On Generalized Signal Waveforms for Satellite Navigation (797942), страница 42
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Wallner 2007]:α rm (c ) d rmn+ 2 (c ) + [β rm (c ) − λmn (c )]d rmn (c ) + γ rm (c ) d rmn− 2 (c ) = 0(4.236)with183GNSS Signal Structure(2m + r + 2)(2m + r + 1)(2m + 2r + 3)(2m + 2r + 5)c 22 (m + r )(m + r + 1) − 2m 2 − 1 2c + (m + r )(m + r + 1)β rm (c ) =(2m + 2r + 3)(2m + 2r − 1)r (r − 1)c2γ rm (c ) =(2m + 2r − 3)(2m + 2r − 1)α rm (c ) =(4.237)If we apply now the following recurrence relation, valid for all Legendre functions in general:(ν − μ + 1) Pυμ+1 (z ) = (2ν + 1) z Pυμ (z ) − (υ + μ ) Pυμ−1 (z )(4.238)and rewrite it in the following matrix form [S. Wallner 2007]:⎛ d 0mn (c )⎞⎞ ⎛ d 0mn (c )⎞⎛ β0 α0⎜ mn ⎟⎟ ⎜ mn ⎟⎜⎜ d 2 (c )⎟⎟ ⎜ d 2 (c )⎟⎜ γ 2 β2 α2⎟⎜⎟⎜⎟⎜(4.239)⋅O O OM ⎟ = λmn ⎜ M ⎟ for m-n even⎟ ⎜⎜mnmn⎜ d 2 k (c )⎟γ 2k β 2k α 2k⎟ ⎜ d 2 k (c )⎟⎜⎟⎜⎟⎜⎟⎜O O O⎠ ⎝ M ⎠⎝⎝ M ⎠⎛ d1mn (c ) ⎞⎞ ⎛ d1mn (c ) ⎞⎛ β1 α 1⎜ mn ⎟⎟ ⎜ mn ⎟⎜⎜ d 3 (c ) ⎟⎟ ⎜ d 3 (c ) ⎟⎜γ 3 β3 α3⎜⎟⎟⎜⋅O OOM ⎟ = λmn ⎜⎜ M ⎟⎟ for m-n odd.
(4.240)⎟ ⎜⎜⎜ d 2mnk+1 (c )⎟γ 2 k +1 β 2 k +1 α 2 k +1⎟ ⎜ d 2mnk+1 (c )⎟⎜⎜⎟⎜⎟⎜OO O⎟⎠ ⎝ M ⎠⎝⎝ M ⎠we can recognize that the problem to solve is in fact the well known Eigenvalue equation withEigenvector d mn (c ) and associated Eigenvalue λmn . According to this, the general prolateangular function is shown to adopt the following form:S mn (c,η ) =∞∑r = 0 ,1*d rmn (c )Pmm+ r (η )(4.241)where Pνμ refers to the Legendre function as defined previously. A very interesting propertyof these functions is that they have a limited domain, i.e. Pνμ : [−1,1] → IR and so do also haveall Prolate Spheroidal Wave Functions. It is important to say that this property cannot only beused in the frequency domain but also in the time domain, what could also be of interest insome applications.
That is to say, that the Prolate Spheroidal Wave Functions could be used toeither define the Power Spectral Density directly or the time chips.Another way of expressing the Prolate Spheroidal Wave Functions is as solution of thefollowing integral equation:λi ψ i (t ) =sin[π β r (t − s )]ψ i (s )dsπ (t − s )-Tc / 2Tc / 2∫(4.242)where Tc represents the chip duration and β r the double-sided bandwidth.
An analyticalexpression is not easy to derive but [F. Antreich and J. A. Nossek, 2007] have shown that184GNSS Signal Structureinteresting signal waveforms can be derived using only a few vectors of the functions base. Itmust be noted though that the proposed solutions do not take into account importantconstraints related to spectrum compatibility and implementation aspects.It is interesting to note that the previous integral equation (4.242) can also be interpreted as aKarhunen-Loève transform.
