On Generalized Signal Waveforms for Satellite Navigation (797942), страница 44
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On the other hand, for the interesting case of non-white interference the output isshown to be:Ψid = TI ∫∞−∞∫∞−∞Gi ( f1 )Gd ( f 2 − f1 ) df1 H ID ( f 2 ) df 22(5.8)For the case that the multiplier output Power Spectral Density is sufficiently flat across theI&D filter, that is for very long pseudorandom noise codes, the computation of the spectralseparation is shown to approximate to the following expression:∞Ψid ≈ TI ∫ Gi ( f 2 ) ⊗ G d ( f 2 ) H ID ( f 2 ) df 2−∞= ∫ Gi ( f )G d ( f )df∞2f =0−∞(5.9)where ⊗ represents the convolution operator. Moreover, if we include the effect of thefiltering at the receiver, the simplified SSC will adopt the following form:κ id = ∫βr / 2−βr / 2Gi ( f )Gd ( f ) H RX ( f ) df2(5.10)where• Gi ( f ) stands for the normalized power spectral density of the aggregate interference,••Gd ( f ) is the normalized power spectral density of the desired signal, andH RX ( f ) is the receiver transfer functionThis equation is not applicable when the desired signal and aggregate interference areline-like.
In the next chapters we will work with this approximation but in chapter 6 we willanalyze the particular case of the C/A code where the short integrations make thisapproximation incorrect.The inner product is not generally applicable when desired signals and interference haveline-like spectra. Therefore, for each particular problem we have to see if the multiplier outputPSD is smooth enough to justify the approximation above.Furthermore, if we normalize the power spectral densities of the desired and interfering193Spectral Separation Coefficientssignals as defined in (4.13), the SSC adopts the following form for the case of an infiniteintegration time, as a function of Doppler:κ id (Δf ) = ∫βr / 2Gi ( f − Δf )Gd ( f ) H RX ( f ) df2−βr / 2(5.11)where Δf indicates the Doppler difference at receiver level between the desired signal and theinterfering signal.
In addition, if an ideal filter of amplitude 1 is considered, we can express(5.11) also as follows:κ id (Δf ) = Gi ( f ) ⊗ Gd ( f ) = ℜG ,G (Δf )id(5.12)Once we have derived a general expression for the Spectral Separation Coefficient and founduseful approximations, we further analyze the cross power spectra terms that we have alreadyanticipated in chapter 4. As we have seen in (5.4), there is a fundamental element of the SSCthat is defined as:Gid ( f ) = Gi ( f )Gd ( f )(5.13)This expression is of great interest since it will help us in finding the main interactionsbetween the interfering spectrum and the desired signal spectrum by simply observing itsbehaviour in the frequency domain. Moreover, if we expand (5.13) expressing the powerspectral density of each of the signals in terms of their autocorrelation, we see that:∞∞∞−∞−∞−∞Gid (ω ) = ∫ ℜ si (τ )e − jωτ dτ ∫ ℜ s d (τ )e − jωτ dτ = ∫∫∞−∞ℜ si (τ 1 )ℜ s d (τ 2 )e − jω (τ 1 +τ 2 ) dτ1dτ 2(5.14)and thereforeℜ si , s d (η ) =12π∞∞∞−∞−∞−∞∫ ∫ ∫ℜ si (τ 1 )ℜ s d (τ 2 )e − jω (τ 1 +τ 2 −η ) dτ1 dτ 2 dω(5.15)which is shown [A.R.
Pratt and J.I.R. Owen, 2003a] to reduce to the simplified form:∞ℜ si , s d (η ) = ∫ ℜ si (τ 1 )ℜ s d (η − τ 1 )dτ1 = ℜ si (η ) ⊗ ℜ s d (η )−∞(5.16)In other words, the autocorrelation function of the cross power is the convolution of theinterfering and desired signals. In addition, applying the Weiner-Kinchine spectrum theoremas shown in [A.R. Pratt and J.I.R.
Owen, 2003a], yields:∞ℜ si , sd (η ) = ∫ Gid ( f ) e j 2πfη df−∞(5.17)Using this expression, we can calculate the cross SSC, namely κ id , in the same manner:∞κ id = ℜ s , s (0) = ∫ ℜ s (τ 1 )ℜ s (− τ 1 )dτ1id−∞id(5.18)194Spectral Separation Coefficients5.1.1SSC between two QPSK signalsThe spectral separation coefficient between two signals that we have derived above appliesfor the case that both signals are in-phase. Although this is a common case, many of theGNSSes today have also navigation signals that are in quadrature with each other, sharing thesame frequency.
