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Starting from the geodetic position estimate p = [ , , h] from the navigation systems, this section presents the equations that would be used to predict the GPS measurements Using the estimated vehicle geodetic position p, the vehicle ECEF posi tion pe can be computed using eqns (29 211) The ECEF position of the i-th satellite pi can also be computed from the ephemeris data, as discussed in Appendix C The computed range to the i-th satellite is R(pe , pi ) = pe pi

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Velocity factor is important in the design of RF transmission lines and antenna systems, when sections of cable, wire, or metal tubing must be cut to specific lengths measured in wavelengths or fractions of a wavelength Taking the velocity factor v, expressed as a ratio, into account, the preceding four formulas become: Lft = 984v / fMHz Lm = 300v / fMHz fMHz = 984v / Lft fMHz = 300v / Lm

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If information is available about the common mode GPS errors, these estimates of the common-mode errors could either be added to the computed ranges or subtracted from the range measurements The following uses the same notation as in Section 952 Assuming that m satellites are available and that measurements from the rst satellite are subtracted from all the others, the di erenced pseudorange and carrier phase measurements can be predicted from the navigation state as i i = = p p R( e , pi ) + i R( e , p1 ) + 1 p p R( e , pi ) + i R( e , p1 ) + 1 + N i N 1

The Electromagnetic Spectrum The entire range of EM wavelengths is called the electromagnetic (EM) spectrum Scientists use logarithmic scales to depict the EM spectrum, as shown in Fig 25-10 The RF spectrum, which includes radio, television, and microwaves, is blown up in this illustration, and is labeled for frequency

for i = 2, , m The same assumptions and comments apply as were discussed following eqns (926-927)

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This section presents the GPS measurement residual equations that would be used by the Kalman lter In this approach, the measurement residuals for the i-th satellite would be computed as i (t) = (t) =

25-10 At A, the EM spectrum from 108 m to 10 12 m,

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which is the measured range di erence between satellites i and 1 minus the predicted di erence For i = 2, , m, the error models for the measurement residuals are i (t) (t)

with each vertical division representing two orders of magnitude (an increase or decrease of the wavelength by a factor of 100) At B, the RF spectrum, with each vertical division representing one order of magnitude (an increase or decrease of the frequency by a factor of 10)

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Radio-wave propagation has been a fascinating science ever since Marconi and Tesla discovered, around the year 1900, that EM fields can travel over long distances without any supporting infrastructure whatsoever Let s look at a few of the factors that affect wireless communications at radio frequencies

The quantities i and i are de ned for i = 1, , m in eqns (883 891) The vector hi is de ned as e h i = hi h 1 e e e

p (pe i ) where hi = pe i as de ned in eqn (812) e p 2 Currently, there is a mismatch between eqns (11159 11160) which is a function of pe = [ x, y, z] and the error state which is a function of p = [ , , h] These two vectors are related according to

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