On Generalized Signal Waveforms for Satellite Navigation (797942), страница 78
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In this case, the explicit expressions for the different SSCcomponents adopt the following form:(SSC1 = n1n2 f c1 f c2 Ξ f c1 , n1 , f c2 , n2 ,0,0n2 −1SSC 2 = 2n1 f c1 f c2 ∑∑ s s Ξ( fn2i' j'i '=1 j '=i ' +1SSC3 = 2n f f1 22 c cSSC 4 = 4 f f1 2c cn1 −1 n1n1 −1 n1∑ ∑ r r Ξ( fi =1 j =i +1i jn 2 −1 n 21c1ci =1 j = i +1 i ' =1 j ' = i ' +1(O.10))(O.11))(O.12), n1 , f c2 , n2 ,0, j '−i ', n1 , f c2 , n2 , j − i,0∑ ∑ ∑ ∑ r r s s Ξ( fi j i' j)1c, n1 , f c2 , n2 , j − i, j '−i ')(O.13)where the Ξ function is defined as follows:⎛ πf ⎞⎛ πf ⎞⎟⎟ sin 2 ⎜⎜sin 2 ⎜⎜1⎟∞n1 f c ⎠n2 f c2 ⎟⎠⎡⎡2πf ⎤2πf ⎤12⎝⎝cos ⎢( j − i )cos ⎢( j '−i ')Ξ( f c , n1 , f c , n2 , j '−i ' , j − i ) = ∫⎥ df221⎥−∞n1 f c ⎦n2 f c2 ⎦(πf )(πf )⎣⎣which is shown to simplify to the following analytical expression:(O.14)380Spectral Separation Coefficient between two generic BCS signals(Ξ f c1 , n1 , f c2 , n2 , j − i, j '−i ')⎡ f 1 (− 1 + k )n − f 2 (− 1 + k )n 3 + f 1 (1 + k )n − f 2 (− 1 + k )n 3 + ⎤2 11 22 11 2ccc⎢ c⎥33⎢⎥1212⎢+ f c (− 1 + k2 )n1 + f c (− 1 + k1 )n2 − 2 f c k2 n1 + f c (− 1 + k1 )n2 + ⎥⎢⎥331212⎢+ f c (1 + k2 )n1 + f c (− 1 + k1 )n2 + 4 f c k2 n1 − f c k1n2 −⎥⎢⎥33⎢− 2 f c1 (1 + k2 )n1 − f c2 k1n2 − 2 f c1k2 n1 + f c2 n2 − f c2 k1n2 −⎥⎢⎥1⎢− 2 f 1 (− 1 + k )n + f 2 k n 3 − 2 f 1n − f 1k n + f 2 k n 3 +⎥2 1cc 1 2c 1c 2 1c 1 233 3 3⎢⎥24 f c ,1 f c , 2 n1 n233⎢⎥1212⎢+ 4 f c k2 n1 + f c k1n2 − 2 f c (1 + k 2 )n1 + f c k1n2 +⎥⎢⎥331212⎢+ f c (− 1 + k2 )n1 − f c (1 + k1 )n2 − 2 f c k2 n1 − f c (1 + k1 )n2 +⎥⎢⎥33⎢+ f c1 (1 + k2 )n1 − f c2 (1 + k1 )n2 + f c1 (− 1 + k 2 )n1 + f c2 (1 + k1 )n2 − ⎥⎢⎥⎢− 2 f 1k n + f 2 (1 + k )n 3 + f 1 (1 + k )n + f 2 (1 + k )n 3⎥c 2 1ccc1 22 11 2⎣⎢⎦⎥(O.15)Next we show some particularized expressions of the function above since they appearrelatively often in the description of the SSC between some of the signals of interest studiedin chapter 5.1.1.
