On Generalized Signal Waveforms for Satellite Navigation (797942), страница 21
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Hegarty, 2003] and[C.J. Hegarty et al., 2005]:+∞T ⎞⎛(4.1)s (t ) = ∑ c k (t ) p⎜ t − k c ⎟N⎠⎝k = −∞where {ck} represents the pseudorandom code symbols (which may be periodic), p(t) is thechip waveform which is non-zero only over the interval support [0, Tc ) refers to the chipperiod and N is the number of equal length divisions of one chip period. In the next chaptersof this chapter we will study different waveforms and analyze them in terms of their spectralefficiency and characteristics.Of all the possible chip waveforms, the binary solutions belong to the most interesting forsatellite navigation.
Nevertheless, technology progresses so fast that it is possible to imaginemore complex non-binary alternative spreading waveforms in a near future. In this chapter wewill discuss the possibilities and potential they could bring to the navigation world.77GNSS Signal Structure4.1.1Autocorrelation and Power Spectral DensityWhen dealing with DSSS signals, two very important characteristics are the autocorrelationfunction and the power spectrum, since they determine the navigation performance of asignal.Let us assume that our signal is stationary in wide sense and can be expressed as follows:s (t ) = ∑ c k (t ) p(t − kTc − θ )(4.2)kwhere θ is a uniformly distributed variable within [0, Tc ) .
As we know, we say that a signal isstationary in wide sense when the first and second moments do not change with the time. Thismeans in other words:andE{s (t )} = E{s(t + τ )} ∀τ ∈ R(4.3)E s (t1 ) s * (t 2 ) = ℜ s (t1 , t 2 ) = ℜ s (t1 + τ , t 2 + τ ) = ℜ s (t1 − t 2 ,0 ) ∀τ ∈ R(4.4){}what is equivalent to saying that the mean is constant and the correlation function does onlydepend on the difference of time between t1 and t2. It must be noted that θ does not vary withtime representing thus an initial random shift in the signal that remains constant over time.Assuming that our signal fulfils the above described properties, we can show that theautocorrelation function is defined thus as:[][] [ℜ s (τ ) = E s (t ) s * (t − τ ) = ∑∑ E c k c n* E p (t − kTc − θ ) p * (t − nTc − θ − τ )k[ℜ s (τ ) = ∑∑ E c k c k*− mmkℜ s (τ ) = ∑∑ ℜ c (m)mk]T1 ∫1Tcc∫Tc0]nTc0p (t − kTc − θ ) p * (t − τ − kTc + mTc − θ )dt(4.5)m=k −np (t − kTc ) p * (t − τ − kTc + mTc )dtm=k −nsince the signal is assumed to be stationary.
We can further simplify this expression as:ℜ s (τ ) =1Tc∑ℜc(m) ℜ p (τ − mTc )(4.6)mAccording to this, the power spectral density of s(t) can be obtained from the FourierTransform of the autocorrelation of s(t), ℜ s (τ ) derived above, according to:⎡1⎤Gs ( f ) = FT [ℜ s (τ )] = FT ⎢ ∑ ℜc (m) ℜ p (τ − mTc )⎥⎣ Tc m⎦12Gs ( f ) = ∑ ℜc (m) P( f ) e − j 2πfmTcTc m(4.7)78GNSS Signal Structurewhere P(f) is the Fourier Transform of the waveform p(t). Moreover, the signal was assumedto be real. Assuming now that the PRN codes show ideal properties – that is random, nonperiodic, identically distributed, equiprobable and independent – then the crosscorrelation canbe approximated as E ck cn* ≈ δ kn or what is equivalent ℜc (m) ≈ δ (m ) , and the power{ }spectrum density simplifies to:Gs ( f ) =P( f )2= f c P( f )Tc2(4.8)For further justification on the use of the Dirac delta in the previous lines, refer to[M.
J. Lighthill, 1958] where additional arguments are provided. This expression is of greatvalue since it will allow us to calculate the power spectral density of the different signals wewill analyze in our work. Indeed, if we can express all the different signal waveforms bymeans of their chip waveform we will be able to use this expression to obtain the powerspectral density. As we will mention later, this is not possible in a general case but fortunatelyit is a good approximation for most of the cases.Moreover, real DSSS signals are not stationary in wide sense. Thus, it is better to work withthe average autocorrelation function in general.
