On Generalized Signal Waveforms for Satellite Navigation (797942), страница 58
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Toextend the result to all the code combinations and have thus the mean correlation for anycomponent signal, we only have to sum over ciNG as shown next:( )( )⎡ ⎛ SiNG c NG + λiNG ⎞⎛ SiNG c NG − λiNG ⎞⎤⎟⎟⎥⎜⎜⎟⎜erferf(7.46)−⋅NG∑⎢ ⎜⎟222N⎠⎦⎝⎠⎝c iNG ⎣This expression can be further simplified if we recall that integrating (7.45) over ciNG andχ iNG =1c NG (that is over all code combinations including that of code i) will be similar except for afactor 2. In fact, it can be shown that⎡ ⎛ S NG c NG + λiNG ⎞⎛ S NG c NG − λiNG ⎞⎤1⎟ − erf ⎜ i⎟⎥ =χ iNG= N NG ⋅ ∑ ⎢erf ⎜⎜ i⎟⎜⎟222ciNG ⎢⎠⎝⎠⎥⎦⎣ ⎝NGNGNGNGNGNG⎡ ⎛S c⎛S c+ λi ⎞− λi ⎞ ⎤1⎟ − erf ⎜ i⎟⎥ =(7.47)= N NG +1 ⋅ 2∑ ⎢erf ⎜⎜ i⎟⎜⎟NG222⎢ ⎝ci ⎣⎠⎝⎠⎦⎥⎡ ⎛ S NG c NG + λiNG ⎞⎛ S NG c NG − λiNG ⎞⎤1⎟ − erf ⎜ i⎟⎥= N NG +1 ⋅ ∑ ⎢erf ⎜⎜ i⎟⎜⎟222⎢ ⎝c NG ⎣⎠⎝⎠⎦⎥NGAs we can see, the last line of the previous equation integrates over c and not ciNG .( )( )( )( )( )( )In the same manner, the correlation between the Gaussian group and the majority voted signalcan be approximated as follows:χG =12N NG⋅∑cNG2π[S (c )]NG⋅e−NG22(7.48)An example confirming the validity of the previous expression is briefly depicted in the nextlines.
Let us imagine that our majority voting signal adopts the following form:c Maj = sign (S G + S NG ) = sign (S G + 5 ⋅ c1 + 10 ⋅ c 2 )(7.49)As we can recognize, the majority voted multiplexed signal consists of the Gaussian groupand two non-Gaussian codes c1 and c2 weighted with 5 and 10 respectively. We analyze nextall possible cases:264Signal Multiplex Techniques for GNSSFor the particular combination (c1 , c2 ) = (− 1,−1) , the mean value of the sum of the Gaussianand non-Gaussian signals S G + S NG will be -15 adopting the probability density function thefollowing form:Figure 7.4.
Probability density function of S G + S NG for (c1 , c2 ) = (− 1,−1)where the area in red indicates the value of the mean correlation for this particularcombination of non-Gaussian codes. Indeed, the mean correlation in this case is shown toadopt the following value:⎡+ 1 ⋅ p(G < 0 ) + ⎤1 ⎢11⎛ 15 ⎞G(7.50)χ c NG = (−1, −1) = ⋅ ⎢+ 1 ⋅ p(G ≥ 15) − ⎥⎥ = ⋅ p{G ∉ (− 15 ,15)} = ⋅ erfc⎜⎟444⎝ 2⎠⎢⎣− 1 ⋅ p(0 ≤ G < 15)⎥⎦where erfc( x ) = 1 − erf (x ) This can also be expressed as a function of the weightings:χ cGNG= ( −1, −1)=⎛ + λ1 + λ21⋅ erfc⎜⎜42⎝⎞⎟⎟⎠ (λ1 ,λ2 )= (5,10 )(7.51)As one can observe, this is the unnormalized correlation between the Gaussian group and themajority voted signal for this particular combination of non-Gaussian codes.
