On Generalized Signal Waveforms for Satellite Navigation (797942), страница 43
Текст из файла (страница 43)
To further provide flexibility in thelocation, the parameters ρ1 and ρ 2 are then introduced.As we can recognize, the location of the different pieces of the chip waveform is not atpositions Tc/n but could adopt any imaginable place determined by ρ1 and ρ 2 . To account forchanges in sign in the general Coded Symbol (CS) sequence, we will further use the generalnotation of chapter 4.2 and define the signal as follows within a sub-chip sc(t). In fact, we candefine our signal for the general case of faded-harmonic CS of four levels as follows:nnT ⎞ n⎛⎛ iT ⎞⎛ iT ⎞p (t ) = ∑ p i pTc / n ⎜ t − i c ⎟ = ∑ p i (t ) pTc / n (t ) ⊗ δ ⎜ t − c ⎟ = pTc / n (t ) ⊗ ∑ p i (t )δ ⎜ t − c ⎟n ⎠ i =1n ⎠n ⎠⎝⎝⎝i =1i =1(4.252)As we can recognize in the expression above, the code symbols start with a delay Tc n wheni = 1 . Thus, an extra shift of half a sub-chip to the left is necessary if we define pTc / n (t )between − Tc 2n and Tc 2n .
As we know this is equivalent to multiplying by exp( jωTc 2n ) .Moreover, if we want our time definition to start at zero and define pTc / n (t ) between 0 andTc n , a shift of one whole sub-chip Tc n to the left would be necessary, what in thefrequency domain is equivalent to multiplying by the factor exp( jωTc n ) . No matter how wedo, the final result is the factor exp( jωTc n ) that we have seen in the Appendix. Using thesymmetric definition around 0, we have:Tc⎧()ϕcos−< t ≤ − ρ2⎪2n⎪ 1− ρ 2 < t ≤ − ρ1⎪⎪pTc / n (t ) = ⎨cos(ϕ ) − ρ1 < t ≤ ρ1⎪ 1ρ1 < t ≤ ρ 2⎪T⎪cos(ϕ )ρ2 < t ≤ c⎪⎩2n(4.253)188GNSS Signal Structureand the Fourier Transform of this generic faded-harmonic CS is thus shown to bePCSFH ( jω ) = Pc ( jω ) ejωTc2nn∑ pk e−jkωTcn(4.254)k =1wherePTc / n ( jω ) = ∫−ρ2−Tc2ncos(ϕ )e− j ωtdt + ∫− ρ1− ρ2e− j ωtdt + ∫ρ1− ρ1cos(ϕ )e− j ωtρ2dt + ∫ e− j ωtρ1dt + ∫Tc2nρ2cos(ϕ )e − jωt dt(4.255)which can be further developed to the following expressionPTc / n ( jω ) =⎤2 ⎡ ⎛ ωTc ⎞sin ⎜⎟ cos(ϕ ) + sin (ωρ1 )[cos(ϕ ) − 1] + sin (ωρ 2 )[1 − cos(ϕ )]⎥ (4.256)⎢ω ⎣ ⎝ 2n ⎠⎦or in the frequency domain:PTc / n ( f ) =1πf⎡ ⎛ πf ⎞⎤⎟⎟ cos(ϕ ) + sin (2πfρ1 )[cos(ϕ ) − 1] + sin (2πfρ 2 )[1 − cos(ϕ )]⎥ (4.257)⎢sin ⎜⎜⎣ ⎝ nf c ⎠⎦Therefore, the power spectral density of any Coded Symbol of four levels as defined above,would present the following formπfkjfcnf cG (f )=ePTc / n ( f )22cos ϕ + (ρ 2 − ρ1 )sin ϕFHCS2n∑p ek =1k−j2πfk 2nf c(4.258)or equivalently,⎡ ⎛ πf ⎞⎤⎟⎟ cos(ϕ ) + sin (2πfρ1 )[cos(ϕ ) − 1] + sin (2πfρ 2 )[1 − cos(ϕ )]⎥f c ⎢sin ⎜⎜nf⎦FH(f )= ⎣ ⎝ c ⎠GCS22cos ϕ + (ρ 2 − ρ1 )sin ϕ2n∑p ek =1k−j2πfk 2nf c(4.259)where the modulating term on the right is common to the MCS definition and therefore all theresults that we obtained in the previous chapters can be used here too.
Finally, it is interestingto see that playing with the parameters ρ1, ρ2 and ϕ, we can select the lobe we want tosuppress and how much we want to attenuate it. Furthermore, it is trivial to see that thistechnique can be further generalized to more lobes if new parameters are introduced,according to the previously discussed scheme.Regarding the multiplex, it is important to note that, as we will see in chapter 7, we cannotapply the Interplex modulation directly since such a signal is not binary and slight changes inthe multiplex scheme are thus required. Nonetheless, the theory that we will derive in chapter7.7.10 on the Modified Interplex could also be applied here too. To show a potentialapplication of this modulation, next figure compares different solutions to implement CBCSusing Faded Harmonics and the Modified Interplex.
For more details on Interplex and theModified Interplex, refer to chapter 7.7.9.189GNSS Signal StructureIn-Phase ComponentCode A1-1-1-11-1Code B1-11-1-11Figure 4.76. Pseudo-random multiplexing of BCS and BOC using Interplex, FadedHarmonics Interplex and modified Interplex for CBCSAs shown in [G.W. Hein et al., 2005], the implementation of any CBCS signal, and inparticular of CBOC, could have also been performed using the Faded Harmonics (FH)Interplex scheme since this one also relies on the sum of two 4-level spread-spectrum signals.However, the analysis of FH-Interplex for CBCS put in evidence two important drawbacks:••The relative power of the Inter-Modulation product is increased, resulting in a loss ofefficiencyThe modulation results in significant distortions on the quadrature signals, for examplethe PRS signal.
