Low Earth orbit (LEO) navigationEdit on GitHubSource: docs/LEO-SIGNAL.md

LEO-PNT signal designs and the multi-band spectrum

The leo-signal scenario kind analyses low Earth orbit (LEO) positioning, navigation and timing (PNT) signals. It is not tied to one system. A signal is a parameterised design (data, not code), so the same kind analyses a published commercial signal, a representative design for a band nobody has published, or any design a scenario writes inline. The spectrum kind draws those designs too, in any band from ultra high frequency (UHF) to C, each band with its own jammers, and a leo-pass band or a leo-pnt-chain run can carry any of them (see Where the designs are used).

Run the bundled examples:

  • kshana example leo-band-trade: a UHF-to-wide-C band trade on representative designs;
  • kshana example xona-pulsar-signals: the published Xona Pulsar X1 and X5 signals;
  • kshana example multi-band-jamming-waterfall: UHF, L, S and C panels under per-band jammers (a spectrum scenario);
  • kshana scenarios/celeste-iod-classical-pilot-signals.toml, from a checkout of the repository (not bundled): the Celeste In-Orbit Demonstration (IOD) bands and signal configuration presented at the European Space Agency (ESA) Navigation Innovation and Support Programme (NAVISP) LEO-PNT workshop, 2026 (see Workshop parameters below).

Code: src/leo_signal.rs (signal designs, the kind), src/navsignal.rs (the closed forms), src/spectrum.rs (the multi-band waterfall), preset files under data/leo-signals/.

A signal design#

Part What it holds
Band centre frequency, transmit bandwidth (the spectrum is zero outside it), and the International Telecommunication Union (ITU) allocation: Radio Navigation Satellite Service (RNSS), Radio Determination Satellite Service (RDSS), Mobile Satellite Service (MSS), Earth Exploration-Satellite Service (EESS), a non-RNSS band, or other
Components any subset of an acquisition component (short and low rate, easy to find), a data component (the navigation message) and a pilot component (data-free, for ranging)
Per component modulation: binary phase-shift keying BPSK(n) at n × 1.023 Mchip/s (n may be a fraction, BPSK(1/3) is 341 kchip/s), binary offset carrier BOC(m,n), multiplexed BOC, or FLAT(<MHz>) for an orthogonal frequency-division multiplexing (OFDM) beacon; share of the power; frequency-division multiple access (FDMA) sub-carrier offsets; code length; data rate
Optional reference and maximum received power, orbit altitude, ranging = false for a Doppler-only signal, notes on what is published and what is assumed

A scenario takes signals from presets = [...] and from inline [[signals]] tables. Without either it analyses the generic-bands preset.

Presets#

One file each under data/leo-signals/, compiled in, each citing its source URL. A value the source does not publish is labelled REPRESENTATIVE in the file and in the report.

Preset Signals Source Stated approximations
xona-pulsar xona-x1 (1593.3225 MHz, 1.023 Mchip/s), xona-x5 (1190.51625 MHz, 10.23 Mchip/s); minimum and maximum received power; 1080 km PUBLIC, arXiv 2509.19551 enhanced Feher quadrature phase-shift keying (EFQPSK) drawn with the rectangular-chip BPSK envelope; code shift keying and overlay codes not modelled; transmit bandwidth = main lobe (not published)
iridium-stl iridium-stl Satellite Time and Location (STL) bursts, quadrature phase-shift keying (QPSK) at 25 ksymbol/s, 780 km, Doppler and timing only PUBLIC, Resilient Navigation and Timing Foundation (RNTF) report carrier at the middle of 1616 to 1626 MHz; received power derived from the published 300 to 2400 times the Global Positioning System (GPS)
starlink-soo starlink-ku-beacon, a 240 MHz OFDM beacon, Doppler only PUBLIC, NAVIGATION 72(1) flat spectrum; channel centre and shell altitude REPRESENTATIVE; the published carrier-to-noise-density ratio (C/N₀) of about 57 dB-Hz is set through the receiver
centispace centispace-l1, centispace-l5, BPSK at 2.046 Mchip/s PUBLIC, PubMed Central PMC10301026 carriers placed at the GPS L1 and L5 carriers ("near" in the source); code length REPRESENTATIVE
generic-c-band generic-c-band-leo, 5020 MHz, BPSK(10) in 20 MHz REPRESENTATIVE, for C-band systems whose parameters are not public (such as TrustPoint) the whole design
generic-bands generic-uhf, generic-l, generic-s, generic-c, generic-c-wide REPRESENTATIVE, allocations from the ITU Radio Regulations; the 465 MHz UHF carrier from the public openRECEIVER survey of LEO-PNT signals the whole design (the UHF bandwidth and modulation too)

