On Generalized Signal Waveforms for Satellite Navigation (797942), страница 70
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PSD of 8-PSK sine-phasedBOC SignalsIn this Appendix we will derive the power spectral density of an 8-PSK signal in sine phasing.As we will see, the same expression derived by [A.R. Pratt and J.I.R. Owen, 2003] is obtainedhere using the more general definition of MCS that we have presented in this thesis.As we saw in chapter 4.5.5.1 the Fourier transform of an arbitrary 8-PSK BOCsin can beexpressed as follows:S BOC8 ( f , f ) ( f ) = λl STOC sin ( f s , f c , ρ l ) + λs STOC sin ( f s , f c , ρ s )scsin(F.1)with13λs = 1 −ρs =42(F.2)11λl =ρl =42This function is graphically shown next:Figure F.1.
Signal waveform of the sine-phased 8-PSK BOC(fs, fc) modulationAs we can recognize, the different amplitudes of the signal waveform correspond to the phasestates of an 8-PSK modulation according to the next figure. Moreover, all the states of theconstellation present the same probability of occurrence.327Power Spectral Density of 8-PSK sine-phased BOC signalsFigure F.2. Constellation points of the sine-phased 8-PSK BOC(fs,fc) modulationFurthermore, the time series of the 8-PSK modulation can be expressed as the sum of thefollowing two functions:Figure F.3. Long Chip S TOCsin ( f s , f c , ρl ) Function required to form 8-PSK BOCsin(fs, fc)Figure F.4.
Short Chip S TOCsin ( f s , f c , ρ s ) Function required to form 8-PSK BOCsin(fs, fc)328Power Spectral Density of 8-PSK sine-phased BOC signalsFor simplicity, we recall again the Fourier transform of the arbitrary TCS(fs, fc,δ ):jπfn−⎡⎤(⎞ f ) = 1 sin ⎢ πf (1 − ρ )⎥ e nf c ∑ sk eS ⎛ nf cTCS⎜⎜ f s =, f c ,δ ⎟⎟πfk =1⎣ nf c⎦2⎝⎠j 2 kπfnf c(F.3)Particularizing now (F.2) in (F.3), we obtain the following expression for the Fouriertransform of the 8-PSK BOC modulation in sine phasing.
As we can see, we have made useof the long and short chip functions S TOCsin ( f s , f c , ρl ) and S TOCsin ( f s , f c , ρ s ) , defined above.S⎧⎡⎩⎣ nf c⎤⎡⎦⎣ nf cjπf−⎤ ⎫ 1 nf c ne ∑ sk ek =1⎦ ⎭ πf( f ) = ⎨λl sin ⎢ πf (1 − ρl )⎥ + λs sin ⎢ πf (1 − ρ s )⎥ ⎬⎞nf⎛BOC8sin ⎜⎜ f s = c , f c ⎟⎟2⎠⎝j 2 kπfnf c(F.4)As a result, the power spectral density of the sine-phased BOC8(fs,fc) will be:⎧⎡ πf⎤⎡ πf⎤⎫⎨λl sin ⎢ (1 − ρl )⎥ + λs sin ⎢ (1 − ρ s )⎥ ⎬f⎣ nf c⎦⎣ nf c⎦⎭G 8 ⎛ nf c ⎞ ( f ) = c ⎩2BOC sin ⎜⎜ f s =, f c ⎟⎟ρ′(πf )2⎠⎝2n∑s ek =1−jk2πfk 2nf c(F.5)or equivalently:2⎧⎡ πf⎤⎡ πf⎤⎫2 ⎛ πf ⎞⎨λl sin ⎢ (1 − ρl )⎥ + λs sin ⎢ (1 − ρ s )⎥ ⎬ sin ⎜⎜ ⎟⎟f⎣ nf c⎦⎣ nf c⎦⎭⎝ fc ⎠G 8 ⎛ nf c ⎞ ( f ) = c ⎩2BOC sin ⎜⎜ f s =, f c ⎟⎟ρ′⎛ πf ⎞(πf )2⎠⎝⎟⎟cos 2 ⎜⎜⎝ nf c ⎠(F.6)where we have used the results of Appendix B for the modulating term.
