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CHAPTER 8 GLOBAL POSITIONING SYSTEM
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where 1 = fc1 and 2 = fc2 are the wavelength of the L1 and L2 carrier signals, respectively, and i is the combined e ect of multipath and receiver noise The subsequent analysis assumes that 1 and 2 are independent 2 Gaussian processes with variance Using the phase observables, the ionospheric delay can be estimated as Ia = f1 f2 2 2 1 1 2 2 f1 f2 (893)
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Direct substitution of eqns (891) and (892) into eqn (893) shows that Ia = f1 f2 2 2 ( 1 2 + N1 1 N2 2 ) f1 f2 f1 f2 = Ia + 1984 ( 2 1 ) + 2 2 (N1 1 N2 2 ) f1 f2 Ia + (894) (895)
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2 Therefore, the variance of Ia at each epoch is approximately 8 ; however, the estimate contains the signi cant constant bias
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2 f1
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f1 f2 2 (N1 1 N2 2 ) f2
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Eqns (889) and (895) have the form Ia Ia
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Ia + n1 Ia + n2 + B
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(896) (897)
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Eqns (896) and (897) are in the exact form considered in Example 54 Therefore, the solution is Ia (k) = Ia (k) B(k) where 1 Bk = Bk 1 + k Ia Ia Bk 1 (898)
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starting at k = 1 with B0 = 0 The estimation error variance for B is 2 1 2 2 2 2 PB = 8(1 + ) k and for Ia (k) is PIa = 8 + , where = k and = 001 is the ratio of the phase noise to the pseudorange noise The desire to estimate the ionospheric delay was a primary motivation for the GPS system to incorporate signals on two frequencies The speci cation of the two frequencies involved a tradeo related to the frequency spacing If the frequency separation was too small, then measurement errors would be signi cantly magni ed as shown in eqn (888) and (894) However, if the frequency separation were too large, then separate antennas would be required to receive the two signals
86 TWO FREQUENCY RECEIVERS
Wide and Narrow Lane Observables
This section develops the equations for the narrow-lane and wide-lane observables These variables are synthesized as linear combinations of the L1 and L2 measurements The interest in the wide-lane signal is that its wavelength is large enough that the wide-lane variable is often used to facilitate the problem of integer ambiguity resolution, see Section 89 The phase measurements of eqns (891 892) can be modeled as 1 N 1 2 N2 = = f1 ( + Ecm + 1 ) c f2 ( + Ecm + 2 ) c f2 Ia c f1 Ia c (899) (8100)
Forming the sum and di erence of eqns (899) and (8100) results in 1 + 2 = f1 f2 f1 f2 + + Ia + (N2 + N1 ) c c c c f2 f1 f1 f2 + + (8101) Ecm + 1 + 2 c c c c f1 f2 f1 f2 Ia + (N1 N2 ) c c c c f2 f1 f2 f1 + (8102) Ecm + 1 2 c c c c
1 2
By de ning the wide and narrow lane wavelengths as w n = = c f1 f2 c , f1 + f2 (8103) (8104)
eqns (8101) and (8102) can be written in meters as ( 1 + 2 ) n + Ecm Ia + (N2 + N1 ) n n n + 1 + 2 1 2 = + Ecm + Ia + (N1 N2 ) w w w + 1 2 1 2 =
(8105)
( 1 2 ) w
(8106)
where w = 154 45 and w = 120 35; therefore, the standard devi 1 34 2 34 ation of the widelane noise (ie, w 1 w 2 ) is approximately 57 times 1 2 the standard deviation of the L1 or L2 phase noise (ie 1 or 2 ) The
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