On Generalized Signal Waveforms for Satellite Navigation (797942), страница 68
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Additionally, since A can also beexpressed as A=jB the expression above simplifies to⎡⎛ nB ⎞⎤⎢2 j sin ⎜ 2 ⎟⎥⎝ ⎠⎦BOC sin ( f s , f c )(f )= − ⎣GMod,e⎛B⎞4 cos 2 ⎜ ⎟⎝2⎠2B=2πfnf c⎛ πf ⎞⎛ nB ⎞sin 2 ⎜⎜ ⎟⎟sin 2 ⎜ ⎟⎝ fc ⎠⎝ 2 ⎠==⎛ πf ⎞⎛B⎞cos 2 ⎜ ⎟⎟cos 2 ⎜⎜⎝ 2 ⎠ B = 2πfnf c ⎟⎠⎝nf(B.11)cOnce a simplified form has been derived for the BCS modulating factor of BOC(fs, fc), wesubstitute in (B.1) yielding the well known expression for the Power Spectral Density that wesaw in chapter 4.3.2.1:⎛ πf ⎞⎛ πf ⎞⎛ πf⎟⎟ sin 2 ⎜⎜ ⎟⎟sin ⎜⎜sin 2 ⎜⎜⎝ nf c ⎠⎝ fc ⎠ = f⎝ fc= fcc2(πf ) cos 2 ⎛⎜ πf ⎞⎟(πf )2⎜ nf ⎟⎝ c⎠2Gnf⎛⎞BOC⎜⎜ f s = c , f c ⎟⎟2⎝⎠Moreover, in chapter 4.3.2.1 we also saw that n = 2 f s⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎞⎟⎟⎟⎟ ⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜⎠ tan 2 ⎛⎜ πf ⎞⎟ = f ⎢ ⎝ f c ⎠ ⎝ nf c ⎠ ⎥c⎜ nf ⎟⎢⎛ πf ⎞ ⎥⎝ c⎠⎟⎟ ⎥⎢ πf cos⎜⎜⎝ nf c ⎠ ⎦⎥⎣⎢2(B.12)f c so that (B.12) can also be expressedas follows:⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟ ⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜f2fcs⎝⎠⎝⎠⎥GBOC( f s , f c ) = f c ⎢⎢⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢2(B.13)This is the well known expression that we find everywhere in the literature.
Now that we havesolved the case of the even BOC modulation in sine phasing, we calculate next its oddcounterpart. For the case of the odd BOC modulation in sine phasing, we have to derive first ageneral expression for any odd n. We will proceed by generalizing over n.312Power Spectral Density of sine-phased BOC signalsFor n = 3, BOCsin(fs , fc) can also be expressed as BCS([+1,-1,+1], fc), such that the generationmatrix will adopt the following form:⎛ s1 s1 {0} s1 s 2 {1} s1 s3 {2}⎞ ⎛1{0} − 1{1} 1{2} ⎞⎟⎜⎟ ⎜M ( [+ 1,−1,+1] ) = ⎜s 2 s 2 {0} s 2 s 3 {1}⎟ = ⎜1{0} − 1{1}⎟⎜s 3 s3 {0}⎟⎠ ⎜⎝1{0} ⎟⎠⎝3Thus, the odd modulating term yields this time:⎡⎞⎛⎛BOC sin ( f s , f c )( f ) = 3 + 2 ⎢− 2 cos⎜⎜ 2πf ⎟⎟ + cos⎜⎜ 2 2πfGMod,o⎝ 3 fc⎝ 3 fc ⎠⎣where o indicates the odd case, and⎛ πf ⎞⎟⎟sin 2 ⎜⎜3fBPSK ( 2 f c )( f ) = fc ⎝ 2 c ⎠Gpulse(πf )(B.14)⎞⎤⎟⎟⎥⎠⎦(B.15)(B.16)In the same manner, for n = 5 , BOCsin(fs, fc)= BOCsin(2fc, fc) what can also be defined as inthe general form BCS([+1,-1,+1,-1,+1], fc) with generation matrix given by:⎛1{0} − 1{1} 1{2} − 1{3} 1{4} ⎞⎜⎟1{0} − 1{1} 1{2} − 1{3}⎟⎜M 5 ( [+ 1,−1,+1,−1,+1] ) = ⎜1{0} − 1{1} 1{2} ⎟⎜⎟1{0} − 1{1}⎟⎜⎜1{0} ⎟⎠⎝Thus, for the case of n = 5 , we will have:⎡⎛BOC sin ( f s , f c )( f ) = 5 + 2⎢− 4 cos ⎜⎜ 2πfGMod,o⎝ 5 fc⎣⎞⎛ 2πf⎟⎟ + 3 cos⎜⎜ 2⎠⎝ 5 fc⎞⎛ 2πf⎟⎟ − 2 cos⎜⎜ 3⎠⎝ 5 fc(B.17)⎞⎛ 2πf⎟⎟ + cos⎜⎜ 4⎠⎝ 5 fc⎞⎤⎟⎟⎥⎠⎦(B.18)If we continue by induction we can find the expression for any odd n:n −1⎡⎤BOC sin ( f s , f c )()(− 1)i (n − i )cos ⎢i 2πf ⎥GMod,fn2=+∑oi =1⎣ nf c ⎦(B.19)As we can recognize, (B.19) is equal to (B.2) except that n is odd now with n ∈ {3,5,7,...}Moreover, it can be shown that for the odd n = 2 f s f c is still valid.
