On Generalized Signal Waveforms for Satellite Navigation (797942), страница 28
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Moreover, λl and λs are the weighting factors114GNSS Signal Structurerequired to shape the chip waveform as shown in Figure 4.27. Appendix F proves that itspower spectral density can be expressed as:⎡ ⎛ πf ⎞ ⎡⎛ πf ⎞⎤ ⎛ πf⎟⎟ ⎢1 + 2 cos ⎜⎜⎟⎟⎥ sin⎜⎜⎢ sin⎜⎜⎝ 8 fs ⎠ ⎣⎝ 4 f s ⎠⎦ ⎝ f c⎢GBOC8 ( f , f ) ( f ) = 2 f c ⎢sin s c⎛ πf ⎞⎢⎟⎟πf cos ⎜⎜⎢⎣⎝ 2 fs ⎠⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(4.111)which is graphically shown in the next figure for the particular case of BOC8(2,2). As wehave already commented in other parts of this thesis, this signal was considered for some timeas a potential alternative for the Galileo E1 OS service given its high level of interoperabilitywith GPS.
For comparison also the original BOC(2,2) is depicted in the next figure.Figure 4.28. Power Spectral Density of BOC(2,2) versus BOC8(2,2)As we can recognize, the 8-PSK BOC(2,2) presents a very similar spectrum to that of theoriginal BOC(2,2), especially at low frequencies, with the interesting advantage that itintroduces additional zeros at 6 and 10 MHz.
Moreover, reducing the power spectrum aroundthe M-Code helps in ensuring higher compatibility with the rest of GPS signals around asidentified in [A.R. Pratt and J.I.R. Owen, 2003b].4.5.5.1.1 ACF of 8-PSK Offset Carrier in sine phasingOnce we have derived the spectral properties of the 8-PSK BOC(2,2) modulation, we spendsome time in the next lines deriving the analytical expression of its ideal autocorrelationfunction for the case of infinite bandwidth.To derive the general form of the ACF of any 8-PSK Offset Carrier in sine phasing we willmake use of the functions derived in chapter 4.3.2.3. To do so, we have to express first thePSD in the oscillation domain of ω in the appropriate form that we saw in previous chapters.According to this, using the spectrum derived in the previous pages, it can be shown that:115GNSS Signal Structure⎡ ⎛ ω ⎞⎡⎛ ω ⎞⎤ ⎛ ω ⎞ ⎤⎟⎟ ⎢1 + 2 cos⎜⎜⎟⎟⎥ sin⎜⎜⎟⎟ ⎥⎢ sin⎜⎜⎝ 16 f s ⎠ ⎣⎝ 8 f s ⎠⎦ ⎝ 2 f c ⎠ ⎥⎢GBOC8 ( f , f ) (ω ) = 2 f c ⎢⎥scsinω ⎛ ω ⎞⎟⎟⎢⎥cos⎜⎜2⎢⎣⎥⎦⎝ 4 fs ⎠2(4.112)which can be further developed and expressed as follows for the particular case of the 8-PSKBOC(2,2):()()()⎡⎛1⎞⎛3 ⎞ ⎤⎢ 8 2 − 9 cos⎜ 4 ωτ ⎟ + − 8 + 4 2 cos(ωτ ) + 3 − 4 2 cos⎜ 4 ωτ ⎟ + ⎥⎠⎝⎠ ⎥⎝⎢⎥1 ⎢⎛5 ⎞⎛1⎞RBOC8 (2, 2 ) (τ ) = 2 ⎢+ − 4 2 + 5 cos⎜ ωτ ⎟ + − 2 2 + 2 cos⎜ ωτ ⎟ +⎥sinτω ⎢⎝4 ⎠⎝2 ⎠⎥⎥⎢⎞⎛7⎛3 ⎞⎥⎢+ 2 2 − 2 cos⎜ ωτ ⎟ + cos⎜ ωτ ⎟ + 8 − 4 2⎝4 ⎠⎝2 ⎠⎦⎣(4.113)where1ΔT=τ=(4.114)m2 fc()(())()From (4.113), we can now obtain an expression for the autocorrelation function in the timedomain in a similar form as we did for the BOC signals of previous chapter: ´RBOC8sin((2, 2 )(τ ) = (8))()()()2 − 9 T1 (τ ) + − 8 + 4 2 T1 (τ ) + 3 − 4 2 T3 (τ ) + − 4 2 + 5 T5 (τ ) +(4)()44+ − 2 2 + 2 T1 (τ ) + 2 2 − 2 T3 (τ ) + T7 (τ ) + 8 − 4 2 S1 (τ )224(4.115)where Tk (τ ) and S k (τ ) were defined in (4.58) and (4.59) respectively.
