On Generalized Signal Waveforms for Satellite Navigation (797942), страница 23
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1{n − 2}⎟... 1{n − 3}⎟⎟...... ⎟1{0}⎟⎠(4.29)87GNSS Signal StructureAccording to this, the modulating function will be:⎧ n −1⎡ 2πf ⎤ ⎫GMod ( f ) = n + 2 ⎨∑ (n − i ) cos ⎢i⎥⎬⎣ nf c ⎦ ⎭⎩ i =1(4.30)Similar to how we will do with the BOC modulation, once the modulating term has beencalculated, the power spectral density can be expressed as follows:⎛ πf ⎞⎟sin 2 ⎜⎜⎡ n −1nf c ⎟⎠ ⎧⎪⎡ 2πf ⎤ ⎤ ⎫⎪BPSK ( nf c )BPSK ( f c )⎝( f ) = fc()GBPSK ( f c ) = Gsubchip pulse ( f )GModnni+−2cos⎨⎢∑⎢i⎥⎥⎬(πf )2 ⎪⎩⎣ nf c ⎦ ⎦ ⎪⎭⎣ i =1(4.31)After some math, it has been shown in Appendix M that (4.31) can be simplified to the wellknown expression that we can find everywhere in the literature:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ fc ⎠(4.32)GBPSK ( f c ) = f c(πf )24.3.2Binary Offset Carrier (BOC)Binary Offset Carrier Signals are a particular case of BCS signals with a representation vectorformed by +1’s and -1’s alternating in a particular defined way.
Two notations[E. Rebeyrol et al., 2005] can be found in the literature to define the BOC signals. Wedescribe them shortly in the following lines.The first model defines the BOC modulation as the result of multiplying the PRN code with asub-carrier which is equal to the sign of a sine or a cosine waveform, yielding so-calledsine-phased or cosine-phased BOC signals respectively as shown in [J.W. Betz, 2001],[L.R.Weill, 2003], [J. Godet, 2001] and [E.
Rebeyrol et al., 2005]. According to thisdefinition, the expression of the sine-phased BOC signal would be:withs(t ) = c(t )sign [sin(2πf S t )](4.33)c(t ) = ∑ ck h (t − kTc )(4.34)kwhere• ck is the code sequence waveform,• fs is the sub-carrier frequency,• and h(t) is the Non Return to Zero (NRZ) code materialization with value 1 over thesupport [0, Tc ) .The second model defines the BOC modulation as follows:s(t ) = ∑ c k pTc (t − kTc )(4.35)k88GNSS Signal Structurewhere pT (t ) describes the chip waveform and is broken up into n rectangular pulses ofcduration Tc n with amplitude ±1.
It is important to note that in this case the sine-phasing orcosine-phasing is considered as part of the chip waveform definition. This convention hasbeen introduced in [A.R. Pratt and J.I.R. Owen, 2003a] and [J.W. Betz, 2001].No matter what definition we choose to describe the BOC modulation in the time domain, theBOC signal is commonly referred to as BOC(fs, fc) where f s = m ⋅ 1.023 and f c = n ⋅ 1.023 sothat generally one only says BOC(m, n) for simplicity. Moreover, unless it is indicated in adifferent way, when we talk about BOC signals we will always mean the sine-phased variant.The parameter Φ is of great interest when analyzing BOC signals. It is defined as two timesthe ratio between the sub-carrier and the chip frequency as follows:Φ=2fsm=2fcn(4.36)As we can see, Φ represents the number of half periods of the sub-carrier that fit in a codechip so that this ratio can be even or odd.
When Φ is even, the two definitions presentedabove for the BOC modulation coincide since we can consider the sub-carrier as included inthe chip waveform. However, when Φ is odd the second definition is not valid any more. Thefollowing example shows this. Indeed, depending on the convention that we adopt to definethe BOC signal we can see that different time series result.Consider the code sequence {1,-1,1, -1,1,1} and a sine-phased waveform with 2 f s f c = 3 . Ifwe employ the first convention, the rectangular sub-carrier that results from taking the sign ofthe sine waveform will be as follows:Figure 4.7.
Sine-phased sub-carrier for the BOC modulationAccordingly, the product of the binary sub-carrier (4.33) with the code sequence results in thefollowing time series:Figure 4.8. Product of the sine-phased sub-carrier and the code sequence {1,-1,1,-1,1,1}where all the transitions have been underlined in red. If we follow now the approach ofdefining the sub-carrier as part of the chip waveform as it is done in the second definition, the89GNSS Signal Structurechip waveform to use will be:Figure 4.9. Chip waveform to represent the sine-phasing according to the seconddefinition of BOCand the resulting time series will be as follows:Figure 4.10. Product of the sine-phased chip waveform and the sequence {1,-1,1,-1,1,1}If we compare now Figure 4.8 and Figure 4.10 we can clearly recognize that the twodefinitions of BOC do not lead to an unique time series representation. Indeed the differenceis a polarity inversion every two bits as identified in [E.
Rebeyrol et al., 2005].We conclude thus that if Φ is odd, a slight modification must be made in the second definitionto account for the effect of the sub-carrier onto the code as shown in [J.W. Betz, 2001].Indeed, the new definition should be for the case of Φ odd as follows:s (t ) = ∑ (− 1) ck pTc (t − k Tc )k(4.37)kresulting then both conventions in the same time series.If we look at the equations above in detail, we can recognize the term (− 1) introduced in thekexpression, what can be interpreted as a modification of the PRN sequence so that all the evenkcode positions would alternate. Indeed, the new code would be then (− 1) c k instead of theoriginal ck. As a conclusion, in the case of Φ odd a modification must be made on the codesequence if we want the sub-carrier to be included in the chip waveform.
