On Generalized Signal Waveforms for Satellite Navigation (797942), страница 50

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Indeed, as explained inchapter 3.6.2, MBOC will be implemented by Galileo as a CBOC signal which is a four levelsequence according to the definition of chapter 3.6.2. Furthermore, a power split of 50/50between data and pilot was considered as well as equal power for the open signals (E1 OS)and the protected signals (PRS). We show the spectra of the in-phase and anti-phasecomponents in the following figure:Figure 6.14.

Averaged PSDs of MBOC(6,1,1/11) in phase and anti-phase channels withSVN 1 that result from taking 10 data bits into account225Spectral Separation Coefficients with data and non ideal codesIf we take a look now at the previous figure, we can clearly see that neither the data channelnor the pilot channel fulfil completely the MBOC spectrum as defined in (4.149) if consideredindependently. Actually we can recognize that the phase signal (data) places power atfrequencies where the pilot (anti-phase) does not. Similarly, the pilot concentrates power atfrequencies where the data does not.

This is especially evident in the small lobes around4.092 and 8.184 MHz, which actually merge into broader lobes when data and pilot arecomputed together. If we add now the data and pilot spectra of the previous figures, we have:Figure 6.15. Averaged SVN 1 PSD of MBOC(6,1,1/11) – Data + Pilot – that results fromtaking 10 bits into account. For comparison the smooth spectrum is shown in redWhen data and pilot are added, we can clearly recognize the MBOC theoretical shape of(4.149). It must be noted that this is only an approximation to prove that the sum of data andpilot results in the agreed MBOC PSD.

Indeed, for a correct calculation of the total MBOCPSD, the pilot channel should be computed using only the secondary code and no data on top.Then we would have to sum them up as we have also done here. However, since the GalileoE1 OS secondary code of the pilot component has a length of 25 bits, the effect is very muchlike that of random data and therefore the resulting PSD is approximately similar to the onethat we have calculated above. In the previous figures we have considered the average of allcombinations of 10 bits but no big qualitative difference can be observed.

For correctnessthough, the anti-phase signal with the pilot should be modulated with the unique sequence of25 bits as defined in [Galileo SIS ICD, 2008].Additionally, to be more accurate, the effect of the signal structure and the multiplex shouldbe taken into account considering the theory of chapter 7.7.The main objective of this chapter was to see and analyze how many data bits must be takeninto account to consider the analytical approach as a good approximation.

As shown inprevious pages, the figures derived above could help in developing simplified average modelsto assess the compatibility among different systems.226Spectral Separation Coefficients with data and non ideal codes6.2.1 Spectral Separation Coefficients for quasi idealcodesIn the previous chapter we have computed the SSCs that result from using the real codes ofGPS and Galileo.

Now we want to go one step further and compare the results obtained abovewith those we would observe in the case that we would use ideal codes of a given length. Itmust be noted that there exist no ideal codes of finite length as this is impossible perdefinition. Indeed, random codes require an infinite length to have ideal properties.Codes are digital sequences that, in order to behave ideally, must be as long and as random aspossible. That is equivalent to saying that they must appear as noise-like as possible.

Onlythen the spreading and dispreading operations will work optimally [G.W. Hein et al., 2006c].However, the codes must remain reproducible. Otherwise, the receiver would not be able toextract the message that was sent. This is the reason why these sequences are said to be nearlyrandom or pseudo-random. As [J. Von Neumann et al., 1951] memorably stated referring tothe possibility of generating codes with finite machines, ”Anyone who considers arithmeticalmethods of producing random digits is, of course, in a state of sin.”Now that we have seen the limitations of our approach, we can set up the model with whichwe will assess the effects of the ideal line structure of a code on the SSCs. In this chapter wewill use what we describe as ideal code of a given length.

This consists basically in using thedesired properties of the spreading sequence of that length in the frequency domain. The codesequence power spectral density will represent then the average of all possible code sequencespectra for the chosen repetition interval.6.2.1.1Signal Model with ideal codesIf we recall the model that we defined in chapter 4.1.1, a DSSS signal that is stationary inwide sense can be expressed as follows:s (t ) = ∑ c k p (t − kTc ) = c(t ) ⊗ p(t )(6.3)kwhere the code sequence waveform s(t) can be seen as the convolution of the PRN codec(t ) with the spreading symbol waveform p(t ) .

Moreover, the spreading symbol waveformp(t) is defined over a specific finite time period Ts, normally equal to the code duration Tc(even case). In the same manner, the code ck is composed of N elements and repeats in realityevery Tp units of time, so that each code element has a duration Δt ofTpΔt =(6.4)NThe spreading symbol is normally designed to have the duration of exactly one code elementΔt , so that in general p (t) = 0 for t ∉ [0, Δt ) . However, if we take a close look at the modeldefined by (6.3), the general spreading symbol waveform covers also the case where the227Spectral Separation Coefficients with data and non ideal codessupport of the function extends beyond the duration of a single code element and beyond asingle repetition of the code sequence.

