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

False — teen spheres sirf $(x,y,z)$ fix kar sakti hain agar tumhara receiver clock perfect ho, lekin usme ek unknown bias $b$ hota hai, jo ek chupi hui 4th unknown hai jiske liye ek 4th equation chahiye (4th satellite). Dekho Trilateration and Multilateration geometry.

False — yeh pseudorange ${\rho }_i=r_i+cb$ measure karta hai, jo clock-bias term $cb$ se contaminated hai; true geometric range $r_i$ tabhi milti hai jab $b$ ko position ke saath solve kiya jaye.

False — agar treat na kiya jaye, toh net $\sim +38\mu \text{s/day}$ drift $\sim 11\text{km/day}$ position error mein convert ho jaata hai; GNSS general relativity ke sabse zyada use hone wale rozana ke confirmations mein se ek hai.

False — special relativity (orbital speed ki wajah se) clock ko slow chalata hai, jabki upar gravitational time dilation ise fast chalata hai; GR effect jeetta hai, isliye net result faster hai.

True — zyada spread-out satellites Dilution of Precision (DOP) ko kam karte hain aur fault detection (RAIM) ke liye redundancy dete hain, isliye fix dono hi taraf se — tight bhi hoti hai aur zyada trustworthy bhi.

False — GPS, Galileo aur BeiDou CDMA use karte hain (shared frequency, unique codes), lekin classic GLONASS FDMA use karta hai, jisme har satellite ko apni thodi si shifted frequency milti hai. Dekho Signal Modulation — CDMA & FDMA.

Mostly true in spirit — iske rows receiver se satellite tak ke line-of-sight unit vectors hain, isliye yeh relative geometry par depend karte hain; kyunki satellites $\sim 20,000$ km door hain, receiver guess mein chhote changes in directions ko bahut kam rotate karte hain, isliye linearization jaldi converge ho jaata hai.

True — jab $b$ pata ho (maano $b=0$), sirf teen unknowns $(x,y,z)$ bachte hain, isliye teen satellites kaafi hain; 4th satellite sirf clock bias solve karne ke liye exist karta hai.

Arithmetic bilkul theek hai lekin $t_{rx}$ ek biased clock pe padhi jaati hai, isliye result $r_i+cb$ hai, na ki $r_i$; "pseudo" label precisely is bias ka naam hai, koi rounding nuisance nahi.

Nahi — DOP sirf satellite ki geometry (${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$) se aata hai; yeh tumhari measurement error ko multiply karta hai, lekin buri geometry mein ek accha receiver bhi poor fix deta hai.

Woh reset nahi karte — oscillator ko ground pe thoda slow tick karne ke liye pre-tune kiya jaata hai taaki orbit mein jaane par sahi rate pe chale; correction built-in hai, roz patch nahi kiya jaata.

Tadad geometry nahi hai — unke line-of-sight vectors almost parallel hote hain, jisse ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ almost singular ho jaata hai, isliye GDOP explode karta hai aur vertical error khaas tor par badh jaata hai.

Ulta hai — observation equations nonlinear hain (square roots), isliye hum pehle Taylor expansion se linearize karte hain; least squares phir overdetermined linear system handle karta hai jab 5+ satellites dikh rahi hon. Dekho Least Squares Estimation.

Light $\sim 30$ cm per nanosecond cover karti hai, isliye timing dominant error budget hai; $1\mu \text{s}$ bias pehle se hi $\sim 300$ m hai — stadium-sized miss.

MEO ek compromise hai: itna high ki thode satellites huge areas cover karein, itna low ki signal delay aur power loss manageable rahe — GEO/IGSO actually aur upar hain aur BeiDou regional boost ke liye use karta hai. Dekho Orbital Mechanics — MEO/GEO/IGSO.

Har phone mein atomic-grade clock rakhna impossibly expensive hai; $b$ ko 4th unknown maankar ek sasta crystal oscillator ko extra satellite se free mein correct kiya ja sakta hai — math hardware ki jagah le leta hai.

Unit vectors $-{\hat{u}}_i$ woh directions hain jisme ek range change tumhe move karta hai, aur constant $1$ column clock bias ka identical effect har pseudorange pe capture karta hai (saari ranges $cb$ se saath badhti hain).

Jaise ek carpenter table ko wide legs se support karta hai, widely separated lines-of-sight ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ ko well-conditioned banate hain, isliye ek chhoti si range error badi position error mein amplify nahi ho sakti.

Observation equation satellite pe radius $r_i$ ki ek sphere centre karti hai; $(x_i,y_i,z_i)$ jaane bina tum jaante ki tum kitne door ho lekin kahan se nahi, isliye sphere ka koi fixed centre nahi hoga.

Buildings sky ka zyada hissa block kar deti hain, isliye ek akela system bahut kam ya badly-clustered satellites dikha sakta hai; GPS+GLONASS+Galileo+BeiDou pool karna visible, well-spread satellites ki tadad badhata hai aur DOP ghataata hai.

Yeh GNSS fixes ko motion models/inertial data ke saath fuse karte hain, noisy fixes ko smooth karte hain aur brief outages (tunnels, canyons) ko bridge karte hain taaki position continuous rahe, na ki jump kare.

Line-of-sight rows us plane mein linearly dependent ho jaati hain, isliye ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ rank kho deta hai — perpendicular direction unconstrained ho jaati hai aur DOP infinity chali jaati hai (koi solution nahi).

Effectively tumhare paas $\sim 3$ independent directions hain, isliye system near-singular hai; ek valid fix print ho sakta hai lekin uska DOP high hai aur answer noise ke liye fragile hai.

Haan, kyunki tumhe pehle se pata nahi ki yeh zero hai; $b$ ek unknown hi rehta hai jo solve kiya jaana hai, isliye chaar equations abhi bhi zaroori hain — ek lucky value variable ko nahi hatati.

Help karta hai: near-vertical line-of-sight height ko strongly constrain karta hai, jo horizon-hugging satellites nahi kar sakti, isliye vertical DOP kam hota hai.

Tumhe ek inter-system time/frame offset account karna hoga (ek extra chhota unknown ya ek jaani-paehchani bias); warna unhe mix karna systematic error inject karega, isliye multi-GNSS receivers explicitly inter-system biases estimate karte hain.

Measured ${\rho }_i$ true $r_i$ se zyada ho jaata hai, residual ${\epsilon }_i$ mein dikhai deta hai; ionospheric/tropospheric delay ko model ya estimate kiya jaata hai, warna yeh fix ko metres tak bias kar deta hai.