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Doppler Lab

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Doppler effect simulation

Doppler lab

The frequency a ground station receives changes as the satellite approaches and recedes. Pick a satellite, a carrier and one of its upcoming passes.

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Satellite
MHz

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Hand calculation

Doppler from a known radial velocity, for checking by hand

km/s
MHz
Doppler shift

-10.204 kHz

Received (first order)

436.989796 MHz

Received (relativistic)

436.989796 MHz

Difference 119.12 mHz

Equations used

Range rate (radial velocity)
ρ˙=ρ⃗⋅v⃗ECEF∣ρ⃗∣,ρ⃗=r⃗sat−r⃗station\dot\rho = \dfrac{\vec\rho\cdot\vec v_{ECEF}}{|\vec\rho|},\qquad \vec\rho = \vec r_{sat} - \vec r_{station}

Positive while the satellite moves away from the station.

First-order Doppler shift
Δf=−ft ρ˙c,fr=ft+Δf\Delta f = -f_t\,\dfrac{\dot\rho}{c},\qquad f_r = f_t + \Delta f
Relativistic longitudinal Doppler
fr=ft1−β1+β,β=ρ˙cf_r = f_t\sqrt{\dfrac{1-\beta}{1+\beta}},\qquad \beta = \dfrac{\dot\rho}{c}

Differs from the first-order value by about f (v/c)^2 / 2, roughly 0.1 Hz at UHF: far below receiver bandwidths.

Uplink pre-compensation
ftx=f0−Δf(f0)f_{tx} = f_0 - \Delta f(f_0)

The station transmits off-frequency so the satellite receives f0.

Doppler rate
d(Δf)dt=−ftc ρ¨\dfrac{d(\Delta f)}{dt} = -\dfrac{f_t}{c}\,\ddot\rho

Largest at closest approach, where the shift crosses zero.