On Generalized Signal Waveforms for Satellite Navigation (797942), страница 79
Текст из файла (страница 79)
Furthermore, itcan be shown that the SSC between an arbitrary BCS signal and an arbitrary cosine-phasedBOC adopts the following form:SSCBCS([r ], f 1 )− BOC ( n , f 2 ) =c(cosc2)⎧n n f f Ξ f , n1 , f , n2 ,0,0 +⎪n2 / 2n 2 / 2 −1⎤⎪ii1 2⎡12()2nff1f,n,f,n,0,2i12(− 1) (n2 / 2 − i )Ξ f c1 , n1 , f c2 , n2 ,0,2i ⎥ ++−Ξ−+∑1 c c ⎢∑12cc⎪i =1⎣ i =1⎦⎪1−n1⎪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 / 2i⎪(− 1) Ξ f c1 , n1 , f c2 , n2 , l ,2i − 1 +⎪∑⎥⎪⎢1n−1⎪⎪Ti =11 2⎥ ⎪⎬⎢⎪+ 4 f c f c ∑ ⎨[r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l × n / 2 −12⎥⎪⎢l =1 ⎪⎪i12()()21n/2if,n,f,n,l,2i+−−Ξ⎥⎪⎢∑212cc⎪⎪⎦⎭⎣ i =1⎩⎩1 21 2 c c1c2c()[([(()] ()]))()()(O.34)385Spectral Separation Coefficient between a generic BCS signal and an arbitrary BOC or BPSK signalEqually, the SSC between an arbitrary BCS signal with code rate f c1 and a BPSK with coderate f c2 is shown to be:()⎧n1n2 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 +cc1 c c ∑212⎪i =1⎪n1 −1=⎨T1 22nff+Ξ f c1 , n1 , f c2 , n2 , l ,0 +2 c c ∑ [ r1 , r2 ,...rn1 ] θ [ r1 , r2 ,...rn1 ], l⎪l =1⎪n1 −1n2 −1⎪⎧⎫T1 2(n2 − i )Ξ f c1 , n1 , f c2 , n2 , l , i ⎬4ff[r,r,...r]θ[r,r,...r],l+⎪∑c c ∑⎨ 1 2n11 2n1l =1 ⎩i =1⎭⎩(SSC BCS([r ], f 1 )− BPSK ( f 2 )cc)[()] ([()])()(O.35)386Spectral Separation Coefficient between a generic BCS signal and the M-CodeO.4Appendix.
SSC between a generic BCS signaland the M-CodeOnce we have derived the analytical expression for the SSC between two generic BCS signalsin Appendix O.1, we show next some particular expressions. For exemplification the SSCbetween an arbitrary BCS( [r ], f c1 ) and the M-Code is presented next. This SSC is shown tobe given by:SSC BCS( [r ], f 1 )− BOC(10,5) = SSC BCS1 − BOC(10,5) = SSC1 + SSC 2 + SSC3 + SSC 4cwhere:(O.36)SSC1 = n1n2 f c1 f c2 Ξ( f c1 , n1 , f c2 , n2 ,0,0 )n2 −1SSC 2 = 2n1 f c1 f c2 ∑∑s s Ξ(fn2i '=1 j '=i ' +1i' j '1c, n1 , f c2 , n2 ,0, j '−i ')n1 −1 n1SSC3 = 2n2 f c1 f c2 ∑ ∑ ri rj Ξ ( f c1 , n1 , f c2 , n2 , j − i,0 )(O.37)i =1 j =i +1SSC 4 = 4 f f1 2c cn2 −1n1 −1 n1∑ ∑ ∑ ∑rr s s Ξ(fn2i '=1 j '=i ' +1 i =1 j =i +1i j i' j'1c, n1 , f c2 , n2 , j − i, j '−i ')As we can recognize, the first sum term is only a variable of n1 and f c1 , if we assume that f c2and n2 are given.
In fact, since the second signal is the M-Code BOC(10,5), we can simplify(O.37) and express the total SSC exclusively as a function only of n1 and f c1 as shown next.Let us begin with the first term of the SSC1:()(SSC1 n1 , f c1 = n1n2 f c1 f c2 Ξ f c1 , n1 , f c2 , n2 ,0,0)fc2 =5.115 MHz,n2 =4(O.38)Similarly, if we observe now the second sum term SSC2, we see that once the chip form of thesecond signal has been fixed, this term only depends on the variable n1.