For simplicity, weexpress the modulating factor above using its exponential equivalent expression:n −1(BOC sin ( f s , f c )( f ) = n + ∑ (− 1) (n − i ) eiA + e−iAGMod,oii =1)A= j(B.20)2πfnf cUsing now the expressions derived above for the sum term Φ ( A) , it can be shown that:n −1Φ ( A) = ∑ (− 1) (n − i ) e iA =i =1ie A (1 − n ) − n e 2 A + (− 1)(eA+ 1)2n +1e A(n +1)=e A (1 − n ) − n e 2 A + e A(n +1)(eA+ 1)2(B.21)Again, this expression is similar to that obtained for the even case, but with a slight313Power Spectral Density of sine-phased BOC signalsdifference. Indeed, since n is odd, the second summand in the numerator has a changed signwith respect to (B.9).
The modulating function is thus shown to present the following form:⎡ n −1⎡ 2πf ⎤ ⎤iBOC sin ( f s , f c )()GMod,fn2=+=⎢∑ (− 1) (n − i )cos ⎢io⎥ ⎥ = { n + Φ ( A) + Φ(− A)}nfπ2f1i=c⎣⎦⎦⎣A= jnf c=n+e (1 − n ) + (− 1) en +1 A ( n +1)A(e(e=n+An)+1A)(− ne2A2+(e+ e − An + 2 (1 − n ) − n e A + e − Ae A + e− A + 2)−A)(1 − n ) + (− 1)n +1 − A ( n +1)e− ne−2 A=(B.22)⎛ πf ⎞⎛ nB ⎞cos 2 ⎜⎜ ⎟⎟cos 2 ⎜ ⎟⎝ fc ⎠⎝ 2 ⎠==⎛ πf ⎞⎛B⎞cos 2 ⎜ ⎟⎟cos 2 ⎜⎜⎝ 2 ⎠ B = 2πfnf c ⎟⎠⎝nf(B.23)(e−A)+12An−⎞⎛ An2⎜e + e 2 ⎟⎟⎜− AnAne +e +2 ⎝⎠==AA 2AA 2− ⎞− ⎞⎛ 2⎛ 2⎜e + e 2 ⎟⎜e + e 2 ⎟⎟⎜⎟⎜⎠⎝⎠⎝2Additionally, since A can also be expressed as A=jB, (B.22) simplifies to⎡⎛ nB ⎞⎤⎢2 cos⎜ 2 ⎟⎥BOC sin ( f s , f c )( f ) = ⎣ ⎝ ⎠⎦GMod,o⎛B⎞4 cos 2 ⎜ ⎟⎝2⎠2B=2πfnf ccOnce we have the BCS modulating factor of an arbitrary odd BOC(fs ,fc) it can be shown thatthe power spectral density is the well known expression we saw in chapter 4.3.2.1:⎡ ⎛ πf ⎞ ⎛ πf⎛ πf ⎞⎛ πf ⎞⎟⎟ cos 2 ⎜⎜ ⎟⎟⎟ cos⎜sin ⎜⎜⎢ sin ⎜⎜nf c ⎠fc ⎠nf c ⎟⎠ ⎜⎝ f c⎝⎝⎝⎢= fc= f(πf )2 cos 2 ⎛⎜ πf ⎞⎟ c ⎢ πf cos⎛⎜ πf ⎞⎟⎢⎜ nf ⎟⎜ nf ⎟⎢⎣⎝ c⎠⎝ c⎠2Gonf⎛BOC⎜⎜ f s = c , f c2⎝⎞⎟⎟⎠⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(B.24)which coincides perfectly with the expressions found in the literature [J.