It is thus necessary to find an expression for this important case too.Fortunately, the results that were obtained in the previous chapter can be reutilized here.Indeed, as shown in Appendix N and by [J.-L. Issler et al., 2003], the SSC between twoQPSK navigation signals sharing the same frequency can be expressed as the sum of 4 SSCsof two BPSK elementary signal components, according to the expression below:()()()(SSC (s I , sQ ) = SSC s ID , sQD + SSC s IP , sQP + SSC s ID , sQP + SSC s IP , sQD)(5.19)where:•SSC(s I , sQ ) is the SSC between the interfering in-phase signal sI and the quadrature•signal sQ.SSC(s ID , sQD ) is the SSC between the data component of the interfering signal and the•data component of the quadrature signal.SSC(s IP , s QP ) is the SSC between the pilot component of the interfering signal and the•pilot component of the quadrature signal.SSC(s ID , s QP ) is the SSC between the data component of the interfering signal and the•pilot component of the quadrature signal.SSC(s IP , s QD ) is the SSC between the pilot component of the interfering signal and thedata component of the quadrature signal.5.2Derivation of analytical expressionsIn the next lines analytical expressions for the SSC with an infinite period of integration, andthus no Doppler error, will be presented.
Moreover, we will further assume ideal codeproperties. Later in chapter 6 we will revisit our assumptions and we will assess the effect ofreal codes on the measured SSC. Equally, the impact of the data on the spectral fine structureof the navigation signals will be investigated.195Spectral Separation Coefficients5.2.1SSC between two generic BCS signalsAs demonstrated in Appendix O, the Spectral Separation Coefficient between two arbitraryBCS signals, namely BCS [r ], f c1 and BCS [s ], f c2 , can be expressed as follows:()()()⎧n1 n 2 f c1 f c2 Ξ f c1 , n1 , f c2 , n 2 ,0,0 +⎪n2 −1 n212⎪+ 2 n f 1 f 21 c c ∑ ∑ s i ' s j ' Ξ f c , n1 , f c , n 2 ,0, j '−i ' +⎪i '=1 j '= i ' +1⎪⎪n1 −1 n1SSC BCS([r ], f 1 ) − BCS([s ], f 2 ) = ⎨1 212cc+2nff2 c c ∑ ∑ ri r j Ξ f c , n1 , f c , n 2 , j − i,0 +⎪i =1 j = i +1⎪1n−n2 n1 −1 n12⎪1 212⎪+ 4 f c f c ∑ ∑ ∑ ∑ ri r j si ' s j ' Ξ f c , n1 , f c , n 2 , j − i, j '−i '⎪⎩i '=1 j '=i ' +1 i =1 j = i +1()()()(5.20)where we have assumed that the receiver integrates for an infinite period of time, there is noDoppler error and the receiver has a front-end with infinite bandwidth.
The functionΞ f c1 , n1 , f c2 , n2 , j − i, j '−i ' is defined in (O.15) of a Appendix O. Moreover, n1 is the length ofthe generation vector r = r1 , r2 ,..., rn1 of the first BCS signal and n2 the length of the(){{generation vector s = s1 , s2 ,..., sn2}} of the second BCS signal.As we commented in chapter 4.7, during the optimization of the Open and Civil Signals inE1/L1, the spectral overlay with the rest of signals sharing the band was one of the mostimportant constraints to take into account. Therefore, the spectral separation coefficient of theselected MBOC signal with the C/A Code, M-Code and PRS was a figure of majorimportance.
Now that we have derived the general expression for the SSC between two BCSsignals, we will particularize the expression to derive some cases of interest. As one canimagine, this reduces time and computation since the function Ξ f c1 , n1 , f c2 , n2 , j − i, j '−i ' can()be saved in tensorial form for all possible input parameters. Then, calculating an SSC wouldbe just a matter of reading from the tensor and combining the outputs according to (5.20).5.2.2SSC between a generic BCS signal and the M-CodeBefore deriving the general expression of the SSC between a generic BCS signal and theM-Code, let us begin with the example of the SSC between BOC(1,1) and the GPS M-Code.As we know, BOC(1,1) can be expressed as BCS([1 -1],1) and BOC(10,5) asBCS([1 -1 1 -1],5).