This effect is known as Inter-Symbol Interference (ISI)and can be observed when the real effects of the satellite transmission filters are considered.The principal effect of finite bandwidth is that the effective impulse response of the spreadingsymbol extends to durations longer than Δt . This is in fact what happens when a finiteimpulse response filter (FIR) is used in the spreading symbol generator.On the other hand, the PRN code is assumed to be binary with c k ∈ {− 1,+1} beingk ∈ [0, N − 1] and repetitive with N code elements. Thus:ck (t ) = ck δ (t − k Δt )(6.5)ck + lN = ck for all integer lAs we can recognize, the PRN code is separated from the chipping waveform in the previousexpressions.

In conclusion, the Fourier Transform of equation (6.3) is shown to beS ( f ) = C k ( f ) P( f )(6.6)and its power spectral density adopts the following formΓ( f ) = S * ( f ) S ( f ) = C k* ( f )C k ( f ) P * ( f ) P( f ) = C k ( f ) P( f )22(6.7)If we normalize for the total power to integrate to one,∫∞−∞Γ ( f ) df = 1(6.8)where the following relationship must remain valid:2Γ( f ) =Ck ( f ) P ( f )∞2∫ Ck ( f ) P( f ) df22(6.9)−∞The formula derived above is of enormous interest because by expressing the satellite signalgenerator function as a function of the power spectral density, the requirement for specificcode families can be bypassed.

In fact, we just have to substitute the code spectrum by itsideal representation in the frequency domain, no matter whether this is realizable or not.Once the expressions for the power spectral density of a signal waveform modulated with anideal code have been derived, it is time to compute the spectrum of the spreading symbol andof the ideal spreading code sequence of length N.For simplicity we will use in the next lines the BPSK signal for exemplification. As can beseen in Appendix O, its normalized power spectral density is shown to be:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ fc ⎠ΓS ( f ) = GBPSK ( f c ) = f c(6.10)(πf )2228Spectral Separation Coefficients with data and non ideal codeswhich has the well known spectrum of the figure below for a chip rate of 1.023 MHz:Figure 6.16. Power Spectral Density of the BPSK(1) modulationAs we mentioned above, we will avoid the use of specific codes in our analysis by taking theproperties that an ideal code of a given length will present.

Indeed, an idealized version of theaperiodic code sequence auto-correlation function would present the following form:Figure 6.17. ACF of an ideal code of finite lengthIt is important to note that while the idealized autocorrelation can not be shown to be theaverage of all possible code sequence choices, except for specific code families, it isrepresentative of the expectation for the behaviour of specific code families. Moreover, this istrue for any code length in general, although there exist lengths where there is no set of codesthat would fulfil the desired properties shown above.In order to account for the continuous repetition of the code sequence, we will employ thecircular autocorrelation function in the next lines.

This provides the correct result if the dataor pseudodata are of the same phase as is the case in the even autocorrelation. On thecontrary, when the data sequence flips its elements, the resulting ACF is then called odd. Theodd correlation is more difficult to compute since it depends on the particular data.As shown by [F. Soualle et al., 2005] the data modulation can cause the chip values over theintegration time to flip resulting thus in the mentioned difference between the even and oddcorrelation. If we further assume that the Doppler shift was perfectly eliminated and there isno data flip, the even crosscorrelation between two codes c1 and c 2 is then shown to be:CC e (τ ) = C1, 2 (τ ) + C1, 2 (τ − N )(6.11)229Spectral Separation Coefficients with data and non ideal codesEqually, for the odd case when the data changes during the coherent integration, thecorrelation adopts the following expression:CC o (τ ) = C1, 2 (τ ) − C1, 2 (τ − N )(6.12)where in both cases C1, 2 (τ ) represents the aperiodic correlation function defined as:⎧ N −1− p⎪ ∑ c1 ( j ) c 2 ( j + p ) 0 ≤ p ≤ N − 1⎪ N −j =1+0 p⎪C1, 2 ( p ) = ⎨ ∑ c1 ( j − p ) c 2 ( j ) 1 − N ≤ p ≤ 0⎪ j =00p ≥N⎪⎪⎩We can see the difference of both in the following figure:(6.13)Figure 6.18.

Difference between the even and odd ACFs (Courtesy of Stefan Wallner)6.2.1.2Even Autocorrelation Function of Quasi Ideal CodesThe normalized periodic even autocorrelation function ACF is shown to beTc⎧ Tc⎫⎫⎧ Tc −τ⎪ ∫ ci (t ) ci (t + τ )dt ⎪ctcttci (t ) ci [(t − Tc ) + τ ]dt ⎪τ()()d++⎪ ∫ ii∫⎪0⎪⎪ 0Tc −τcirc ⎪γ circ (τ ) = lim ⎨⎬⎬ = Tlim⎨TTccTc →∞c →∞22⎪⎪⎪⎪∫0 ci (t ) dt ⎪∫0 ci (t ) dt⎪⎪⎪⎭⎩⎩⎭(6.14)where circ stands for circular.

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