AltBOC for Dummies (797920), страница 2
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International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.5FsFigure 7: SSB subcarrier PSD.(7)(6)s(t ) = [c A (t ) + cB (t )]SCB ,cos (t ) + j[cB (t ) − c A (t )]SCB ,sin (t )« 3 state Code » : (CA + CB)Carrier : In Phase (I)Subcarrier : Cos phasing« 3 state Code » : (CA - CB)Carrier : In Quadrature (Q)Subcarrier : Sin phasingFigure 8 : breakdown of expression (7) Alt-BOC can be viewed as the sum of 2 signalsCACB11CA+ CB CA- CB Fresnel plot2Phasing00Ts/4 Ts/2 3Ts/4Ts412-1-1-22404321-1021132-110-24133Table 2 : Values taken by the Alt-BOC modulation as a function of Ca and Cb chip, and time.It is interesting to notice that there is no such thingas cosine or sine Alt-BOC, as Alt-BOC has bothcosine and sine phased signals in it.Expression (7) shows that the 2-code Alt-BOC isthe sum of two “3-state codes” (Ca+Cb) and (CbCa) modulating a BOC sub-carrier.
Notice thatthese 2 signals are both in carrier phase quadratureand sub-carrier phase quadrature, and when one ofthese code values zero the other one doesn’t andvice-versa.The resulting modulation has 4 phase plots and hasoscillations at the sub-carrier frequency either alongthe vertical or the horizontal line, as shown in Table2SUBCARRIER VARIANTSThe Alt-BOC is based on a BOC subcarrier. In factthis signal is part of a broader family that we couldcall “Alt-OC” or Alternate Offset Carrier.
Indeed,the carrier doesn’t have to be binary.For example, it could have been possible to base themodulation on pure sine and cosine Subcarrier, thatis the Lineary Offset Carrier (LOC) subcarrier :BOC IS Alt-BOC !It turns out that the standard BOC is a special caseof Alt-BOC. If the 2 PN codes of an alternativeBOC are made identical, the signal becomes a «classical » BOC.More precisely, If :Ca = Cb,then the resulting signal is the cosine BOC,and if :Ca = - Cb,then the resulting signal is the sine BOC.ION GNSS 21st. International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.SC L ,cos (t ) = cos(2πf s t )(8)SC L ,sin (t ) = sin (2πf s t )(9)The resulting SSB “Linear” subcarrier would havebeen the complex exponential :SC L , SSB (t ) = e j 2πf st964(10)such a modulation can’t be used with HPA workingat saturation.The PSD of (10) has a single harmonic at +fs,precisely what we would expect from an ALT-OCSSB sub-carrier.
Note that (5) is equivalent to (10)sampled at a rate of 4fs.Expression (7) remains valid with LOC subcarrier.This is only an example, Alt-OC signal could havebeen derived using 3-state or 4-state or n-state subcarrier.2312DIFFERENT CODE RATE4− 2There are 2 codes on each frequency, each of whichare in phase quadrature, so the baseband expressionbecomes :s (t ) = [c A (t ) + j.c' A (t )] ⋅ SC 4,SSB * (t )+ [c B (t ) + j.c' B (t )] ⋅ SC 4, SSB (t )Figure 9 : Modulation constellation for the 4-codeAlt-BOC(11)The signal will have oscillations at the sub-carrierfrequency, either along the horizontal and verticalaxis of the constellation, or along the diagonals.The vertical and horizontal oscillations have higheramplitudes than the diagonal ones.
Moreover, theytake a zero amplitude half the time.40− 2State1 or 31282Ts/2− 2576After some computations not detailed here, theresulting modulation constellation is shown infigure 9. This constellation has 9 plots, the “0” plotis possible and correspond to nothing beingtransmitted. The envelope is clearly not constant so28254-CODES ALT-BOC30− 2A nicety of Alt-BOC is that code rates don’t have tobe the same on the lower and on the upperfrequencies. It would have been perfectly feasibleto have different code rates on E5a and E5b.Ts3Ts/22TsState5 or 776State2 or 8234− 212082Ts/2− 2576State4 or 6Figure 10 : Oscillating nature of the 4 code Alt-BOCION GNSS 21st.
International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.965Ts3Ts/22Ts2√22√22Ts/2Ts3Ts/2Ts/22TsTs3Ts/22Ts-2-2√2-2√2Ts/8Figure 11 : How the constant envelope was madeAnd the modulation can be described using themodulation table that can be found in the GalileoICD and is shown in table 3 in a graphical wayMAKING THE ENVELOPPE CONSTANTIf you could make the oscillation along thehorizontal and vertical axis look like the one alongthe diagonals, then the envelope would be constant.Well, let’s do it then, and this is how the constantenvelope Alt-BOC with 4 codes was invented ! (fig.11)The modulation now has a constant envelope andpresents a 8-PSK constellation (Figure 12)The mathematical formulation found in the GalileoICD came only later showing that in the process aninter-modulation product was generated.23124− 228− 2The coordinate of each plot k are given by :2⋅ejkπ4k ∈ {1,2,3,4,5,6,7,8}5(12)76Figure 12 : resulting 8-PSK constellationTable 3 : Graphical representation of the ALT-BOC modulation tableION GNSS 21st.
