On Generalized Signal Waveforms for Satellite Navigation (797942), страница 77
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Wallner et al., 2005]:Figure M.1. Histogram of the Doppler Frequency Offsets for GPS and Galileo E1/L1Once we have defined mathematically the shape of the different sources of interference, wecan now compute the degradation suffered by a receiver due to other signals.
As we said atthe beginning, the interference of one system onto another one is given by the reduction of theeffective C N 0 , which can be expressed as follows:⎛C ⎞C⎟⎟ =⎜⎜⎝ N 0 ⎠eff N 0 + I Total(M.4)where N0 refers to the noise floor. Furthermore, we assume a noise value of -201.5 dBW/Hzfor all the purposes. In addition, the degradation of the effective C N 0 due to intra-systeminterference can be thus expressed as follows:C⎛ C ⎞IN0⎟⎟ == 1 + IntraΔ⎜⎜CN0⎝ N0 ⎠N 0 + I Intra(M.5)Equally, for the case of the degradation caused by the inter-system interference we have thefollowing expression:CI Inter + I Interop⎛ C ⎞N 0 + I Intra⎟⎟ =(M.6)Δ⎜⎜= 1+CN 0 + I Intra⎝ N0 ⎠N 0 + I Intra + I Inter + I Interop372Interference ModelNow that we have the necessary mathematical expressions to calculate the degradationsuffered by a receiver according to this simplified model, we only need to obtain the power ofthe desired and interfering signals at each point of the earth.
To do that, we have to simulatethe satellite positions and movements and account for the attenuations of the signal from thesatellite to the receiver. Indeed, once we know the minimum received powers as given in[Galileo SIS ICD, 2008], [GPS ICD 200], [GPS ICD-705, 2005] and [GPS ICD-800, 2006]we obtain the transmission power Pj of each satellite as done by [S. Wallner et al., 2005] andwe can derive the power received at user level at every point of the earth according to:C j = Pj + G j − Adist − Aatm − Apol + Guserwhere••••••(M.7)Pj depicts the transmission power from satellite j,Gj is the satellite antenna gain,Adist is the attenuation due to the distance between the satellite and the user,Apol is the attenuation of the signal due to the mismatch losses in the polarization,Aatm is the attenuation of the signal due to the atmosphere, andGuser is the receiver antenna gain.The satellite antenna gain Gj is a function of the Off-Boresight Angle α as defined next:Figure M.2.
Definition of Off-Boresight Anglebeing the typical satellite antenna gain as shown in the following figure:Figure M.3. Assumed Typical Satellite Antenna GainAdditionally, the signal attenuation due to the free-space losses Adist is given by:Adist⎛ c ⎞⎟⎟= ⎜⎜⎝ 4π d fc ⎠2(M.8)where,• c is the speed of light• d is the distance between the satellite and the user• and fc is the carrier frequency373Interference Model374Spectral Separation Coefficients between two QPSK signalsNAppendix. SSC between two QPSKsignalsIn the next lines the SSC between two QPSK signals is derived. Let us assume for simplicitythat the desired QPSK signal adopts the following form [J.-L. Issler et al., 2003]:[]Pd d d (t ) c dD (t ) + j c dP (t )where,•(N.1)Pd is the amplitude of the desired useful signal,•d d (t ) is the data modulating the useful PRN data-code in phase,•ccD (t ) is the useful data-code, and•c dP (t ) is the PRN code of the useful pilot-channel.Moreover, let us assume the following receiver model as done by [J.-L.