Indeed, the square ofthe absolute value of the sum term is common to the usual BOC modulation in sine phasing.This term was derived in (B.11) for the even case. Additionally, the correction factor ρ’ isshown to be:ρ + ρs(F.7)ρ′ = l= 0 .52According to this,2⎧ 1⎡ 3πf ⎤ ⎛1 ⎞ ⎡ πf ⎤ ⎫2 ⎛ πf ⎞sin ⎢⎟ sin ⎢⎨⎥ ⎬ sin ⎜⎜ ⎟⎟⎥ + ⎜1 −2 ⎣ 4nf c ⎦ ⎝2 ⎠ ⎣ 4nf c ⎦ ⎭⎝ fc ⎠G 8 ⎛ nf c ⎞ ( f ) = 2 f c ⎩2BOC sin ⎜⎜ f s =, f c ⎟⎟⎛ πf ⎞(πf )2⎠⎝⎟⎟cos 2 ⎜⎜⎝ nf c ⎠(F.8)and since in the case of the sine-phased TOC modulation n = 2 f s f c , we can also express itas follows:2⎧ 1⎡ 3πf ⎤ ⎛1 ⎞ ⎡ πf ⎤ ⎫2 ⎛ πf ⎞sin ⎢⎟ sin ⎢⎨⎥ ⎬ sin ⎜⎜ ⎟⎟⎥ + ⎜1 −2 ⎣ 8 fs ⎦ ⎝2 ⎠ ⎣8 fs ⎦⎭⎝ fc ⎠GBOC8 ( f , f ) ( f ) = 2 f c ⎩2scsin⎛ πf ⎞(πf )⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠(F.9)which can be further simplified as:329Power Spectral Density of 8-PSK sine-phased BOC signals⎡ ⎛ πf ⎞ ⎡⎛ πf ⎞⎤ ⎛ πf⎟⎟ ⎢1 + 2 cos⎜⎜⎟⎟⎥ sin⎜⎜⎢ sin⎜⎜⎝ 8 fs ⎠ ⎣⎝ 4 f s ⎠⎦ ⎝ f c⎢GBOC8 ( f , f ) ( f ) = 2 f c ⎢sinsc⎛ πf ⎞⎟⎟⎢π f cos⎜⎜⎢⎣⎝ 2 fs ⎠⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(F.10)For the case of the odd sine-phased BOC8(fs,fc) modulation, only the modulating term changeswith respect to the above derived expression.
Thus, since this is common to that of anyBOCsin modulation with n odd, the general expression is shown to be:⎡ ⎛ πf ⎞ ⎡⎛ πf ⎞⎤ ⎛ πf⎟⎟⎥ cos⎜⎜⎟⎟ ⎢1 + 2 cos⎜⎜⎢ sin ⎜⎜⎝ 4 f s ⎠⎦ ⎝ f c⎝ 8 fs ⎠ ⎣⎢GBOC8 ( f , f ) ( f ) = 2 f c ⎢sin s c⎛ πf ⎞⎢⎟⎟πf cos⎜⎜⎢⎣⎝ 2 fs ⎠⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(F.11)These expressions coincide perfectly with equivalent expressions that can be found in[A.R. Pratt and J.I.R. Owen, 2003].330Power Spectral Density of 8-PSK cosine-phased BOC signalsGAppendix. PSD of 8-PSK cosine-phasedBOC SignalsWe derive in this Appendix the power spectral density of a generic 8-PSK BOC modulationwith cosine phasing in an analogue way as we have done with its sine-phased counterpart.Indeed any 8-PSK signal can be expressed as a linear combination of TCS signals in thedomain of the Fourier Transform as we have seen in Appendix F. This will be of course alsothe case for the 8-PSK BOC cosine-phased modulation, although as we will see next, thelinear combination to build here is a little bit more complex since we need UTCS.As we have seen in Appendix F, any sine-phased 8-PSK BOC modulation can be expressed asthe sum of two sine-phased TOC signals.
