On Generalized Signal Waveforms for Satellite Navigation (797942), страница 73
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In the next figure we show the two analyzedversions of the AltBOC modulation, namely the general expression and the constant envelopemodified version with a sub-carrier frequency of 15.345 MHz and a code frequency of10.23 MHz.Figure I.4. Power Spectral Density of the general AltBOC(15,10) and the modifiedconstant envelope AltBOC(15,10)Equally, for an AltBOC(10,10) – thus with Φ even – the difference between the generalAltBOC and the constant envelope solution is shown to be minimum:347Power Spectral Density of the AltBOC ModulationFigure I.5. Power Spectral Density of the general AltBOC(10,10) and the modifiedconstant envelope AltBOC(10,10)It is interesting to note that the power spectral densities of the general AltBOC modulationand the modified constant envelope AltBOC have similar shapes and only minor differencescan be observed in the high order lobes.
Indeed, the main and first side lobes are nearlyidentical being the differences lower than 1 dB. The only differences can be observed in thehigh frequency components and come from the extra terms that are needed to achieveconstant envelope. Nonetheless in real implementations these would be filtered afteramplification and thus for most of the bandwidths of interest we can state that there is noqualitative difference between both solutions regarding the spectrum.
In terms ofimplementation, however, it is clear that the constant envelope solution is superior.348Power Spectral Density of the CBCS ModulationJAppendix. PSD of the CBCS modulationIn the next lines the power spectral density of the CBCS modulation will be derived. CBCS isa specific implementation of the MBOC modulation which receives for this particular case thename of CBOC. Thus, all the derivations of this chapter also apply for the CBOCimplementation of MBOC that Galileo has selected for the E1 Open Service (OS).The Composite Binary Coded Symbols modulation, or CBCS for short, is defined as thesuperposition of a BOC signal with a BCS by means of a modified and optimized Interplexscheme.
This last sentence is of great importance because while CBCS specifies the way thesignals are multiplexed at payload level, MBOC is more generic and does not say anythingabout how the time stream should look like.In the most general case, CBCS([s], fc, ρ) represents the superposition of a BOC(fc, fc) with aBCS([s], fc) in such a way that the BCS component has a percentage ρ of power with respectto the total power of the multiplexed signal. Furthermore, the vector [s] indicates the symbolsthat constitute the subchips of the BCS signal.
Next figure depicts schematically the principle:Figure J.1. CBCS chip waveform as a superposition of a BOC signal and a BCS signalUnlike BPSK, BOC or BCS, the CBCS signal is formed by 4-level sub-carriers. As we haveshown in chapter 4.8.4, other multiplex techniques such as the FH-Interplex could have alsoperformed CBCS. However, with important drawbacks. In order to avoid them, a new schemewas proposed in [CNES, 2005]. This has been analyzed in chapter 7.7.9. For facility in thederivations we recall again the mathematical definition of CBCS in the time domain:⎤⎡ cD (t )cosθ1 sBOC( f c , f c ) (t ) + cosθ 2 sBCS([ s ], f c ) (t ) + ⎥⎢⎥⎢ 2cP (t )⎢s (t ) = A1 ⎢+cosθ1 sBOC( f c , f c ) (t ) − cosθ 2 sBCS([ s ], f c ) (t )) + ⎥⎥2⎥⎢⎛ sin θ1 + sin θ 2 ⎞⎥⎢+ j sPRS (t ) ⎜⎟ + sIM (t )⎥⎦⎢⎣2⎠⎝[][](J.1)349Power Spectral Density of the CBCS Modulation⎛ sin θ1 − sin θ 2 ⎞sIM (t ) = − j cD (t ) cP (t ) sPRS (t ) ⎜⎟2⎝⎠where,•(J.2)•A1 is the amplitude of the modulation envelope, sum of the OS data and pilot signals,PRS and the Inter-Modulation product IM,θ1 and θ 2 describe the angular distance between the points of the 8-PSK modulation•as depicted in Figure J.2,sBOC( f c , f c ) (t ) represents the BOC(1,1) modulation with a chip rate f c ,•sBCS([ s ], f c ) (t ) represents the BCS([s],1) modulation with subchips vector given by [s]and chip rate f c ,•sPRS (t ) is the PRS modulation BOCcos(15,2.5),••sIM (t ) is the Inter-Modulation product signal, andcD (t ) and cP (t ) are the PRN codes for the data and pilot channel of the OS.The equation above is graphically shown in the figure below.
We can recognize thatcompared with the BOC(1,1) Interplex baseline, two new phase states have appeared toaccount for the new BCS modulation waveform. Moreover, the quadrature component,namely PRS in the case of Galileo, presents a PSD that is not affected by the waveformstransmitted on the in-phase component.Figure J.2. Oscillation of the BOC and BCS signals in CBCSIt is also of interest to note that thanks to the introduction of the additional BCS, there willalways be OS signal being emitted at any time for any combination of code chips.
