On Generalized Signal Waveforms for Satellite Navigation (797942), страница 57
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As shown by[R. S. Orr and B. Veytsman, 2002], if all the chips of the signals to multiplex were weightedas in a linear multiplexer in proportion to the square root of their power allocation, (7.24)would not reflect in general the desired power distribution. In fact, those signals with smallamounts of power could become suppressed relative to more powerful codes.Let us assume that we want to majority vote two signals with power ratios 20 % and 80 % ofthe total power respectively.
This means that one signal will be four times stronger than theother one. As we mentioned in the previous lines, a linear multiplexer would assigncoefficients 1 and 2 to the weak and strong signals according to:cMaj = c1 + 2 ⋅ c2(7.25)259Signal Multiplex Techniques for GNSSIf we show now the possible chip combinations of this scheme and the correlation betweeneach of the two codes and the majority voted code, we have:Table 7.3.
Chip combinations and correlation for linear majority combining of themajority rule cMaj = c1 + 2 ⋅ c2Code 1++--Code 2+-+-Majority Voting Codeχ Code 1 – MV Code+-+-+--+χ Code 2 – MV Code++++As we can recognize, while the mean correlation of code 1 with the majority voting signal iszero, code 2 presents a perfect correlation of 100 %.As we can see, the weakest signal is not reflected at all in the majority voted signal at the endas the information from code 1 has gone lost in the majority voted signal.
This small signalsuppression or capture is a well known result of non-linear signal processing operation andreflects indeed the fact that no coalition of minority stockholders can ever outvote a 51 %majority interest as graphically expressed by [R. S. Orr and B. Veytsman, 2002].With the previous example, we have demonstrated that a faithful representation of acommanded power distribution can not be achieved in the most general case using a linearmultiplexer. Indeed, the signal weight cannot be the square root of the power allocation unlessthe signals to multiplex were Gaussian distributed.[R.
S. Orr and B. Veytsman, 2002] have derived a set of equations that give an elegantsolution to this problem. According to the algorithm presented in their work, the crosscorrelation between the majority voted code and a particular component code is constraint bya set of equations in such as way that the power allocated to the particular signal is equal tothe square of the value of the corresponding correlation between this signal and the majorityvote signal.
In addition, a coefficient of proportionality is also introduced in the model inorder to control the efficiency or multiplexing losses common to each code. The solution ofthe equations provides the appropriate weighting of each signal maximizing the efficiency forthe desired power ratio. The model is extremely non-linear and therefore, to minimize theresolution of the algorithm, [R.
S. Orr and B. Veytsman, 2002] propose to assign each of thecomponent codes to one of two groups, designated as Gaussian and non-Gaussian.The components assigned to the Gaussian group G are typically small in power but numerousin number. As one can imagine, the division between Gaussian and not Gaussian (NG) is notalways so straightforward. Normally, the criterion to define a group of signals as Gaussian isthat the weighted sum of their chips, with the weight being proportional to the square root ofthe power allocation, will have a power that is less than a specified fraction of the total power,260Signal Multiplex Techniques for GNSStypically 5 % to 10 %.
It must be underlined that, in spite of existing well defined statisticaltests to decide on the Gaussianness of a group of signals, the determination of this decisionthreshold is relatively flexible and up to the designer. Indeed, the threshold value for this testis a parameter that permits some flexibility. [R.
S. Orr and B. Veytsman, 2002] have shownthat still in cases where the Gaussian group does not ideally behave as it should according totheory, the algorithm delivers good solutions.Taking (7.24), the majority voted signal will be formed in this case as follows:GNG⎛ N G G N NG NG ⎞GNG⎜c Maj = sign S + S= sign ⎜ ∑ λi ci + ∑ λ j c j ⎟⎟j =1⎝ i =1⎠()(7.26)where G refers to the Gaussian group of signals, NG to the non-Gaussian group and N Gand N NG are respectively the number of Gaussian and non-Gaussian signals.The commanded power distribution is described by a set of non-decreasing ratios {Ri } with0 ≤ i ≤ N NG , where the lowest ratio, R0 , describes the power of the Gaussian group and isnormalized to 1 as shown by [R.
S. Orr and B. Veytsman, 2002]. Accordingly, the remainingsignals 1 ≤ i ≤ N NG will represent the non-Gaussian group. Following this notation, Ri wouldindicate that the non-Gaussian signal ci has a power Ri times that of the Gaussian group.The Gaussian group signals have assigned weighting factors that are equal to the root squareof the power allocation. Therefore, if all the N G signals had the same power and given thatthe whole power of the Gaussian group is normalized to unity, each code of the group wouldbe allocated a power 1 N G being consequently the weighting factor of all the signals in theN G .
The composite Gaussian group S G is normalized to have power 1Gaussian group 1and mean zero being therefore its probability density function defined as follows.( )1 −e2πfS G S G =(S )G 2(7.27)2Since in the next lines the probability of S G to be in an arbitrary region − x < S G < x willappear relatively often, it is worth to recall the value of this probability()p−x<S <x =∫Gx−x( )f S G S dS = ∫GGx−x1 −e2π(S )G 22dS G =x2π ∫20G 2e − (S ′ ) dS ′G(7.28)This result can be further simplified, at least regarding the notation, if we express it in termsof the mathematical error function, which is defined as follows:erf (x ) =2π ∫x02e − t dt(7.29)According to this, the probability that the Gaussian group variable S G is between –x and xcan also be expressed as follows:261Signal Multiplex Techniques for GNSS()⎛ x ⎞p − x < S G < x = erf ⎜⎟⎝ 2⎠(7.30)As we have emphasized in previous lines, the collective power ratio R0 of the Gaussian signalcodes must be unity.