For the case of the cosine-phased 8-PSK BOCmodulation however, five UPSK functions are needed. UPSK is the unilateral version ofTPSK, which is a particular case of UTCS with BPSK-like shape. Indeed, an arbitrary8-PSK BOCcos can be defined as:5S BOC8cos( )( f , f ) f = ∑ λi S UTCS( f , ρ )scci =1(G.1)iwhereλ1 = −1 λ2 = 1 −ρ1 = 17ρ2 =812λ3 =125ρ3 =8λ4 =123ρ4 =81λ5 = 1 −1ρ5 =82(G.2)being the chip waveform for the even case as follows:Figure G.1. Signal waveform of the cosine-phased 8-PSK BOC(fs, fc) modulationUsing now the general expression of the Fourier transform of an UTCS signal as given by(E.1), since BOC8cos ( f s , f c ) can be expressed as the linear combination of UTCS signals, itsFourier transform will adopt the following form:jπfρ⎧⎪ 5⎛ πf ⎞ nf c i ⎫⎪ 1⎟⎟ eS 8 ⎛ nf c ⎞ ( f ) = ⎨− ∑ λi sin ⎜⎜ ρi⎬BOC cos ⎜⎜ f s =, f c ⎟⎟nfi =1⎪⎪⎭ πfc⎝⎠2⎠⎝⎩n∑sk =1ke−j 2 kπfnf c(G.3)331Power Spectral Density of 8-PSK cosine-phased BOC signalsyielding the power spectral density of the cosine-phased BOC8(fs,fc) thus:⎛ πf ⎞⎟⎟ e− ∑ λi sin⎜⎜ ρinfi1=c⎝⎠fG 8 ⎛ nf c ⎞ ( f ) = c2BOC cos ⎜⎜ f s =, f c ⎟⎟ρ′(πf )2⎠⎝5jπfρ i 2nf cn∑s ek =1−jk2πfk 2nf c(G.4)This can be further simplified as shown next:⎛ πf ⎞⎟⎟ eλi sin ⎜⎜ ρi∑nf=1ic⎝⎠fG 8 ⎛ nf c ⎞ ( f ) = c2BOC cos ⎜⎜ f s =, f c ⎟⎟ρ′(πf )2⎠⎝5jπfρ i 2nf c⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ fc ⎠⎛ πf ⎞⎟⎟cos 2 ⎜⎜⎝ nf c ⎠(G.5)since the square of the absolute value of the sum term, namely the modulating factor, iscommon to the even sine-phased BOC modulation, whose expression was derived in (B.11).Additionally, the correction factor ρ ′ is shown to be this time 1 2 too.
Consequently,πf7πf⎛ πf ⎞ j nf c ⎛1 ⎞ ⎛ 7πf ⎞ j 8 nf c⎟⎟ e⎟e− sin ⎜⎜+ ⎜1 −+⎟ sin ⎜2 ⎠ ⎜⎝ 8nf c ⎟⎠⎝⎝ nf c ⎠5πf23πf⎛ 3πf ⎞ j 8 nf c⎛ 5πf ⎞ j 8 nf c11⎟e⎟⎟ e+sin ⎜⎜+sin ⎜⎜+2 ⎝ 8nf c ⎟⎠2 ⎝ 8nf c ⎠πfGnf⎛BOC8cos ⎜⎜ f s = c , f c2⎝⎞⎟⎟⎠( f ) = 2 fc1 ⎞ ⎛ πf ⎞ j 8 nf c⎛⎟e+ ⎜1 −⎟ sin ⎜2 ⎠ ⎜⎝ 8nf c ⎟⎠⎝⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ f c ⎠ (G.6)⎛ πf ⎞⎟⎟cos 2 ⎜⎜nf⎝ c⎠(πf )2Moreover, since n = 2 f s f c , the previous expression can be further simplified as follows:πf7πf⎛ πf ⎞ j 2 f s ⎛1 ⎞ ⎛ 7πf ⎞ j 16 f s⎟⎟ e⎟e− sin ⎜⎜+ ⎜1 −+⎟ sin ⎜2 ⎠ ⎜⎝ 16 f s ⎟⎠⎝⎝ 2 fs ⎠5πf23πf⎛ 5πf ⎞ j 16 f s⎛ 3πf ⎞ j 16 f s11⎟⎟ e⎟e+++sin ⎜⎜sin ⎜⎜2 ⎝ 16 f s ⎟⎠2 ⎝ 16 f s ⎠πfGBOC8cos( fs , fc )( f ) = 2 fc1 ⎞ ⎛ πf ⎞ j 16 f s⎛⎟e+ ⎜1 −⎟ sin ⎜2 ⎠ ⎜⎝ 16 f s ⎟⎠⎝(πf )2⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ f c ⎠ (G.7)⎛ πf ⎞⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠We can further simplify this expression a little bit more yielding finally:⎡ ⎧⎪⎛ πf ⎞⎤⎛ πf ⎞⎫⎪ ⎛ πf⎛ πf ⎞ ⎡⎟⎟⎥ + 2 sin 2 ⎜⎜⎟⎟⎬ sin ⎜⎜⎟⎟ ⎢1 − 4 − 2 2 sin 2 ⎜⎜⎢ ⎨− 1 + cos⎜⎜⎢ ⎪⎩⎝ 8 f s ⎠⎦⎝ 4 f s ⎠⎪⎭ ⎝ f c⎝ 8 fs ⎠ ⎣GBOC8 ( f , f ) ( f ) = 2 f c ⎢sccos⎛ πf ⎞⎢⎟⎟πf cos⎜⎜⎢2f⎝ s⎠⎣()⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(G.8)332Power Spectral Density of 8-PSK cosine-phased BOC signalsOnce we have obtained the expression for the even 8-PSK cosine-phased BOC modulation,we can easily derive the form for the odd one too using the results from other parts of thisthesis.
Indeed, the main advantage of expressing our signal as a linear combination of UTCSis that we can also derive the odd version using a modulating factor as suggested in this work.Recalling thus the definition of even and odd in chapter 4.3.2, if for the even form we had avector [-1,+1,+1,-1] for the odd case we should have [-1,1,1,-1,-1,1]. However, since we areexpressing our signal with a sine-phased function whose generating vector is [-1,1], for theodd case we should then take [-1,1,-1]. Therefore, to calculate the expression for n odd, wesimply have to look at the modulating term of the odd sine-phased BOC modulation.Additionally, since the sine-phased is a linear combination of sine-phased TOC modulations,n will adopt the value n = 2 f s f c .
Taking into account all these considerations, the powerspectral density of GBOC8 ( f , f ) ( f ) for n odd is shown to adopt the following form:cossc⎡ ⎧⎪⎛ πf⎢ ⎨− 1 + cos⎜⎜⎢⎪⎝ 8 fsGBOC8 ( f , f ) ( f ) = 2 f c ⎢ ⎩s ccos⎢⎢⎣()⎞⎡⎛ πf ⎞⎤⎛ πf⎟⎟ ⎢1 − 4 − 2 2 sin 2 ⎜⎜⎟⎟⎥ + 2 sin 2 ⎜⎜⎠⎣⎝ 8 f s ⎠⎦⎝ 4 fs⎛ πf ⎞⎟⎟πf cos⎜⎜2f⎝ s⎠⎞⎫⎪ ⎛ πf⎟⎟⎬ cos⎜⎜⎠⎪⎭ ⎝ f c⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(G.9)333Power Spectral Density of 8-PSK cosine-phased BOC signals334Equivalent C/N0 in presence of RF interferenceHAppendix. Equivalent C/N0 in presence ofInterferenceThis Appendix derives some expressions of interest for the Equivalent Carrier to Noise Ratioin presence of RF interference. As we know, one of the main effects of RF interference is toreduce the Cd N 0 of the desired signal d, as shown next:⎛ Cd ⎞⎟⎟ =⎜⎜⎝ N 0 ⎠effCdN0∫βr / 2−βr / 2C∫ −βr / 2 Gd ( f ) df + Ni0βr / 2Gd ( f ) df∫βr / 2−βr / 2(H.1)Gi ( f ) Gd ( f ) dfwhere the subindex i refers to the interfering signal and d to the desired signal.
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