Mathematical derivation of non-linear Doppler positioning equations using multi-satellite passes from commercial mega-constellations.
Medium Earth Orbit (MEO) GPS satellites orbit at 20,200 kilometers, appearing to move slowly across the sky over 12 hours. Consequently, their Doppler shift is small (±4 kHz) and changes at a sluggish fraction of a hertz per second.
In contrast, Low Earth Orbit (LEO) satellites at 500 kilometers orbit Earth every 90 minutes. A receiver observes Doppler shifts exceeding ±50 kHz with instantaneous frequency rates-of-change (df/dt) reaching up to 60 Hz/s. By measuring this steep S-curve over just 30 seconds of an overhead pass, LEOGeolocation pinpoints the receiver's location on Earth with remarkable precision. ``` +-----------------------+ Ku-Band Downlink +-----------------------+ | LEO Satellite #1 | --------------------------------> | LEOGeolocation SDR | | Speed: 7.5 km/s | df/dt = -45.2 Hz/sec | Carrier Tracking Loop | +-----------------------+ +-----------------------+ | v Kalman Filter Solver Centimeter 3D Coordinate ```
Because Doppler positioning relies on the physical carrier frequency shift rather than the data payload, LEOGeolocation functions even with encrypted commercial communication signals, creating a completely independent alternative to GPS.