Issler et al., 2003]:Figure N.1. SSC Receiver ModelAnd that the interfering received QPSK signal presents the following form:[]Pi d i (t ) ciD (t ) + j ciP (t )with:•(N.2)Pi is the amplitude of the interfering signal,•d i (t ) is the data modulating the interfering data-code,•ciD (t ) is the interfering data-code,•ciP (t ) is the PN code of the interfering pilot-channel.As we can see, the replica signals generated by the receiver will be:c dD (t − Δτ ) − j c dP (t − Δτ )(N.3)and− c dD (t − Δτ ) − j c dP (t − Δτ )(N.4)375Spectral Separation Coefficients between two QPSK signalswhere•cdD (t − Δτ ) is the PRN code replica of the data-channel,•and cdP (t − Δτ ) the PRN code replica of the pilot-channel.Furthermore, the phase shift between the desired signal and the replica is given by e jϕ whilethe phase shift with the interfering signal is e jθ with θ = 2 π fd t + θ0, where fd is the Dopplerfrequency shift. Equally, if the power ratio between the interfering signal and the desiredsignal is expressed as follows:Pi(N.5)Θ=Pdthe value of the signal s1 (t ) at point 1 can thus be expressed as follows:⎤⎡ ⎡d d (t ) cdD (t ) d d (t − Δτ ) cdD (t − Δτ ) + j cdP (t ) d d (t − Δτ ) cdD (t − Δτ ) ⎤ jϕe⎥⎢⎢⎥DPPP⎥⎢ ⎣⎢− j d d (t ) cd (t ) cd (t − Δτ ) + cd (t ) cd (t − Δτ )⎦⎥s1 (t ) = Pd ⎢⎥⎡d i (t ) ciD (t ) d d (t − Δτ ) cdD (t − Δτ ) + j ciP (t ) d d (t − Δτ ) cdD (t − Δτ ) ⎤ jθ jϕ ⎥⎢⎥e e ⎥⎢+ Θ ⎢− j d (t ) c D (t ) c P (t − Δτ ) + c P (t ) c P (t − Δτ )⎢⎥⎦iidid⎣⎦⎣(N.6)with e jΩ = e jθ e jϕ .
Moreover, if we integrate (N.6), the signal at the output of the correlator atpoint 2 will be then:⎤⎡ ⎡ cdD (t ) cdD (t − Δτ )dt + j d d (t − Δτ ) cdP (t ) cdD (t − Δτ ) dt ⎤∫T⎥⎢ ⎢T∫⎥c⎢⎢ c⎥ e jϕ ⎥⎥⎢ ⎢− j d d (t ) ∫ cdD (t ) cdP (t − Δτ ) dt + ∫ cdP (t ) cdP (t − Δτ ) dt⎥⎥⎢ ⎢⎣⎥⎦TcTc⎥⎢⎤⎡⎥⎢DDs2 (t ) = Pd ⎢⎥⎢d i (t ) d d (t − Δτ ) ∫ ci (t ) cd (t − Δτ ) dt +⎥⎥⎢⎥⎢Tc⎥⎢⎢jθ jϕ ⎥PD⎥e e ⎥⎢+ Θ ⎢+ j d d (t − Δτ ) ∫ ci (t ) cd (t − Δτ ) dtTc⎥⎢⎥⎢⎥⎢⎥⎢PPDP⎢− j d i (t ) ∫ ci (t ) cd (t − Δτ ) dt + ∫ ci (t ) cd (t − Δτ ) dt ⎥⎥⎢TcTc⎦⎣⎦⎣what can be further simplified to:[]⎡ ℜc D (Δτ ) + ℜc P (Δτ ) e jϕ⎤d⎢ d⎥s2 (t ) = Pd ⎢⎡d i (t ) d d (t ) ℜciD , c dD (Δτ ) + j d d (t ) ℜciP , c dD (Δτ )⎤ ⎥⎢+ Θ ⎢⎥ e jΩ ⎥⎢⎥⎦ ⎥⎦⎢⎣− j di (t )ℜciD , c dP (Δτ ) + ℜciP , c dP (Δτ )⎣(N.7)(N.8)As we can see, we have assumed that we integrate over the duration of the data bits and thusthe bits have no impact on the SSC computation.