The Karhunen-Loève is a representation of a stochastic processby means of a linear combination of orthogonal functions on a finite support of definition.The coefficients are random variables and the expansion basis takes a form that depends onthe specific problem. For example, for the Wiener process the basis functions are sinusoidals.If we take a close look at equation (4.242), we can recognize the similarity of:sin[π β r (t − s )](4.243)π (t − s )to the Dirichlet kernel.
This kernel can be further expressed in terms of Legendreprolynomials which are in fact the basis functions of the Prolate Spheroidal Wave Functionsas we saw in previous pages. It is also interesting to see that the integral equation aboveimplicitly describes the problem of maximizing the power of a signal in the frequency domainwhen this signal is also highly localized in time. As a matter of fact, the continuous PSWFsare highly confined simultaneously in time and frequency, what makes them also of greatinterest for modeling periodic phenomena if they are used as wavelet functions.Related to the discussion of previous chapters on signals with good spectral confinement, it isof interest to note that while the Fourier and Shannon bandwidths are normally used toestimate what the effective bandwidth of a process is, also other figures such as the Campbellbandwidth equally provide valuable information.
The Campbell bandwidth indicates theprocess spectral entropy and is defined as follows:∞BCampbell = e4.8.4− ∫ G ( f ) log [G ( f )]df-∞(4.244)Faded Harmonics (FH) Interplex ModulationFaded Harmonics (FH) is an Interplex modulation that uses non-binary signals to bettercontrol the spectral emissions of the open signals of Galileo. For more details on Interplexrefer to chapter 7.7. Using tertiary modulation waveforms is a way of achieving this objective.Tertiary Offset carrier signals have been introduced in [A.R.
Pratt and J.I.R. Owen, 2003] andare conceptually equivalent to the TCS and TOC modulations that we described in chapter4.5. The main driver of the work presented in [A.R. Pratt and J.I.R. Owen, 2003] was to findan alternative signal to the original BOC(2,2) baseline of 2002. that would overlay themilitary signals less while it should perform as good as the reference BOC(2.2) signal. Theproposed solution received the name of 8-PSK BOC(2,2) or BOC8(2,2). The idea behind, aswe already pointed out in chapters 4.5.1 and 4.5.2, is to suppress specific secondary lobes by185GNSS Signal Structurezeroing one pair of harmonics in the sub-carrier spectra. This can be easily accomplished bybuilding up multilevel signals which can be expressed as the sum of other elementary subcarriers.
Indeed, as shown in Appendix F, by properly selecting the parameters ρ l and ρ s wecould completely suppress any specific secondary lobe.We can generalize the results that we obtained in chapter 4.5.4 if we realize that any BOCsub-carrier is indeed a periodic signal and can thus be expressed as a Fourier series:∞ss (t ) = ∑ bk sin (2 π k f c t )(4.245)k =1where1bk =Tc∫Tc / 20ejk2πtTcT2π1 c jk Tc te jkπtdt −edt=[1 − cos(kπ )] = 2∫T Tc / 2jkπjkπ(4.246)for k odd.
Furthermore, since c k − c k* = jbk , if we define the Fourier sum between 1 andinfinity, the odd coefficients in sine will adopt the following form:bk =4kπ(4.247)while all the cosine terms are zero.If we simply subtract now to the BOC signal above a sub-carrier of frequency and amplitudeequal to that of the harmonic we want to suppress, for example the nth harmonic, thatharmonic of the sub-carrier will be eliminated. This means, a faded harmonic sub-carrier thathas suppressed the nth harmonic would ideally look as follows:ssn (t ) = ss (t ) −44sin (2 π n f c t ) = ∑ bk sin (2 π k f c t ) −sin (2 π n f c t )nπnπk(4.248)Such a signal would not be practical since it has an infinite number of levels due to the sineterm on the right. Therefore a rectangular approximation to the sine function is moreappropriate and instead of subtracting a sinusoid we will subtract another BOC signal withfrequency and amplitude equal to that of the lobe we want to annulate as shown next:4 n4[ bk − n +1 + bn + k −1 ] sin (2 π n f c t ) (4.249)sˆsn (t ) = ss (t ) −ss (t ) = ∑ bk sin (2 π k f c t ) −nπnπkThis is shown graphically next.