It is important to notethat the time unit is the chip.Figure 4.29 shows graphically the shape of the autocorrelation function for the particular caseof the BOC(2,2) modulation.Figure 4.29. Autocorrelation Function of BOC8(2,2)116GNSS Signal StructureWe analyze in the next chapter the ACF of the cosine-phased version of the 8-PSK BOC(fs, fc)modulation. Later we will compare it with its sine-phased counterpart that we have juststudied. As an example, we will take the Galileo PRS since this service will make use of thecosine phased BOCcos(15,2.5).4.5.5.28-PSK Offset Carrier in cosine phasing or 8-PSK BOCcos(fs, fc)If we express the cosine-phased 8-PSK modulation as a linear combination of TOC signals aswe did with its sine-phased counterpart in (4.110), the representation in the time domain is notso easy to accomplish.
Therefore, as shown in Appendix E, an alternative expression isemployed that is based on a linear combination of unilateral TCS symbols (UTCS).According to this, the power spectral density of a generic even cosine-phased 8-PSK BOCmodulation is shown to adopt the following form:⎡⎡⎛ πf ⎞ ⎡⎛ πf ⎞⎤⎛ πf ⎞ 2 ⎛ πf ⎞⎤ ⎛ πf⎟⎟ ⎢1 − 4 − 2 2 sin 2 ⎜⎜⎟⎟⎥ + 8 sin 2 ⎜⎜⎟⎟ cos ⎜⎜⎟⎟⎥ sin⎜⎜⎢ ⎢− 1 + cos⎜⎜⎢ ⎢⎣⎝ 8 fs ⎠ ⎣⎝ 8 f s ⎠⎦⎝ 8 fs ⎠⎝ 8 f s ⎠⎥⎦ ⎝ f cGBOC8 ( f , f ) ( f ) = 2 f c ⎢cos s c⎛ πf ⎞⎢⎟πf cos⎜⎜⎢2 f s ⎟⎠⎝⎣()⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦(4.116)If we particularize now the expression above for the case of the cosine 8-PSK BOC(15,2.5)modulation, we can recognize interesting spectral properties compared with those of thetypical BOCsin(15,2.5) case.
We show the resulting spectrum in the next figure and compare itwith other similar alternatives that were object of study during the design of the GalileoSignal Plan.Figure 4.30. Power Spectral Density of some studied PRS alternatives1172GNSS Signal StructureAs we can recognize in the figure above, the cosine-phased 8-PSK BOC(15,2.5) modulationbehaves similar to the cosine-phased BOC(15,2.5) in the inner part of the spectrum, while onthe outer part it is slightly better regarding the spectral isolation, what would be interesting ifwe consider the effect on the GLONASS signals.
In addition, BOCsin(15,2.5) and8-PSK BOCsin(15,2.5) are also shown for comparison.A figure of great interest in analyzing the spectral compatibility between signals in a sharedband is the Cross Power Spectral Density (CPSD). The Cross Power Spectral Density isbasically the product of the power spectral densities of two signals and gives an idea of howmuch they overlap with each other.