To normalize theexpression, we only have to multiply by + λ1 + λ2 resulting thusρ cGNG= ( −1, −1)=⎛ + λ1 + λ21⋅ + λ1 + λ2 erfc⎜⎜42⎝⎞⎟⎟⎠ (λ1 ,λ2 )= (5,10 )(7.52)In the same manner, for the combination of non-Gaussian codes (c1 , c2 ) = (− 1,+1) , the meanvalue of the sum of the Gaussian and non-Gaussian signals S G + S NG will be 5 in this case,yielding the mean unnormalized correlation:⎡+ 1 ⋅ p(G ≥ 0 ) + ⎤1 ⎢11⎛ 5 ⎞Gχ c NG = (−1, +1) = ⋅ ⎢+ 1 ⋅ p(G < −5) − ⎥⎥ = ⋅ p{G ∉ (− 5,5)} = ⋅ erfc⎜⎟444⎝ 2⎠⎢⎣− 1 ⋅ p(− 5 ≤ G < 0 )⎥⎦(7.53)or again for any arbitrary two weighting factors:χ cGNG= ( −1, +1)=⎛ − λ1 + λ 21⋅ erfc ⎜⎜42⎝⎞⎟⎟⎠(7.54)so that the normalized expression will be:265Signal Multiplex Techniques for GNSSρ cGNG= ( −1, +1)=⎛ − λ1 + λ 21⋅ − λ1 + λ 2 erfc ⎜⎜42⎝⎞⎟⎟⎠(7.55)In the same manner, the mean normalized correlation for the code combination(c1 , c2 ) = (+ 1,−1) will be:⎛ + λ1 − λ 2 ⎞1⎟(7.56)ρ cG =(+1, −1) = ⋅ + λ1 − λ 2 erfc ⎜⎜⎟NG4and for (c1 , c2 ) = (+ 1,+1) ,ρ cGNG= ( +1, +1)=2⎝⎛ − λ1 − λ 21⋅ − λ1 − λ 2 erfc ⎜⎜42⎝⎠⎞⎟⎟⎠(7.57)Grouping now all the previous normalized correlations, the mean value will be then:ρ⎛ S NG ⎞1NG⎟= ⋅ ∑ S erfc ⎜⎜ 2 ⎟4 c NG⎝⎠Gc NG(7.58)where S NG = 5 ⋅ c1 + 10 ⋅ c2 in this particular example.
Moreover, the erfc function can be wellapproximated as follows when the argument is higher than 3 (as it is the case in Majority Votecombinations) [M. Abramovitz and I.A. Stegun, 1965]:erfc( x ) ≈1x πe−x2(7.59)For our particular case, this implies that the correlation between the Gaussian group and themajority voted signal can be approximated for any generic non-Gaussian code combinationc NG as follows:ρ cG =NG12N NG⋅∑c NG2π[S (c )]NG⋅e−NG2(7.60)2which is the expression presented in (7.48).Dividing now (7.47) by (7.60) and squaring, the power ratio of code ciNG will adopt thefollowing form:⎧⎪ χ NG ⎫⎪Ri = ⎨ iG ⎬⎪⎩ ρ c NG ⎪⎭2( )( )⎧ ⎡ ⎛ S NG c NG + λNG ⎞⎛ S NG c NG − λiNG ⎞⎤ ⎫i⎪ ∑ ⎢erf ⎜⎜ i⎟⎟ − erf ⎜⎜ i⎟⎟⎥ ⎪22π ⎪ c NG ⎣ ⎝⎠⎝⎠⎦ ⎪= ⎨⎬NGNG 2[S (c )]8⎪−⎪∑NG e 2⎪⎪c⎩⎭2(7.61)which coincides with the formula derived by [R.
S. Orr and B. Veytsman, 2002]. Once wehave the ratio of the power of any code ciNG of the non-Gaussian group with respect to theGaussian group, the power loss factor of all the signals multiplexed can be expressed in termsof the losses of the Gaussian group multiplied by the total power of the signal since all thepower ratios are normalized to the power of the Gaussian group:( )⎛ N⎞L = ⎜⎜1 + ∑ Ri ⎟⎟ ρ cGNGi =1⎝⎠NGor simplified:NG[S NG (c NG )]−⎛ N⎞⎡ 122= ⎜⎜1 + ∑ Ri ⎟⎟ ⎢ N NG ⋅ ∑⋅e⎢πNG2i =1c⎝⎠⎣22⎤⎥⎥⎦2(7.62)266Signal Multiplex Techniques for GNSS⎛ N⎞2⎜ 1 + ∑ Ri ⎟ ⎡[S NG (c NG )] 2 ⎤−2⎥L = ⎜⎜ N NGi =−11 ⎟⎟ ⎢∑ e(7.63)⎥2π ⎢c NG⎜⎜⎟⎟ ⎣⎦⎝⎠which also coincides with the expression derived by [R.