We can also express equation (N.8) as:s2 (t ) = Pd (Y + Θ Z )(N.9)376Spectral Separation Coefficients between two QPSK signalswith:[](N.10)Y = ℜc D (Δτ ) + ℜc P (Δτ ) e jϕand[dd]Z = d i (t ) d d (t ) ℜc D , c D (Δτ ) + j d d (t ) ℜc P , c D (Δτ ) − j d i (t )ℜc D , c P (Δτ ) + ℜc P , c P (Δτ ) e jΩidididid(N.11)In addition, since the SSC can also be interpreted as the mean power of the cross-correlationfunction as defined in chapter 5.1.1, and we are interested in the SSC between the interferingsignal and the desired signal, we have:12SSC(si , sd ) = lim ∫ N dt(N.12)Tc → ∞ Tcwhat can be further simplified as follows:122SSC(si , sd ) = lim ∫ ℜc D , c D (Δτ ) + ℜc P , c D (Δτ ) + ℜc D , c P (Δτ ) − ℜc P , c P (Δτ ) dtididididTc → ∞ Tc Tcor equivalently:1⎧DDPP∫ ℜciD ,cdD (Δτ ) ℜciP ,cdD (Δτ )dt⎪SSC ci , cd + SSC ci , cd + 2 Tlimc →∞ Tc Tc⎪SSC(si , sd ) = ⎨⎪+ SSC c D , c P + SSC c P , c D − 2 lim 1 ℜ D P (Δτ )ℜ P P (Δτ )dtididci , c dci , c dTc → ∞ T ∫⎪c Tc⎩{[(] [)(()]}(N.13))()(N.14)where the two integral terms are shown to be much smaller than the others.
As a conclusion,we can simplify as follows [J.-L. Issler et al., 2003]:()()()(SSC (si , sd ) = SSC ciD , cdD + SSC ciP , cdP + SSC ciD , cdP + SSC ciP , cdD)(N.15)which is the expression suggested in chapter 5.1.1.377Spectral Separation Coefficients between two QPSK signals378Spectral Separation Coefficient between two generic BCS signalsOAppendix. Analytical expressions tocompute SSCsIn the next lines we derive analytical expressions to compute the Spectral SeparationCoefficients (SSC) between two generic BCS signals using the theory derived in chapters 4and 5.
Later, particular expressions of interest will be obtained.O.1Appendix. SSC between two generic BCSsignalsAs we have seen in chapter 5.1, the Spectral Separation Coefficient (SSC) between two BCSsignals can be approximated when the integration time tends to infinity:SSC BCS([r ], f 1 ) − BCS([s ], f 2 ) = ∫ccβr / 2−βr / 2GBCS(r , f 1 )( f )GBCS(s , f 2 )( f )df ≡ SSCBCS1 − BCS2cc(O.1)where the power spectral density of each MCS signal was derived in (4.23) for the mostgeneral case. If we further assume that we work with binary sequences, the expression can befurther simplified as follows:⎛ πf ⎞sin 2 ⎜⎜ i ⎟⎟n −1 n⎡2πf ⎤ ⎫⎝ nfc ⎠ ⎧n +(O.2)GBCS([s ], f i ) ( f ) = f ci2si s j cos⎢( j − i ) i ⎥ ⎬⎨∑∑2cnfc ⎦ ⎭(πf ) ⎩ i =1 j =i +1⎣and thus the product of the two power spectral densities will adopt the following form⎧2 ⎛ πf ⎞⎟⎪ sin ⎜⎜1⎟n1 −1 n1⎡2πf ⎤ ⎫⎝ n1 f c ⎠ ⎧n +⎪f12ri rj cos ⎢( j − i ) 1 ⎥ ⎬ ×⎨∑∑1c2⎪⎪n1 f c ⎦ ⎭(πf ) ⎩ i =1 j =i +1⎣GBCS(r , f 1 )( f )GBCS(s , f 2 ) = ⎨cc⎞⎪2 ⎛ πf⎟⎜⎜sin2n 2 −1 n 2⎪n2 f c ⎟⎠ ⎧⎡2πf ⎤ ⎫2⎝⎪× f c()+−nssji2cos''⎨∑∑2''ij⎥⎬⎢n2 f c2 ⎦ ⎭(πf )2 ⎩⎪⎩i ' =1 j ' = i ' +1⎣Moreover, we can further expand this expression as follows:GBCS([r ], f )( f )GBCS([s ], f ) ≡ GBCS ( f )GBCS ( f ) =1c2c1(O.3)2⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟n1 f c ⎠n2 f c2 ⎟⎠1 2⎝⎝n1n2 f c f c+(πf )2(πf )2⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟n1 f c ⎠n2 f c2 ⎟⎠⎡2πf ⎤1 2⎝⎝+ 2n1 f c f c ∑ ∑ si ' s j 'cos ⎢( j '−i ')⎥+22n2 f c2 ⎦(πf )(πf )i '=1 j '=i ' +1⎣n2 −1n2⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟n1 f c ⎠n2 f c2 ⎟⎠⎡2πf ⎤1 2⎝⎝+ 2n2 f c f c ∑ ∑ ri rjcos ⎢( j − i )⎥+22n1 f c1 ⎦(πf )(πf )i =1 j =i +1⎣n1 −1 n1⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟n1 f c ⎠n2 f c2 ⎟⎠⎡⎡2πf ⎤2πf ⎤1 2⎝⎝cos ⎢( j '−i ')cos ⎢( j − i )+ 4 f c f c ∑ ∑∑ ∑ ri rj si ' s j '⎥222⎥n2 f c ⎦n1 f c1 ⎦(πf )(πf )i =1 j =i +1 i '=1 j '=i ' +1⎣⎣n1 −1 n1 n2 −1n2(O.4)379Spectral Separation Coefficient between two generic BCS signalsAfter integrating in the receiver bandwidth we can express our SSC as sum of other fourterms yielding:SSC BCS1 − BCS2 = ∫βr / 2−βr / 2GBCS1 ( f )GBCS2 ( f )df = SSC1 + SSC 2 + SSC 3 + SSC 4(O.5)where⎛ πf ⎞⎛ πf ⎞⎟⎟ sin 2 ⎜⎜sin 2 ⎜⎜1⎟2 ⎟β/2nfnfr⎝ 2 c ⎠ df⎝ 1 c ⎠SSC1 = n1n2 f c1 f c2 ∫−βr / 2(πf )2(πf )2(O.6)⎛ πf ⎞⎛ πf ⎞⎟⎟ sin 2 ⎜⎜sin 2 ⎜⎜1⎟βr / 2n2 f c2 ⎟⎠n1 f c ⎠⎡2πf ⎤1 2⎝⎝SSC 2 = 2n1 f c f c ∑ ∑ si ' s j ' ∫cos ⎢( j '−i ')df222⎥−βr / 2nf(πf )(πf )i ' =1 j ' = i ' +12 c ⎦⎣n 2 −1 n 2⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟βr / 2n2 f c2 ⎟⎠n1 f c ⎠⎡2πf ⎤1 2⎝⎝SSC 3 = 2n2 f c f c ∑ ∑ ri rj ∫cos ⎢( j − i ) 1 ⎥ df22−βr / 2n1 f c ⎦(πf )(πf )i =1 j = i +1⎣n1 −1 n1(O.7)(O.8)Finally,SSC 4 =⎛ πf ⎞⎛ πf ⎞⎟ sin 2 ⎜⎜⎟sin 2 ⎜⎜1⎟βr / 2n2 f c2 ⎟⎠n1 f c ⎠⎡⎡2πf ⎤2πf ⎤1 2⎝⎝= 4 f c f c ∑ ∑∑ ∑ ri rj si ' s j ∫cos ⎢( j '−i ')cos ⎢( j − i )⎥⎥ df222−βr / 2n2 f c ⎦n1 f c1 ⎦(πf )(πf )i =1 j =i +1 i '=1 j '=i ' +1⎣⎣n1 −1 n1 n2 −1n2(O.9)Furthermore, it can be shown that an analytical expression can be found for the fourintegrations above when β r → ∞ .
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