The ATOMIC "zero-clock" polynomial ephemeris is a navigation-message model, not a signal design, and lives with the navigation-message work, not here.

What is computed per signal#

  • Band-limited power spectral density (PSD) and, per component, the fraction of its power the transmit band passes. For a BPSK component at the centre the closed form η = (2/π)[Si(πBT_c) − sin²(πBT_c/2)/(πBT_c/2)] sits beside the numeric value (90.3 % for the main lobe).
  • Root-mean-square (RMS, Gabor) bandwidth of the tracked component inside the band, with the BPSK closed form β² = [B/2 − sin(πBT_c)/(2πT_c)] / (π² T_c η).
  • Code-tracking thermal-noise jitter of a delay lock loop (DLL) against C/N₀ and early-late correlator spacing, from the band-limited early-late formula of Betz and Kolodziejski (2009): σ² = B_L(1 − ½B_L T) ∫ G sin²(πfΔ) df / [(2π)² (C/N₀) (∫ f G sin(πfΔ) df)²], coherent by default, with the non-coherent squaring loss on request. The report adds the unlimited-band textbook form (Kaplan and Hegarty) and the vanishing-spacing bound set by the Gabor bandwidth. The ranging accuracy is that jitter in metres at the reference C/N₀ and spacing.
  • Acquisition: the largest satellite Doppler on an overhead pass at the elevation mask (f v R_E cos(el) / ((R_E + h) c), circular orbit, no Earth rotation) plus the oscillator and user terms; Doppler bins of 2/(3T); code bins over one period; the detection probability of a square-law detector (generalised Marcum Q) at a per-cell false-alarm probability, centred and at the worst-case straddle; and the mean acquisition time of a serial and a code-parallel single-dwell search (Holmes).
  • Compatibility: the spectral separation coefficient (SSC) of the band-limited signal into GPS L1 coarse/acquisition (C/A), Galileo E1, GPS L5, Galileo E5a, E5b and the E5 alternative BOC (AltBOC) signal over each global navigation satellite system (GNSS) receiver band, the C/N₀ loss it causes at its received power, and the SSC and C/N₀ loss of each GNSS signal into the LEO tracked component.
  • Jammer tolerance: the jammer-to-signal power ratio (J/S) of a continuous-wave (CW) tone at the carrier, flat noise over the transmit band and noise matched to the tracked component that brings the tracked component to the tracking threshold, from the spectrum kind's own SSC code (spectrum::Jammer::ssc).

C/N₀ convention: the sweep and the reference are the total in-band C/N₀. A component's share enters the tracking and jammer formulas as 10 log₁₀(s/η) (its share of the unfiltered power over the in-band fraction), the same convention the spectrum kind uses for a plain band, whose modulation's power outside the receiver band is lost the same way.

The band trade#

[trade] puts every signal beside a reference signal: first-order ionospheric group delay 40.3·TEC/f² for a slant total electron content (TEC) you give (1 TEC unit, TECU, = 10¹⁶ electrons/m²), free-space path loss (FSPL) at a slant range, ranging accuracy at equal C/N₀ and at equal effective isotropic radiated power (EIRP) (C/N₀ minus the free-space-loss difference, isotropic antennas at both ends), and the jammer tolerance at the equal-EIRP C/N₀.

scenarios/leo-band-trade.toml (50 TECU, 1000 km, 45 dB-Hz, 0.5-chip coherent spacing, 1 Hz loop, 20 ms):

