On Generalized Signal Waveforms for Satellite Navigation (797942), страница 74
Текст из файла (страница 74)
Equally,for the pilot channel we would have:⎧⎪cos 2 (θ1 ) GBOC( f c , f c ) ( f ) + cos 2 (θ 2 ) GBCS([ s ], f c ) ( f ) +⎨*⎪⎩− 4 f c cos(θ1 ) cos(θ 2 ) Re S BOC ( f c , f c ) ( f ) S BCS([ s ], f c ) ( f )OSGP ( f ) =cos 2 (θ1 ) + cos 2 (θ 2 ) − 2r cos(θ1 ) cos(θ 2 ){}(J.25)Another expression of interest is the power spectral density of the data and pilot channels withrespect to the total OS power. Equally interesting is also to obtain the power of the353Power Spectral Density of the CBCS ModulationBOC(fc, fc) and BCS([s], fc) component with respect to the total OS power.
We derive next thecorresponding expressions.Let us study first the power of the data channel with respect to the total OS channel. Indeed, ifwe divide (J.13) by (J.18), we obtain the next relationship:PDOSPOS D + OS P=∞∫∫∞−∞−∞GDOS ( f ) dfGOS D + OS P ( f ) df=1 cos 2 (θ1 ) + cos 2 (θ 2 ) + 2r cos(θ1 )cos(θ 2 )2cos 2 (θ1 ) + cos 2 (θ 2 )(J.26)Equally, the percentage of pilot OS power with respect to the total OS power is:PPOSPOS D + OS P=∫∫∞−∞∞−∞GPOS ( f ) dfGOS D + OS P ( f ) df=1 cos 2 (θ1 ) + cos 2 (θ 2 ) − 2r cos(θ1 )cos(θ 2 )2cos 2 (θ1 ) + cos 2 (θ 2 )(J.27)To calculate the total power on the BOC(fc, fc) signal, we can use the equation (J.1) and thus,cos 2 (θ1 )PBOC( f c , f c ) = A12(J.28)2so that the percentage of BOC(fc, fc) power with respect to the total OS power will adopt thefollowing form:PBOC( f c , f c )cos 2 (θ1 )=1− ρ =(J.29)POS D + OS Pcos 2 (θ1 ) + cos 2 (θ 2 )If we repeat now for the BCS component,PBCS([ s ], f c ) = A12cos 2 (θ 2 )2(J.30)and normalizing (J.30) to the total OS power, we have the percentage ρ of power on the BCScomponent:PBCS([ s ], f c )cos 2 (θ 2 )=ρ=(J.31)POS D + OS Pcos 2 (θ1 ) + cos 2 (θ 2 )In the same manner, the useful power of the PRS for the quadrature signal can be easilyobtained from the signal definition shown at the beginning of the Appendix.
In fact:PPRSA12(sin θ1 + sin θ 2 )2=4(J.32)Equally, for the Inter-Modulation Product we can derive a similar expression:A12(sin θ 2 − sin θ1 )2PIM =(J.33)4Finally, if we sum up the power of all the desired signals plus the Inter-Modulation term, wefind as expected that the CBCS modulation has constant envelope of amplitude A1:POS + PPRS + PIM = A12(J.34)It is interesting to note that while for the BOC(fc, fc) Interplex the inter-modulation poweronly depends on one modulation index, namely m, in the case of CBCS both indexes θ1 and354Power Spectral Density of the CBCS Modulationθ 2 have to be considered. This means in other words that fixing the IM power is easier withCBCS than it would be if only a BOC(fc, fc) were transmitted.
The result is thus a moreefficient control of the IM power since we have more degrees of freedom to play.One final but important comment is that Interplex with only BOC(fc, fc) can be easilydescribed taking θ 2 equal to π/2.Once the most important equations describing CBCS have been derived, we study next howto calculate the multiplex parameters when we fix the percentage of power on the BCScomponent, the power split between data and pilot and the power split between the differentsignals. Moreover, it is important to note that the expressions derived above were obtained forinfinite bandwidth differing thus the results slightly when filtering effects are considered.According to the [Galileo SIS ICD, 2008] the power split between the OS data and pilotsignals shall be 50/50 while the open signals and PRS should have the same power levels.
Theresulting equations system to solve is then:⎧cos 2 (θ 2 )=ρ⎪22⎪ 2 cos (θ1 ) + cos (θ 2 )1⎪ A1cos 2 θ1 + cos 2 θ 2 = POS =⎨2⎪ 22A⎪ 1 (sin θ + sin θ )2 = P = 112PRS⎪ 42⎩()(J.35)where the first equation indicates the amount of power that is moved from BOC(fc, fc) to theBCS signal and the second and third equations represent the power ratios between thedifferent signals. We can also observe that the equations above do not depend on the specificBCS vector [s] since they only account for power relationships. Nonetheless, as it could beexpected, the real BCS vector plays indirectly an outstanding role in assessing if one signal iscompatible with the rest of signals in the band or not.