W. Betz, 1999],[A.R. Pratt and J.I.R. Owen, 2003a] and [E. Rebeyrol et al., 2005].Furthermore, since n = 2 f s f c , the previous expression can also be shown as follows:Gonf⎛⎞BOC⎜⎜ f s = c , f c ⎟⎟2⎝⎠⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟ ⎥⎢ cos⎜⎜ ⎟⎟ sin ⎜⎜f2fcs⎝⎠⎝⎠⎥= fc ⎢⎢⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢2(B.25)314Power Spectral Density of cosine-phased BOC signalsCAppendix. PSD of cosine-phased BOCsignalsThe derivation of the power spectral density of the BOC modulation in cosine phasing is alittle bit more complicated, but it can be accomplished in a similar way as we have done withits sine-phased counterpart.
If we recall (4.54), the Power Spectral Density of the evencosine-phased BOC is shown to be:⎛ πf ⎞⎟sin 2 ⎜⎜⎡n / 2nf c ⎟⎠ ⎪⎧⎡⎡ 2πf ⎤ ⎤ ⎪⎫2πf ⎤ n / 2 −1iie⎝()()21cos21+−−+ ∑ 2 (− 1) (n / 2 − i )cos ⎢2iGBOC cos ( f s , f c ) = f cni⎨⎢∑⎢⎥⎥⎥⎬2nf c ⎦ i =1(πf ) ⎪⎩⎣⎣ nf c ⎦ ⎦ ⎪⎭⎣ i =1(C.1)or equivalently,Genf⎛⎞BOC cos ⎜⎜ f s = c , f c ⎟⎟4⎝⎠BPSK ( nf c )BOC cos ( nf c( f )GMod,= Gpulsee4, f c )(f )(C.2)where n ∈ {4, 8,12,16...}.
In order to use the results obtained in the previous Appendixes, weBOC cos ( nf c 4 , f c )( f ) in the brackets using the Euler´s formula:will expand the modulation term GMod,e⎡2πf ⎤ ecos ⎢(2 i − 1)⎥=nf c ⎦⎣2 πfj ( 2 i −1) nfj 2iBOC cos ( nf cAccording to this, GModBOC cos ( nf cGMod,e4, f c )c2 πf+e22 πf− j ( 2 i −1) nf− j 2ic(C.3)2 πf⎡ 2πf ⎤ e nf c + e nf ccos ⎢2 i⎥=nf2c⎣⎦4, f c )( f ) can be expressed as follows:⎡n / 2⎡⎤⎣ i =1⎣nf c ⎦(C.4)n / 2 −1⎡⎤⎤i =1⎣nf c ⎦ ⎦( f ) = n + 2⎢∑ (− 1)i cos ⎢(2 i − 1) 2πf ⎥ + ∑ 2(− 1)i (n / 2 − i )cos ⎢2i 2πf ⎥ ⎥or equivalently,BOC cos ( nf cGMod,e4, f c )( f ) = n + ∑ (− 1)i [e− Ae2iA + e Ae− 2iA ] + ∑ (− 1)i (n − 2i )[e2iA + e− 2iA ]n/2n / 2 −1i =1i =1(C.5)(C.6)A= j2 πfnf cwhat can also be expressed as:[BOC ( nf cGMod,e]⎧ n/2i− A 2 iAA − 2 iA+⎪+ ∑ (− 1) e e + e ei =1⎪⎪ n / 2 −14, f c )( f ) = n + ⎨+ n ∑ (− 1)i e 2iA + e− 2iA −i =1⎪n/2−1⎪i2 iA− 2 iA−⎪ ∑ 2 i (− 1) e + e⎩ i =1[][(C.7)]A= j2 πfnf cDecomposing the different terms of the sum, we have:BOC ( nf cGMod,e4, f c )( f ) = n + Φ1 ( A) + Φ 2 ( A) − Φ 3 ( A)(C.8)315Power Spectral Density of cosine-phased BOC signalswhere,n/2[Φ1 ( A) = Φ1+ ( A) + Φ1− ( A) = ∑ (− 1) e − Ae 2iA + e Ae − 2iAi]i =1n / 2 −1[Φ 2 ( A) = Φ +2 ( A) + Φ −2 ( A) = n ∑ (− 1) e 2iA + e − 2iAi](C.9)i =1∑ 2i(− 1) [en / 2 −1Φ 3 ( A) = Φ 3+ ( A) + Φ 3− ( A) =i2 iA+ e − 2iA]i =1As we can observe, Φ1− ( A) = Φ1+ (− A) remaining this identity true also for the other twosummands Φ −2 ( A) and Φ 3− ( A) .