Moreover, for simplicity and to avoid confusion in the notation, thefollowing change of notation is proposed:f c ,1 = f c1(O.16)f c , 2 = f c2In fact, it can be shown thatΞ( f c ,1 , n1 , f c , 2 , n2 ,0,0) =()− ( f c ,1n1 + f c , 2 n2 ) f c2,1n12 − 4 f c ,1 f c , 2 n1n2 + f c2, 2 n22 + f c ,1n1 − f c , 2 n26 f c3,1 f c3, 2 n13n233(O.17)Equally, the following identity is shown to be also true:()Ξ f c1 , n1 , f c2 , n2 , j − i,0 = Ξ( f c ,1 , n1 , f c , 2 , n2 , k1 ,0) =⎡⎤⎢⎥33⎢2 f c3,1n13π 3 2k13 − (k1 − 1) − (k1 + 1)⎥⎢⎥33⎢ ⎛ f c , 2 (1 + k1 )n2 − f c ,1n1 + f c , 2 (− 1 + k1 )n2 + f c ,1n1 + ⎞⎥⎟⎥⎢ ⎜⎜331⎢+ + f (1 − k )n + f n + f (− 1 + k )n + f n + ⎟⎥⎜⎟⎥c,2c ,1 1c,2c ,1 11 21 23 33 3 312π f c ,1 f c , 2 n1 n2 ⎢ ⎜⎟⎥⎢ ⎜ − 2( f n k + f n )3⎟c,2 2 1c ,1 1⎢ ⎝⎠⎥⎢⎥3⎢⎥k133 3 31−⎢− 2 f c ,1 f c , 2 n1 n2⎥f c ,1n1 f c , 2 n2⎣⎦()(O.18)and its symmetric version:381Spectral Separation Coefficient between two generic BCS signals()Ξ f c1 , n1 , f c2 , n2 ,0, j '−i ' = Ξ( f c ,1 , n1 , f c , 2 , n2 ,0, k 2 ) =⎡⎤⎢⎥33⎢2 f c3,1n13 2k23 − (k2 − 1) − (k 2 + 1)⎥⎢⎥33⎢ ⎛ f c ,1 (1 + k 2 ) n1 − f c , 2 n2 + f c ,1 (− 1 + k2 ) n1 + f c , 2 n2 + ⎞⎥⎟⎥⎢ ⎜⎜331⎢+ + f (1 − k ) n + f n + f (1 + k ) n + f n − ⎟⎥⎟⎥⎜c ,1c,2 2c ,1c,2 2212133 3 312 f c ,1 f c , 2 n1 n2 ⎢ ⎜⎟⎥⎢ ⎜ − 2( f n k + f n )3⎟c ,1 1 2c, 2 2⎢ ⎝⎠⎥⎢⎥3⎢⎥k133 3 3− 2⎢− 2 f c ,1 f c , 2 n1 n2⎥f c ,1n1 f c , 2 n2⎣⎦()(O.19)It is interesting to note the symmetry that (O.18) and (O.19) present, since the one expressioncan be expressed in terms of the other by just inverting the role of fc,1 and n1 with that of fc,2and n2.
Finally, the general analytical expression for the SSC between two BCS sequences canbe expressed as follows:()⎧n1n2 f c1 f c2 Ξ f c1 , n1 , f c2 , n2 ,0,0 +⎪n 2 −1 n 212⎪+ 2 n f 1 f 21 c c ∑ ∑ si ' s j ' Ξ f c , n1 , f c , n2 ,0, j '−i ' +⎪i ' =1 j ' = i ' +1⎪⎪n1 −1 n1=⎨1 2122nff+2 c c ∑ ∑ ri r j Ξ f c , n1 , f c , n2 , j − i ,0 +⎪i =1 j = i +1⎪n−1n 2 n1 −1 n12⎪1 212⎪+ 4 f c f c ∑ ∑ ∑ ∑ ri rj si ' s j ' Ξ f c , n1 , f c , n2 , j − i, j '−i '⎪⎩i ' =1 j ' = i ' +1 i =1 j = i +1(SSC BCS1 − BCS 2)((O.20))()382Self Spectral Separation Coefficient of a generic BCS signalO.2Appendix.
Self SSC of a generic BCS signalAs we have seen in chapter 5.2.3, the self SSC of a generic BCS sequence is shown to be:SSC BCS1 −BCS2⎧n 2 f c2 Ξ( f c , n, f c , n,0,0 ) +⎪n −1n⎪+ 2nf 2c ∑ ∑ ri ' r 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 rj si ' s j ' Ξ( f c , n, f c , n, j − i, j '−i ')i '=1 j '=i ' +1 i =1 j =i +1⎩(O.21)where the fourth term of the SSC sum can be further expanded as follows:n −1SSC 4 (n ) = ∑nn −1n∑ ∑ ∑ r r s s Ξ( f , n, f , n, j − i, j '−i')i '=1 j '=i ' +1 i =1 j =i +1i j i' j'ccn −1 n⎧()ssssss...