In fact, depending on how the specificBCS sequence looks like, the spectral overlapping with the other signals around will bedifferent, determining thus the maximum amount of power ρ that can be put on its BCS partin order not to interfere.Until now we have analyzed the case when the BCS signal is on both the data and pilotchannels as this is the baseline of Galileo for the OS. Nevertheless, for some specificapplications, allocating the high frequency components (thus the BCS signal) only on the pilotchannel could be of interest.
Indeed, the GPS implementation of MBOC, namely TMBOC,goes in this direction.In order to have all the power of the BCS signal only on the pilot channel and still maintainconstant envelope, the expression of the CBCS modulation has to be generalized.Accordingly, (J.1) and (J.2) can be slightly modified as shown next:355Power Spectral Density of the CBCS Modulation⎧k1 d D (t ) cD (t ) sBOC(1,1) (t ) + cP (t ) {k2 sBOC(1,1) (t ) − k3 sBCS([ s ],1) (t ) }+s (t ) = ⎨⎩+ j d PRS (t ) cPRS (t ) { k 4 sPRS (t ) + sIM (t )}(J.36)where the constants k1 , k 2 , k 3 and k 4 are calculated from••••the power split between data and pilot,the relationship of powers between OS and PRS,the percentage of power on the BCS signal with respect to the total OS power underthe constraint that the phase points are on the unit circle, andaccounting for the different filter losses of the signals due to bandlimiting.Mathematically, all these conditions can be expressed as followsk32=ρk12 + k22 + k32k22 + k32=ξk12(J.37)k42= 10 β / 10k12 + k22 + k32where:• ρ is the percentage of power on the BCS signal,• ξ indicates the percentage of power that falls onto the pilot channel with respect tothe data channel.
Thus, if we have a power split of 50/50, ξ = 1 and for 75/25, ξ = 3 ,• and β indicates the power difference between PRS and OS in dB, accounting for thedifferent filter losses of both signals due to satellite bandlimiting.It should be noted that this calculation ignores the effect of the correlation between theBOC(fc, fc) and BCS([s], fc), which is introduced by virtue of the satellite bandlimiting.Additionally, since all the phase points have to be on the unit circle, we have:(k1 + k 2 + k 3 )2 + (k 4 + IM1 )2 = 1 ⎫⎪(k1 + k 2 − k 3 )2 + (k 4 + IM 2 )2 = 1 ⎪⎪(k1 − k 2 + k 3 )2 + (k 4 + IM 3 )2 = 1 ⎪(k1 − k 2 − k 3 )2 + (k 4 + IM 4 )2 = 1 ⎪⎪⎬ ⇒ s (t )(− k1 + k 2 + k 3 )2 + (k 4 + IM 5 )2 = 1⎪(− k1 + k 2 − k 3 )2 + (k 4 + IM 6 )2 = 1⎪⎪(− k1 − k 2 + k 3 )2 + (k 4 + IM 7 )2 = 1⎪⎪(− k1 − k 2 − k 3 )2 + (k 4 + IM 8 )2 = 1⎪⎭=1(J.38)since the real component of the signal takes 8 values with equal probability, given as shownin the following table:356Power Spectral Density of the CBCS ModulationTable J.1.
Value of the signal s(t) as a function of the different code inputsC D (t )DD (t )C P (t )+1+1+1Re{s (t )}k1 + k 2 + k3+1+1-1k1 + k 2 − k 3+1-1+1k1 − k 2 + k 3+1-1-1k1 − k 2 − k 3-1+1+1− k1 + k 2 + k 3-1+1-1− k1 + k 2 − k 3-1-1+1− k1 − k 2 + k 3-1-1-1− k1 − k 2 − k 3Additionally, the IM component IM = {IM1 , IM 2 ,..., IM 8 } must take the appropriate value tobring the phase plots to the unit circle. Note that this is true independently of the BCScomponent is in phase or in anti-phase. Finally, an extra constraint comes from the necessarycondition that the Inter-Modulation signal has zero mean:< s IM (t ) > =1 8∑ IM i = 08 i =1(J.39)If we put all the conditions together, we can see that we have totally twelve equations, namely(J.37), (J.38) and (J.39), and twelve unknowns to find, namely IM , k1 , k2 , k3 and k 4 .Unfortunately, while (J.36) covers more cases than (J.1) and (J.2), in general it is not possibleto find an explicit expression for the coefficients k1 , k2 , k3 and k 4 .We show next with an example how the parameters of the CBCS multiplex could be obtainedfor the hypothetical case that the CBOC implementation of MBOC would allocate the wholeBOC(6,1) component on the pilot OS signal.
