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title	tags	summary
Doppler shift and range rate	doppler, frequency-offset, clock, ranging, tracking	How the motion of a satellite changes the frequency it appears to be received at, and how that shift is turned into a position.

Doppler shift and range rate

A satellite is observed at a frequency that differs from the one it transmits, because it is moving. That difference is the principal observable in a tracking system, and turning it into a position is the whole of ranging.

Definition

Doppler shift is the change in observed frequency caused by relative motion along the line of sight. Range rate is the rate of change of range, and it is the same physical quantity: one is a frequency, the other a velocity.

For a satellite moving at v_r radially, at range R, the received frequency shift is


\frac{\Delta f}{f} = -\frac{v_r}{c}

approaching raises the frequency. The non-relativistic form is adequate for orbital speeds; the relativistic correction is about one part in 10^{13} and matters only for a clock discipline experiment.

Mathematics

The two-way shift

A signal that travels to the satellite and back is shifted twice, so the observed fractional shift is


\frac{\Delta f}{f} = -\frac{2 v_r}{c}

with v_r the range rate. The factor of two is why the same satellite appears to shift twice as much when tracked in two-way mode, and a range model that ignores it has a factor-of-two error that is easy to miss.

Range from accumulated shift

Integrating the two-way shift gives the range,


R(t) = R_0 - \int \frac{2 c \, v_r}{f_0} \, dt

which is why a tracking system measures range as the integral of a frequency rather than as a timing. The accumulator drifts with any error in the reference frequency, which is why the reference clock is the limiting element in the whole chain.

Range rate from the carrier

Alternatively the carrier phase is differenced directly, giving range rate without an accumulator. This is the common approach and it is noisy, because phase noise differentiates into range rate noise amplified by the carrier frequency. The trade is a clean velocity against a degraded one.

Clock offset

Separately from the motion, the transmitted frequency may not equal the nominal one. A frequency offset between the spacecraft clock and the reference is indistinguishable from a range rate over a short arc, because both move the phase. Separating them needs a second observable, typically a range measurement from a different station, or an upload that is self-referenced.

Unit conventions

Frequencies in hertz; a shift is often expressed in parts per billion.
Range rate in metres per second or kilometres per second.
Range in kilometres, with the carrier frequency in hertz, so the factor 2c/f carries its units.
The factor of two applies to two-way; one-way is the single case.

Applicability

Doppler is insensitive to noise that is common to the uplink and the downlink, which is why it is the observable of choice in a jamming environment: a jammer must be coherent to spoof it.

The shift is large enough to measure and small enough to alias. If the estimate of range rate is off by more than half a carrier cycle per sample interval, the tracking wraps and the recovered position is wrong by an arbitrary amount. This is a real failure mode in practice and it is why the sample rate is bounded relative to the dynamics.

A one-way system needs a stable reference on the spacecraft; a two-way system needs a stable reference on the ground. Either way the clock, not the antenna, is usually the limit.

Standards

Range and Doppler conventions: the tracking and data link messages specified by the CCSDS Space Link Protocol
Reference oscillator discipline: the same documents, in the time and frequency parts

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