NavMuseum

Radio laboratory

The radio navigation laboratory

Up to four transmitters, one receiver, four ways to measure. Move anything, change what you measure, and watch the fix and its uncertainty respond.

Every radio navigation system turns a measurement into a curve on the chart and a fix into the place where curves cross. This laboratory keeps one world fixed while you change what you measure: a bearing gives a line, a range gives a circle, a time difference gives a hyperbola, and a phase difference gives a whole family of hyperbolas, only one of which is yours.

Figure 1 The laboratory. Drag the transmitters (brass) or the receiver P, switch the measurement, and press Take a measurement for a fresh draw of seeded noise. The sage ellipse is the 1σ uncertainty of the least-squares fix; the close-up shows it at full scale. Share copies a link to this exact setup.#

What to try

  • The same place, four answers. Leave P where it is and step through Bearing, Range, Time difference and Phase. The geometry stays put; only the shape of each constraint changes.
  • Break the geometry. In Time difference mode with three transmitters, drag P out past one transmitter along the line joining it to S₁, the baseline extension. The hyperbolas meet at a shallow angle and the ellipse stretches into a long sliver. Turn on the geometry map to see where that happens before you go there.
  • Two crossings. With two transmitters in Range mode, the circles meet twice. The solver picks the crossing nearer P, which stands in for a navigator's dead reckoning. Add a third transmitter and the ambiguity goes away.
  • Distance hurts bearings. In Bearing mode, a 1° error is a 1.7 km sideways error at 100 km and 5 km at 300 km. Move P far from every transmitter and watch the ellipse grow even though the noise setting has not changed.
  • Slip a lane. In Phase mode, raise the timing noise, which here sets how good the coarse lane-identification measurement is, until the readout reports a slip. The fix jumps by a whole lane: precise, confident, and wrong.
  • Redundancy. Add a fourth transmitter in any mode. With more measurements than unknowns, the least-squares fix no longer passes exactly through every curve, and the ellipse shrinks.

The laboratory uses a flat map, signals traveling in straight lines at the speed of light, perfectly synchronized transmitters and independent Gaussian errors. Real systems had to deal with Earth's curvature, ground-wave delays over land, skywave contamination at night, and correlated errors. The chapter The invisible lighthouse tells that story; the time-difference mode here is the idealized core of LORAN, and the phase mode is the idealized core of Decca.