In order to acceleratethe computation of SSCs we can calculate in advance this term as a function of n1 given thatthe second signal is BOC-like, according to the following expression:[]SSC 2 (n1 , f c1 ) = 2n1 f c1 f c2 − 3 Ξ( f c1 , n1 , f c2 , n2 ,0,1) + 2 Ξ( f c1 , n1 , f c2 , n2 ,0,2 ) − Ξ ( f c1 , n1 , f c2 , n2 ,0,3)(O.39)where we can easily recognize that the coefficients of the terms in brackets, namely -3, +2 and-1, correspond to the values of the generation matrix of BOC(10,5) shown in (5.22)The third sum term, namely SSC3, depends on the signal structure of the first signal.
Thus, forevery k1 = j − i and every n1 we will compute the value as a function of k1 and n1. Wedevelop now this term to find an expression with which we can calculate everything moreeasily in a numerical way:n1 −1 n1SSC3 (n1 , f c1 ) = 2n2 f c1 f c2 ∑ ∑ ri rj Ξ( f c1 , n1 , f c2 , n2 , j − i,0)(O.40)i =1 j =i +1Indeed,387Spectral Separation Coefficient between a generic BCS signal and the M-Code((SSC3 n1 , f c1))⎧[r1 , r2 ,...rn1 ][0, r1 , r2 , r3 ...rn1 −1 ] T Ξ f c1 , n1 , f c2 , n2 ,1,0 +⎪T12⎪+ [r1 , r2 ,...rn1 ][0,0, r1 , r2 ...rn1 − 2 ] Ξ f c , n1 , f c , n2 ,2,0 +1 2⎪=2n2 f c f c ⎨T12⎪+ [r1 , r2 ,...rn1 ][0,0,0, r1...rn1 − 3 ] Ξ f c , n1 , f c , n2 ,3,0 + ...⎪12T⎪⎩+ [r1 , r2 ,...rn1 ][0,0,0,0...r1 ] Ξ f c , n1 , f c , n2 , n1 − 1,0 + ... f 2 = 5.115 MHz, n = 4c2((()))(O.41)As we can see, we can express the double sum of (O.40) as the single sum of n1-1 terms, whatfor the numerical calculations simplifies significantly the problem.
As a conclusion, the thirdterm of the SSC is only a function of n1. Moreover, we can further simplify as follows:([(n1 −1)SSC3 n1 , f c1 = 2n2 f c1 f c2 ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l)] Ξ( fTl =11c, n1 , f c2 , n2 , l ,0)f c2 =5.115 MHz,n2 =4(O.42)In a similar way, making use of the operator θ , we can now develop the fourth SSC term tohave a more compact expression as follows:⎧− 3Ξ ( f c1 , n1 , f c2 , n2 , l ,1) + ⎫n1 −1⎪T⎪SSC 4 (n1 , f c1 ) = 4 f c1 f c2 ∑ [r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], l ⎨+ 2Ξ( f c1 , n1 , f c2 , n2 , l ,2 ) − ⎬l =1⎪⎪12⎩− Ξ( f c , n1 , f c , n2 , l ,3) ⎭ 2[()]f c =5.115 MHz,n2 =4(O.43)or equivalently:n1 −1 n1⎧12⎪(s1s2 + s2 s3 + s3 s4 )∑ ∑ ri rj Ξ f c , n1 , f c , n2 , j − i,1 +i =1 j =i +1⎪n1 −1 n1⎪⎪= 4 f c1 f c2 ⎨+ (s1s3 + s2 s4 )∑ ∑ ri rj Ξ f c1 , n1 , f c2 , n2 , j − i,2 +i =1 j =i +1⎪n1 −1 n1⎪⎪+ s1s4 ∑ ∑ ri rj Ξ f c1 , n1 , f c2 , n2 , j − i,3i =1 j =i +1⎩⎪f 2 =5.115 MHz,n =4((SSC 4 n1 , f c1))(())c2(O.44)If we recall now all the expressions derived in the preceding lines and group them together,we can see that the SSC between a generic BCS signal and the M-Code will be:388Spectral