This does notrepresent a real problem from a theoretical point of view but it is important to note thatdepending on which convention is used, the receiver must be adapted consequently becauseotherwise it would suffer from non desired losses [E. Rebeyrol et al., 2005].Once the two definitions of BOC have been presented, it seems that the first one representsbetter the original definition of the BOC signal since no exception in the definition must be90GNSS Signal Structuremade depending on whether the figure Φ = 2fsis even or odd. Nonetheless, the secondfcconvention allows for easier and more tractable derivations in some cases and thus bothconventions will be indistinctively used in this thesis.Moreover, we have shown that the second convention is also correct as long as the PRN codeis correspondingly modified. Since in this chapter we will derive expressions for smoothspectra and assume consequently that the PRN code shows ideal properties, also the modifiedcode version should present similar ideal properties and we can directly consider the subcarrier as included in the code materialization.
This will considerably simplify the derivationsas we will see. Consequently we can use (4.8) to calculate the power spectral density of BOCas shown in the different Appendixes. This is actually not only valid for the BOC modulation,but for all the signals that can be expressed as shown in (4.2).Last but not the least, it must be noted that for non-ideal codes or very short codes this is nottrue any more.
We will analyze these effects in chapter 6.2.2.It is important to note that the conclusions derived above for the sine-phased BOC modulationcan easily be extended to the cosine case and to any BCS signal in general. In fact, also for aBCS signal we can distinguish between even and odd BCS signals in a similar manner as wedid above. However, the examples might not be so easy to analyze in this case.4.3.2.1Binary Offset Carrier with sine phasing: BOCsin(fs , fc)As we saw in chapter 4.3.2 the Binary Offset Carrier Modulation can be expressed as a BCSsequence with a vector that is formed by alternating +1 and -1 a number of times f s f c .
Inthe next lines we will derive the general expression of the power spectral density. To do so,we recall (4.26) and we build up the corresponding matrix to calculate GMod ( f ) as defined inthe previous chapter. As we will see next, the matrix shows symmetry properties that willallow to simplify the problem considerably.To start, let us analyze the particular case of BOC(fc, fc). As we saw in the preceding lines,BOC(fc, fc), also known as BOCsin(fc, fc), can be expressed as BCS([1,-1], fc) and presents thusthe following matrix:⎛ s s {0} s1s2 {1}⎞ ⎛1{0} − 1{1}⎞⎟=⎜⎟(4.38)M 2 ( [1,−1] ) = ⎜⎜ 1 1s2 s2 {0}⎟⎠ ⎜⎝1{0}⎟⎠⎝According to this⎛BOC ( f c , f c )( f ) = 2 − 2 cos⎜⎜ 2πfGMod⎝ 2 fc⎛ πf ⎞⎞⎟⎟⎟⎟ = 4 sin 2 ⎜⎜⎝ 2 fc ⎠⎠(4.39)adopting the pulse term of the PSD the following form:91GNSS Signal Structure⎛ πf ⎞⎟sin 2 ⎜⎜2 f c ⎟⎠BPSK ( 2 f c )⎝( f ) = fcGpulse(πf )2(4.40)Finally, according to (4.26), the power spectral density of this particular case would be:⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎥⎢ sin⎜⎜ ⎟⎟ sin ⎜⎜f c ⎠ ⎝ 2 f c ⎟⎠ ⎥BPSK ( 2 f c )BOC ( f c , f c )⎝⎢( f )GMod( f ) = fc ⎢GBOC( f c , f c ) ( f ) = Gpulse⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f c ⎠ ⎦⎥⎣⎢2(4.41)Let us now extend this expression to any BOCsin(fs, fc) generalizing on f s f c withf s f c being an integer, namely the number of times that the pair {1,-1} repeats.
Indeed, oncewe have found the expression GMod ( f ) with f s f c = 1 , we calculate for f s f c = 2 in thesame manner:BOCsin(fs, fc)= BOCsin(2fc, fc) or BCS([1,-1,1,-1],fc)(4.42)where the definition matrix is shown to be:⎛ s1s1{0} s1s2 {1} s1s3 {2} s1s4 {3} ⎞ ⎛1{0} − 1{1} 1{2}⎜⎟ ⎜s2 s2 {0} s2 s3 {1} s2 s4 {2}⎟ ⎜1{0} − 1{1}⎜4=M ( [1,−1,1,−1] ) = ⎜s3 s3 {0} s3 s4 {1} ⎟ ⎜1{0}⎜⎟ ⎜⎜⎜⎟s4 s4 {0}⎠ ⎝⎝and thus⎡⎛⎞⎛⎞⎛BOC ( 2 f c , f c )( f ) = 4 + 2 ⎢− 3 cos⎜⎜ 2πf ⎟⎟ + 2 cos⎜⎜ 2 2πf ⎟⎟ − cos⎜⎜ 3 2πfGMod⎝ 4 fc ⎠⎝ 4 fc ⎠⎝ 4 fc⎣− 1{3}⎞⎟1{2}⎟− 1{1} ⎟⎟1{0}⎟⎠⎞⎤⎟⎟⎥⎠⎦(4.43)(4.44)It is interesting to note that the term in the brackets resembles a Fourier series until term n-1.If we continue now by induction we can see that the expression for any n adopts the followingform:n −1⎛⎞BOC ( nf c 2, f c )( f ) = n + 2∑ (− 1)i (n − i ) cos⎜⎜ i 2πf ⎟⎟(4.45)GModi =1⎝ nf c ⎠where n ∈ {2, 4, 6, 8...}.
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