Global Positioning System Study Guide

📖 Core Concepts GPS (Global Positioning System) – A U.S.‑owned, satellite‑based hyperbolic navigation system that provides user‑position and precise time anywhere with line‑of‑sight to ≥4 satellites. Pseudorandom (Gold) Code – Unique binary sequence each satellite transmits; the receiver aligns its locally generated copy to measure signal arrival time. Pseudo‑range – Measured distance: \[ pi = di + b \] where \(pi\) = observed range to satellite i, \(di\) = true geometric range, \(b\) = receiver clock bias (time error × speed of light). Clock Bias – The dominant unknown because the receiver’s oscillator is far less stable than the satellite atomic clocks; it is solved together with the three spatial coordinates. Relativistic Corrections – Satellite clocks run 38 µs faster per day due to combined special‑relativistic (velocity) and general‑relativistic (gravitational) effects; GPS continuously pre‑corrects this offset. Geometric Dilution of Precision (GDOP) – A scalar that quantifies how satellite geometry amplifies all other error sources; lower GDOP → better accuracy. Coordinate Frames – Raw solution is in an Earth‑centered Cartesian (ECEF) frame; it is later converted to latitude, longitude, altitude using the WGS‑84 ellipsoid. --- 📌 Must Remember Constellation: 24 – 32 satellites, altitude ≈ 20 200 km, 6 orbital planes inclined ≈ 55°, spaced 60° in right ascension. Minimum Satellites: 4 → 3‑D position + clock bias; 3 → position only if receiver clock is perfectly synchronized. Frequencies: L1 = 1575.42 MHz (C/A & P), L2 = 1227.60 MHz (P, later L2C), L5 = 1176 MHz (civilian safety‑of‑life). Typical Accuracy: Post‑Selective‑Availability civilian receiver ≈ 5 m. Survey‑grade RTK / carrier‑phase ≈ 2 cm or better. Relativistic Offset: +38 µs/day; without correction → ≈ 10 km position error per day. GPS Time: Continuous, no leap seconds, offset from UTC/TAI by a constant; receiver must apply the offset to obtain UTC. Error Sources: Satellite clock & ephemeris error, ionospheric & tropospheric delay, multipath, receiver noise, GDOP. Selective Availability (SA): Discontinued 1 May 2000; historic civilian accuracy 100 m, now 5 m. --- 🔄 Key Processes Signal Acquisition – Antenna captures L‑band signal; receiver searches for PRN codes, locks onto at least four satellites. Ephemeris Download – Decode subframes 2‑3 (≈ 18–30 s) to obtain precise orbital parameters. Pseudo‑range Computation – Align code, measure time‑of‑arrival, multiply by speed of light \(c\). Form Navigation Equations – For each satellite i: \(pi = \sqrt{(x-xi)^2+(y-yi)^2+(z-zi)^2}+b\). Solve for \((x,y,z,b)\) – Four‑satellite case: Linearise & iterate (Gauss–Newton) or use Bancroft closed‑form. >4 satellites: Apply least‑squares (or weighted LS) to minimise residuals. Apply Corrections – Relativistic clock correction (pre‑applied by satellite). Ionospheric model (Klobuchar) or dual‑frequency ionosphere‑free combination. Tropospheric model (standard). Convert to Geodetic Coordinates – Transform ECEF \((x,y,z)\) to latitude, longitude, altitude (WGS‑84). Output Position & Time – Provide position, velocity (from Doppler), and UTC‑corrected time. --- 🔍 Key Comparisons Three‑satellite trilateration vs. Four‑satellite hyperboloid method 3‑sat: assumes perfect receiver clock → intersect three spheres → two possible points (one on Earth). 