Moreover, following the matrix notation of chapter 4.3, we can furtherexpress our signals of study as follows:⎛1{0} − 1{1}⎞2⎟M BOC(1,(5.21)1) ( [+ 1,−1] ) = ⎜⎜1{0}⎟⎠⎝and for the M-Code we have equally:196Spectral Separation Coefficients⎛1{0} − 1{1} 1{2} − 1{3}⎞⎜⎟{}{}{}101112−⎜⎟M M2 -Code ( [+ 1,−1,+1,−1] ) = ⎜1{0} − 1{1}⎟⎜⎟⎜1{0}⎟⎠⎝(5.22)From the matrices above we can extract the following parameters:s1 = [+1 - 1]s 2 = [+1 - 1 + 1 - 1]f c1 = 1.023 MHz n1 = 2f c2 = 5.115 MHz n2 = 4(5.23)and therefore, we can express the SSC between BOC(1,1) and M-Code as follows:∞SSC BOC(1,1)− BOC(10,5) = ∫ GBOC(1,1) ( f )GBOC(10,5) ( f ) df(5.24)−∞with(())⎧⎛ − 3Ξ f c1 , n1 , f c2 , n 2 ,0,1 + ⎞ ⎫⎜⎟ ⎪⎪1 2121 212⎪n1 n 2 f c f c Ξ f c , n1 , f c , n 2 ,0,0 + 2n1 f c f c ⎜ + 2Ξ f c , n1 , f c , n 2 ,0,2 − ⎟ −⎪⎜⎟ ⎪⎪⎜ − Ξ f c1 , n1 , f c2 , n 2 ,0,3⎟⎪⎝⎠ ⎪SSC BOC(1,1)− BOC(10,5) = ⎨⎬12⎛ − 3Ξ f c , n1 , f c , n 2 ,1,1 + ⎞ ⎪⎪⎜⎟⎪1 2121 212⎜⎟ ⎪⎪− 2n 2 f c f c Ξ f c , n1 , f c , n2 ,1,0 − 4 f c f c ⎜ + 2Ξ f c , n1 , f c , n 2 ,1,2 − ⎟ ⎪⎪⎜ − Ξ f c1 , n1 , f c2 , n 2 ,1,3⎟ ⎪⎝⎠ ⎭⎩(5.25)As it can be shown, the theoretical value obtained from the expression above is in this caseSSC BOC (1,1) −BOC (10,5) = −83.1091 dB-Hz.
We can graphically show the validity of the result()()(((())))obtained above if we numerically compute the SSC between BOC(1,1) and the M-Codeincreasing the bandwidth of integration progressively. As we can see, our analytical valuematches very well the results of the simulations.Figure 5.2. Spectral Separation Coefficient between BOC(1,1) and M-Code as a functionof the Integration Bandwidth. It must be noted that the PSD of both signals isnormalized to 1 W in an infinite bandwidth. The analytical value is shown in red197Spectral Separation CoefficientsIn general, the SSC between an arbitrary signal and the M-Code can be further expressed asfollows, according to (O.46) in Appendix O:SSC BCS([r ], f 1 )-BOC(10,5 )c⎧ 1 2 ⎡n2 Φ(n1 ,0,0) +⎤⎪n1 f c f c ⎢⎥+⎪⎣2 [− 3 Φ (n1 ,0,1) + 2 Φ (n1 ,0,2) − Φ (n1 ,0,3)]⎦=⎨n −1⎤⎪2 f 1 f 2 1 r θ r , l T ⎡n2 Φ(n1 , l ,0) +⎢+ 2 {− 3Φ(n , l ,1) + 2 Φ(n , l ,2) − Φ (n , l ,3)}⎥⎪ c c ∑l =1111⎣⎦⎩[ ( )](5.26)where Φ (n, l1 , l2 ) = Ξ ( f c , n, f c , n, l1 , l2 ) as shown in detail in the Appendix.5.2.3Self SSC of a generic BCS signalAfter having obtained an analytical expression for the SSC between a BCS signal and theM-Code we solve next the case of the SSC of a generic BCS signal with itself.
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