Furthermore, if we look in detail at (C.9), we can see that itcan be simplified again using the methodology of previous Appendixes. Indeed,n/2Φ ( A) = ∑ (− 1) e+1ii =1−A(e ) = e ∑ (− e ) = e2A i−An/22A i(− e )2 A n / 2 +1i =1what can also be expressed as follows:n/2()( )n / 2 +12A− − e2 A− e2 A−A e=e− e2 A − 1e2 A + 1(C.10)−Ae − 1]( ) e[e + −e 1] = 2[cosh( A)Φ1+ ( A) = ∑ (− 1) e − A e 2 A =iii =1An(C.11)An−AAIt must be noted, that according to the definition of the BOC modulation in cosine phasing asn 2 +1a BCS signal, n ∈ {4,8,12,...} and the term (− 1)can be further simplified since n 2 + 1 willalways be odd.
In the same manner:) [ee(n/2ii =1and consequently,n/2] [− An] [e[Φ1 ( A) = Φ1+ ( A) + Φ1− ( A) = ∑ (− 1) e(2i −1) A + e − (2i −1) A =i](C.12)−1e − An − 1=A+ e − A 2 cosh ( A)Φ1− ( A) = Φ1+ (− A) = ∑ (− 1) i e A e −2 A =i =1+ e − An − 2e A + e− AAn](C.13)Furthermore, Φ 2 ( A) is shown to simplify to:n / 2 −1⎡ e 2 A + e An e −2 A + e − An ⎤i(C.14)Φ 2 ( A) = Φ +2 ( A) + Φ −2 ( A) = n ∑ (− 1) e 2iA + e − 2iA = − n ⎢+2A1 + e − 2 A ⎥⎦i =1⎣ 1+ e[or equivalently,(eΦ ( A) = − n2A]) ()[+ e −2 A + e An + e − An + 2 + e A(n − 2 ) + e − A(n − 2 )(e2A+e)]−A 2(C.15)For the third sum term, namely Φ3(A), we have to solve first the following intermediateproblem:Φ ( A) =+3n / 2 −1∑ 2 i (− 1) ei =1i2 iA=∑ 2 i (− e )n / 2 −12A i(C.16)i =1To do so, we define the following auxiliary function:n / 2 −1n / 2 −1(− e 2 A ) n / 2 + e 2 A = (− 1)n / 2 e An + e 2 Aif ( A) = ∑ (− 1) e 2iA = ∑ (− e 2 A ) i =− e2A −1− e2A −1i =1i =1(C.17)316Power Spectral Density of cosine-phased BOC signalsSince n/2 is even, we can further simplify the expression above as follows:f ( A) =n / 2 −1ei∑ (− 1) e2iA = −i =1+ e2 Ae2 A + 1An(C.18)being the derivative of f ( A) the function Φ 3+ ( A) as shown next:Φ 3+ ( A) =(2 − n )e A(n + 2 ) − n e An − 2 e 2 Adf ( A) n / 2 −1i= ∑ 2 i (− 1) e 2iA =2dAe2 A + 1i =1((C.19))In an analogue way, substituting A by -A in (C.19) we can see thatΦ 3− ( A) = Φ 3+ (− A) =and therefore,Φ 3 ( A) = Φ 3+ ( A) + Φ 3− ( A) =n / 2 −1∑ 2 i (− 1) ei− 2 iA=(2 − n )e− A(n + 2 ) − n e− An − 2e−2 A(ei =1−2 A+ 1)2(C.20)(2 − n )e A(n + 2 ) − n e An − 2e2 A + (2 − n )e− A(n + 2 ) − n e− An − 2e−2 A(e2A)+1(e2−2 A)+12(C.21)what can be further simplified according to:Φ 3 ( A) = Φ 3+ ( A) + Φ 3− ( A) =(2 − n )(e An + e − An ) − n [e A(n − 2 ) + e− A(n − 2 ) ] − 4(C.22)2+ e− ANow that we have calculated all the sum terms Φ1 ( A) , Φ 2 ( A) and Φ 3 ( A) , we can have a(eA)simplified expression for the modulating term of the power spectral density of thecosine-phased BOC modulation.
