UNIT 1 - introduction to GPS

1. GPS SIGNAL
Each GPS satellite transmit two signal for positioning purposes:

L1 signal (carrier frequency of 1,575.42 MHz). Modulated onto the L1 carrier are two pseudo-
random noise (PRN) ranging codes and the navigation (broadcast) message. The codes (used to
determine the pseudo-ranges) are (a) the 1 millisecond-long C/A-code (chipping rate about 1
MHz); (b) the weeklong segment of the P-code (chipping rate about 10 MHz). The navigation
(broadcast) message includes orbital information (ephemeris), the offset dt of the satellite clock
from the GPS system time, information on the health of the satellite and the expected accuracy of
the range measurements (UERE). The message contains also almanac data for other satellites
(used by the receiver to determine the location of each satellite). For receivers that track the
weeklong P-code, the broadcast message includes a special hand-over word (HOW), that tells the
receiver where in the P-code to start searching.

L2 signal (carrier frequency of 1,227.60 MHz) is modulated by the P-code and the navigation
message – the C/A code is not present.

Figure 1. How the different components of the GPS signal are combined. After Langley (1990)

The PRN codes are unique for each satellite and the correlation between any pair of codes is very
low. This allows all satellites to share the same carrier frequency.
There are basically two methods to deny civilians full use of the GPS system:

Selective Availability (SA) - adding noise to the clock and ephemeris in the navigation message
(SA has been turned off in 2000)

Anti-spoofing (AS)- the P-code is encrypted (Y-code) and available to military users only.

Ref. – Langley R. (1990), Why is the GPS signal so complex?, GPS World, May/June, p. 56.
 (After Hoffmann-Wellenhof et al. (1997), GPS: Theory and Practice, 4th Ed., Springer.)

2. GPS OBSERVABLES
The pseudorange. The GPS receiver measures the distance (pseudorange) between the satellite and
the antenna by measuring the time the signal takes to propagate from the satellite to the receiver. The
pseudorange is this time offset multiplied by the speed of light.
The pseudorange is biased by the lack of time synchronization between the clock in the GPS
satellite and the clock in the GPS receiver. Other bias effects include the ionosphere and troposphere
delay, multipath and receiver noise. The equation for the pseudorange observable is

p = ρ + c × ( dt − dT ) + dion + dtrop + ε p

where p is the pseudorange, ρ is the geometric range to the satellite, c is the speed of light, dt and dT
are the offsets of the satellite and receiver clock from the GPS time, dion and dtrop the delays imparted by
the ionosphere and troposphere and εp represents the effect of multipath and receiver noise. The
receiver coordinates are hidden in the geometric range ρ.

Figure 2. How the pseudorange is measured, after Langley (1998), in GPS for Geodesy, p. 151.

Carrier phase. A more precise observable than the pseudo-range is the phase of the received carrier
with respect to the phase generated by an oscillator in the GPS receiver. The difference between the
received carrier and the receiver generated one is called the carrier beat phase. The problem is that the
GPS receiver cannot distinguish one cycle of a carrier from another. The receiver measures the
fractional phase, and keeps track of changes to the phase. The initial phase is undetermined, or
ambiguous, by an integer number of cycles N.
If we convert the carrier beat phase into an equivalent distance by multiplying by the carrier
wavelength λ, we get

Φ = ρ + c × ( dt − dT ) + λ × N + dion + dtrop + ε p

which is very similar to the pseudorange expression, the major difference being the presence of the
ambiguity term λ × N .

Linear combinations. We can form what are known as between-receivers (or between- satellites)
differences to obtain new observable with significantly reduced errors.

Figure 3. Linear combinations. After Langley (1993).

The between-receivers single difference (two different receivers tracking the same satellite) -
eliminates the satellite clock offset

∆Φ = ∆ρ − c × ∆dT + λ × ∆N + ∆dion + ∆dtrop + ∆ε p

The between-satellites single difference (one receiver tracking two satellites) - eliminates the
receiver clock offset

∇Φ = ∇ρ − c × ∇dt + λ × ∇N + ∇dion + ∇dtrop + ∇ε p

The double difference (we can difference either the between receivers or the between-satellite
difference pairs) - eliminate both the receiver and satellite clock offset

∆∇Φ = ∆∇ρ + λ × ∆∇N + ∆∇dion + ∆∇dtrop + ∆∇ε p


The LC (ionosphere free) combination. The linear combination of the L1 and L2 phase
measurements reduces the effect of the ionosphere, but may amplify other sources of error

fL21 fL22
Φ LC = Φ − Φ L2
fL21 − fL22 fL21 − fL22
L1

Wide-lane and narrow-lane combinations (applied for ambiguity resolution)

Φ L1 Φ L 2 Φ Φ
φWL = − λ NL = L1 + L 2 m
λ L1 λ L 2 λ L1 λ L 2
c c
λWL = ≈ 0.86 m λ NL = ≈ 0.11 m
f L1 − f L 2 f L1 + f L 2

Ref. – Langley R. (1993), The GPS observables, GPS World, April, p. 52.

3. ERROR BUDGET.
Both systematic errors (biases) and random noise affect the code pseudoranges p and phase
pseudoranges Φ. The error sources can be classified into three groups (see Table 6.1)

(After Hoffmann-Wellenhof et al. (1997), GPS: Theory and

Practice, 4th Ed., Springer.)

UERE stands for User Equivalent Range Error. After

Langley (1997)
 Orbital Errors /Clock Bias/Measurement Noise: As mentioned earlier, GPS signals contain
information about ephemeris (orbital position) errors, and about the rate of clock drift for the
broadcasting satellite. The data concerning ephemeris errors may not exactly model the true satellite
motion or the exact rate of clock drift. Distortion of the signal by measurement noise can further
increase positional error. The disparity in ephemeris data can introduce 1-5 meters of positional error,
clock drift disparity can introduce 0-1.5 meters of positional error and measurement noise can
introduce 0-10 meters of positional error (see Table 6.2).

(After Hoffmann-Wellenhof et al. (1997), GPS: Theory and Practice, 4th Ed., Springer.)

Signal propagation: The ionosphere and troposphere both refract the GPS signals. This causes the
speed of the GPS signal in the ionosphere and troposphere to be different from the speed of the GPS
signal in space. Therefore, the distance calculated from "Signal Speed x Time" will be different for the
portion of the GPS signal path that passes through the ionosphere and troposphere and for the portion
that passes through space.

Multipath: A GPS signal bouncing off a reflective surface prior to reaching the GPS receiver
antenna is referred to as multipath. Because it is difficult to completely correct multipath error, even in
high precision GPS units, multipath error is a serious concern to the GPS user.

Selective Availability (turned off in January 2000): Ephemeris errors should not be confused with
Selective Availability (SA), which is the intentional alteration of the time and ephemeris signal by the
Department of Defense. SA can introduce 0-70 meters of positional error. Fortunately, positional errors
caused by SA can be removed by differential correction.

Dilution of Precision (DOP). The UERE is mapped into the computed position by a geometrical
factor called DOP. The DOP is a mathematical function involving the relative coordinates of the
receiver and the satellite and can be easily computed for a particular satellite arrangement. The more
spread out the satellites are in the sky, the smaller the DOP value. A typical value for the horizontal
dilution of precision (HDOP), assuming that a receiver is processing the signals of 4 satellites only, is
2.0.

Ref. - Langley, R. B. (1997), The GPS error budget. GPS World , Vol. 8, No. 3, pp. 51-56.