On Generalized Signal Waveforms for Satellite Navigation (797942), страница 27
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Owen, 2003b], Tertiary Offset Carrier Signals are3-level signals, similar in the form to the BOC signals, but with a dwell time ρ of value 0 ineach sub-carrier half cycle. As shown by [A.R. Pratt and J.I.R. Owen, 2003b], these signalshave appeared in response to the ever demanding needs of compatibility and interoperabilityof satellite navigation, in particular in the E1/L1 band.109GNSS Signal StructureAccording to this definition, if ρ 2 is a rational number it will be possible to express it bymeans of a fraction and the denominator of the reduced form of this fraction will be half ofthe minimum required length n of the MCS vector that would define the modulation using theMCS notation of chapter 4.2.
As we can observe from our assumption that ρ 2 is a rationalnumber, the length n would be finite too. Indeed, if ρ were irrational, n would have to beinfinite to represent the signal using the MCS definition. However, the Generalized MCSdefinition would be in this case more appropriate and the TOC modulation could be seen as aparticular case of GMCS. In this chapter we concentrate on the case when ρ 2 is rational.A straightforward approach to derive the general power density of the TOC signals is basedon (4.96).
Indeed, Tertiary Offset Carrier signals are a particular case of TCS with the samedefinition vector as that of the BOC signals. Thus, we can use the derived expressions for themodulating factor obtained in chapter 4.3.2 to give the general expression of a genericTOC(fs, fc, ρ). Again, we will distinguish between TOC signals in sine phasing, namelyTOCsin(fs, fc, ρ), and TOC signals in cosine phasing or TOCcos(fs, fc, ρ). We analyze both next.4.5.2.1Tertiary Offset Carrier in sine phasing : TOCsin(fs, fc, ρ)Assuming that ρ is a rational number, the sine-phased TOC signal fulfils the condition thatρ/2 = 2m/n and is shown to follow the pattern below. Indeed, for the particular case of thesine-phased TOC(fs, fc, ρ), the TCS vector would adopt the following form:0,0,0,0,0,0,0,…,1,1,1,1,1,1,1,1…,0,0,0,0,0,0,0…,-1,-1,-1,-1,-1,-1,-1…,0,0,0,0,0,0,0,…mn/2-2m2mn/2-2mm(4.97)It is clear to see that the concept can be easily generalized to any sine-phased TOC(fs, fc, ρ)without great difficulties as far as f s f c delivers an integer number.
We would simply have toextend the figure above by the factor f s f c .Recalling (4.96) and simplifying the terms in the GMod brackets for the case of a binary code,it can be shown that the PSD of a generic TOCsin(fs, fc, ρ) signal adopts the following form:⎡ πf⎤(1 − ρ )⎥sin 2 ⎢n −1⎛ f ⎞⎣ nf c⎦ ⎧n + 2 (− 1)i (n − i ) cos⎛⎜ i 2πf ⎞⎟⎫ (4.98)GTOCsin (n , f c , ρ ) = ⎜⎜ c ⎟⎟⎨∑⎜ nf ⎟⎬(π f )2i =1⎝1− ρ ⎠c ⎠⎭⎝⎩If we go now one step further using the results obtained in (B.11) for the sine-phased BOCmodulation, the power spectral density is shown to simplify to the following expression:⎡ πf⎤⎛ ⎞(1 − ρ )⎥ sin 2 ⎜⎜ πf ⎟⎟sin 2 ⎢⎛ f ⎞⎣ nf c⎦⎝ fc ⎠(4.99)GTOCsin (n , f c , ρ ) = ⎜⎜ c ⎟⎟2(π f )⎝1− ρ ⎠2 ⎛ πf ⎞⎟⎟cos ⎜⎜nf⎝ c⎠110GNSS Signal StructureWe can also write it in a more compact way as follows,2sincen = 2 fs⎡ ⎛ πf ⎞ ⎡ πf⎤⎤(1 − ρ )⎥ ⎥⎢ sin⎜⎜ ⎟⎟ sin ⎢f2f⎛ f ⎞⎦⎥GTOCsin ( f s , f c , ρ ) = ⎜⎜ c ⎟⎟ ⎢ ⎝ c ⎠ ⎣ s⎢⎥⎛ πf ⎞⎝1− ρ ⎠⎟⎟π f cos⎜⎜⎢⎥⎝ 2 fs ⎠⎣⎢⎦⎥f c , what coincides perfectly with the formulas(4.100)derivedin[A.R.
