On Generalized Signal Waveforms for Satellite Navigation (797942), страница 56
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and R.S. Orr, 1998]. Interlacing is an intelligent way to achieve non-uniformeffective power distribution among the codes as we will see in the next chapters.7.4.4Majority Voting: Scalar Combination with UniformWeightingLet us assume, as we also did in previous lines, that we want to multiplex an odd number of2N+1 statistically independent binary signals using majority logic. Furthermore, let us assumethat the codes are statistically balanced, so that the chip values can be modelled asindependent, identically distributed binary random variables.
According to this, the majorityvoted signal that will result of multiplexing the 2N+1 individual signals can be expressed as:⎛ 2 N +1 ⎞cMaj = Maj(c1 , c2 , c3 ,..., c2 N +1 ) = sign ⎜ ∑ ci ⎟(7.8)⎝ i =1 ⎠where the Majority operator Maj indicates the sign of the majority of the signals. This signalreceives the name of majority voted signal and is the one that will be transmitted instead ofthe 2N+1 signals. As one can imagine, in order the correlation losses not to be very high, themajority vote signal should somehow represent each of the 2N+1 individual signals that formit. To measure how true this assumption is, the correlation between the majority voted signalcMaj and a particular reference code has to be calculated.The result of a single chip correlation, denoted χ in the following lines, equals +1 or -1depending on the coincidence of the majority voting chip and the replica chip of the particularcode we correlate with, assuming perfect alignment.
In fact, the majority chip matches thereference chip (thus χ = +1 ) if and only if at least N chips from the other remaining 2N codesalso match it [J.J. Spilker Jr. and R.S. Orr, 1998]. Otherwise the correlation will be -1.According to this, the average correlation between any particular code ci and the majorityvoting signal cMaj will be⎧⎪+ 1 ⋅ p(ci = +1, cMaj = +1) + 1 ⋅ p (ci = −1, cMaj = −1) −⎪⎩− 1 ⋅ p(ci = +1, cMaj = −1) − 1 ⋅ p(ci = −1, cMaj = +1)χ =⎨(7.9)what can also be expressed as follows:255Signal Multiplex Techniques for GNSS⎧⎪+ 1 ⋅ p(ci = +1) ⋅ p(cMaj = +1 ci = +1) + 1 ⋅ p(ci = −1) ⋅ p(cMaj = −1 ci = −1) −(7.10)⎪⎩− 1 ⋅ p(ci = +1) ⋅ p(cMaj = −1 ci = +1) − 1 ⋅ p(ci = −1) ⋅ p (cMaj = +1 ci = −1)χ =⎨We assume thatp(ci = +1) = p(ci = −1) =12(7.11)Furthermore, it is trivial to see thatp (cMaj = +1 ci = +1) = pN2 N (+ 1) + p N2 N+1 (+ 1) + ...
+ p22NN (+ 1)(7.12)where p N2 N (+ 1) is the probability that exactly N codes out of 2N adopt the value +1. As it canbe shown, this probability is also equal to that of having N codes out of 2N with value -1 andadopts the following form:1 ⎛ 2N ⎞⎟(7.13)p N2 N (+ 1) = p N2 N (− 1) = 2 N ⎜⎜2 ⎝ N ⎟⎠being thus the following identity true:p (cMaj = +1 ci = +1) = pp (cMaj = −1 ci = −1)(7.14)In the same manner, it can be shown that:p (cMaj = −1 ci = +1) = p (cMaj = +1 ci = +1) = pN2 N+1 (− 1) + p N2 N+ 2 (− 1) + ... + p22NN (− 1)(7.15)If we further develope (7.10), we can see that the mean correlation χ between any particularcode ci and the majority voting code cMaj simplifies to:or equivalently⎧ 1⎪+ 2 ⋅⎪⎪+ 1 ⋅⎪χ =⎨ 2⎪− 1 ⋅⎪ 2⎪ 1⎪− ⋅⎩ 2χ=[p2NN(+ 1) + pN2 N+1 (+ 1) + ... + p22NN (+ 1) ] +[p2NN(− 1) + pN2 N+1 (− 1) + ...
+ p22NN (− 1) ] −[p2NN +1(− 1) + p (− 1) + ... + p (− 1) ] −[p2NN +1(+ 1) + pN2 N+ 2 (+ 1) + ... + p22NN (+ 1) ]2NN +2(7.16)2N2N1 2N11 ⎛ 2N ⎞⎟pN (+ 1) + pN2 N (− 1) = pN2 N (+ 1) = 2 N ⎜⎜222 ⎝ N ⎟⎠(7.17)This expression can be further simplified using an approximation based on the Stirling boundsof the factorial function as shown by [W. Feller, 1957]:χ* =e−18NπN≈χ =122 N⎛ 2N ⎞⎜⎜⎟⎟⎝ N ⎠(7.18)which is a good approximation even for low values of N.256Signal Multiplex Techniques for GNSSIf we normalize now the amplitude of the majority code to the summed power of thecomponent codes, that is 2 N + 1 , the normalized mean correlation ρ between any particularcode ci and the majority voting code c Maj will be12N + 1 ⎛ 2N ⎞2N + 1 −8N⎜⎟eρ=≈22 N ⎜⎝ N ⎟⎠πN(7.19)The problem of this implementation of the majority voting is that since all the signals areequally weighted in power, there appear relatively large majority combining losses per signal,resulting in relatively poor overall power efficiencies.
