111 lines
4.0 KiB
Markdown
111 lines
4.0 KiB
Markdown
---
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title: Doppler shift and range rate
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tags: doppler, frequency-offset, clock, ranging, tracking
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summary: How the motion of a satellite changes the frequency it appears to be received at, and how that shift is turned into a position.
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---
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# Doppler shift and range rate
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A satellite is observed at a frequency that differs from the one it
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transmits, because it is moving. That difference is the principal observable
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in a tracking system, and turning it into a position is the whole of
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ranging.
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## Definition
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**Doppler shift** is the change in observed frequency caused by relative
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motion along the line of sight. **Range rate** is the rate of change of
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range, and it is the same physical quantity: one is a frequency, the other a
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velocity.
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For a satellite moving at $v_r$ radially, at range $R$, the received
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frequency shift is
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$$
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\frac{\Delta f}{f} = -\frac{v_r}{c}
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$$
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approaching raises the frequency. The non-relativistic form is adequate for
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orbital speeds; the relativistic correction is about one part in $10^{13}$ and
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matters only for a clock discipline experiment.
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## Mathematics
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### The two-way shift
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A signal that travels to the satellite and back is shifted twice, so the
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observed fractional shift is
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$$
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\frac{\Delta f}{f} = -\frac{2 v_r}{c}
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$$
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with $v_r$ the range rate. The factor of two is why the same satellite
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appears to shift twice as much when tracked in two-way mode, and a range
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model that ignores it has a factor-of-two error that is easy to miss.
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### Range from accumulated shift
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Integrating the two-way shift gives the range,
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$$
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R(t) = R_0 - \int \frac{2 c \, v_r}{f_0} \, dt
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$$
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which is why a tracking system measures range as the integral of a
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frequency rather than as a timing. The accumulator drifts with any error in
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the reference frequency, which is why the reference clock is the limiting
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element in the whole chain.
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### Range rate from the carrier
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Alternatively the carrier phase is differenced directly, giving range rate
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without an accumulator. This is the common approach and it is noisy, because
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phase noise differentiates into range rate noise amplified by the carrier
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frequency. The trade is a clean velocity against a degraded one.
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### Clock offset
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Separately from the motion, the transmitted frequency may not equal the
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nominal one. A **frequency offset** between the spacecraft clock and the
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reference is indistinguishable from a range rate over a short arc, because
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both move the phase. Separating them needs a second observable, typically a
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range measurement from a different station, or an upload that is
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self-referenced.
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## Unit conventions
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- Frequencies in hertz; a shift is often expressed in parts per billion.
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- Range rate in metres per second or kilometres per second.
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- Range in kilometres, with the carrier frequency in hertz, so the factor
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$2c/f$ carries its units.
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- The factor of two applies to two-way; one-way is the single case.
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## Applicability
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Doppler is insensitive to noise that is common to the uplink and the
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downlink, which is why it is the observable of choice in a jamming
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environment: a jammer must be coherent to spoof it.
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The shift is large enough to measure and small enough to alias. If the
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estimate of range rate is off by more than half a carrier cycle per sample
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interval, the tracking wraps and the recovered position is wrong by an
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arbitrary amount. This is a real failure mode in practice and it is why the
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sample rate is bounded relative to the dynamics.
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A one-way system needs a stable reference on the spacecraft; a two-way system
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needs a stable reference on the ground. Either way the clock, not the
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antenna, is usually the limit.
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## Standards
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- Range and Doppler conventions: the tracking and data link messages
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specified by the CCSDS Space Link Protocol
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- Reference oscillator discipline: the same documents, in the time and
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frequency parts
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## See also
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- [[link-budget]] for whether there is enough signal to measure at all
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- [[interference]] for what a coherent jammer does to this measurement
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- [[tt-and-c]] for the full measurement chain |