Pratt and J.I.R. Owen, 2005] for the even case.In the same manner, we can also obtain the expression for the power spectral density of theodd sine-phased TOC modulation as follows:⎡ ⎛ πf ⎞ ⎡ πf⎤⎤(1 − ρ )⎥ ⎥⎢ cos⎜⎜ ⎟⎟ sin ⎢f2f⎛ f ⎞⎦⎥GTOCsin ( f s , f c , ρ ) = ⎜⎜ c ⎟⎟ ⎢ ⎝ c ⎠ ⎣ s⎢⎥−ρ1⎛ πf ⎞⎝⎠⎟⎟πf cos⎜⎜⎢⎥⎢⎣⎥⎦⎝ 2 fs ⎠2(4.101)Next figure shows the evolution of the signal in the time domain:Figure 4.25. Chip waveform of TOCsin(1,1) for a dwell time ρ4.5.2.2Tertiary Offset Carrier in cosine phasing : TOCcos(fs, fc, ρ)In a similar way, the PSD of any TOCcos(fs, fc, ρ) can be expressed as follows,⎡ πf⎤(1 − ρ )⎥sin 2 ⎢⎛ f ⎞⎣ nf c⎦GGTOC cos (n , f c , ρ ) = ⎜⎜ c ⎟⎟Mod BOC cos ( n , f c )2−1ρ()fπ⎝⎠(4.102)where,BOC cos ( nf cGMod4, f c )⎧⎪⎡n/2⎡2πf= ⎨n + 2 ⎢∑ (− 1) i cos ⎢(2i − 1)nf c⎪⎩⎣⎣ i =1⎤ n / 2−1⎛ 2πfi⎥ + ∑ 2(− 1) (n / 2 − i ) cos⎜⎜ 2 i⎦ i =1⎝ nf c⎞⎤ ⎫⎪⎟⎟⎥ ⎬⎠⎦ ⎪⎭(4.103)111GNSS Signal StructureAs shown in (C.26), this modulating term can be simplified for any n, yielding the followingexpression for the even cosine-phased TOC modulation:⎡ πf⎤⎛ ⎞⎛⎞(1 − ρ )⎥ 4 sin 2 ⎜⎜ πf ⎟⎟ sin 2 ⎜⎜ πf ⎟⎟sin 2 ⎢⎛ f ⎞⎣ nf c⎦⎝ fc ⎠⎝ nf c ⎠(4.104)GTOC cos (n , f c , ρ ) = ⎜⎜ c ⎟⎟2(π f )⎝1− ρ ⎠2 ⎛ 2πf ⎞⎟⎟cos ⎜⎜⎝ nf c ⎠what can also be written in a more compact way as follows,2⎡⎛ πf ⎞ ⎛ πf ⎞ ⎡ πf⎤⎤⎟⎟ sin ⎢(1 − ρ )⎥ ⎥⎢ 2 sin⎜⎜ ⎟⎟ sin ⎜⎜⎛ f ⎞⎝ fc ⎠ ⎝ 4 f s ⎠ ⎣ 4 f s⎦⎥GTOC cos (n , f c , ρ ) = ⎜⎜ c ⎟⎟ ⎢⎢⎥⎛ πf ⎞⎝1− ρ ⎠⎟⎟π f cos⎜⎜⎢⎥⎝ 2 fs ⎠⎣⎢⎦⎥where we have made the change n = 4 f s f c as already seen in chapter 4.3.2.2.(4.105)In the same manner, the expression for the power spectral density of the odd cosine-phasedTOC modulation is shown to present the following form:⎡⎛ πf⎢ 2 cos⎜⎜⎛ f ⎞⎝ fcGTOC cos (n , f c , ρ ) = ⎜⎜ c ⎟⎟ ⎢⎝1− ρ ⎠ ⎢⎢⎢⎣⎞ ⎛ πf ⎞ ⎡ πf⎤⎤⎟⎟ sin ⎜⎜⎟⎟ sin ⎢(1 − ρ )⎥ ⎥⎠ ⎝ 4 fs ⎠ ⎣ 4 fs⎦⎥⎥⎛ πf ⎞⎟⎟π f cos⎜⎜⎥⎥⎦⎝ 2 fs ⎠2(4.106)Finally, it is important to comment regarding the time representation of the cosine-phasedTOC modulation, that this is similar to the sine-phased version that we studied in the previouschapter, but shifted by a quarter of the phase.4.5.3Tertiary Phase Shift Keying TPSKTertiary Phase Shift Keying signals or TPSK for short are also a particular case of TertiaryCoded Symbols with a modulation vector that consists of only ones.
