transmit signal. The results for the standard interplex will vary for each power mode, but remain the same for the considered HPA region of operation. The results of the scalable interplex vary for each power mode and HPA operation region. Figures 4 and 5 show the TOA estimation error SD with respect to the receiver bandwidths for all HPA operating regions and power modes I and II, respectively. Tables 5, 6, 7, and 8 report the relative performance changes in the TOA estimation error SD for five selected receiver bandwidths for the eight HPA and power mode combinations. The base value was the TOA estimation error SD of the standard interplex. A decrease in TOA estimation error SD corresponds to an improvement in ranging accuracy; conversely, an increase in TOA estimation error SD corresponds to a degradation in ranging performance.

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FIGURE 4

TOA estimation error SD for all HPA models and power mode I for varying receiver bandwidths

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FIGURE 5

TOA estimation error SD for all HPA models and power mode II for varying receiver bandwidths

Across all power modes, the results generally show that the TOA estimation error SD for BOC_cos-R(15, 2.5) has a decreasing trend for higher power modes while the values for the other components show an inreasing trend due to the different power allocation. Interestingly, the scalable interplex operation states appear to favor the first and last component in comparison to the other two components. BOC_cos-R(15, 2.5) experiences reductions in its TOA estimation error SD ranging from 9% to 18% compared with the standard interplex result. The BPSK-RC(1, 0.22) experiences a ranging performance increase in a range from 1% to 5%. However, the performance of BPSK-RC(1, 0.22) decreases for some HPA and power mode combinations.

The results for power mode I shown in Figure 4 demonstrate that BPSK-RC(1, 0.22) would benefit in all HPA scenarios. However, CBOCs would deteriorate in ranging performance when using HPA 1 as well as HPA 4 for CBOC(6, 1, 1/11,−). Moreover, for any other HPA model, the scalable interplex found operating states that improved the ranging performance. In Figure 5, we can observe that, for power mode II, BPSK-RC(1, 0.22) would experience a decrease in ranging performance only in case of HPA 4. The CBOCs would also deteriorate in ranging performance for HPA 4. The application of any other HPA model results in improved ranging performance for all signals when scalable interplex is applied instead of the standard interplex.