In fact,⎧n +⎪− AnAn⎪+ e + e − 2 +⎧n +e A + e− A⎪+ Φ ( A ) + ⎪⎪AnA( n − 2 )−2 A− An2ABOC ( nf c 4 , f c )+ e − A( n − 2 )( f ) = ⎪⎨ 1= ⎨+ − n (e + e ) − n (e + e ) − 2n − n eGMod,e+2⎪+ Φ 2 ( A ) − ⎪(e A + e− A )⎪⎩− Φ 3 ( A)⎪⎪ (n − 2 )(e An + e − An ) + n e A(n − 2 ) + e − A(n − 2 ) + 4⎪+(e A + e− A )2⎩[][[]](C.23)If we further develop it, we obtain:BOC ( nf cGMod,e4, f c )( f ) = (eAn)() ((e + e )) ()+ e − An e A + e − A − 2 e An + e − An − 2 e A + e − A + 4A−A 2(C.24)or equivalently,2BOC ( nf cGMod,eAn−⎛ An2⎞ ⎛ A −A ⎞⎜e − e 2 ⎟ ⎜e 2 − e 2 ⎟⎜⎟ ⎜⎟4, f c )⎠ ⎝⎠(f )= ⎝A−A 2e +e(2(C.25))A= j2πfnfcFinally, since A=jB, we can simplify this expression as follows:317Power Spectral Density of cosine-phased BOC signals2BOC ( nf cGMod,e⎡⎛ Bn ⎞⎤ ⎡⎛ B ⎞⎤⎢2 j sin ⎜ 2 ⎟⎥ ⎢2 j sin ⎜ 2 ⎟⎥⎝⎠⎦ ⎣⎝ ⎠⎦4, f c )(f )= ⎣2[2 cos(B )]2B=2πfnf c⎛ πf ⎞⎛ πf ⎞⎟4 sin 2 ⎜⎜ ⎟⎟ sin 2 ⎜⎜fc ⎠nf c ⎟⎠⎝⎝=⎛ 2πf ⎞⎟⎟cos 2 ⎜⎜nf⎝ c ⎠(C.26)so that the power spectral density of BOCcos( f s = nf c 4 , f c ) is shown to be:GBOCcos ( f s =nf⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞⎟⎟ 4 sin 2 ⎜⎜ ⎟⎟ sin 2 ⎜⎜⎟⎟⎟sin 2 ⎜⎜sin 2 ⎜⎜ ⎟⎟ sin 4 ⎜⎜fc ⎠nf c ⎟⎠nf c ⎠fc ⎠nf c ⎠⎝⎝⎝⎝⎝(C.27)= 4 fc4c , f c ) = f c(πf )222 ⎛ 2πf ⎞2 ⎛ 2πf ⎞⎟⎟(πf ) cos ⎜⎜ ⎟⎟cos ⎜⎜⎝ nf c ⎠⎝ nf c ⎠Finally since n = 4 f s f c , it is trivial to see that the expression of the Power Spectral Densityof an arbitrary cosine-phased BOC reduces in the even case to:BPSK ( nf c )BOC cos ( nf c( f )GMod,GBOC cos ( f s , f c ) = Gpulsee⎡⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎥⎢ 2 sin ⎜⎜ ⎟⎟ sin ⎜⎜fc ⎠4 f s ⎟⎠ ⎥4, f c )⎝⎝⎢( f ) = fc ⎢⎥⎛ πf ⎞⎟⎟πf cos⎜⎜⎢⎥⎝ 2 fs ⎠⎣⎢⎦⎥2(C.28)Once we have obtained the expression for the even BOC modulation in cosine phasing, wecalculate its odd counterpart next.For the case of the odd BOC modulation in cosine phasing, we have to derive a generalexpression for any n.
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