In fact, for the case of three transmittedsignals, the majority vote multiplexing is shown to result in a 1.25 dB multiplexing loss (whatcorresponds to an efficiency of approximately 75 %). It is important to keep in mind thatwhen three signal are majority voted, N adopts the value 1 in the previous equations( 3 = 2 ⋅ 1 + 1 ). Next table exemplifies the possible chip combinations for majority combiningof three codes and the correlation between each of the three codes and the majority voted codeTable 7.2.
Chip combinations and correlation for majority combining of three codesCode 1++++----Code 2++--++--Code 3+-+-+-+-Majority Voting Codeχ Code 1 – MV Code+++-+---+++--+++χ Code 2 – MV Codeχ Code 3 – MV Code++-++-+++-++++-+As we can see, the unnormalized mean correlation between any particular code and themajority voting code adopts the value + in 18 cases and – in 6 cases. Probabilisticallyspeaking, this implies that:3p (χ = + ) =4(7.20)1p (χ = − ) =4so that the unnormalized mean correlation will be χ = 1 2 , resulting in an apparent loss of6.02 dB.
If we consider now the power of the three codes, the total loss will be ρ = 3 2which corresponds to the 1.25 dB we mentioned some lines above. This result coincidesperfectly with the predictions of the theory developed in previous equations.The correlation power loss factor L(N) is defined by [J.J. Spilker Jr. and R.S. Orr, 1998] as thefraction of power of any code in the majority voted signal measured at the correlator outputand is shown to adopt the following form:L( N ) = ρ 2(7.21)or expressed in dB,257Signal Multiplex Techniques for GNSS⎡⎛ 2 N ⎞⎤⎟⎟⎥LdB ( N ) = −20 log10 ⎢2 − 2 N 2 N + 1 ⎜⎜⎝ N ⎠⎦⎣(7.22)It is interesting to analyze this equation when the number of equal-weighted inputs increases.In fact, when N → ∞ , (7.19) shows that the achievable per-code correlation asymptoticallyapproaches 2 π , so that the correlation losses will increase as the number of signals tomultiplex increases, but will never be higher than 1.96 dB.
In fact, when the receiverperforms a correlation among all the possibilities, some of the received chips will be wrongbut limited in number according to the above derived expressions. The following figure showsthe losses as a function of the number of multiplexed signals. To compute the curve, the exactformula of the majority losses was employed. However, the difference with respect to theStirling approximation is minimum even for low numbers of signals.Figure 7.2.
Majority voting lossesAnother important drawback of this simple implementation of the majority vote multiplexingis that it is difficult to control the relative power levels between the different multiplexedsignals without incurring in additional losses as shown in [P.A. Dafesh et al., 2006]. Indeed,in the previous derivations all codes or signals are assigned the same power levels.Furthermore, this multiplexing technique does not provide sufficient spectral separation andhas limited inherent flexibility in adjusting the amplitude of generated harmonics, being allthese great disadvantages.A way to achieve an arbitrary weighting of the power of the signals to multiplex is using astatistical mix of majority vote rules operating on appropriately chosen subsets of the inputchips of each signal [P.A.
Dafesh et al., 2006]. Here, the power distribution is realised playingwith the relative frequency of use of the various majority vote rules. As we can imagine, aparticular power distribution can be accomplished with different majority vote rules and thusthe optimum of all the possible solutions will be that one with the smallest multiplexinglosses. This will be further clarified in the next chapters.To conclude this chapter it is important to mention that the correlation loss can also beinterpreted as the additional fraction of transmit power ΔP that is required to neutralize thereceiver performance loss, so that it can be defined as follows:ΔP = L−1(7.23)258Signal Multiplex Techniques for GNSS7.4.5Majority Voting: Scalar Combination with NonUniform WeightingIn the previous chapter we have analyzed the simplest realization of the majority votingtechnique, namely the uniformly weighted version, where all the signals contribute with thesame power to the majority voted signal.
As we saw there, this solution presents limitationswith respect to the flexibility to meet the power requirements in normal GNSS systems. Tocope with this, this chapter describes the non-uniform solution.The non-uniform weighting can be likened to shareholder voting as graphically described by[R. S. Orr and B. Veytsman, 2002]. In fact, based on a targeted power allocation, each of thesignals to multiplex is allocated a number of votes, which may be fractional in the mostgeneral case. Then, at each chip epoch, the transmitted majority voted value is selected bytaking the sign of the sum of the weighted chips of each individual code. Without loss ofgenerality, the codes or channels are assumed to be binary.
This is shown in the expressionnext proposed by [R. S. Orr and B. Veytsman, 2002]:⎛ N⎞c Maj = sign ⎜ ∑ λi ci ⎟(7.24)⎝ i =1⎠where λi is the number of votes allocated to the i-th of the N signals, ci represents the chipvalue of the i-th signal and cMaj is the majority voted chip value. As we can see in theexpression above, this generalized form of majority voting also includes the particular case of(7.8) where all the weights are equal. Moreover, it is easy to recognize that this adaptation ofmajority logic enables a constant-envelope multiplexing of an arbitrary distribution of chipsynchronous CDMA signals. In addition, there is no constraint on the number of signals thatcan be multiplexed so that also an even number of codes could be majority voted using thisgeneral approach. One final comment on the previous equation is that the weighting factorsmust be selected in such a way that the summation is different than zero at any time.As we can read from (7.24), the key to achieve an efficient multiplexing is the correctselection of the weighting factors so that the resulting composite signal does indeed reflect thedesired power distribution among the various user signals.
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