Exercises — GNSS — GPS, GLONASS, Galileo, BeiDou

• ${\rho }_i$ = pseudorange to satellite $i$: the distance you think you measured, using your imperfect clock.
• $r_i$ = the true geometric distance (straight line) from you to satellite $i$.
• $cb$ = the clock-bias distance: your clock error $b$ (seconds) turned into metres by multiplying by $c$.
• ${\epsilon }_i$ = leftover errors (atmosphere bending, receiver noise).

Four. The unknowns are your three coordinates $x,y,z$ and your clock bias $b$ — that is 4 unknowns. Each satellite gives one equation, so you need at least 4 equations ⇒ 4 satellites.

CDMA (Code Division Multiple Access). Every satellite shares the one frequency but carries a unique pseudo-random code; the receiver correlates the incoming signal against each known code to pick out one satellite at a time.

Distance = speed × time: $\rho =ct=(3\times 10^8)(70\times 10^{-3})=2.1\times 10^7\text{m}=21,000\text{km}.$ Why: $\rho =ct$ is just "how far light goes in that time." $21,000$ km sits inside the MEO band (~19,000–24,000 km, see Orbital Mechanics — MEO/GEO/IGSO), so the sanity check passes.

$cb=(3\times 10^8)(2\times 10^{-6})=600\text{m}.$ Why: the bias term in the observation equation is literally $cb$. A single microsecond is $300$ m; two of them, $600$ m. This is why timing — not geometry — dominates raw GNSS error, and why $b$ must be solved for, never assumed zero.

With $b=0$ the pseudoranges are true distances, so you are on both circles: $x^2+(y-20000{)}^2=25000^2,(x-30000{)}^2+(y-20000{)}^2=25000^2.$ Subtract the two equations — the $(y-20000{)}^2$ terms cancel: $x^2-(x-30000{)}^2=0\Rightarrow 60000x-30000^2=0\Rightarrow x=15000\text{km}.$ Put $x=15000$ back into circle 1: $(15000{)}^2+(y-20000{)}^2=25000^2\Rightarrow (y-20000{)}^2=25000^2-15000^2=20000^2.$ So $y-20000=\pm 20000\Rightarrow y=40000$ km or $y=0$ km. Look at the figure: two circles meet at two points. On Earth you discard the physically impossible one (e.g. the point above the satellites, or off the planet). This is the 2D shadow of why one extra satellite is needed to kill ambiguity.

An uncorrected clock error $b$ becomes a range error $cb$: $cb=(3\times 10^8)(38\times 10^{-6})=1.14\times 10^4\text{m}\approx 11.4\text{km per day.}$ Why: the same $\rho =r+cb$ machinery — a timing error of $38\mu \text{s}$ is a distance error of $\approx 11$ km. That is why the satellite oscillator is pre-detuned on the ground: the fix is baked in before launch. This blends Special Relativity — Time Dilation (clock slows) and General Relativity — Gravitational Time Dilation (clock speeds up), the GR term winning.

Special relativity slows the moving clock by fraction $\frac{v^2}{2c^2}$ per unit time. Over one day ($86400$ s $=8.64\times 10^{10}\mu \text{s}$): $\frac{v^2}{2c^2}=\frac{3874^2}{2(3\times 10^8{)}^2}=8.34\times 10^{-11}.$ $\Delta t_{SR}=-(8.34\times 10^{-11})(8.64\times 10^{10}\mu \text{s})\approx -7.2\mu \text{s/day.}$ Net: $+45+(-7.2)\approx +37.8\approx +38\mu \text{s/day}$. ✓ Why: SR (motion) and GR (weaker gravity high up) push in opposite directions; the gravitational speed-up wins, leaving the famous $+38$.

Each row of the geometry matrix $\mathbf{H}$ is the unit line-of-sight vector from you to a satellite (plus a $1$ for the clock column). If all four satellites lie in nearly the same direction, those unit vectors are nearly parallel. Parallel rows make ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ almost singular (its inverse blows up). Since $\text{GDOP}=\sqrt{\mathrm{tr}(({\mathbf{H}}^{\mathsf{T}}\mathbf{H}{)}^{-1})},$ a near-singular matrix makes GDOP huge, and ${\sigma }_{\text{position}}=\text{DOP}\times {\sigma }_{\text{measurement}}$ then amplifies a tiny range error into a giant position error. Vertical suffers most because no satellite is overhead or below to "pin" your height — look at the figure: with all sight-lines skimming the horizon, moving you up or down barely changes any range. See Dilution of Precision (DOP) and Least Squares Estimation.

Widely spread supports resist wobble because a small push no longer collapses the structure — mathematically, spread-out directions make the geometry matrix well-conditioned so errors don't amplify. A phone achieves it by using all four constellations at once (GPS + GLONASS + Galileo + BeiDou), so more satellites are visible in more directions, lowering DOP — crucial in urban canyons.

(a) Unknown correction vector $\Delta \mathbf{x}=(\Delta x,\Delta y,\Delta z,c\Delta b)$ — three position nudges plus the clock-bias nudge (in metres).

(b) For satellite $i$, its row of $\mathbf{H}$ is $(-{\hat{u}}_{ix},-{\hat{u}}_{iy},-{\hat{u}}_{iz},1)$, where ${\hat{u}}_i$ is the unit vector pointing from your guessed position toward the satellite. That $1$ in the last column says "a clock error shifts all pseudoranges equally."

(c) With the measured-minus-predicted residuals $\Delta \mathbf{\rho }$: $\Delta \mathbf{x}=({\mathbf{H}}^{\mathsf{T}}\mathbf{H}{)}^{-1}{\mathbf{H}}^{\mathsf{T}}\Delta \mathbf{\rho }.$

(d) We iterate because the true equations are nonlinear (square roots). The Taylor linearization is only accurate near the guess. So we: guess → linearize → solve for $\Delta \mathbf{x}$ → update the guess → repeat until $\Delta \mathbf{x}$ shrinks below a tolerance (usually 2–3 loops). This is exactly Least Squares Estimation wrapped in a loop; for a moving receiver you feed it into a Kalman Filter together with an Inertial Navigation System (INS).

Yes. The altitude constraint is a free extra equation ("your height is 0"). Count: unknowns are still $x,y,z,b$ = 4. Equations = 3 pseudoranges + 1 altitude = 4. Four equations, four unknowns ⇒ solvable. This is "altitude aiding," a real trick used by ships, aircraft with barometers, and receivers with a known-terrain database.

Number of unknowns in a standard GNSS fix ::: Four — $x,y,z$ and clock bias $b$. Range error from a $1\mu \text{s}$ clock error ::: $300$ m. Which relativity effect makes the GPS clock run fast ::: General relativity (weaker gravity up high) — it outweighs the special-relativity slowdown. What a high GDOP means physically ::: Satellites are bunched together, so small range errors blow up into large position errors. How 3 satellites can still give a fix ::: Add an external constraint (e.g. known altitude) as the 4th equation.