Indeed, the Cross Power Spectral Density between thestudied signal and the M-Code is shown in the next figure. We will talk about this figure morein detail in the chapter 5.Figure 4.31. Cross Power Spectral Density of PRS alternatives with GPS M-CodeAs we can clearly see, while for an offset of less than 20 MHz both analyzed solutions showmore or less the same values, BOC8cos (15,2.5) seems to be slightly better above 20 MHz. Toconclude our analysis on the cosine-phased 8-PSK modulation, we derive in the next linessome expressions of interest for the autocorrelation function.4.5.5.2.1 ACF of 8-PSK Offset Carrier in cosine phasingFor the particular case of BOC8cos (15,2.5) , the power spectral density is shown to be expressedas follows in the ω domain.118GNSS Signal StructureGBOC8cos(15, 2.5 )(ω ) =()⎡⎤⎞⎛ 29⎞⎛ 19⎞⎛ 17⎢4 2 cos⎜ 96 ωτ ⎟ − 4 2 cos⎜ 32 ωτ ⎟ + − 60 2 + 62 cos⎜ 48 ωτ ⎟ +⎥⎠⎝⎠⎝⎠⎝⎢⎥⎢⎥⎞⎛7⎞⎛ 31 ⎞⎛ 1⎢+ − 4 + 4 2 cos⎜ ωτ ⎟ + − 38 + 36 2 cos⎜ ωτ ⎟ + 50 − 52 2 cos⎜ ωτ ⎟ +⎥⎠⎝ 16⎠⎝ 48 ⎠⎝ 24⎢⎥⎢⎥⎛ 7⎞⎛ 17⎞⎛ 35 ⎞⎢+ 4 2 − 8 cos⎜ ωτ ⎟ + 60 2 − 58 cos⎜ ωτ ⎟ − 4 2 cos⎜ ωτ ⎟ +⎥⎝ 96⎠⎝ 48 ⎠⎝ 96⎠⎢⎥⎢⎥⎛ 41 ⎞⎛1 ⎞⎛ 11⎞⎢+ − 10 + 12 2 cos⎜ ωτ ⎟ + 4 − 4 2 cos⎜ ωτ ⎟ + 4 2 − 8 cos⎜ ωτ ⎟ +⎥⎝ 48 ⎠⎝8 ⎠⎝ 32⎠⎢⎥⎢⎥552⎛⎞⎛⎞⎛⎞⎢+ − 4 + 4 2 cos⎜ ωτ ⎟ + − 70 + 68 2 cos⎜ ωτ ⎟ + 80 − 32 2 cos⎜ ωτ ⎟ +⎥⎝ 24⎠⎠⎝3⎠⎝ 16⎢⎥⎢⎥131⎛⎞⎛⎞⎛ 23 ⎞⎢+ 4 2 − 8 cos⎜ ωτ ⎟ − 40 cos 2 ⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ +⎥⎠⎝ 96⎠⎝6⎠⎝ 96⎢⎥⎢⎥111319⎢+ 4 2 cos⎛⎜ ωτ ⎞⎟ + 20 2 − 22 cos⎛⎜ ωτ ⎞⎟ + 8 − 4 2 