Separation Coefficient between a generic BCS signal and the M-Code(SSC BCS( [r ], f 1 )− BOC(10,5)c)⎧⎧n2Ξ f c1 , n1 , f c2 , n2 ,0,0 +⎫⎪⎪⎪⎪ 1 2 ⎪ ⎡− 3Ξ f c1 , n1 , f c2 , n2 ,0,1 + ⎤ ⎪⎥⎬ +⎪n1 f c f c ⎨ ⎢12⎪+ 2 ⎢+ 2Ξ f c , n1 , f c , n2 ,0,2 −⎥ ⎪⎪⎥⎪⎪ ⎢− Ξ f 1 , n , f 2 , n ,0,3⎪12cc⎦⎭⎩ ⎣⎪=⎨⎧n2 Ξ f c1 , n1 , f c2 , n2 , l ,0 +⎫⎪⎪⎪⎪12n −1⎪+ 2 f 1 f 2 1 [r , r ,...r ] θ [r , r ,...r ], l T ⎪ ⎡− 3 Ξ f c , n1 , f c , n2 , l ,1 + ⎤ ⎪⎥⎬⎨ ⎢1 2c c ∑ 1 2n1n1⎪+ 2⎢+ 2 Ξ f c1 , n1 , f c2 , n2 , l ,2 −⎥ ⎪l =1⎪⎪⎥⎪⎪ ⎢− Ξ f 1 , n , f 2 , n , l ,3⎪c1c2⎦⎭⎩ ⎣⎩((()))([()]((())))(O.45)In order to simplify the notation a little bit more, we can do Φ(n, l1 , l 2 ) = Ξ( f c , n, f c , n, l1 , l 2 )and we can further simplify the expression as follows:(SSCBCS( [r ], f 1 )− BOC(10,5) n1 , f c1c)⎧ 1 2 ⎡n2Φ(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 ⎡n2Φ(n1 , l ,0 ) +∑cc⎢⎥⎪l =1⎣+ 2{− 3 Φ(n1 , l ,1) + 2 Φ(n1 , l ,2) − Φ(n1 , l ,3)}⎦⎩[ ( )](O.46)389Spectral Separation Coefficient between a generic BCS signal and the M-Code390Power of a generic BCS signal within a Bandwidth βrO.5Appendix.
Power of a generic BCS signalwithin a Bandwidth βrIn this Appendix analytical expressions for the power that falls within a bandwidth βr arederived. Recalling (4.26), the power spectral density of a generic BCS signal is defined as:⎛ πf ⎞⎟⎟sin 2 ⎜⎜n −1 nnfωT ⎤ ⎫⎡⎝ c ⎠ ⎧n +GBCS([s ], f c ) ( f ) = f c2si s j cos ⎢( j − i ) c ⎥ ⎬ = GSubchip pulse ( f )GMod ( f )⎨∑∑2n ⎦⎭(πf ) ⎩ i =1 j =i +1⎣(O.47)and integrating the PSD of a generic BCS in a bandwidth βr, the total power that comesthrough the filter can be expressed as follows:⎛ πf ⎞⎟⎟sin 2 ⎜⎜βrβrn −1 nnfωT ⎤ ⎫⎡c⎝⎠ ⎧n +P = ∫ 2β r GBCS([s ], f c ) ( f )df = ∫ 2β r f c2si s j cos ⎢( j − i ) c ⎥ ⎬ df = P1 + P2⎨∑∑2−−n ⎦⎭(πf ) ⎩ i =1 j =i +1⎣22(O.48)or equivalently:⎛ πf ⎞⎛ πf ⎞⎟⎟⎟sin 2 ⎜⎜sin 2 ⎜⎜βrβrn −1 nnf c ⎠nf c ⎟⎠ωT ⎤⎡⎝⎝22P = nf c ∫ β rdf + 2 f c ∑ ∑ si s j ∫ β rcos ⎢( j − i ) c ⎥ df(O.49)22−−n()()ππff⎦⎣11==+iji22where the first integration, namely P1 , is shown to converge to the following expression:βr2P1 =−with∫βr⎛ πf ⎞⎟⎟sin 2 ⎜⎜⎝ nf c ⎠ df =(πf )2⎡⎛ πβ ⎞⎤2 ⎢− 1 + cos⎜⎜ r ⎟⎟⎥⎝ f c n ⎠⎦⎣βπ2⎛ πβ ⎞2π Si ⎜⎜ r ⎟⎟⎝ fc n ⎠+fcn(O.50)2Si ( z ) = ∫z0sin (t )dtt(O.51)Moreover, it can also be shown that in the limit, when the integration bandwidth tends toinfinity, (O.50) simplifies to:⎛ πf ⎞βr⎟sin 2 ⎜⎜2nf c ⎟⎠1⎝(O.52)lim P1 = lim ∫df =2β r →∞β r →∞nf c(πf )βr−In fact,βr2−∫βr22⎛ πf ⎞⎟⎟sin 2 ⎜⎜⎝ nf c ⎠ df = 1nπf c(πf )2nf c β r2π∫nf β− c r2πsin 2 ( x )dxx2where using partial integration, it can be shown that:b2bsin 2 ( x )1 21 2sin ( x )()()dx=−sinb+sina+dx∫a x 2∫bax2a(O.53)(O.54)391Power of a generic BCS signal within a Bandwidth βrfor two generic integration limits.