This makesthe modulation more efficient and reduces the IM power consequently.Let us now look at the data and pilot channels of the Open Service in detail. In fact, recallingthe CBCS time definition, we can easily separate the data and pilot channels as follows:c (t )sD (t ) = A1 D cosθ1 sBOC( f c , f c ) (t ) + cosθ 2 sBCS([ s ], f c ) (t )2(J.3)cP (t )sP (t ) = A1cosθ1 sBOC( f c , f c ) (t ) − cosθ 2 sBCS([ s ], f c ) (t )2[][]350Power Spectral Density of the CBCS ModulationAs we have shown in the introduction of chapter 4.1.1, the autocorrelation of a signal that isstationary in wide sense adopts the following form:1ℜ s (τ ) = E s (t ) s * (t − τ ) = ∑ ℜ c (m) ℜ p (τ − mTc )(J.4)Tc mAccording to this, if we use the notation of chapter 4, the autocorrelation function of the datachannel can be expressed as[]ℜOSD (τ ) =1⎧1cos 2 (θ1 )∑ ℜc D (m )ℜ BOC( f c , f c ) (τ − mTc ) + cos 2 (θ 2 )∑ ℜc D (m )ℜ BCS([ s ], f c ) (τ − mTc ) +⎪A ⎪ TcTcmm=⎨BOC ( f c , f c )([ s ], f c )4 ⎪2 cos(θ ) cos(θ )(t − kTc − θ ) pTBCS(t − τ − jTc − θ )12 ∑∑ E cD , k cD , j E pTcc⎪⎩kj(J.5)BOC ( f c , f c )BCS([ s ], f c )and pTcrepresent the chip waveforms of BOC(fc, fc) and BCS([s], fc)where pTc21[] []correspondingly, following the notation of chapter 4.1.
This formulation can be furtherdeveloped if the expectation operator is expressed in integral form as shown next:ℜOSD (τ ) =1⎧1cos 2 (θ1 )∑ ℜc D (m )ℜBOC( f c , f c ) (τ − mTc ) + cos 2 (θ 2 )∑ ℜc D (m )ℜ BCS([ s ], f c ) (τ − mTc ) +⎪TcmmA2 ⎪ Tc= 1 ⎨Tc4 ⎪1( fc , fc )([ s ], f c )2 cos(θ1 ) cos(θ 2 )∑ ℜc D (m ) ∑ ∫ pTBOC(t − kTc − θ ) pTBCS(t − τ − kTc + mTc − θ )dtcc⎪⎩Tc k 0mor equivalently:(J.6)ℜOSD (τ ) =1⎧1cos 2 (θ1 )∑ ℜcD (m )ℜ BOC( fc , fc ) (τ − mTc ) + cos 2 (θ 2 )∑ ℜcD (m )ℜ BCS([ s ], fc ) (τ − mTc ) +⎪TcA ⎪Tcmm=⎨14 ⎪+ 2 cos(θ1 ) cos(θ 2 )∑ ℜcD (m ) ℜ BOC( fc , fc ) / BCS([ s ], fc ) (τ − mTc )⎪⎩m Tc(J.7)If we further assume that the data codes show ideal properties, then ℜ cD (m ) = δ (m ) and the21autocorrelation of the data channel yields then:1⎧12()cos(θ)τcos 2 (θ 2 ) ℜ BCS([ s ], fc ) (τ )ℜ+1BOC(f,f)cc⎪2TcA1 ⎪TcℜOS⎨D (τ ) =4 ⎪ 2+ cos(θ1 ) cos(θ 2 ) ℜ BOC( fc , fc ) / BCS([ s ], fc ) (τ )⎪⎩ Tc(J.8)We can repeat now the same steps for the pilot channel and arrive to a similar expression forthe pilot autocorrelation:1⎧1cos 2 (θ1 ) ℜ BOC( fc , fc ) (τ ) + cos 2 (θ 2 ) ℜ BCS([ s ], fc ) (τ )2 ⎪TcA1 ⎪Tc(J.9)ℜOS⎨P (τ ) =4 ⎪ 2− cos(θ1 ) cos(θ 2 ) ℜ BOC( fc , fc ) / BCS([ s ], fc ) (τ )⎪⎩ Tc351Power Spectral Density of the CBCS ModulationComparing the autocorrelation of the pilot OS with that of the data channel, we can recognizethat there is only a sign difference in the cross-correlation term, which is in phase for the datachannel and in anti-phase for the pilot channel.