Signal Band Iono delay ΔFSPL vs L Ranging, equal C/N₀ C/N₀, equal EIRP Ranging, equal EIRP CW J/S tolerance
generic-uhf 465 MHz, BPSK(5) 93.19 m (×6.57) −8.17 dB 0.221 m 53.17 dB-Hz 0.086 m 42.1 dB
generic-l 1191.795 MHz, BPSK(10) 14.19 m 0 0.111 m 45.00 dB-Hz 0.111 m 45.0 dB
generic-s 2492.028 MHz, BPSK(5) 3.24 m (×0.229) +6.41 dB 0.185 m 38.59 dB-Hz 0.388 m 41.7 dB
generic-c 5020 MHz, BPSK(10) 0.80 m (×0.056) +12.49 dB 0.111 m 32.51 dB-Hz 0.465 m 43.4 dB
generic-c-wide 5100 MHz, BPSK(50) 0.77 m (×0.055) +12.63 dB 0.022 m 32.37 dB-Hz 0.093 m 50.3 dB

The trade reads both ways. At equal C/N₀, ranging accuracy is set by the chip rate and the band (wide C ranges five times finer than L). At equal radiated power the free-space loss charges the higher bands in full: C band loses 12.5 dB of C/N₀ and ranges four times worse than L, UHF gains 8.2 dB, while carrying 6.6 times the ionospheric delay. Wide C buys its accuracy back with bandwidth. These rest on the representative designs and on isotropic antennas; a directional user antenna at C band changes the equal-EIRP columns.

Xona Pulsar#

scenarios/xona-pulsar-signals.toml at the published minimum received powers against a 290 K antenna and a 2 dB noise figure (floor −201.98 dBW/Hz):

X1 X5
Reference C/N₀ 53.78 dB-Hz 57.08 dB-Hz
Ranging accuracy (0.5 chip, coherent, 1 Hz) 0.402 m 0.028 m
Maximum satellite Doppler, 1080 km 33.2 kHz 24.8 kHz
Doppler × code bins (1 ms, 0.5 chip) 104 × 2046 78 × 20 460
Mean code-parallel acquisition time 0.15 s 0.37 s

X5 sits at 1190.51625 MHz, inside the Galileo E5 band: its SSC is −91.2 dB/Hz into Galileo E5a and GPS L5, −103.2 dB/Hz into E5b and −86.2 dB/Hz into the AltBOC signal, and one X5 satellite at its minimum power raises a GNSS receiver's noise density by 0.0001 to 0.005 dB. The X1 Doppler, 33.2 kHz, is inside the 32 to 34 kHz the paper reports (the VALIDATED Doppler row below).

The multi-band spectrum#

The spectrum kind's [[bands]] take, besides the five named GNSS bands (gps-l1ca, galileo-e1, gps-l2c, gps-l5, galileo-e5a):

  • signal = "<preset signal>": a preset design, drawn with every component, band-limited to its transmit bandwidth, received at the preset's reference power or signal_power_dbw. C/N₀ and J/S refer to the tracked component (tracked_power_dbw in the report);
  • centre_mhz with modulation and signal_power_dbw: any custom band.

[[panels]] add waterfalls over other frequency ranges on the same timeline and colour scale, and a wideband (barrage) jammer joins CW, narrowband, chirp and matched noise. A plain band behaves exactly as before: the bundled l-band-waterfall-jamming example still gives 43.48 dB-Hz nominal and 3.23 dB-Hz minimum for L1 C/A (pinned in tests/leo_signal_reference.rs).

scenarios/multi-band-jamming-waterfall.toml (60 s, one row per second):

Band Nominal C/N₀ After its own jammer Jammer
generic-uhf (465 MHz) 49.41 dB-Hz 34.38 dB-Hz from 10 s CW at the carrier, −120 dBW
gps-l1ca 43.48 dB-Hz 3.38 dB-Hz from 20 s (lost) 16 MHz chirp, 50 mW at 100 m
galileo-e5a 46.98 dB-Hz 38.76 dB-Hz from 30 s 40 MHz barrage at 1185 MHz, −118 dBW
xona-x5 54.51 dB-Hz 46.29 dB-Hz from 30 s the same barrage
generic-s (2492.028 MHz) 49.23 dB-Hz 41.69 dB-Hz from 40 s 2 MHz noise, −128 dBW
generic-c-band-leo (5020 MHz) 47.41 dB-Hz 40.35 dB-Hz from 50 s matched noise, −124 dBW

Each jammer takes only its own band; the test holds every other band at its nominal C/N₀ until its own jammer starts.