Moreover, since∞∞sin 2 ( x )sin 2 (x )dx=2∫ x2∫0 x 2 dx−∞and given the fact that using complex integration we have∞sin ( x )π∫0 x dx = 2the integration limit can be simplified as follows⎛ πf ⎞nf c β rβr⎟⎟sin 2 ⎜⎜2π2nfsin 2 (x )1⎝ c ⎠ df = 1 limlim P1 = lim ∫dx =22∫β r →∞β r →∞β→∞nπf c rxnf c(πf )βrnf c β r−−2(O.55)(O.56)(O.57)2πsince only the integral term of (O.54) is different than zero when a → 0 and b → ∞ .Equally, we can find an explicit expression for the second integration of (O.49):⎛ πf ⎞⎟⎟sin 2 ⎜⎜βrnfc⎝⎠ cos ⎡ 2πkfTc ⎤ dfP2 = ∫ 2β r2⎢⎣ n ⎥⎦−(πf )2(O.58)which can be further simplified as follows:⎡⎤⎛ βr k π ⎞⎛ β π ⎞ ⎛ β kπ ⎞⎟⎟ + 2 f c n cos⎜⎜ r ⎟⎟ cos⎜⎜ r⎟⎟ +⎢− 2 f c n cos⎜⎜⎥⎝ fc n ⎠⎝ fc n ⎠ ⎝ fc n ⎠⎢⎥⎢⎡⎤⎡⎤⎥⎡⎤⎢ β r (k − 1)π Si ⎢ β r π (k − 1) ⎥ − 2β r k π Si ⎢ β r π k ⎥ + β r (1 + k )π Si ⎢ β r π (1 + k ) ⎥ ⎥fc nfcn⎢⎣⎦⎣ fc n ⎦⎣⎦ ⎥⎦P2 = ⎣2βr fc n π(O.59)where the change k = j − i was made for simplicity in the notation.
If we calculate now thelimit when the bandwidth of integration tends to infinity, we obtain the simplified expression:⎛ πf ⎞⎟⎟sin 2 ⎜⎜βrnfc⎝⎠ cos ⎡ 2 π k f Tc ⎤ df = k − 1 − 2 k + k + 1(O.60)lim P2 = lim ∫ 2β r2⎥⎦⎢⎣β r →∞β r →∞ −2nfn()πfc2Moreover, since k ≥ 1 always, it can be shown that the following relationship for P2 is validfor any k in the limit when βr tends to infinity:k −1 − 2 k + k +12nf ck − 1 − 2k + k + 1=02nf c=(O.61)Finally, combining now (O.50) and (O.59) above, we have the following analyticalexpressions for the integration of the PSD of a BCS in a finite bandwidth βr:P = P1 + P2 = ∫βr2−βrGBCS ( f )df(O.62)2and thus:392Power of a generic BCS signal within a Bandwidth βr⎧⎡⎛ πβ ⎞⎤⎛ πβ ⎞2 ⎢− 1 + cos⎜⎜ r ⎟⎟⎥ 2 π Si⎜⎜ r ⎟⎟⎪⎝ f c n ⎠⎦⎪⎣⎝ fc n ⎠+⎪fcnβr+⎪nf c2π⎪P=⎨⎡⎤⎛ β kπ ⎞⎛ β π ⎞ ⎛ β kπ ⎞⎪⎟⎟ + 2 f c n cos⎜⎜ r ⎟⎟ cos⎜⎜ r ⎟⎟ +− 2 f c n cos⎜⎜ r⎢⎥⎪2 n−1 n⎝ fcn ⎠⎝ fcn ⎠ ⎝ fcn ⎠⎢⎥ss⎪+2 ∑ ∑ i j⎢⎥⎪ β r nπ i=1 j =i+1 ⎢ β (k − 1)π Si ⎡ β r π (k − 1) ⎤ − 2 β k π Si ⎡ β r π k ⎤ + β (1 + k )π Si ⎡ β r π (1 + k ) ⎤ ⎥rrr⎢⎥⎢⎥⎢⎥⎪fc nfcn⎢⎣⎣⎦⎣ fcn ⎦⎣⎦ ⎥⎦ k = j −i⎩(O.63)Combining now (O.52) and (O.60) for the infinite bandwidth case we have:limβ r →∞∫βrβ r GBCS ( f ) df = 1 +2−21 n −1 n∑ ∑ si s j [ k − 1 − 2 k + 1 + k ] = 1n i =1 j = i +1(O.64)As we expected since the definition of the PSD is normalized to infinite bandwidth.393Power of a generic BCS signal within a Bandwidth βr394BibliographyBibliography[M.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