cos⎛⎜ ωτ ⎞⎟ +⎥⎢⎥⎝ 96⎠⎝ 16⎠⎝ 32⎠⎢⎥⎢+ 4 2 − 8 cos⎛⎜ 1 ωτ ⎞⎟ + 4 2 cos⎛⎜ 13 ωτ ⎞⎟ + 82 − 84 2 cos⎛⎜ 5 ωτ ⎞⎟ + 4 2 cos⎛⎜ 25 ωτ ⎞⎟ − ⎥⎢⎥⎠⎝ 32⎝ 48 ⎠⎠⎝ 96⎠⎝ 96⎢⎥⎢− 4 2 cos⎛⎜ 1 ωτ ⎞⎟ + 4 2 − 8 cos⎛⎜ 17 ωτ ⎞⎟ + 88 2 − 176 cos⎛⎜ 1 ωτ ⎞⎟ +⎥⎢⎥⎠⎝ 32⎠⎝ 12⎠⎝ 96⎢⎥⎢+ − 26 + 28 2 cos⎛⎜ 11 ωτ ⎞⎟ + − 64 2 + 160 cos⎛⎜ 1 ωτ ⎞⎟ + − 90 + 92 2 cos⎛⎜ 1 ωτ ⎞⎟ −⎥⎥1 ⎢⎝ 16⎠⎝3 ⎠⎝ 48 ⎠= 2⎢⎥+τω ⎢⎛ 23 ⎞⎛ 43 ⎞⎛ 53 ⎞⎛5⎞ ⎥− 4 2 cos⎜ ωτ ⎟ + 14 − 12 2 cos⎜ ωτ ⎟ − 4 2 cos⎜ ωτ ⎟ + − 16 2 + 40 cos⎜ ωτ ⎟ −⎢⎝ 32⎠⎝ 48 ⎠⎝ 96⎠⎝6⎠ ⎥⎢⎥⎢− 4 2 cos⎛ 85 ωτ ⎞ + 2 − 4 2 cos⎛ 15 ωτ ⎞ − 4 2 cos⎛ 5 ωτ ⎞ +⎥⎜⎟⎜⎟⎜⎟⎢⎥⎝ 96⎠⎝ 16⎠⎝ 96⎠⎢⎥⎢+ − 80 + 40 2 cos⎛ 7 ωτ ⎞ + − 144 + 72 2 cos⎛ 1 ωτ ⎞ − 4 2 cos⎛ 83 ωτ ⎞ +⎥⎜⎟⎜⎟⎜⎟⎢⎥⎝4⎠⎝ 96⎠⎝ 12⎠⎢⎥⎢⎥⎛ 23 ⎞⎛ 37⎞⎛ 15⎞⎢+ − 54 + 52 2 cos⎜⎝ 48 ωτ ⎟⎠ + 18 − 20 2 cos⎜⎝ 48 ωτ ⎟⎠ + 4 2 cos⎜⎝ 32 ωτ ⎟⎠ +⎥⎢⎥⎢⎥⎞⎛ 37⎞⎛ 77⎞⎛ 3⎞⎛9⎢+ 46 − 44 2 cos⎜ 16 ωτ ⎟ + 8 − 4 2 cos⎜ 32 ωτ ⎟ + 4 2 cos⎜ 96 ωτ ⎟ − 4 2 cos⎜ 96 ωτ ⎟ + ⎥⎠⎝⎠⎝⎠⎝⎠⎝⎢⎥⎢⎥⎞⎞⎛ 5⎞⎛ 7⎛ 7⎢+ 4 − 4 2 cos⎜ ωτ ⎟ + − 86 + 84 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ +⎥⎠⎝ 32⎝ 48 ⎠⎠⎝ 24⎢⎥⎢⎥⎛ 35⎞⎛ 29⎞⎛ 55 ⎞⎢+ 4 2 − 8 cos⎜ ωτ ⎟ + 4 2 − 8 cos⎜ ωτ ⎟ + 30 − 28 2 cos⎜ ωτ ⎟ +⎥⎠⎝ 96⎠⎝ 48 ⎠⎝ 32⎢⎥⎢⎥⎛5⎞⎛ 19⎞⎛ 25 ⎞⎢+ − 112 + 56 2 cos⎜ ωτ ⎟ − 4 2 cos⎜ ωτ ⎟ + 44 2 − 42 cos⎜ ωτ ⎟ +⎥⎝ 12⎠⎝ 96⎠⎝ 48 ⎠⎢⎥⎢⎥⎛ 9⎞⎛ 41 ⎞⎛ 47⎞⎢+ 4 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ +⎥⎝ 32⎠⎝ 96⎠⎝ 96⎠⎢⎥⎢⎥⎞⎛ 31 ⎞⎛ 7⎞⎛ 59⎢+ 4 2 cos⎜ ωτ ⎟ − 4 2 cos⎜ ωτ ⎟ + 4 2 cos⎜ ωτ ⎟⎥⎢⎣⎥⎦⎠⎝ 32⎠⎝ 32⎠⎝ 96(())()(()())(()())())(())()(()()))(()(()())())())))())(()()(()(((())()(((()()()())119GNSS