Frequency design of LEO-based navigation augmentation signals for dual-band ionospheric-free ambiguity resolution

Abstract

Due to the spectrum congestion of current navigation signals in the L-band, it is difficult to apply for another two proper frequencies in this band for future low earth orbit (LEO)-based navigation augmentation systems. A feasible frequency scheme of using the combined frequencies in the L, S and C bands is proposed. A high-efficiency modulation scheme, termed continuous phase modulation, is adopted to make full use of the very limited spectrums and satisfy the radio frequency compatibility with the existing navigation systems, radio astronomy, and microwave landing systems. The high propagation loss in the S and C bands is absent for LEO, as the power margin owing to the short-distance propagation has compensated the frequency-dependent attenuation. Besides, for high-precision positioning, we consider the specific integer ratios between frequencies and propose a strategy for LEO precise point positioning (PPP) ambiguity resolution (AR) by directly fixing the L + S or L + C dual-band ionospheric-free (IF) ambiguity. Based on the simulated data, the quality of fractional cycle biases (FCBs) and the performance of PPP AR are analyzed. After removing the FCBs, 100.0, 99.7 and 71.7% of the fractional parts are within ± 0.15 cycles for GPS narrow-lane, LEO L + S dual-band IF and LEO L + C dual-band IF float ambiguities. At user stations, the convergence time of GPS PPP in static mode can be significantly shortened from 17.9 to within 2.5 min with the augmentation of 5.44 LEO satellites. Furthermore, compared with ambiguity-float solutions, the positioning accuracy of GPS AR + LEO AR solutions in east, north and up components is improved from 0.008, 0.008 and 0.027 m to 0.002, 0.003 and 0.011 m for 10-min sessions, respectively, and the fixing rate after time to first fix is almost 100%.

Appendices

Appendix

PSD expressions of CPM signals

The autocorrelation function of a CPM signal can be expressed as:

$$\Re \left( \tau \right) = \frac{1}{T}\int_{0}^{T} {\prod\limits_{{i = \left\lceil {1 - L} \right\rceil }}^{{\left\lfloor {{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}} \right\rfloor + 1}} {\frac{1}{M} \cdot \frac{{\sin \left\{ {2\pi hM \cdot \left[ {q\left( {t + \tau - iT} \right) - q\left( {t - iT} \right)} \right]} \right\}}}{{\sin \left\{ {2\pi h \cdot \left[ {q\left( {t + \tau - iT} \right) - q\left( {t - iT} \right)} \right]} \right\}}}dt} }$$

(24)

where $T$ is the symbol duration, and $\tau$ is the correlation time. $L$ is the pulse length. $M$ is the modulation order indicating that the data are $M$-ary symbols. $h$ is the modulation index; only if $h$ > 1, spectrum splitting can appear, and the larger the index is, the farther the distance between two main lobes, otherwise, the power spectra has only one main lobe. Note that though a longer $L$ and a bigger $M$ can effectively decrease the amplitude of side lobes, sometimes the feature of spectrum splitting may lose even if $h$ > 1. $q\left( t \right)$ is the phase response function depends on the shape of the corresponding frequency pulse, for a rectangular pulse, we have

$$q\left( t \right) = \left\{ \begin{gathered} 0,t \le 0 \hfill \\ \frac{t}{2LT},0 < t \le LT \hfill \\ \frac{1}{2},t > LT \hfill \\ \end{gathered} \right.$$

(25)

while for a raised-cosine pulse, we have

$$q\left( t \right) = \left\{ \begin{gathered} 0,t \le 0 \hfill \\ \frac{t}{2LT} - \frac{1}{4\pi }\sin \left( {\frac{2\pi t}{{LT}}} \right),0 < t \le LT \hfill \\ \frac{1}{2},t > LT \hfill \\ \end{gathered} \right.$$

(26)

where $t$ is the time. Due to the smoother waveform, the raised-cosine pulse contributes to a stronger spectrum roll-off in side lobes than the rectangular one. Then, the PSD of a CPM signal derived from Fourier transformation of $\Re \left( \tau \right)$ is written as:

$$\begin{gathered} G_{{{\text{CPM}}}} \left( f \right) = 2\left[ {\int_{0}^{{\left( {1 - s} \right)T}} {\Re \left( \tau \right)\cos \left( {2\pi f\tau } \right)d\tau } } \right. \hfill \\ \quad \quad \quad \quad + \frac{{1 - \psi \left( {jh} \right)\cos \left( {2\pi fT} \right)}}{{1 + \psi^{2} \left( {jh} \right) - 2\psi \left( {jh} \right)\cos \left( {2\pi fT} \right)}}\int_{{\left( {1 - s} \right)T}}^{{\left( {2 - s} \right)T}} {\Re \left( \tau \right)\cos \left( {2\pi f\tau } \right)d\tau } \hfill \\ \quad \quad \quad \quad - \frac{{\psi \left( {jh} \right)\sin \left( {2\pi fT} \right)}}{{1 + \psi^{2} \left( {jh} \right) - 2\psi \left( {jh} \right)\cos \left( {2\pi fT} \right)}}\left. {\int_{{\left( {1 - s} \right)T}}^{{\left( {2 - s} \right)T}} {\Re \left( \tau \right)\sin \left( {2\pi f\tau } \right)d\tau } } \right] \hfill \\ \end{gathered}$$

(27)

with

$$\psi \left( {jh} \right) = {{\sin \left( {M\pi h} \right)} \mathord{\left/ {\vphantom {{\sin \left( {M\pi h} \right)} {\left[ {M\sin \left( {\pi h} \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {M\sin \left( {\pi h} \right)} \right]}}$$

(28)

$$s = \left\lfloor {1 - L} \right\rfloor$$

(29)

where $f$ is the frequency. The parameters $T$, $M$, $L$, $h$ and $q\left( t \right)$ codetermine the spectral characteristics, and the specific configurations for proposed CPM signals are given in Table

Table 6 Specific configurations for proposed CPM signals

6.