Question bank — GNSS — GPS, GLONASS, Galileo, BeiDou

False — three spheres would fix $(x,y,z)$ only if your receiver clock were perfect, but it carries an unknown bias $b$, a hidden 4th unknown that needs a 4th equation (4th satellite). See the Trilateration and Multilateration geometry.

False — it measures the pseudorange ${\rho }_i=r_i+cb$, contaminated by the clock-bias term $cb$; the true geometric range $r_i$ only falls out once $b$ is solved alongside position.

False — untreated, the net $\sim +38\mu \text{s/day}$ drift maps to $\sim 11\text{km/day}$ of position error; GNSS is one of the most-used everyday confirmations of general relativity.

False — special relativity (orbital speed) makes the clock run slow, while gravitational time dilation up high makes it run fast; the GR effect wins, so the net is faster.

True — more spread-out satellites lower the Dilution of Precision (DOP) and give redundancy for fault detection (RAIM), so you get both a tighter and a more trustworthy fix.

False — GPS, Galileo and BeiDou use CDMA (shared frequency, unique codes), but classic GLONASS uses FDMA, giving each satellite its own slightly shifted frequency. See Signal Modulation — CDMA & FDMA.

Mostly true in spirit — its rows are line-of-sight unit vectors from receiver to satellite, so they depend on relative geometry; because satellites are $\sim 20,000$ km away, small changes in the receiver guess barely rotate these directions, which is why the linearization converges fast.

True — with $b$ known (say $b=0$), only three unknowns $(x,y,z)$ remain, so three satellites suffice; the 4th satellite exists purely to solve the clock bias.

The arithmetic is fine but $t_{rx}$ is read on a biased clock, so the result is $r_i+cb$, not $r_i$; the label "pseudo" is precisely this bias, not a rounding nuisance.

No — DOP comes only from satellite geometry (${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$); it multiplies your measurement error, but a great receiver in bad geometry still gets a poor fix.

They don't reset it — the oscillator is pre-tuned to tick slightly slow on the ground so that once in orbit it runs at the correct rate; the correction is built in, not patched daily.

Count is not geometry — their line-of-sight vectors are nearly parallel, making ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ nearly singular, so GDOP explodes and vertical error especially blows up.

Backwards — the observation equations are nonlinear (square roots), so we linearize by Taylor expansion first; least squares then handles the overdetermined linear system when 5+ satellites are seen. See Least Squares Estimation.

Light covers $\sim 30$ cm per nanosecond, so timing is the dominant error budget; a $1\mu \text{s}$ bias is already $\sim 300$ m — a stadium-sized miss.

MEO is a compromise: high enough that few satellites cover huge areas, low enough that signal delay and power loss stay manageable — GEO/IGSO are actually higher and used by BeiDou for regional boost. See Orbital Mechanics — MEO/GEO/IGSO.

An atomic-grade clock in every phone is impossibly expensive; treating $b$ as a 4th unknown lets a cheap crystal oscillator be corrected for free by the extra satellite — the math replaces the hardware.

The unit vectors $-{\hat{u}}_i$ are the directions along which a range change moves you, and the constant $1$ column captures the clock bias's identical effect on every pseudorange (all ranges grow by $cb$ together).

Like a carpenter bracing a table with legs spread wide, widely separated lines-of-sight make ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ well-conditioned, so a small range error cannot amplify into a large position error.

The observation equation centres a sphere of radius $r_i$ on the satellite; without knowing $(x_i,y_i,z_i)$ you'd know how far you are but not from where, so the sphere has no fixed centre.

Buildings block much of the sky, so a single system may show too few or badly-clustered satellites; pooling GPS+GLONASS+Galileo+BeiDou raises the count of visible, well-spread satellites and lowers DOP.

They fuse GNSS fixes with motion models/inertial data, smoothing noisy fixes and bridging brief outages (tunnels, canyons) so position stays continuous rather than jumping.

The line-of-sight rows become linearly dependent in that plane, so ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ loses rank — the perpendicular direction is unconstrained and DOP goes to infinity (no solution).

Effectively you have $\sim 3$ independent directions, so the system is near-singular; a valid fix may still print but its DOP is high and the answer is fragile to noise.

Yes, because you don't know it's zero ahead of time; $b$ remains an unknown to be solved, so four equations are still required — a lucky value doesn't remove the variable.

It helps: a near-vertical line-of-sight strongly constrains height, which horizon-hugging satellites cannot, so it lowers vertical DOP.

You must account for an inter-system time/frame offset (an extra small unknown or a known bias); otherwise mixing them injects a systematic error, which is why multi-GNSS receivers estimate inter-system biases explicitly.

The measured ${\rho }_i$ grows beyond the true $r_i$, showing up in the residual ${\epsilon }_i$; ionospheric/tropospheric delay is modelled or estimated, otherwise it biases the fix by metres.