On Generalized Signal Waveforms for Satellite Navigation (797942), страница 45
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Indeed, thisspectral separation coefficient is of great importance too as it gives an idea of how robust asignal is to interference coming from the same family of signals or from adapted jammers thatmatch its power spectral density.After some manipulation of the original SSC between two arbitrary BCS sequences, it can beshown that the self SSC adopts the following form:SSC SelfBCS = SSC BCS([r ] , f 1 ) − BCS([s ] , f 2 )ccr = s , f c1 = f c2 , n1 = n2 = n⎧n 2 f c2 Ξ ( f c , n, f c , n,0,0 ) +⎪n −1n⎪+ 2nf 2c ∑ ∑ s i ' s j ' Ξ ( f c , n, f c , n,0, j '−i ') +⎪i '=1 j ' = i ' +1⎪n −1 n=⎨22nf+c ∑ ∑ ri r j Ξ ( f c , n, f c , n, j − i,0 ) +⎪i =1 j = i +1⎪n −1nn −1 n⎪2⎪+ 4 f c ∑ ∑ ∑ ∑ ri r j s i ' s j ' Ξ ( f c , n, f c , n, j − i, j '−i ')i ' =1 j ' = i ' +1 i =1 j = i +1⎩(5.27)which is only a function of n, that is the length of the generation vector r = s , and the chipform of the particular BCS sequence as we can clearly recognize.
Moreover, as shown inAppendix O.2, the different sum terms of the SSC simplify to the following expression:⎧n 2 f c2 Ψ (n,0,0 ) +⎪Selfn −1 ⎧n −1SSC BCS⎫([s ], f c ) = ⎨TTT2⎪4 f c ∑ ⎨ns [θ (s , l1 )] Ψ (n, l1 ,0 ) + ∑ s [θ (s , l1 )] s [θ (s , l 2 )] Ψ (n, l1 , l 2 )⎬l1 =1 ⎩l 2 =1⎭⎩(5.28)We show in the next figure as an example the value of the self SSC of the C/A Code as afunction of the integration bandwidth. As we can see, the theoretical prediction matchesperfectly with the results of the simulations even for narrow receiver bandwidths.198Spectral Separation CoefficientsFigure 5.3. Self Spectral Separation Coefficient of C/A Code as a function of theIntegration Bandwidth.
The analytical value is shown in red5.2.4SSC between a generic BCS signal and a sinephased BOC signal(As shown in Appendix O.3, the SSC between a generic BCS signal BCS [r ], f c1sine-phased BOC with chip rate fSSC()BCS [r ], f c1 − BOCsin ( f s2 =2cn2 f c2 2, fc )2and subcarrier frequency f = n f2s22 c) and a2 is:=⎧n1 n2 f c1 f c2 Ξ( f c1 , n1 , f c2 , n2 ,0,0) +⎪n2 −1⎪+ 2n f 1 f 2 (− 1)i (n − i ) Ξ ( f 1 , n , f 2 , n ,0, i ) +1 c c ∑212cc⎪i =1(5.29)⎪n1 −1⎨T1 212⎪+ 2n2 f c f c ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l Ξ ( f c , n1 , f c , n2 , l ,0) +l =1⎪n1 −1n2 −1⎪⎫⎧Ti1 212⎪+ 4 f c f c ∑ ⎨[r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l ∑ (− 1) (n2 − i ) Ξ( f c , n1 , f c , n2 , l , i )⎬l =1 ⎩i =1⎭⎩5.2.5[()][()]SSC between a generic BCS signal and a cosinephased BOC signalEqually, the SSC between a generic BCS signal and a cosine-phased BOC is shown