4‑sat: solves for clock bias simultaneously → intersect three hyperboloids → unique Earth‑bound solution. Iterative Gauss–Newton vs. Bancroft Closed‑Form Iterative: higher final accuracy, especially with poor geometry; requires more computation. Bancroft: single‑step matrix inversion, very fast; may be less accurate when GDOP is high. L1 C/A (1.023 Mcps) vs. L5 (10.23 Mcps) L1 C/A: widely available, modest accuracy (5 m). L5: higher chip rate, wider bandwidth → 30 cm accuracy for capable receivers. GPS vs. Other GNSS (GLONASS, Galileo, BeiDou) GPS: 24‑sat global, mature civil signals (L1, L2C, L5). GLONASS: FDMA instead of CDMA, similar accuracy. Galileo: adds E5a/E5b, tighter integrity. BeiDou: regional + global, includes B1I/B2I. --- ⚠️ Common Misunderstandings “Three satellites are enough for a 3‑D fix.” – Only true if the receiver clock is already synchronized; otherwise a fourth satellite is required. “GPS time equals UTC.” – GPS time does not include leap seconds; you must add the current offset to obtain UTC. “Ionospheric delay is negligible for civilian receivers.” – It can contribute several meters of error; models or dual‑frequency measurements are needed for high precision. “Selective Availability still degrades accuracy.” – SA was turned off in 2000; any mention of 100 m errors is historical. “More satellites always mean better accuracy.” – Geometry matters; a cluster of satellites can increase GDOP despite high satellite count. --- 🧠 Mental Models / Intuition Hyperboloid Picture: Each pair of satellites defines a hyperboloid of constant distance‑difference; the receiver sits where three such hyperboloids intersect. Insphere Analogy: Imagine spheres centered on each satellite with radii equal to measured pseudo‑ranges; the receiver is the tiny sphere (radius = clock bias × c) that touches all larger spheres simultaneously. GDOP as Pyramid Base: Satellites spread out like a wide base of a pyramid; the wider the base, the more stable (lower GDOP) the apex (your position). --- 🚩 Exceptions & Edge Cases Multipath‑Dominated Environments (urban canyons, indoors) → large pseudorange errors; aided GNSS or inertial dead‑reckoning may be required. Low‑Elevation Satellites (< 15°) → severe ionospheric/tropospheric delays and higher GDOP; many receivers mask them out. Receiver Clock Drift – Low‑cost crystal oscillators can introduce tens of meters of bias if not quickly corrected by the navigation solution. Insufficient Satellites (< 4) – Use Assisted GPS (A‑GPS), dead‑reckoning, or rely on a nearby base‑station (RTK) for a fix. --- 📍 When to Use Which Standard civilian navigation → L1 C/A (available on all receivers). High‑accuracy survey → L5 or L2C + RTK carrier‑phase corrections. Fast time‑to‑first‑fix on smartphones → Assisted GPS (download almanac/ephemeris via cellular data). Quick rough position → Bancroft closed‑form (good for initial estimate). Precise positioning under poor geometry → Iterative Gauss–Newton with weighted LS and ionosphere‑free combination. When satellite geometry is tight (high GDOP) → Delay fix, wait for better geometry or use augmentation (WAAS, EGNOS). --- 👀 Patterns to Recognize High GDOP → satellites clustered in a small sky sector; expect larger position error. Sudden pseudorange bias shift across all satellites → receiver clock offset change (often after warm‑up). Consistent range error on low‑elevation satellites → ionospheric/tropospheric delay not fully corrected. Repeated loss of lock on a single satellite → multipath or obstruction specific to that satellite’s line‑of‑sight. Ephemeris vs. Almanac – Ephemeris updates every 30 s; almanac only gives coarse orbit, useful for acquisition but not precise positioning. --- 🗂️ Exam Traps Distractor: “Three satellites can determine latitude, longitude, and altitude.” – Wrong unless clock is perfect; exam answer should be four satellites. Distractor: “GPS time automatically includes leap seconds.” – Incorrect; you must apply the known offset to obtain UTC. Distractor: “Selective Availability is still active, limiting civilian accuracy to 100 m.” – Out‑of‑date; SA has been disabled since 2000. Distractor: “Ionospheric error can be ignored for L5 because of its higher frequency.” – Not true; while L5 reduces ionospheric impact, a residual error remains and must be modeled for cm‑level work. Distractor: “More satellites always lower GDOP.” – GDOP depends on geometry, not raw count; a set of satellites all over one horizon can increase GDOP. ---