International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.966Desired codess E 5 (t ) =SSB subcarrier functions at Fs(eE 5a− I (t ) + j eE 5a−Q (t )) [scE 5−S (t ) − j scE 5−S (t − Ts,E 52 21(eE 5b− I (t ) + j eE 5b−Q (t )) [scE 5−S (t ) + j scE 5−S (t − Ts,E 52 21(eE 5a−I (t ) + j eE 5a−Q (t )) [scE 5−P (t ) − j scE 5−P (t − Ts,E 52 21(eE 5b− I (t ) + j eE 5b−Q (t )) [scE 5− P (t ) + j scE 5− P (t − Ts ,E 52 214 )] +2 ⋅SC8, SSB * (t )4 )] +2 ⋅SC8, SSB (t )4 )] +2 ⋅SC 38, SSB * (t )4)]2 ⋅SC 38, SSB (t )SSB subcarrier function at -3*FsInter-modulation productFigure 13 : Interpretation of the Galileo SIS ICD formulasMATHEMATICAL FORMULATIONFrom expression (15) and (16), the Fresnel plot ofthe “single” and “product” sub-carrier can be easilyderived and are shown in figure 14 and 15.The formulation found in the Galileo SIS ICD wasfirst published in [3].
Figure 13 explains how thisformulation is to be understood. We see that in theprocess of making the envelope constant, an intermodulation product was generated.Let’s define the SSB “single” and “product” subcarriers as :SC8,SSB (t ) =12P1SC 38, SSB (t ) =Note that the[sc (t ) + j sc (t − TE 5− S12 P3[scE 5− SE 5− P(t ) +s, E54)]j sc E 5− P (t − Ts , E 5 4 )]3(13)⎧ SC8,SSB (t ) = e 8 4 )⎨( i −1)Ts i⋅Ts⎩t mod Ts ∈ 8 , 8[⎧SC 38,SSB (t ) = e j ( 8 −i⋅ 4 )⎨( i −1)Ts i⋅Ts⎩ t mod Ts ∈ 8 , 85π3π[]Single SCProduct SC3()4P (2 − 2 )P1= 2 + 23=1(16)Power85,36 %467328514,64 %4Figure 15 : Fresnel plot of the “product”subcarrierTable 4 : Amplitude & power of the subcarrier“single” and “product”.ION GNSS 21st. International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.7Figure 14 : Fresnel plot of the “single” subcarrier,to be compared with figure 6.And the power of each subcarrier is given in table 4Power exsp.86(15)]15normalization.
8 stands for the number of states.These 2 sub-carriers can be expressed in a verysimple expression similar to (5) :+i⋅π4(14)2P1 and 2P3 factors are there forj (π2967The inter-modulation product hasits main lobe at 3Fs, that is 30 MHz away fromE5a and E5b, the cross modulation product willnot interfere with the desired signal.In fact, when we made theenvelope constant, all we did was transform the3rd and 5th harmonic of the desired signal subcarrier into inter-modulation product.The constant envelope Alt-BOCsignal is therefore as good as the non constantenvelope Alt-BOC signalThe constant envelope Alt-BOC succeedscompletely in doing what it is meant to do, that isproviding QPSK modulated codes at +fs and –fsfrom the center frequency.Of course, if we compare the PSD of the full signal,we see some differences between the constant andnon constant envelope Alt-BOC (figure 17)especially for the central secondary lobe and thethird harmonic.
This might come from complexinteraction between the codes and the intermodulation product. But the same kind ofdifference can be seen if we compare Alt-BOC withBOC, or BOC sin with BOC cosine. Remarkably,the fundamental harmonic has the same shape inboth modulation.PSD PROPERTIESThese 2 sub-carriers have some remarkable PSDproperties that can be guessed from the fresnel plot.The “single” SSB sub-carrier SC8,SSB is a finerapproximation of the Linear SSB sub-carrier(expression (10)) that the binary SSB sub-carrier(expression (5)) so you would expect it to havemost of its energy in its fundamental harmonic at+fs.The “product” SSC sub-carrier SC38,SSB circle 3times around the origin of the Fresnel plot in thereverse direction, so you would expect itsfundamental harmonic to be at -3fs and it indeed is.In fact, a remarkable property is that the single SSBsub-carrier only has harmonics at +fs, -7fs and +9fs,while the “product” SSB sub-carrier only hasharmonics at -3fs and -5fs, as described in table 5and figure 16.Freq-5fs-3fs-fs+fs+3fs+5fsSC8, SSB00094,9600SC8, SSB * 0094,9600061,500022,200061,500SC 38,SSBSC 38,SSB * 22,2Table 5 : PSD table, for each harmonics of the subcarrier, expressed in % of total power.E5aE5bSC8,SSBSC8,SSB*SC38,SSBSC38,SSB*-9Fs-7Fs-5Fs-3Fs-Fs+Fs3Fs5Fs7Fs9FsFigure 16 : Subcarrier PSDFigure 17 : full signal PSD enveloppe for constantand non constant envelope Alt-BOC (from [11])And yet another remarkable property is that if wemultiply the power of the fundamental harmonic ofthe single carrier by its relative power in theconstant Alt-BOC modulation, we find the exactpower of the 2-code Alt-Boc modulation shown intable 1, that is :ALT-BOC AFTER FILTERINGIt seems that the transmitted bandwidth at payloadwill be bellow 90MHz.