to be:=SSCn f2()BCS [r ], f c1 − BOCcos ( f s2 =2 c4, f c2 )⎧n1 n2 f c1 f c2 Ξ( f c1 , n1 , f c2 , n2 ,0,0) +⎪n2 / 2n2 / 2 −1⎤ii⎪1 2⎡1212()()()()()+−Ξ−+−−Ξ2nff1f,n,f,n,0,2i121n/2if,n,f,n,0,2i∑∑cccccc112212⎢⎥+⎪i =1i =1⎣⎦⎪n1 −1⎪T1 212⎪+ 2n2 f c f c ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l Ξ( f c , n1 , f c , n2 , l ,0) +⎨l =1⎪⎧⎡ n2 / 2⎤⎫i⎪(− 1) Ξ( f c1 , n1 , f c2 , n2 , l ,2i − 1) +⎪∑⎢⎥⎪n−11⎪⎪Ti =11 2⎢⎥ ⎪⎬⎪+ 4 f c f c ∑ ⎨[r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l × n / 2−12⎢⎥⎪l =1 ⎪i⎪12()()()+−−Ξ21n/2if,n,f,n,l,2i⎢⎥⎪∑2c1c2⎪⎪⎩⎣ i =1⎦⎭⎩(5.30)[()][()]199Spectral Separation Coefficients5.2.6SSC between a generic BCS signal and BPSKFinally, the SSC between an arbitrary BCS signal and BPSK simplifies as shown next:SSC BCS([r ], f 1 )− BPSK ( f 2 )cc⎧n1 n2 f c1 f c2 Ξ( f c1 , n1 , f c2 , n2 ,0,0) +⎪n2 −1⎪+ 2n f 1 f 2 (n − i ) Ξ( f 1 , n , f 2 , n ,0, i ) +1 c c ∑2c1c2⎪i =1⎪n1 −1=⎨T1 212⎪+ 2n2 f c f c ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l Ξ( f c , n1 , f c , n2 , l ,0) +l =1⎪n1 −1n2 −1⎪⎫⎧T1 212⎪+ 4 f c f c ∑ ⎨[r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l ∑ (n2 − i ) Ξ ( f c , n1 , f c , n2 , l , i )⎬l =1 ⎩i =1⎭⎩[()][()](5.31)5.2.7MBOC Theoretical SSCsAs shown in chapter 3.7, the selection of MBOC(6,1,1/11) was the result of long researchworks carried out by Working Group A members of the EU and US.
Given the importancethat the spectral separation of MBOC played at that time, we show next the analyticalexpressions for the SSC between an arbitrary BCS signal and MBOC(6,1,1/11). The driver ofstudying this variable is that if a new signal is proposed in the future for this band, the spectralseparation with the MBOC(6,1,1/11), as well as with the rest of signals, will have to becarefully studied to make sure that the new signal would be compatible with the GalileoE1 OS and GPS L1C.
It is important to note that this potential signal does not necessarilyhave to be from GPS and Galileo, but could be from any other system.MBOC(6,1,1/11) is a particular case of the MBCS signals analyzed in chapter 3.6.2 andrepresents the multiplexing of BOC(1,1) and BOC(6,1) with 1/11 of power on the highfrequency BOC component. As we have shown in chapter 4, while the CBOC implementationreaches this percentage modifying the amplitude of the modulation in the time domain,TMBOC obtains the desired percentage by repeating the BOC(6,1) component in the timedomain correspondingly.To calculate the SSC between a given signal and MBOC, we calculate separately the SSCwith BOC(1,1) and BOC(6,1) given that the multiplexed GPS L1C and Galileo E1 OS signalsare a lineal combination of both.