As shown in chapter 4.7, the power ratiobetween the BCS signal, in this particular case BOC(6,1), and the total OS power is 1/11 atgeneration. This means that in reality the power after filtering in the satellite will be slightlylower on BOC(6,1). Moreover, let us assume that the PRS power would be 2 dB above the OSpower at user level and that the effect of filtering in the satellite is also taken into account.This assumption is different from the baseline when OS and PRS have the same power.If we solve now for CBOC(6,1,1/11) with all the BOC(6,1) power on the pilot channel, withequal power for pilot and data, with 1/11 of the OS power in the BOC(6,1) beforebandlimiting, the composite signal may be defined by⎧⎛⎛ 2π t ⎞⎛ 2π t ⎞ P ⎧2π t ⎞⎫D⎟⎟⎬ +⎟⎟ − 0.1688 sign ⎜⎜ sin⎟⎟ + cOS ⎨0.3581sign ⎜⎜ sin⎪0.3959 d D cOS sign ⎜⎜ sinTc ⎠TT/6cc⎠⎭⎝⎠⎝⎝⎪⎩s (t ) = ⎨⎧⎪⎫⎪⎛2π t ⎞⎪⎟⎜+jdcst+0.7915signsin()⎨⎬PRSPRSIM⎪⎟⎜ T⎪⎩⎪⎭c PRS / 6 ⎠⎝⎩(J.40)357Power Spectral Density of the CBCS Modulationwhere the signal sIM (t) is given by:sc IM (t ) =∞∑ai = −∞i8rect (t − iTc / 8)(J.41)adopting a i the following values:8Table J.2.
Values of the Inter-Modulation Signal (IM) to achieve a constant envelopei01234567a-4.0614.10-11.9393.10-21.8690.10-11.9985.10-11.9985.10-11.8690.10-11.9393.10-2-4.0614.10-1We show the Inter-Modulation signal next graphically:Figure J.3. Inter-Modulation Signal necessary to have a constant envelope whenBOC(6,1) is only on the pilot channelThe phase states of the constellation are equally shown in the next figure. As we can see, themain effect is that the number of states has duplicated, what is of course a clear drawback.
Inaddition, it is important to realize that this implementation of CBOC is not compliant with theMBOC spectrum definition since a cross term appears as shown in chapter 4.7.5.4.Figure J.4. CBOC 16-PSK modulation that results when all the BOC(6,1) component isplaced on the pilot channel358Cramér Rao Lower BoundKAppendix. Cramér Rao Lower BoundThe mean-squared error for any estimate of a nonrandom parameter has a lower bound,known in the literature as the Cramér-Rao lower bound or CRLB in short[J.-A. Avila-Rodriguez et al., 2006a].
The Cramér-Rao lower bound defines the ultimateaccuracy of any estimation and shows the minimum code pseudorange variance we wouldhave with the best possible receiver implementation. Indeed, the Cramér-Rao lower bound isnothing else than a different way of expressing the Gabor bandwidth which sets the physicallimit of a signal for a given bandwidth. This last one is also known in the literature as the rootmean square bandwidth.In our particular case, we are interested in finding the bound of the matrix:([ ])()TE εε T = E ⎡ θˆ − θ θˆ − θ ⎤⎢⎣⎥⎦(K.1)being ε the code delay error, θ the real code delay value and θˆ its estimation.
As it can beshown, the Cramér-Rao lower bound is deeply related to the Fisher information matrix F inthe following form:[ ]E εε T ≥ b(θ )b T (θ ) + [I + ∇ θ (b )]F −1 [I + ∇ θ (b )]T(K.2)where b(θ) is the bias of the estimate θ and F is the already mentioned Fisher matrix definedby means of the Hessian matrix as follows{}F = − E ∇ θ ∇ θT [ln [P ( x , y / θ )]](K.3)It must be noted that according to this general definition, the Cramér-Rao lower bound alsoapplies for the case of biased estimates in contrast to the way it is widely used in the generalliterature where the Cramér-Rao lower bound is understood as the minimum unbiasedvariance estimate bound.Looking now at the bound of every parameter in particular, the Cramér-Rao lower bound canbe expressed as follows:2TE ⎡ θˆ − θ ⎤ ≥ bi2 (θ ) + [I + ∇θ (b )]F −1 [I + ∇θ (b )] ii(K.4)⎢⎣⎥⎦(){}and for the case the estimator is unbiased, the bound simplifies then to2E ⎡ θˆ − θ ⎤ ≥ F −1 ii⎢⎣⎥⎦()[ ](K.5)The most important conclusion that can be drawn from observing the equation above is thatfor the unbiased case, the estimator is not necessary for the computation of the bound.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