Therefore, the intermodulation product at +/-3fs and +/-5Fs is mostlyfiltered out and the ground user receives only thefundamental harmonic of the sub-carrier, that is anALT-LOC signal.Furthermore, a ground user using a receiver withfrom 20 to 40 MHz of bandwidth centred on eitherE5a or E5b will mostly see a 10 Mchip/s QPSKcode.And this is precisely what the user need !0.9496 x 0,8536 = 0,8106What this all means is :Making the envelope constantdoes not cause any loss of desired signal power inthe main lobes.The energy that was originally inthe harmonic of the BOC sub-carrier is now splitbetween the “single” and “product” sub-carrier(figure 16)ION GNSS 21st. International Technical Meeting of theSatellite Division, 16-19, September 2008, Savannah, GA.968become roughly the Alt-Loc signal expressed in(19) :-30GioveA E5 PSDTheoretical E5 DSP-40s (t ) = [c A (t ) + cB (t )]cos(2πf st )POWER (dBm)-50(19)+ j[cB (t ) − c A (t )]sin (2πf st )-60-70This means that this signal can optimally be trackedusing an Alt-LOC replica instead of an Alt-BOCreplica.Taking advantage of the properties of expression(19) which are mostly the same than for expression(7), we propose a correlator architecture that willisolate the “sum” (CΣ=Ca+Cb) and “delta” (CΔ=CbCa) codes-80-90-100-60-40-200204060Frequency (MHz)Figure 18 : signal PSD enveloppe transmitted byGiove A’s NSGU (3 dB bandwidth is 72 MHz)SINGLE LOBE TRACKINGIIThis is what Alt-BOC was designed for, as itprecisely puts 2 BPSK codes in quadrature at eachfrequency E5a and E5b.
Any classical BPSK orQPSK tracking method can be used at either E5aans E5b. The first secondary lobe is alsotransmitted so it can be possible to use a 40 MHzfront-end associated with narrow correlationtechniques.If tracking both E5a and E5b at the same time ondifferent channel, bear in mind that the observedpseudo-range won’t be the same due to ionosphericdelay that is different on E5a and E5b.I()()Σ↓I QΣ = RΣΣ (τˆ ) ⋅ cos(θˆ) ⋅ sin(φˆ)Σ↓I QΔ = − RΔΔ (τˆ ) ⋅ sin(θˆ) ⋅ cos(φˆ)Σ↓C Σ (τˆ )QSC cos()φˆNCOcarrierC Σ (τˆ )()SCsin φˆNCOSubcarrierQIΣ = RΣΣ (τˆ ) ⋅ sin(θˆ) ⋅ cos(φˆ)-Σ↓QIΔ = − RΔΔ (τˆ ) ⋅ cos(θˆ) ⋅ sin(φˆ)Σ↓QQΣ = RΣΣ (τˆ ) ⋅ sin(θˆ) ⋅ sin(φˆ)Σ↓QQΔ = RΔΔ (τˆ ) ⋅ cos(θˆ) ⋅ cos(φˆ)C Δ (τˆ )QQ()sin θˆIn particular, the desired signal can be expressed ina way similar to (7) :C Δ (τˆ )NCOcodeFigure 19 : Correlator architectureNext we do :(17)⎧ I I = I IΣ + QQΔ = 2(Raa (τ ) + Rbb (τ )) ⋅ cos(φ ) ⋅ cos(θ )⎪ I = Q − I = 2(R (τ ) + R (τ )) ⋅ cos(φ ) ⋅ sin(θ )(20)⎪ QIΣQΔaabb⎨(()())()=−=+⋅⋅⎪QI I QΣ QIΔ 2 Raa τ Rbb τ sin φ cos(θ )⎪⎩QQ = QQΣ + I IΔ = 2(Raa (τ ) + Rbb (τ )) ⋅ sin (φ ) ⋅ sin(θ )Where Raa and Rbb are the auto-correlation functionsof the pilot codes, θ is the carrier tracking phaseerror, φ is the sub-carrier phase error.We obtain trigonometric expression of the carrierand sub-carrier phase tracking error.It is therefore possible to track the carrier and thesub-carrier.
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