+++⎪ 1 2 2 3 3 4∑∑ ri rj Ξ( fc , n, fc , n, j − i,1) +i =1 j = i +1⎪n −1 n⎪⎪SSC 4 (n ) = ⎨+ (s1s3 + s2 s4 + s3 s5 + ...)∑ ∑ ri rj Ξ( f c , n, f c , n, j − i,2 ) +i =1 j = i +1⎪n −1 n⎪⎪+ (s1s4 + s2 s5 + s3 s6 + ...)∑ ∑ ri rj Ξ( f c , n, f c , n, j − i,3) + ...i =1 j = i +1⎩⎪(O.22)(O.23)or equally:⎧⎡(r1r2 + r2 r3 + r3r4 + ...) Ξ( f c , n, f c , n,1,1) + ⎤⎪⎥⎢⎪(s1s2 + s2 s3 + s3 s4 + ...) ⎢(r1r3 + r2 r4 + r3r6 + ...) Ξ( f c , n, f c , n,2,1) + ⎥ +⎪⎢⎣(r1r4 + r2 r5 + r3r7 + ...) Ξ( f c , n, f c , n,3,1) + ...⎥⎦⎪⎪⎡(r1r2 + r2 r3 + r3r4 + ...)Ξ( f c , n, f c , n,1,2 ) + ⎤⎪⎥⎢⎪+ (s1s3 + s2 s4 + s3 s5 + ...) ⎢(r1r3 + r2 r4 + r3r6 + ...)Ξ( f c , n, f c , n,2,2 ) + ⎥ +SSC 4 (n ) = ⎨⎢⎣(r1r4 + r2 r5 + r3r7 + ...)Ξ( f c , n, f c , n,3,2 ) + ...⎥⎦⎪⎪⎡(r1r2 + r2 r3 + r3r4 + ...)Ξ( f c , n, f c , n,1,3) + ⎤⎪⎥⎢⎪+ (s1s4 + s2 s5 + s3 s6 + ...) ⎢(r1r3 + r2 r4 + r3r6 + ...)Ξ( f c , n, f c , n,2,3) + ⎥ +⎪⎢⎣(r1r4 + r2 r5 + r3r7 + ...)Ξ( f c , n, f c , n,3,3) + ...⎥⎦⎪⎪⎩+ ...(O.24)This expression can be further simplified using the scalar and shift operator θ.
Indeed:⎧⎪[r1 , r2 ,...rn ] [θ ( [r1 , r2 ,...rn ], l1 )] T × ⎫⎪SSC 4 (n ) = 4 f ∑∑ ⎨Ξ( f c , n, f c , n, l1 , l2 )T⎬l1 =1 l2 =1 ⎪⎩× [ s1 , s2 ,...sn ] [θ ( [ s1 , s2 ,...sn ], l2 )] ⎪⎭2cn −1 n −1(O.25)where the operator θ indicates a linear shift of the vector s by l elements to the rightaccording to the following notation:383Self Spectral Separation Coefficient of a generic BCS signaln1 −1 n1n1 −1i =1 j = i +1l =1∑[(∑ ri rj Ξ( fc1, n1, f c2 , n2 , j − i,1) = ∑[r1, r2 ,...rn1 ] θ [r1, r2 ,...rn1 ], l)] Ξ( fT1c, n1 , f c2 , n2 , l ,1)(O.26)or in its expanded form:()⎧[r1 , r2 ,...rn1 ][0, r1 , r2 , r3 ...rn1 −1 ] T si s j Ξ f c1 , n1 , f c2 , n2 ,1,1 +⎪T12n1 −1 n1⎪⎪+ [r1 , r2 ,...rn1 ][0,0, r1 , r2 ...rn1 − 2 ] si s j Ξ f c , n1 , f c , n2 ,2,1 +12∑∑ ri rj Ξ f c , n1, fc , n2 , j − i,1 = ⎨+ [r , r ,...r ][0,0,0, r ...r ] T s s Ξ f 1, n , f 2 , n ,3,1 + ...i =1 j = i +1⎪ 1 2 n11n1 − 3i jc1c2⎪T12⎪⎩+ [r1 , r2 ,...rn1 ][0,0,0,0...r1 ] si s j Ξ f c , n1 , f c , n2 , n1 − 1,1 + ...(O.27)Summarizing, the different terms of the Self SSC of a BCS signal can also be expressed as:()((()))SSC1 (n ) = n 2 f c2 Ξ( f c , n, f c , n,0,0)n −1SSC 2 (n ) = 2nf c2 ∑ [r1 , r2 ,...rn ] [θ ( [r1 , r2 ,...rn ], l )] Ξ( f c , n, f c , n,0, l )Tl =1n −1SSC3 (n ) = 2nf c2 ∑ [r1 , r2 ,...rn ] [θ ( [r1 , r2 ,...rn ], l )] Ξ( f c , n, f c , n, l ,0)T(O.28)l =1⎧⎪[r1 , r2 ,...rn ] [θ ( [r1 , r2 ,...rn ], l1 )]T ×⎫⎪SSC 4 (n ) = 4 f ∑∑ ⎨Ξ ( f c , n, f c , n, l1 , l2 )T⎬l1 =1 l 2 =1 ⎪⎩× [r1 , r2 ,...rn ][θ ( [r1 , r2 ,...rn ], l2 )] ⎪⎭2cn −1 n −1Thus, grouping all the contributions of the SSC and since [r1 , r2 ,...rn ] = [ s1 , s2 ,...sn ] we canexpress the