Signal Structure()()()⎡⎤⎞⎛ 49⎞⎛3⎞⎛1⎢+ − 92 2 + 94 cos⎜ 16 ωτ ⎟ + 76 2 − 74 cos⎜ 16 ωτ ⎟ + 4 2 − 8 cos⎜ 96 ωτ ⎟ +⎥⎠⎝⎠⎝⎠⎝⎢⎥⎢⎥⎞⎛ 67⎞⎛ 61 ⎞⎛ 17⎢+ − 4 + 4 2 cos⎜ ωτ ⎟ + 4 2 cos⎜ ωτ ⎟ − 4 2 cos⎜ ωτ ⎟ +⎥⎠⎠⎝ 96⎠⎝ 96⎝ 24⎢⎥⎢⎥⎛ 47⎞⎛ 43 ⎞⎛5 ⎞⎢+ − 6 + 4 2 cos⎜ ωτ ⎟ + 4 2 cos⎜ ωτ ⎟ + 4 − 4 2 cos⎜ ωτ ⎟ +⎥⎝ 48⎠⎝ 96⎠⎝8 ⎠⎢⎥⎢⎥⎛ 13⎞⎛ 25 ⎞⎛ 21 ⎞⎢+ − 4 + 4 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ +⎥⎝ 24⎠⎝ 96⎠⎝ 32⎠⎢⎥⎢⎥⎛ 91 ⎞⎛ 27⎞⎛1⎞ ⎛1 ⎞⎛1⎞ ⎛2 ⎞⎢+ 4 2 cos⎜ ωτ ⎟ + 4 2 − 8 cos⎜ ωτ ⎟ − 32 cos⎜ ωτ ⎟ cos⎜ ωτ ⎟ − 16 cos⎜ ωτ ⎟ cos⎜ ωτ ⎟ − ⎥⎝ 96⎠⎝ 32⎠⎝6⎠ ⎝3 ⎠⎝6 ⎠ ⎝3 ⎠⎢⎥⎢⎥15111⎛⎞ ⎛⎞⎛⎞ ⎛⎞⎛⎞⎢− 8 cos⎜ ωτ ⎟ cos⎜ ωτ ⎟ − 24 cos⎜ ωτ ⎟ cos⎜ ωτ ⎟ + 176 − 80 2 cos⎜ ωτ ⎟ +⎥1 ⎢⎝6⎠ ⎝6⎠⎝6 ⎠ ⎝2 ⎠⎝6 ⎠⎥+ 2⎢⎥τω293 ⎞797⎛⎞⎛⎞⎛⎞⎛⎢+ 34 − 36 2 cos⎜ ωτ ⎟ + − 4 + 4 2 cos⎜ ωτ ⎟ + 8 − 4 2 cos⎜ ωτ ⎟ + − 4 + 4 2 cos⎜ ωτ ⎟ + ⎥⎢⎝8 ⎠⎝ 48 ⎠⎠ ⎥⎝8⎠⎝ 96⎢⎥⎢+ 4 − 4 2 cos⎛⎜ 19 ωτ ⎞⎟ + 8 − 4 2 cos⎛⎜ 89 ωτ ⎞⎟ +⎥⎢⎥⎠⎝ 96⎠⎝ 24⎢⎥⎢+ 4 − 4 2 cos⎛⎜ 11 ωτ ⎞⎟ + 4 2 − 8 cos⎛⎜ 65 ωτ ⎞⎟ + 4 − 4 2 cos⎛⎜ 23 ωτ ⎞⎟ + 4 2 − 8 cos⎛⎜ 71 ωτ ⎞⎟ +⎥⎢⎥⎠⎠⎝ 96⎠⎝ 24⎠⎝ 96⎝ 24⎢⎥⎢+ − 76 2 + 78 cos⎛⎜ 11 ωτ ⎞⎟ + 8 − 4 2 cos⎛⎜ 73 ωτ ⎞⎟ +⎥⎢⎥⎝ 48 ⎠⎝ 96⎠⎢⎥⎢+ − 16 + 8 2 cos⎛⎜ 11 ωτ ⎞⎟ + 8 − 4 2 cos⎛⎜ 95 ωτ ⎞⎟ + 4 2 − 8 cos⎛⎜ 13 ωτ ⎞⎟ +⎥⎢⎥⎝ 96⎠⎝ 32⎠⎝ 12⎠⎢⎥⎢+ − 68 2 + 66 cos⎛ 13 ωτ ⎞ + − 48 2 + 120 cos⎛ 1 ωτ ⎞ + 24 2 − 48 cos⎛ 3 ωτ ⎞ + − 48 2 + 120 ⎥⎜⎟⎜⎟⎜⎟⎢⎣⎥⎦⎝ 48 ⎠⎝2⎠⎝4⎠()()()((())())(()()()()()()((())()((())()())))(())()()(4.117)withτ=1ΔT=m2 fc(4.118)From (4.117), we can derive an expression for the autocorrelation function in the timedomain.