Where the designs are used#

A design is looked up by name wherever a band can name one:

  • a spectrum band (signal = "<design>", above): multi-band-jamming-waterfall, leo-resilience-js-margin and the leo-resilience-multiband-diversity and leo-resilience-gnss-jammed-leo-carries campaigns;
  • a leo-pass band (signal = "<design>"): the band takes the design's centre frequency (unless frequency_mhz is given), transmit bandwidth and tracked chip rate, splits its EIRP across the components by their power shares, and reports the tracked component's C/N₀ and band-limited code-tracking jitter at every epoch (LEO-PASS.md);
  • the leo-pnt-chain kind, which hands a design's power split and chip rate to the pass and its carrier and chip rate to the fused fix (leo-pnt-end-to-end, xona-pulsar-end-to-end; LEO-PNT.md).

Workshop parameters#

scenarios/celeste-iod-classical-pilot-signals.toml holds the Celeste IOD frequency bands and the "Classical Pilot" signal configuration #1 as presented at the ESA NAVISP LEO-PNT workshop, 2026, with a shape check of one design against the measured spectrum shown there. It is one of the Celeste IOD preset's files (the others are src/celeste_iod.rs and the other scenarios/*celeste-iod*.toml); no Celeste signal specification is public. Its components carry their assumptions (power splits, code lengths, FDMA offsets, and a representative modulation for the bands whose modulation was not presented). The shape check is MODELLED consistency: it compares shapes, not calibrated levels.

It is withheld with the rest of the preset: delete src/celeste_iod.rs and the scenarios/*celeste-iod*.toml files. Nothing compiles the scenario in: the command-line interface lists it as a repository-only scenario (kshana example names it and explains, rather than printing it), and a test refuses any include_str! of it. Every other scenario, test and oracle runs without it; the test that reads it passes with a note when it is absent, and the README's scenario-file count still counts it.

Evidence#

Claim Label Oracle
BPSK power in band (90.3 % in the main lobe); band-limited early-late jitter reducing to the textbook coherent and non-coherent forms and to its Gabor bound; BPSK self-SSC 2/(3 R_c) VALIDATED Kaplan & Hegarty, Betz & Kolodziejski 2009, Betz 2001, Abramowitz & Stegun Table 5.1
Band-limited Gabor bandwidth closed form; offset BPSK SSC at non-zero offset; AltBOC unit area (all within the row above, as cross-checks) internal consistency derived here, checked against quadrature
Maximum LEO Doppler VALIDATED Xona Pulsar X1 32 to 34 kHz (arXiv 2509.19551); Iridium ±36 kHz (RNTF)
leo-signal designs, acquisition, compatibility, jammer tolerance, trade, shape checks MODELLED the validated closed forms, the detector identity P_d(0) = P_fa, Holmes's mean time, the spectrum kind's own SSC chain
Multi-band spectrum MODELLED reduction to the unchanged spectrum and jamming chains

Not modelled#

  • The exact spectra of EFQPSK, code shift keying and OFDM.
  • Spreading-code line structure and multiple-access cross-correlation.
  • Transmit and receive filters beyond an ideal brick wall: no out-of-band emission.
  • Multipath, quantisation and automatic gain control.
  • Earth rotation in the Doppler bound (up to about 0.46 km/s of range rate).
  • Ionospheric delay beyond first order, and the split of electron content above and below a LEO satellite: the slant TEC is an input.
  • Atmospheric, rain and polarisation losses.
  • Multi-dwell acquisition logic.
  • Aggregate interference beyond a per-satellite count.

References#