General expressions were derived in Appendix O.In the case of BOC(6,1), the BCS vector is shown to be s = [1,-1,1,-1,1,-1,1,-1,1,-1,1,-1] andfrom its M generation matrix, it can be shown that the SSC between an arbitrary BCS signalBCS( [r ], f c1 ) and BOC(6,1) can be expressed as follows:SSC BCS ([r ], f 1 )− BOC(6,1) = SSC 1 + SSC 2 + SSC 3 + SSC 4c(5.32)200Spectral Separation Coefficientswhere:()(SSC 1 n1 , f c1 = n1 n 2 f c1 f c2 Ξ f c1 , n1 , f c2 , n 2 ,0,0SSC 2 n1 , f c1()()()() () () (] [θ ([r , r ,...r ], l )] Ξ( f(((n1 −1l =1())()⎡− 11 Ξ f c1 , n1 , f c2 , n 2 ,0,1 + 10 Ξ f c1 , n1 , f c2 , n 2 ,0,2 − 9 Ξ f c1 , n1 , f c2 , n 2 ,0,3 + ⎤⎥⎢+ 8 Ξ f c1 , n1 , f c2 , n 2 ,0,4 − 7Ξ f c1 , n1 , f c2 , n 2 ,0,5 + 6 Ξ f c1 , n1 , f c2 , n 2 ,0,6 − ⎥1 2⎢= 2n1 f c f c ⎢⎥121212⎢− 5 Ξ f c , n1 , f c , n 2 ,0,7 + 4 Ξ f c , n1 , f c , n 2 ,0,8 − 3 Ξ f c , n1 , f c , n 2 ,0,9 + ⎥⎥⎦⎢⎣+ 2 Ξ f c1 , n1 , f c2 , n 2 ,0,10 − Ξ f c1 , n1 , f c2 , n 2 ,0,11SSC 3 n1 , f c1 = 2n 2 f c1 f c2 ∑ [r1 , r2 ,...rn1SSC 4 n1 , f c1)T12n1)))1c, n1 , f c2 , n 2 , l ,0((()))))))))([()]((((((((()))(5.33)withf c2 =1.023MHzn2 = 12(5.34)In a similar way the parameters for the SSC with BOC(1,1) could be derived.5.2.8)⎧− 11 Ξ f c1 , n1 , f c2 , n 2 , l ,1 + 10 Ξ f c1 , n1 , f c2 , n 2 , l ,2 −⎫⎪⎪1212⎪− 9 Ξ f c , n1 , f c , n 2 , l ,3 + 8 Ξ f c , n1 , f c , n 2 , l ,4 − ⎪⎪⎪1212n1 −1T ⎪− 7 Ξ f c , n1 , f c , n 2 , l ,5 + 6 Ξ f c , n1 , f c , n 2 , l ,6 − ⎪1 2= 4 f c f c ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l ⎨⎬1212l =1⎪− 5 Ξ f c , n1 , f c , n 2 , l ,7 + 4 Ξ f c , n1 , f c , n 2 , l ,8 − ⎪⎪− 3 Ξ f 1 , n , f 2 , n , l ,9 + 2 Ξ f 1 , n , f 2 , n , l ,10 − ⎪c1c2c1c2⎪⎪12⎪⎭⎪⎩− Ξ f c , n1 , f c , n 2 , l ,11Analytical Power of a generic BCS signal in a givenBandwidth βrOnce we have the general expression for the SSC of any possible combination of signals, weconcentrate now on another figure: namely the power that falls in a given bandwidth.
Indeed,sometimes SSCs are not computed in an infinite but in a finite bandwidth and the powerspectral densities are normalized to the transmission bandwidth. As shown in Appendix O.5,the general expression of the power of a BCS signal is shown to be given by:⎧⎡⎛ πβ ⎞⎤⎛ πβ ⎞2 ⎢− 1 + cos⎜⎜ r ⎟⎟⎥ 2 π Si⎜⎜ r ⎟⎟⎪⎪⎝ f c n ⎠⎦⎣⎝ fcn ⎠+⎪βrfcn⎪nf c+2π⎪⎪⎡⎛ β π ⎞ ⎛ β kπ ⎞ ⎤⎛ β r kπ ⎞⎪⎟⎟ + 2 f c n cos⎜⎜ r ⎟⎟ cos⎜⎜ r ⎟⎟ + ⎥⎢− 2 f c n cos⎜⎜P=⎨⎝ fcn ⎠ ⎝ fcn ⎠ ⎥⎝ fcn ⎠⎢⎪⎢⎥n −1 n⎪⎡ β π (k − 1)⎤⎡ β r πk ⎤2⎢s s β r (k − 1)π Si ⎢⎪+⎥ − 2β r kπ Si ⎢⎥+ ⎥2 ∑ ∑ i j⎢⎥fnfnβπni=1j=i+1c⎣⎦⎣ c ⎦⎪r⎥⎢⎪⎡ β r π (1 + k )⎤⎢⎥⎪⎥⎢+ β r (1 + k )π Si ⎢⎥⎪fcn ⎦⎣⎣⎦ k = j −i⎩(5.35)201)Spectral Separation CoefficientsIn the next pages the spectral separation coefficients of different signals of interest are shownin detail.