S. Orr and B. Veytsman, 2002].NGOnce we have derived the general expressions for the losses of the majority vote multiplexand thus the efficiency of the modulation for a targeted power distribution, the next step is tofind the optimum weighting factors. This problem is basically a minimization exercise thatconsists in finding the weighting factors that minimize the total losses subject to the envisagedpower division between the different signals. This can be expressed as follows:{L}minNG{λ }(7.64)subject to {Ri }iwhere⎛ N⎜ 1 + ∑ Ri⎜L = ⎜ N NGi =−11π⎜⎜ 2⎝NGand⎞22⎟⎡[S NG (c NG )] ⎤−⎟⎢2⎥e⎟ ⎢∑NG⎥⎟⎟ ⎣ c⎦⎠( )( )⎧ ⎡ ⎛ S NG c NG + λ NG ⎞⎛ S NG c NG − λiNGi⎟ − erf ⎜ i⎪ ∑ ⎢erf ⎜⎜ i⎟⎜22π ⎪ c NG ⎢⎣ ⎝⎠⎝Ri = ⎨NGNG 2[S (c )]8⎪−⎪∑NG e 2c⎩(7.65)⎞⎤ ⎫⎟⎥ ⎪⎟⎠⎥⎦ ⎪⎬⎪⎪⎭2(7.66)[R. S.
Orr and B. Veytsman, 2002] have proposed an efficient way to solve this problemwhere the set of weighting factors λiNG is efficiently calculated.{ }7.4.6GeneralizedMajorityCyclostationary SolutionsVoting(GMV):In the previous chapter different majority vote solutions have been derived for the case thatthe weighting rule applied to each instantaneous set of component channels is constant overtime. Furthermore, theory was presented to derive the weighting factors required to obtain adesired power distribution.
However, as shown by [R. S. Orr and B. Veytsman, 2002], notalways the targeted power allocation of the different services can be accomplished on thebasis of an stationary approach and a cyclostationary solution is required then. In this case,the weighted majority voting rules exhibit time variation varying the weighting coefficientsover time. The time variation is applied periodically over the largest available processinginterval as shown by [R. S.
Orr and B. Veytsman, 2002], being this interval normally theshortest data symbol of any of the component codes of the multiplexing. The cyclostationarypower allocation can be further tuned by averaging different weighting schemes over time.267Signal Multiplex Techniques for GNSS7.4.7Generalized Majority Voting (GMV): Sub-MajorityVotingAs shown by [R. S. Orr and B.
Veytsman, 2002], the stationary solutions that result fromapplying the theory of previous chapters to the commanded powers of the majority votedsignal present a quantized behaviour in the sense that the achieved gains do not automaticallychange when sufficiently small changes in the vote allocation are realized. Indeed, a breakpoint only occurs when a slight change in the vote allocation permits some coalition of votesto dominate in a situation when they previously could not. In this sense, if an accurateallocation of the powers on the different signals is required, this can only be achieved on thebasis of a cyclostationary solution as introduced in the previous lines.Based on this idea, a simple way of decreasing the effective power allocated to a particularchannel is to omit this code from a certain number of majority votes resulting in the so-calledsub-majority voting.
When this occurs, certain codes or signals do not participate in the submajority voting, changing thus the weightings of the different codes or channels over the time(thus a cyclostationary solution). It must be noted that while the weighting might changerelatively often, it will however be constant for a relatively long period of time, in the order ofthe length of a bit.
This can also be understood as time-multiplexing different signalsaccording to a predefined scheme. In the following lines we analyze the implementationproposed by [G. L. Cangiani et al., 2002]. To illustrate the functioning of this cyclostationarysolution, we take as an example the case when three signal codes are majority voted.As shown by [G. L. Cangiani et al., 2002], when three signals are combined usingGeneralized Majority Voting (GMV) on a sub-majority voting basis, there are four possibleelements to consider: the majority vote of the three chips and the three individual chipsthemselves. As it can be demonstrated, if one of the codes is smaller than the other two, thetransmission of solo chips from that code will never result in an efficient solution. Accordingto this, if the targeted power distribution is {G1 , G2 , G3 }, being the gains listed in nondecreasing order, the majority vote Maj{c1 , c 2 , c3 }and the solo chips c 2 and c3 should betransmitted the following fractions of time [R.
S. Orr and B. Veytsman, 2002]:2t Maj =G 2 + G3t c2 =t c3 =G2 − 1G 2 + G3(7.67)G3 − 1G 2 + G3where the three time fractions sum to unity as one could expect. The resulting signal withinterlacing of the majority vote of c1 , c 2 and c3 and the solo chips c 2 and c3 will be referred toas { Maj(c1 , c 2 , c3 ), c 2 , c3 } in following chapters.268Signal Multiplex Techniques for GNSSThe interpretation of the time fractions is as follows. Let us assume that each data bit contains100 chips and that the commanded power distribution is {G1 , G2 , G3 } = {1,4,9} .
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