Now that we have derived the expressions forthe data and pilot autocorrelations of the Open Service, the Power Spectral Densities of bothchannels can be obtained in the following form:222⎧ 2A12 ⎪cos (θ1 ) f c S BOC ( f c , f c ) ( f ) + cos (θ 2 ) f c S BCS([ s ], f c ) ( f ) +G (f)=⎨**4 ⎪+ 2 f cos(θ )cos(θ ) Sc12BOC ( f c , f c ) ( f ) S BCS ([ s ], f c ) ( f ) + S BOC ( f c , f c ) ( f ) S BCS ([ s ], f c ) ( f )⎩OSD{}(J.10)which can be simplified as shown next:222⎧ 2A12 ⎪cos (θ1 ) f c S BOC ( f c , f c ) ( f ) + cos (θ 2 ) f c S BCS([ s ], f c ) ( f ) +OSGD ( f ) =⎨*4 ⎪+ 4 f cos(θ )cos(θ ) Re Sc12BOC ( f c , f c ) ( f ) S BCS ([ s ], f c ) ( f )⎩{}(J.11)or equivalently:22A12 ⎧⎪cos (θ1 ) GBOC( fc , fc ) ( f ) + cos (θ 2 ) GBCS([ s ], fc ) ( f ) +G (f)=⎨*4 ⎪+ 4 f c cos(θ1 ) cos(θ 2 ) Re S BOC( f , f ) ( f ) S BCS([ s ], fc ) ( f )c c⎩OSD{}According to this, the power of the data channel will beA2POS D = 1 cos 2 θ1 + cos 2 θ 2 + 2r cos(θ1 )cos(θ 2 )4[](J.12)(J.13)where, the cross-correlation between BOC(fc, fc) and BCS([s], fc) is defined as follows:*⎫⎧⎪ S BOC( f c , f c ) ( f ) S BCS([ s ], f c ) ( f ) + ⎪⎬ df∫− ∞ ⎨⎪+ S *()()fSf⎪⎭BCS([ s ], f c )⎩ BOC( f c , f c )(J.14)If we solve now for the pilot channel, it can be shown that the power spectral density of the1r=Tc1∫Tc sBOC( f c , f c ) (t ) sBCS([ s ], f c ) (t )dt = Tc∞pilot OS will be:222⎧ 2A12 ⎪cos (θ1 ) f c S BOC ( f c , f c ) ( f ) + cos (θ 2 ) f c S BCS([ s ], f c ) ( f ) +G (f)=⎨*4 ⎪− 4 f cos(θ )cos(θ ) Re Sc12BOC ( f c , f c ) ( f ) S BCS ([ s ], f c ) ( f )⎩OSP{}(J.15)or expressed in terms of the power spectral density,A2G (f)= 14OSPsuch that⎧⎪cos 2 (θ1 ) GBOC( fc , fc ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], fc ) ( f ) +⎨*⎪⎩− 4 f c cos(θ1 ) cos(θ 2 ) Re S BOC( fc , fc ) ( f ) S BCS([ s ], fc ) ( f )POS P ={[}]A12cos 2 (θ1 ) + cos 2 (θ 2 ) − 2r cos(θ1 )cos(θ 2 )4(J.16)(J.17)If we sum up now the power spectral densities of the data and pilot channels as given by(J.12) and (J.16), we obtain the general power expression for the power of the composite OS:[]A12GOS D + OS P ( f ) =cos 2 (θ1 ) GBOC( f c , f c ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], f c ) ( f )2(J.18)352Power Spectral Density of the CBCS ModulationThus, the total power of the OS signal, with data and pilot together, will be:A2POS D + OS P = 1 cos 2 (θ1 ) + cos 2 (θ 2 )2[](J.19)adopting the normalized expression of the OS power spectral density the following form:GOS D + OS P ( f ) =[A122]A12cos 2 (θ1 ) GBOC( f c , f c ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], f c ) ( f )2∞2⎡cos 2 (θ ) ∞ G()()+fdfcosθ1BOC(f,f)2∫−∞∫− ∞ GBCS([ s ], f c ) ( f )df ⎤⎥⎦cc⎢⎣(J.20)Since the phase states are sitting on the circle and the power spectral densities of the data andpilot channels are normalized to infinite bandwidth, we can express the normalized powerspectral density of the OS signal as follows:GOS D + OS P ( f ) =cos 2 (θ1 ) GBOC( f c , f c ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], f c ) ( f )(J.21)cos 2 (θ1 ) + cos 2 (θ 2 )If we have a close look at the expression above, we can see that we can express the percentageof power that falls on the BCS signal as follows:cos 2 (θ 2 )ρ=(J.22)cos 2 (θ1 ) + cos 2 (θ 2 )Thus,GOS D + OS P ( f ) = (1 − ρ )GBOC( f c , f c ) ( f ) + ρ GBCS([ s ], f c ) ( f )(J.23)This means, that the total OS power spectral density can be defined as the linear combinationof the PSDs of the two waveforms composing the CBCS signal, namely BOC(fc, fc) andBCS([s], fc), weighted by the percentage ρ of power that is put on the BCS component.If we divide now the expressions of the data and pilot power spectral densities given in (J.12)and (J.16) by the integrated data and pilot power, we obtain the normalized expressions:⎧⎪cos 2 (θ1 ) GBOC( f c , f c ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], f c ) ( f ) +⎨*⎪+ 4 f c cos(θ1 ) cos(θ 2 ) Re S BOC ( f c , f c ) ( f ) S BCS([ s ], f c ) ( f )OSGD ( f ) = ⎩cos 2 (θ1 ) + cos 2 (θ 2 ) + 2r cos(θ1 ) cos(θ 2 ){}(J.24)where r indicates the correlation between BOC(fc, fc) and BCS([s], fc) for zero offset.
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