On Generalized Signal Waveforms for Satellite Navigation (797942), страница 65
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Cangiani et al., 2002] to which ofthe three interplex modulator inputs s1 , s 2 or s 3 during the period in which the particulartargeted gains apply. Furthermore, the intervote logic has to decide which three codes aremajority voted and where the two remaining codes input the interplex multiplexer such as toobtain the maximum global efficiency.Since intervote bases on interplex to combine all the signals, we recall the general expressionof the interplex multiplexing for the I and Q channels as presented in (7.106) of chapter 7.7.2:296Signal Multiplex Techniques for GNSSI=P1 P2 s 2 (t ) + P1 s3 (t )P1 P2 s 2 (t ) + P1 s3 (t )=(1 + P1 )(P1 + P2 )P s (t ) − P2 s1 (t ) s 2 (t ) s3 (t ) P1 s1 (t ) − P2 s1 (t ) s 2 (t ) s3 (t )=Q= 1 1(1 + P1 )(P1 + P2 )P12 + P1 P2 + P1 + P2P12 + P1 P2 + P1 + P2(7.144)where s1 (t ) , s 2 (t ) and s 3 (t ) are the input interplex components and the powers are allexpressed in terms of P3 with P3 = 1 .
Furthermore, the efficiency of the interplex modulationwas shown to be:η Interplex =P1 (1 + P1 + P2 )(1 + P1 )(P1 + P2 )(7.145)As we mentioned above, all the permutations of five codes and all locations of the majorityvote have to be tested to find the best global efficiency. Indeed, analysis shows that although aparticular three-code combination could result in an optimum GMV solution for threeparticular codes, it is not always true that placing these three codes as input of the interplexand the other two uncombined in the other two interplex inputs would result in the best globalintervote efficiency. This means in other words, that all possible 5!= 120 combinations have tobe tried to select the optimum configuration.Fortunately, the number of efficiency evaluations significantly reduces if symmetryconsiderations are taken into account.
In fact, a careful look at (7.144) reveals that only threelogical possibilities for the majority vote placement are to be considered, namely s1 , s 2 or s 3 .⎛5⎞In this manner, the number of combinations to try is of only 3⎜⎜ ⎟⎟ = 30 . Moreover, positions⎝ 3⎠s 2 and s 3 are interchangeable so that the order does not affect the efficiency formulas as wewill also prove next.
According to this, only two placement options are possible for themajority vote signal:•Majority Vote on I: As we have mentioned, placing the majority vote signal on eithers 2 or s 3 is arbitrary in terms of the total efficiency. Therefore, the default majority•vote signal will be taken to be s 2 in the following pages.Majority Vote on Q: In this case, the majority vote signal will be s1 .7.8.3Intervoting with Majority Vote on IIn the case that the majority voted signal is placed on I, the generic transmitted binary signals{c1 , c 2 , c3 , c4 , c5 } can be assigned as follows [G. L.
Cangiani et al., 2002]:s1 = c 4s 2 = { Maj(c1 , c 2 , c3 ), c 2 , c3 }(7.146)s 3 = c5297Signal Multiplex Techniques for GNSSwhere the majority vote was arbitrarily placed on s 2 as we discussed above. Furthermore, thenotation s 2 = {Maj(c1 , c 2 , c3 ), c 2 , c3 } indicates that s 2 is an interlace of the majority vote ofc1 , c 2 and c3 with solo chips of c 2 and c3 , as described in chapter 7.4.7.We further assume that the code set {c1 , c 2 , c3 , c 4 , c5 } has as target commanded powerdistribution {g1 , g 2 , g 3 , g 4 , g 5 } . According to this, the normalized interplex gain of s1 and s 3should be then g 4 and g 5 respectively. Taking now (7.144) into consideration this implies:g 4 = g (s1 ) =P12(1 + P1 )(P1 + P2 )P1g 5 = g (s 3 ) =(1 + P1 )(P1 + P2 )so thatP1 = g 4 g 5(7.147)(7.148)Recalling the efficiency expression of the general interplex modulation as given in (7.107),the total efficiency of Intervoting with majority vote on I will be then as follows:Iη Intervote=P1 (1 + P1 + P2ηGMV )(1 + P1 )(P1 + P2 )(7.149)It is important to note that this expression is slightly different to that presented in (7.107) forthe general interplex case.