Next figure shows thechip form of such a signal:Figure 4.26. Chip waveform of TPSK (1) for a dwell time ρ112GNSS Signal StructureAccording to the definition of previous chapters, it can be shown that the power spectraldensity adopts the following form in this case:⎡ πf⎤sin 2 ⎢ (1 − ρ )⎥n −1⎛ f ⎞⎣ nf c⎦ ⎪⎧n + 2⎡ (n − i )cos⎛⎜ i 2πf ⎞⎟⎤ ⎫⎪(4.107)GTPSK ( f c , ρ ) = ⎜⎜ c ⎟⎟⎨⎢∑⎜ nf ⎟⎥ ⎬(π f )2⎪⎩⎝1− ρ ⎠c ⎠⎦ ⎪⎝⎣ i =1⎭As we already saw in chapter 4.3.1, the term in the brackets can be simplified after somemath, and thus the expression for the power spectral density is shown to simplify to:⎡ πf⎤⎛ πf ⎞sin 2 ⎢ (1 − ρ )⎥ sin 2 ⎜⎜ ⎟⎟⎛ f ⎞⎣ nf c⎦⎝ fc ⎠(4.108)GTPSK ( f c , ρ ) = ⎜⎜ c ⎟⎟2(π f )⎝1− ρ ⎠2 ⎛ πf ⎞⎟⎟sin ⎜⎜⎝ nf c ⎠Or more explicitly:⎡ ⎡π f⎤ ⎛(1 − ρ )⎥ sin ⎜⎜ πf⎢ sin ⎢nf⎛ f ⎞⎦ ⎝ fcGTPSK ( f c , ρ ) = ⎜⎜ c ⎟⎟ ⎢ ⎣ c⎛ πf ⎞⎝1− ρ ⎠ ⎢π f sin ⎜⎜ ⎟⎟⎢⎝ nf c ⎠⎣⎢4.5.4⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎦⎥2(4.109)Generic m-PSK Coded SymbolsAll the modulations that we have analyzed so far in chapter 4.5 are tertiary.
However, TCSsignals are a particular case of a greater family of signals known as m-PSK Coded Symbols.In addition, m-PSK Coded Symbols are a particular case of MCS.m-PSK Coded Symbols have as spreading symbol an integer number m of equal-lengthsegments with a castle-like shape of 2 log 2 (m ) − 1 amplitude levels. Moreover, it can easily beshown that m-PSK Coded Symbols can be expressed as a linear combination of TCS orUTCS signals, where the UTCS signals are Unilateral TCS waveforms, analyzed inAppendix E. A particular case of m-PSK signal that is especially interesting in navigation isthe m-PSK Offset Carrier Modulation. We describe this signal waveform in detail in thefollowing chapters.4.5.5m-PSK Offset Carrier or m-PSK BOCThe m-PSK Offset Carrier modulation was discussed in [A.R.
Pratt and J.I.R. Owen, 2003b]where it was defined as m-PSK BOC modulation. In this thesis we will use the generalizednotation to define such a signal. Additionally, we will distinguish between sine-phasing andcosine-phasing. According to this, a sine-phased Offset Carrier with sub-carrier frequency fsand code frequency fc corresponds to an m-PSK BOCsin(fs, fc) in our notation. It is importantto recall that whenever we refer to a BOC, we mean the sine-phased version by default.Otherwise we will indicate it.113GNSS Signal StructureThe interest of this modulation lies in the fact that it allows for a very accurate spectrumcontrol. Given the importance that this topic has had during the design of the Galileo E1 OSsignals, specific configurations like 8-PSK BOC(2,2), were seriously considered in the past.In next chapter, general expressions for the power spectral densities will be derived for thiscase. As we will see, since the m-PSK Offset Carrier signals can be expressed as a linearcombination of TCS and UTCS, we can use the expressions derived above in our derivations.4.5.5.18-PSK Offset Carrier in sine phasing or 8-PSK BOCsin(fs, fc)The m-PSK BOC modulation can be expressed as a linear combination of TOCs with theircorresponding amplitudes.
For the case of the BOC8(fs , fc), Figure 4.27 shows in detail howthe castle chip construction of the chip waveform would look like.Figure 4.27. Time domain representation of a sine-phased BOC8(fs , fc)It must be noted that m refers here to the ratio between the sub-carrier frequency and the coderate according to the figure above. The amplitudes of the different parts of the chip result fromprojecting the phase points of an 8-PSK modulation as graphically explained in Figure F.2 ofthe Appendix.
Moreover, as we derive in Appendix F, all the points of the constellationpresent the same probability of occurrence.Following thus the time definition of Appendix F, the Fourier transform of any8-PSK sine-phased BOC signal can be expressed as follows,S BOC8sin( fs , fc )( f ) = λl STOCsin( f s , fc ,ρl )+ λs STOC sin ( f s , f c , ρ s )(4.110)where ρ l and ρ s represent the length of the zero support of the long and short sine-phasedTOC modulations as defined in chapter 4.5.2.1.
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