It is important to note that the time variable is expressed in chips.120GNSS Signal StructureRBOC 8cos(15, 2.5 )(τ ) =()()()+ (50 − 52 2 )T (τ ) + (4 2 − 8)T (τ ) + (60 2 − 58 )T (τ ) − 4 2 T (τ ) + (− 10 + 12 2 )T+ (4 − 4 2 )T (τ ) + (4 2 − 8)T (τ ) + (− 4 + 4 2 )T (τ ) + (− 70 + 68 2 )T (τ ) +(τ ) + (8 − 4 2 )T (τ ) + 4 2 T (τ ) ++ (80 − 32 2 )T (τ ) + (4 2 − 8)T (τ ) − 40 M= 4 2 T29 (τ ) − 4 2 T17 (τ ) + − 60 2 + 62 T19 (τ ) + − 4 + 4 2 T 1 (τ ) + − 38 + 36 2 T31 (τ ) +963248716)174879618(24113223()4835965242396⎛1 1⎞⎜ , ⎟⎝6 6⎠(4148(τ ) +5163196)1196()+ 20 2 − 22 T13 (τ ) + 8 − 4 2 T19 (τ ) + 4 2 − 8 T 1 (τ ) + 4 2 T13 (τ ) + 82 − 84 2 T 5 (τ ) +1632()96(96)(48)+ 4 2 T25 (τ ) − 4 2 T 1 (τ ) + 4 2 − 8 T17 (τ ) + 88 2 − 176 T 1 (τ ) + − 26 + 28 2 T11 (τ ) +3232961216()()()+ (− 16 2 + 40 )T (τ ) − 4 2 T (τ ) + (2 − 4 2 )T (τ ) − 4 2 T (τ ) + (− 80 + 40 2 )T (τ ) ++ (− 144 + 72 2 )T (τ ) − 4 2 T (τ ) + (− 54 + 52 2 )T (τ ) + (18 − 20 2 )T (τ ) + 4 2 T (τ ) ++ (46 − 44 2 )T (τ ) + (8 − 4 2 )T (τ ) + 4 2 T (τ ) − 4 2 T (τ ) + (4 − 4 2 )T (τ ) ++ (− 86 + 84 2 )T (τ ) + (8 − 4 2 )T (τ ) + (4 2 − 8)T (τ ) + (4 2 − 8)T (τ ) ++ (30 − 28 2 )T (τ ) + (− 112 + 56 2 )T (τ ) − 4 2 T (τ ) + (44 2 − 42 )T (τ ) + 4 2 T (τ ) ++ (8 − 4 2 )T (τ ) + (8 − 4 2 )T (τ ) + 4 2 T (τ ) − 4 2 T (τ ) + 4 2 T (τ ) ++ (− 92 2 + 94 )T (τ ) + (76 2 − 74 )T (τ ) + (4 2 − 8)T (τ ) + (− 4 + 4 2 )T (τ ) ++ 4 2 T (τ ) − 4 2 T (τ ) + (− 6 + 4 2 )T (τ ) + 4 2 T (τ ) + (4 − 4 2 )T (τ ) ++ (− 4 + 4 2 )T (τ ) + (8 − 4 2 )T (τ ) + (8 − 4 2 )T (τ ) + 4 2 T (τ ) + (4 2 − 8)T (τ ) −(τ ) − 16 M (τ ) − 8 M (τ ) − 24 M (τ ) + (176 − 80 2 )T (τ ) +− 32 M+ − 64 2 + 160 T1 (τ ) + − 90 + 92 2 T 1 (τ ) − 4 2 T23 (τ ) + 14 − 12 2 T43 (τ ) − 4 2 T53 (τ ) +34856859614748512254819967323166796⎛1 1⎞⎜ , ⎟⎝ 6 3⎠2132⎛1 5⎞⎜ , ⎟⎝6 6⎠172443962596932599649964748⎛1 2⎞⎜ , ⎟⎝6 3⎠559631321167242932479615323796532132471237487796354896596234833241964815168396916619632589196273216⎛1 1⎞⎜ , ⎟⎝6 2⎠()()()()+ (4 − 4 2 )T (τ ) + (8 − 4 2 )T (τ ) + (4 − 4 2 )T (τ ) + (4 2 − 8)T (τ ) + (4 − 4 2 )T+ (4 2 − 8)T (τ ) + (− 76 2 + 78 )T (τ ) + (8 − 4 2 )T (τ ) + (− 16 + 8 2 )T (τ ) ++ (8 − 4 2 )T (τ ) + (4 2 − 8)T (τ ) + (− 68 2 + 66 )T (τ ) + (− 48 2 + 120 )T (τ ) ++ (24 2 − 48 )T (τ ) + (− 48 2 + 120 )S (τ )+ 34 − 36 2 T29 (τ ) + − 4 + 4 2 T3 (τ ) + 8 − 4 2 T79 (τ ) + − 4 + 4 2 T7 (τ ) +4819248968996719611241148959634659673961332813482324(τ ) +1112121(4.119)121GNSS Signal StructureThe autocorrelation is shown graphically in the next figure.Figure 4.32.
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