Self SSC of an arbitrary BCS signal as follows:⎧⎪⎪n 2 f c2 Ξ( f c , n, f c , n,0,0 ) +⎪n−1⎪T(O.29)SSCSelf BCS = ⎨4nf c2 ∑ [ s1 , s2 ,...sn ] [θ ([ s1 , s2 ,...sn ], l )] Ξ ( f c , n, f c , n, l ,0 ) +l =1⎪⎪n−1 n−1 ⎧[ s , s ,...s ] [θ ([ s , s ,...s ], l )] T × ⎫⎪1 21n⎪4 f c2 ∑∑ ⎪⎨ 1 2 n⎬Ξ ( f c , n, f c , n, l1 , l2 )T⎪⎪⎭l1 =1 l2 =1 ⎪[()]×[s,s,...s][s,s,...s],lθ12122nn⎩⎩or equivalently:⎧n 2 f c2Ξ( f c , n, f c , n,0,0 ) +⎪⎫⎧n [ s1 , s2 ,...sn ] [θ ([ s1 , s2 ,...sn ], l1 )] T Ξ ( f c , n, f c , n, l1 ,0) +⎪⎪⎪⎪⎪n−1SSCSelf BCS = ⎨ 2 ⎪TT⎤⎡n −1 ⎧4f⎫⎬⎪ c ∑ ⎨⎢ ⎪[ s1 , s2 ,...sn ] [θ ([ s1 , s2 ,...sn ], l1 )] ×⎪l1 =1 ⎪ ∑ ⎨⎬ Ξ ( f c , n, f c , n, l1 , l2 )⎥ ⎪⎪⎥⎪⎪⎩⎢⎣ l2 =1 ⎪⎩× [ s1 , s2 ,...sn ] [θ ([ s1 , s2 ,...sn ], l2 )] ⎪⎭⎪⎩⎦⎭(O.30)Moreover, to simplify the notation a little bit more we will do s = [ s1 , s2 ,...sn ] and assuming aparticular code frequency, we do the change Ψ (n, l1 , l2 ) = Ξ( f c , n, f c , n, l1 , l2 ) .
According to this,we can simplify the notation as follows, for a particular chip rate fc:⎫⎧n s [θ (s , l1 )]T Ψ (n, l1 ,0) +n −1 ⎪⎪SSCSelf BCS = n 2 f c2 Ψ (n,0,0) + 4 f c2 ∑ ⎨⎡ n−1⎤⎬Tl1 =1 ⎪ ⎢∑ s [θ (s , l1 )] s [θ (s , l2 )] Ψ (n, l1 , l2 )⎥ ⎪⎦⎭⎩⎣ l2 =1(O.31)384Spectral Separation Coefficient between a generic BCS signal and an arbitrary BOC or BPSK signalO.3Appendix.
SSC between a generic BCS and anarbitrary BOC or BPSKThe general expression for the SSC between a generic BCS signal and a sine or cosine-phasedBOC modulation adopts a similar form to the expressions derived in the previous chapters.Indeed, this SSC is a particular case of the general SSC derived in (O.20). Interestingly, theSSC between a generic BCS and an arbitrary BOC signal shares SSC1 and SSC3 with the SelfSSC of that BCS signal, being only different by the terms SSC2 and SSC4.Indeed, for the case of the sine-phased modulation these are shown to adopt the general:n2 −1SSC 2 (n1 ) = 2n1 f c1 f c2 ∑ (− 1) (n2 − i ) Ξ( f c1 , n1 , f c2 , n2 ,0, i )ii =1[ (⎧SSC 4 (n1 ) = 4 f c1 f c2 ∑ ⎨[r1 , r2 ,...rn1 ] θ [r1 , r2 ,...rn1 ], ll =1 ⎩n1 −1) ] ∑ (− 1) (n − i )Ξ ( fTn2 −1i2i =11c⎫, n1 , f c2 , n2 , l , i )⎬⎭(O.32)Grouping together all the terms we have then:SSC BCS([r ], f 1 )− BOCc2sin ( n2 , f c )(=)⎧n1n2 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 +cc1 c c ∑212⎪i =1(O.33)⎪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 2(− 1)i (n2 − i )Ξ f c1 , n1 , f c2 , n2 , l , i ⎬+4ff[r,r,...r]θ[r,r,...r],l⎪∑c c ∑⎨ 1 2n1n11 2l =1 ⎩i =1⎭⎩()[()] ([()])()where the fourth term SSC4 resembles the expressions that we saw in Appendix B for thederivation of the power spectral density of the sine-phased BOC modulation.
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