In the example, we have taken BOC(1,1) and we havesuppressed the 9th and 11th harmonic respectively. These harmonics were selected given theirproximity to the M-Code. As we can clearly recognize, the spectrum is very close to that ofthe original BOC(1,1) with the only difference that those components of the correspondingfaded harmonics are lower. As one can expect, this will clearly contribute to reaching a betterspectral separation with the rest of signals around. Moreover, this effect is more important thehigher the sub-carrier frequency we want to cancel in the original spectrum.186GNSS Signal StructureFigure 4.74. Harmonic fading of BOC(1,1) using a 4-level sub-carrierThis is indeed the idea behind the faded-harmonics modulation.
As we can also recognizefrom the expression above, the number of harmonics or lobes that are suppressed is directlyrelated to the number of levels of the resulting sub-carrier. For exemplification, we show nextthe power spectral density that an even sine-phased BOC signal with suppressed lobes wouldhave if only one harmonic is eliminated. Generalizing the result to more signals and withmore cancelled lobes is trivial.⎡ ⎛ πf ⎞ ⎛ πf ⎞⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟⎟⎟ ⎥sin ⎜⎜ ⎟⎟ sin ⎜⎜⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜f2ff2nf4ncscs⎠−⎝ ⎠ ⎝⎠⎥GBOC( f s , f c ) = f c ⎢ ⎝ ⎠ ⎝⎢nπ⎛ πf ⎞⎛ πf ⎞ ⎥⎟⎟⎟⎟ ⎥πf cos⎜⎜⎢ πf cos⎜⎜⎢⎣⎝ 2 fs ⎠⎝ 2nf s ⎠ ⎥⎦This can also be expressed as follows:⎡ ⎛ πf⎢ sin ⎜⎜fnGBOC( f s , fc ) = f c ⎢ ⎝ c⎢ πf⎢⎣⎢⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎦⎥2⎡ ⎛ πf ⎞⎛ πf ⎞ ⎤⎟⎟⎟⎟ ⎥sin ⎜⎜⎢ sin ⎜⎜⎝ 2nf s ⎠ ⎥⎢ ⎝ 2 fs ⎠ − 4⎢ ⎛ πf ⎞ nπ⎛ πf ⎞ ⎥⎟⎟⎟⎟ ⎥πf cos⎜⎜⎢ cos⎜⎜⎝ 2nf s ⎠ ⎦⎥⎣⎢ ⎝ 2 f s ⎠2(4.250)2(4.251)where the super index n indicates that we have suppressed the n–th lobe.
It must be noted thatthe lobe n to suppress must be a multiple of the sub-carrier fs of the signal. Another interestingcomment is that not only the lobe we want to suppress is eliminated but also other secondarylobes or harmonics are slightly attenuated. Nonetheless this effect is practically negligible.As we mentioned above, we can generalize this procedure to more lobes and harmonics tosuppress.
The only problem is that the number of levels grows very quickly as moreharmonics are eliminated. As we can imagine, this could make the implementation moredifficult in the end. A solution to this problem has been proposed in chapter 4.5.5 and by[L. Ries and J.-L. Issler 2003] based on the principle that completely removing one lobe or afew of them is sometimes not absolutely necessary since attenuating them by some dBs mightbe enough. Indeed, if we work with four level signals but do not constrain ourselves to187GNSS Signal Structuredividing the code chip waveform in equal length segments as done in chapter 4.5, we gain adegree of freedom that compensates that one that we lose by fixing the number of possibleamplitude levels to only four. Let us imagine a four-level signal as depicted below:Figure 4.75. Four-Level Waveform to realize the fading effectIn spite of being the segments in principle of any arbitrary length, we create however areference framework by dividing the chip in n subchips.
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