To calculate them, the general SSC definition as well as the expressions derived inprevious chapters will be employed to study the effects of filtering and normalization withrespect to the transmission bandwidth. It must be noted that the bandwidth must be readexactly as it appears on the table. Thus 24 MHz must be interpreted as 24.00 MHz. For therest a factor 1.023 is explicitly multiplying.5.2.9Efficiency ParametersTogether with the SSC derivations of previous chapters, we present here some other figures ofinterest that help in understanding the power distribution profile of a particular signal.As we saw in (4.13), sometimes it is of interest to normalize the power spectral density to thetransmission bandwidth βT to compensate for the power that goes lost outside the transmissionbandwidth.
As we saw there, to keep the transmitted power constant after applying filtering,the power has to be raised by the corresponding factor:ε =∫βT2β− T2G ( f )df(5.36)This parameter indicates how much of the PSD normalized in an infinite bandwidth fallswithin a given bandwidth and thus the correction that must be made to have a normalizedpower spectral density of 1 W within the transmission bandwidth. The more energy the signalconcentrates at low frequencies, the less significative will be the necessary correction as table5.1 next clearly shows for different signals. As we can recognize, the parameter ε isequivalent to P in (5.35).Once we have realized the necessary correction on the normalized PSD to accomplish thedesired power emission, we can see that in general the receiver bandwidth will be muchnarrower than the emission bandwidth, resulting thus in an additional reduction of power atuser level versus that emitted by the satellite.
This is a figure of great importance for thecorrect design of a system, since as we have seen, all the power specifications are given atuser level and correspondingly the powers must be adjusted in the satellite. To the object ofanalyzing this effect for different signals, we define an additional efficiency parameter:η=∫βR2β− R2G ( f )dfwhereG( f )G(f )=∫βT2−βTG ( f )df(5.37)(5.38)2This parameter indicates how much power falls within the receiver bandwidth. In the nexttable, the above defined efficiency parameters are calculated for the GPS and Galileo signalsin E1/L1. For simplicity, the same bandwidths as in [S.
Wallner et al., 2005] are assumed.202Spectral Separation CoefficientsTable 5.1. Efficiency Parameters of GPS L1 signalsBPSK(1)BPSK(10) BOC(10,5)BOC(1,1)MBOCTX BW [MHz]30.6930.6930.6930.6930.69RX BW [MHz]24.0024.0024.0024.0024.00εη-0.0294-0.3101-0.8023-0.0890-0.1545-0.0077-0.1216-0.3727-0.0222-0.0207Table 5.2. Efficiency Parameters of Galileo E1 signalsBOC(1,1) MBOCBOCcos(15,2.5)TX BW [MHz]40.9240.9240.92RX BW [MHz]24.0024.0040.92εη-0.0665-0.0989-1.1069-0.0448-0.0762-∞As we can recognize, Galileo presents better values than GPS due to the wider transmissionand receiver bandwidth that we have assumed. These are indeed standard figures that can befound today in the hardware specifications of any receiver and signal generator manufacturer.Furthermore, for GPS a bandwidth of 30.69 MHz was selected so as to have the main lobe ofthe widest signal inside, namely the M-Code.
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