This means that the commanded distribution power allocations of theGaussian signals P1G , P2G ,..., PNGG must be normalized as follows:Pi G =Pi GNG∑Pi =1(7.31)Giand the corresponding weighting coefficients will thus adopt the following form:λGi = Pi G1≤ i ≤ NG(7.32)For the non-Gaussian group codes, the power ratios of the N NG non-Gaussian signal codes areequally determined as shown next [R. S.
Orr and B. Veytsman, 2002]:Ri =Pi NGNG∑Pi =11 ≤ i ≤ N NG(7.33)GiAs we can see, the ratio Ri indicates the relative power between the non-Gaussian code i andthe power of the Gaussian group.In line with the derivations of chapter 7.4.4, we have to derive now the correlation equationsto find the optimum weighting factors. Indeed, the power allocated to each non-Gaussiansignal should be the square of the correlation between that code and the majority voted signal.NGSince each chip in the multiplex can adopt two values, there is a total of 2 N possibilities atany time for the N NG non-Gaussian chips [R. S. Orr and B. Veytsman, 2002]. Let us definec NG as a combination of N NG non-Gaussian chips such thatN NG( ) ∑λS NG c NG =j =1NGjc NGj(7.34)( )and SiNG c NG equals (7.34) except for the exclusion of the i-th chip,SNGiN NG(c ) = ∑ λNGj =1, j ≠ iNGjc NGj(7.35)If we recall (7.24), the Majority Vote (MV) signal in its general form is shown to be:[ ( )( )][( )]cMaj = sign S G c G + S NG c NG = sign S G + S NG c NG(7.36)( )If we have a look at a particular non-Gaussian code ciNG , the value of the sum S G + S NG c NGcan adopt the two following values:S +SGNG(cNG⎧S G + S iNG (c NG ) + λiNG) = ⎨ G NG NG NG⎩ S + S i (c ) − λifor ciNG = +1for ciNG = −1(7.37)262Signal Multiplex Techniques for GNSSbeing thus the correlation between the replica desired signal ciNG and cMaj as follows:((((⎧+ 1 ⋅ p ciNG⎪NG⎪+ 1 ⋅ p ciχi = ⎨NG⎪− 1 ⋅ p ci⎪− 1 ⋅ p c NGi⎩) [= −1) ⋅ p[S= +1) ⋅ p[S= −1) ⋅ p[S((c(c(c)] ()]⋅ p(S)]⋅ p(S)]⋅ p(S((c(c(c))− λ)+ λ)− λ)< 0) +< 0) −≥ 0)= +1 ⋅ p SiNG c NG ⋅ p S G + SiNG c NG + λiNG ≥ 0 +NGiNGiNGiNGNGNGG+SG+ SiNGG+ SiNGNGiNGNGNGNGiNGiNGi(7.38)[ ( )] indicates the probability that the Non-Gaussian sum adopts a particularwhere p SiNG c NGvalue determined by c NG .
Furthermore, the sign function is defined as:⎧ 1 for x ≥ 0sign ( x ) = ⎨⎩− 1 for x < 0(7.39)It is important to note, that the coefficients must be selected such that the sum of all theweighted signals is never zero. Moreover, the probability to have a specific combination ofNon-Gaussian codes is shown to adopt the following form:[ ( )]p S iNG c NG =12(7.40)N NG −1If we further develope (7.38), we have a mean correlation:1⎧ 1GNGNG− λiNG +⎪1 ⋅ 2 ⋅ N NG −1 ⋅ p S ≥ − S i c2⎪11⎪GNGNG+ λiNG −⎪⎪+ 1 ⋅ 2 ⋅ 2 N NG −1 ⋅ p S < − S i cχi = ⎨⎪− 1 ⋅ 1 ⋅ 1NG ⋅ p S G < − S NG c NG − λ NG −ii⎪2 2 N −1⎪⎪− 1 ⋅ 1 ⋅ 1NG ⋅ p S G ≥ − S iNG c NG + λiNG⎪⎩2 2 N −1which can also be expressed as:(( )((( ))( ))(( ))(( )))(( )(7.41))1⎧ 1GNGNG+ λiNG + N NG ⋅ p S G > S iNG c NG − λiNG −⎪⎪ N NG ⋅ p S ≤ S i c2χi = ⎨2⎪− 1NG ⋅ p S G > S NG c NG + λ NG − 1NG ⋅ p S G ≤ S NG c NG − λ NGiiii⎪⎩ 2 N2N(( ))(( ))(7.42)If we have a look now at the next figure representing the probability density function of theGaussian group,Figure 7.3.
Probability density function of the Gaussian sum S G263Signal Multiplex Techniques for GNSSit is clear to recognize thatand⎛ S NG (c NG ) + λiNGp (S G ≤ S iNG (c NG ) + λiNG ) − p (S G > S iNG (c NG ) + λiNG ) = erf ⎜⎜ i2⎝( )⎛ S NG c NG − λiNGp S G > S iNG c NG − λiNG − p S G ≤ S iNG c NG − λiNG = −erf ⎜⎜ i2⎝(( )) (( ))⎞⎟⎟⎠(7.43)⎞⎟ (7.44)⎟⎠Therefore, the mean correlation can be simplified to:χi =12NNG( )⎡ ⎛ S iNG c NG + λiNG⋅ ⎢erf ⎜⎜2⎣⎢ ⎝( )⎞⎛ S NG c NG − λiNG⎟⎟ − erf ⎜ i⎜2⎠⎝⎞⎤⎟⎟⎥⎠⎦⎥(7.45)The expression derived above corresponds to a particular combination of N NG − 1 nonGaussian chips ciNG where all non-Gaussian codes, except for code ciNG , were considered.
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