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GNSS Signal Model

Signal Model and Modulation

Signal Model

\[ \begin{align} s(t) = \underbrace{\sqrt{2 P_I} x_I (t) D_I (t) \cos \left(2 \pi f_c t + \phi \right)}_{\text{In-phase}} + \underbrace{\sqrt{2 P_Q} x_Q (t) D_Q (t) \cos \left(2 \pi f_c t + \phi \right)}_{\text{Quadrature}} \\ \end{align}, \]

where,

Properties Definition
\(P_I\) In-phase signal power (W)
\(P_Q\) Quadrature signal power (W)
\(x_I(t)\) In-phase ranging code. Modulation with the C/A-code, \(x_I(t)\) using BPSK modulation: \(x(t) = \begin{cases} \exp \left( j0 \right) &= +1, \quad \text{if code bit = 0} \\ \exp\left(j \pi \right) &= -1, \quad \text{if code bit = 1} \end{cases}\)
\(x_Q(t)\) Quadrature ranging code (Modulation by the P(Y) code, i.e., military usage)
\(D_I(t)\) In-phase navigation signal
\(D_Q(t)\) Quadrature navigation signal
\(f_c\) Carrier Frequency (Hz)
\(\phi\) Carrier phase offset (rad)

Modulation

modulation modulation

Figure 1 A factor graph representation (Factor Graph Tutorial, ION, GNSS+ 2023)

GPS Signals

GPS Frequencies

Carrier Band L1 L2 L3 L4 L5
Center Frequency 1575.42MHz 1227.60MHz 1381.05MHz 1379.91MHz 1176.45MHz
Carrier Wavelength (\(c / f_0\)) 19cm 24cm 25cm
Quadrature I/Q I/Q I/Q I/Q I/Q
Legacy Usage C/A, P(Y) P(Y)
Modern Usage C/A, P(Y), M, L1CD, L1CP P(Y), M, L2C I5-code, Q5-code

GPS Navigation Signals

LNAV L2 CNAV L5 CNAV CNAV-2 MNAV
Definition Legacy navigation L2 civilian navigation L5 civilian navigation Civilian navigation Military navigation
Data
  1. GPS date, time and the satellite status
  2. Ephemeris. Received in 18-36 seconds and is valid up to 4 hours
  3. Almanac (up to 32 satellites). Received in 12.5 minutes and is valid for up to 2 weeks
  1. GPS date, time and the satellite status
  2. Ephemeris. Received in 18-36 seconds and is valid up to 4 hours
  3. Almanac (up to 64 satellites). Received in 12.5 minutes and is valid for up to 2 weeks
  1. GPS date, time and the satellite status
  2. Ephemeris. Received in 18-36 seconds and is valid up to 4 hours
  3. Almanac (up to 64 satellites). Received in 12.5 minutes and is valid for up to 2 weeks
 Unknown
Properties Modulated onto C/A and P(Y) codes at 50b/s (very slow)
  1. Forward error correction (FEC) via convolution codes with rate 1/2
  2. Modulated onto L2C code at 50b/s. Navigation data at 25b/s (50% FEC overhead)
  3. Improved data structure (packet based instead of frame based)
  4. Better interoperability with other constellations and has excess bandiwdth to include more data in future
  1. Forward error correction (FEC) via convolution codes with rate 1/2
  2. Modulated onto L5 code at 50b/s. Navigation data at 25b/s (50% FEC overhead)
  3. Improved data structure (packet based instead of frame based)
  4. Better interoperability with other constellations and has excess bandiwdth to include more data in future
 Modulated onto L1C code at 100b/s, navigation data at 50b/s (50% FEC overhead), and has similar data structure and FEC to L2 CNAV and L5 CNAV Packet based and uses FEC

GPS Ranging Signals

C/A P(Y) L2C L5 L1C M
Definition Coarse / Acquisition code Precision code Second civilian signal Third civilian signal Fourth civilian signal Military code
Type Legacy Legacy Modernized Modernized Modernized Modernized
Frequency Band L1 L1 and L2 L2 L5 L1 L1 and L2
Usage Civilian Military Civilian Civilian Civilian Military
Properties
  1. Short code at 1.023Mb/s
  2. Repeats every 1 ms (1023 code bits, i.e., chips)
  1. Long code at 10.23Mb/s
  2. Main P-code repeats every 37 weeks (2.289214E14 chips)
  3. Each satellite transmits a different segment of the main code (6.187104E12 chips, repeats every 1 week)
  4. P-code is XORed with a secret cryptographic W-code to give P(Y)
  5. P(Y) is robust to interference and spoofing
  1. Contains two distinct codes multiplexed at 1.023Mb/s: CM, CL
  2. CM (civil-moderate code): Medium code at 511.5kb/s, repeats every 20ms (10230 chips), and has a new navigation signal (L2 CNAV) modulated onto it
  3. CL (civil-long-code): Long code at 511.5kb/s, repeats every 1.5 seconds (767250 chips), no navigation data modulation (dataless sequence), and improved SNR
  4. 2.7dB higher SNR than C/A at 2.3dB less power
  5. As of 2021, broadcasting from 23 GPS satellites
  1. Contains two codes transmitted in quadrature (I/Q) at 10.23Mb/s
  2. I5-code (in-phase code): Medium code at 10.23Mb/s, repeats every 1 ms (10230 chips), has a new navigation signal (L5 CNAV), has a synchronization sequence modulated onto it (10 bit Neuman-Hofman code)
  3. Q5-code (quadrature code): Medium code at 10.23Mb/s, repeats every 1 millisecond (10230 chips), no navigation data modulation (dataless sequence), has a synchronization sequence modulated onto it (20-bit Neuman-Hofman code)
  4. 3dB higher power and 10x higher bandwidth than other civil signals
  5. As of 2022, broadcasting from 17 GPS satellites
  1. Contains two codes transmitted in quadrature (I/Q) at 1.023Mb/s
  2. L1CD (in-phase data signal): Repeats every 10 milliseconds (10230 chips), Binary Offset Carrier (BOC) modulation (not BPSK), has a new navigation signal (CNAV-2) modulated onto it
  3. L1CP (quadrature pilot signal): Repeats every 10ms (10230 chips), Multiplexed Binary Offset Carrier (MBOC) modulation (not BPSK), additional overlay code (L1CO) modulated onto it for better acquisition
  4. 1.5dB higher power and improved tracking compared to C/A
  5. As of 2021, broadcasting from 4 GPS satellites
  1. Improves the security, performance and robustness of legacy P(Y)
  2. 5.115Mb/s bitrate, BOC modulation
  3. Has new navigation signal (MNAV) modulated onto it
  4. Block III and later satellites will broadcast the M code from a dedicated directional high gain antenna in addition to the full-earth antenna. Providing 20dB signal gain in selective regions