These distortions may induce unacceptable losses on the receiver, aswell as an increased spreading of the PRS signal in adjacent frequency bands.190Spectral Separation Coefficients5. Spectral Separation Coefficient (SSC)5.1DefinitionThe Spectral Separation Coefficients, or SSCs for short, are a very powerful figure to indicateand measure the degree of interference that a signal suffers due to other signals sharing theband. This degradation is evaluated in terms of reduction of the Signal to Noise plusInterference Ratio (SNIR) of the desired signal, measured at the output of a receiver’scorrelator that uses an ideal matched filter to receive the navigation signals as described by[J.-L.
Issler et al., 2003], [J.W. Betz, 2001b] and [A.R. Pratt and J.I.R. Owen, 2003a]. Indeed,the methodology to compute the SSCs relies on the idea of measuring the power of thedesired signal and its reduction due to all the interfering signals, at the correlator’s output.This can graphically be shown in the following schemeFigure 5.1. SSC Correlator Model for the Spectral Separation Coefficients calculationswhere sd (t) refers to the desired signal, si (t) to the interfering signal and s*d (t) is the matchedspreading waveform of the desired signal.
Moreover, the sum module represents theintegration and dump function. This is the general model that we can find in the literature[J.J. Spilker, 1997b]. It is also important to note that this model implies that the receiver doesnot make use of any cancellation technique.Another interesting interpretation of the SSC is to see it as the mean power of thecrosscorrelation function between the desired and the interfering signal[J.-L. Issler et al., 2003]. For this reason, also expressions for the crosscorrelation function inthe frequency and time domain are derived in the next lines.As shown in [A.R. Pratt and J.I.R. Owen, 2003a], the SSC provides a measure of the noisepower output by a receiver when certain signals are incident at its input.
As one can imagine,the better the isolation of a signal with the rest of signals in the band, the lower will be theequivalent noise caused by them, resulting thus the concept of cross power spectral densityhere of special interest.If we take a closer look at Figure 5.1, it is possible to see that the power spectrum at theoutput of the correlator of a filter H (ω ) matched to the desired signal – being thus H (ω )equal to the conjugate of the desired signal spectrum S d* (ω ) – can be expressed as follows:191Spectral Separation CoefficientsSo (ω ) ={}Pi Si (ω ) + Pd S d (ω ) H (ω ) ={ P S (ω ) +ii}Pd S d (ω ) S d* (ω )(5.1)where Pi refers to the power level of the interfering signal and Pd that of the desired signal.Moreover, in a general case a protection filter can also be included in the expression as shownin [A.R.
Pratt and J.I.R. Owen, 2003a]. It is also important to see that the spectral definitionabove encompasses both the spreading codes as well as the spreading waveform of the desiredand interfering signals, being thus valid for the most general case.The power spectral density can thus be expressed as follows:{}} S (ω ){Po (ω ) = S o (ω ) = Pi S i (ω ) + Pd S d (ω ) + 2 Pi Pd Re S i (ω ) S d* (ω )2222d(5.2)what can be further simplified yielding:{Po (ω ) = So (ω ) = Pi Si (ω ) + Pd S d (ω )222} S (ω )d2= [Pi Gi (ω ) + Pd Gd (ω )] S d (ω ) (5.3)2if we assume that the interfering signal and the desired signal are uncorrelated, averaging thusthe cross-spectrum term to zero.
This can be achieved either by means of the signal structureor the code structure being thus this assumption also true even if we work with signals of thesame family, as long as all the codes are ideally orthogonal. As we can imagine, this is a verystrong assumption since especially for the case of intra-system interference of signals withshort codes this approximation could lead to wrong results as we will show in chapter 6.2.2.Nonetheless, we will still consider it as valid in this chapter.Moreover, it is important to note that the functions Gi(ω) and Gd(ω) are both normalizedPower Spectral Densities as those defined in chapter 4.1.1. Thus, the total power at the outputof the correlator will be the integral of the output spectrum, expressed as follows:∞∞∞−∞−∞−∞Po = ∫ Po ( f )df = Pi κ id + Pd κ dd = Pi ∫ Gi ( f )Gd ( f )df + Pd ∫ Gd2 ( f )df(5.4)where we can recognize the spectral separation coefficient κ dd of the desired signal with itselfand between the desired signal and the interfering signal κ id .
These are defined as:∞κ dd = ∫ Gd2 ( f )df−∞∞κ id = ∫ Gi ( f )Gd ( f )df(5.5)−∞where the receiver and transmitter are assumed to have infinite bandwidth. If this is not thecase, either the integration limits require adjustment or corrections to the SSCs must be madeto include the effects of the finite bandwidths of receiver and transmitter, as we saw in (4.13).We can obtain the same expressions reasoning in a different way. If we assume that theundesired signal is stochastic and Gaussian with a normalized power spectral density Gi ( f )and that the desired signal can also be characterized adequately as an independent stochastic192Spectral Separation Coefficientsprocess with normalized Gd ( f ) , then the multiplier output will be Gaussian and it is shown tobe the convolution of both power spectral densities, according to:∞S m ( f ) = ∫ Gi ( f1 )Gd ( f − f1 )df1(5.6)−∞such that after the Integrate and Dump (I&D) operator we will have:S m ( f ) TI H ID ( f )2(5.7)where H ID ( f ) = sinc(π f TI ) being TI the coherent integration time.According to this, if the interference is Additive White Gaussian Noise (AWGN) then the2I&D output will be N 0 TI H ID ( f ) and thus the correlation sum variance will adopt thevalue N 0 TI .
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