Indeed, the power allocated on the signal s 2 in the numerator hasbeen corrected by the efficiency of the majority vote multiplex as not all the power allocatedon s 2 corresponds in reality to the useful binary signals c1 , c 2 and c3 . In fact, as shown in(7.68), part of the power goes lost in the majority voting reducing the efficiency from 1 toηGMV . For the denominator, however, the whole power allocated on s 2 has to be consideredincluding also the non-desired multiplex losses.As we have mentioned above, signal s 2 is the majority vote of c1 , c 2 and c3 . According tothis, for the targeted power gains g1 , g 2 and g 3 , the power gain of s 2 should beP1P2g + g 2 + g3g (s2 ) == 1= P2 g5(7.150)(1 + P1 )(P1 + P2 )ηGMVor equivalently,g + g 2 + g3(7.151)P2ηGMV = 1g5This can be further simplified if we recall thatg + g 2 + g3(7.152)ηGMV = 12g 2 + g3Consequently,(g (s 2 ) =or in a similar formP2()g2 + g3(=)2g2 + g3g5= P2 g 5)(7.153)2(7.154)298Signal Multiplex Techniques for GNSSUsing now equations (7.149), (7.148), (7.154) and (7.151) together, the total intervoteefficiency when the majority vote is placed on I will be:g4 ⎛gg + g 2 + g3 ⎞⎜⎜1 + 4 + 1⎟⎟ggg()1PPPη++I5 ⎝55⎠12 GMV(7.155)η Intervote== 12(1 + P1 )(P1 + P2 ) ⎛ g ⎞⎛⎜ g⎞gg+23⎟⎜⎜1 + 4 ⎟⎟ 4 +⎜⎟g5 ⎠ g5g5⎝⎝⎠()which can be further simplified and perfectly coincides with the expression shown by[G.
L. Cangiani et al., 2002]:Iη Intervote=g 4 ( g1 + g 2 + g 3 + g 4 + g 5 )2(g 4 + g5 )⎡⎢ g 2 + g3 + g 4 ⎤⎥⎣⎦()(7.156)It is interesting to note from the expression above that the position of s 2 and s 3 isinsignificant what reinforces the comments we made in chapter 7.8.2 as we anticipated thatpositions s 2 and s 3 are arbitrarily interchangeable. This is also valid for the case ofIntervoting with Majority Vote on Q as we show next.7.8.4Intervoting with Majority Vote on QIn the previous lines we have derived the analytical expression for the intervote efficiencywhen the majority vote signal is placed on I. In this chapter, we will repeat the same exercisefor the case that the majority signal is placed on Q.In this case, the transmitted signals can be assigned as follows [G. L.
Cangiani et al., 2002]:s1 = {Maj(c1 , c 2 , c3 ), c 2 , c3 }s2 = c4(7.157)s 3 = c5Since the code set {c1 , c 2 , c3 , c 4 , c5 } has as target commanded power distribution{g1 , g 2 , g 3 , g 4 , g 5 } , the normalized interplex gain of s2 and s3 should be then g 4 and g5respectively. Taking (7.144) into consideration, this implies:g 4 = g (s2 ) =P1P2(1 + P1 )(P1 + P2 )P1g 5 = g (s3 ) =(1 + P1 )(P1 + P2 )(7.158)so thatP2 = g 4 g 5(7.159)Recalling now the expression for the efficiency of the intervote multiplex on I, the totalefficiency with majority vote on Q adopts a slightly different form:299Signal Multiplex Techniques for GNSSQη Intervote=P1 (1 + P1ηGMV + P2 )(1 + P1 )(P1 + P2 )(7.160)As we can recognize, the power allocated on signal s1 in the numerator has been corrected bythe efficiency of the majority vote multiplex following the same logic as in the case ofmajority vote on I.
On the other hand, the whole power allocated on s1 is considered for thedenominator including also the non-desired multiplex losses as explained in previous chapter.Signal s1 is the majority vote of c1 , c 2 and c3 . According to this, for the targeted power gainsg1 , g 2 and g 3 , the power gain of s1 will be:P12g + g 2 + g3g (s1 ) == 1= P1 g5(1 + P1 )(P1 + P2 )ηGMVorg1 + g 2 + g3g5P1ηGMV =This can be further simplified as follows:g (s1 ) =or equivalently(g2 + g3(P =(7.161))2g2 + g31(7.162)= P1 g 5(7.163))2(7.164)g5Putting now (7.160), (7.159), (7.151) and (7.154) together, the total intervote efficiency whenthe majority vote is placed on Q will be then:(Q=η IntervoteP1 (1 + P1ηGMV + P2 )=(1 + P1 )(P1 + P2 ) ⎛⎜1+⎜⎝g 2 + g3g5() ⎛⎜1 + g + g21⎜⎝) (g 2 + g3 ⎞⎟⎛⎜⎟⎜g5⎠⎝22+ g3g5g 2 + g3g5+)2g4 ⎞⎟g5 ⎟⎠g ⎞+ 4⎟g5 ⎟⎠(7.165)which can be simplified coinciding with the expression of [G.
L. Cangiani et al., 2002]:ηQIntervote=((g2 + g3⎡ g + g23⎢⎣) (g2)21+ g2 + g3 + g4 + g5 )(+ g5 ⎤⎡ g2 + g3⎥⎦ ⎢⎣)2+ g4 ⎤⎥⎦(7.166)In the previous lines we have derived the efficiency of the intervote multiplex for the case thatthe majority vote signal is either on I or on Q for a particular code set {c1 , c 2 , c3 , c 4 , c5 } withtarget commanded power distribution { g1 , g 2 , g 3 , g 4 , g5 } .
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