According to this, the average autocorrelationfunction can be expressed as follows:1ℜ s (τ ) =TcTc1∫t =0ℜ s (t + τ ,τ )dt = Tc+∞Tc+∞∑ ∑ E [c c ] ∫ p(t − nT ) p(t − τ − kT ) dtk = −∞ n = −∞knc(4.9)ct =0where p(t ) was assumed to be real. This expression can be further developed as follows:ℜ s (τ ) =1Tc+∞+∞∑∑k = −∞ n = −∞Tc (1− n )E [ck cn ]∫ p(t ) p[t − τ − (k − n )T ]dtc(4.10)t = − nTcwhere ℜ s (τ ) was derived some lines above in (4.6). Again, assuming the codes show idealproperties, the crosscorrelation will be E [c k c n ] ≈ δ kn .Observing the equation above we can clearly recognize that the average autocorrelationfunction for a DSSS signal is equal to the aperiodic autocorrelation function of the chipwaveform under the assumption that the codes are ideally random.
This is a very importantresult since we will base most of the derivations of this chapter on it. Later we will relax thisstrong assumption on the ideality of the codes and see how the results differ when actual nonideal codes are employed to modulate the chip waveform.Once we have derived the expression for the average autocorrelation function (ACF) of ageneric DSSS signal, the first question that arises is how the ACF of an ideal waveformshould look like in order to be able to accomplish the best possible precise ranging in satellitenavigation.
As shown in the following figure, the sharper the peak of the autocorrelationfunction, the more precise the ranging will be with this waveform. Similar works with bandlimited signals have been carried out by [F. Antreich and J. A. Nossek, 2007]. An79GNSS Signal StructureE-L discriminator with 0.1 chips of spacing was employed in this example.Figure 4.2. Example of the relationship between ACF and multipath performanceThe following conclusions can be obtained from the figure above:••••••••If an E-L tracker is used, there is a high correlation between the multipathperformance of a signal and the first derivative of the autocorrelation function.The first plateau (in yellow in the figure) or peak of the multipath envelopes isdetermined by the spacing of the correlator, by the location of the first secondary peakof the ACF and by the slope of the ACF around the main peak.The steeper the slope, the better as it reduces the plateau of the multipath envelopes.The nearer to the main peak the inversion of the sign of the secondary peak takesplace, the better the multipath since the envelope is obligated to fall to 0 (blue).An inversion in the slope of the ACF forces the multipath envelopes to pass throughzero.
Therefore the closer this inversion is to the main peak, the lower the multipathenvelopes will be for short multipath (blue).No inversion in the slope makes the multipath envelope to move/keep the height of theplateau (green).A change in the slope (not in the sign) makes the multipath envelope tend to a newplateau (yellow).The sensitivity of a signal to multipath is lower the higher the chipping rate is sincethe effect of multipath is invisible to the receiver once the multipath signal comesfrom a given distance that depends on the particular discriminator.As a conclusion, the steeper the main peak of the autocorrelation and the more ripples thishas, the better the potential performance of the signal will be. The counterpart is that thehigher the number of elements in one chip, the higher will be the number of ripples of theautocorrelation so that the receiver might have problems to track or acquire the correct peak.From (4.10) we can deduce that a signal with a sharp autocorrelation function can begenerated by selecting p(t) with good aperiodic correlation properties.
In the ideal case, p(t)80GNSS Signal Structureshould be a Dirac delta.The expression for the chip waveform p(t) is given byn −1T ⎞⎛p (t ) = ∑ pi pTc / n ⎜ t − i c ⎟n⎠⎝i =0(4.11)According to this, each chip waveform is broken up into n rectangular pulses of duration Tc nwith amplitudes defined by the sequence {pi}. Furthermore, pTc / n (t ) represents the shape ofeach of the rectangular pulses the chip waveform is broken up into. In principle {pi}, definedas MCS (Multilevel Coded Symbols) sequence in this thesis, could adopt any real value,although for satellite navigation a bi-phase signal with {pi} ∈ {+1,-1} is typical.Finally, the power spectral density of the DSSS signal can be obtained as the FourierTransform of the autocorrelation function derived above, according toGs ( f ) = ∫∞τ = −∞ℜ s (τ ) e − j 2πfτ dτ(4.12)The interesting thing about the derivations above is that since the autocorrelation function wasexpressed in a tailored way using the general formulation of (4.11), the power spectrum canalso be tailored shaping thus the attributes of the desired signal as we wish.
This is of greatinterest for navigation applications, since on the one hand, as we mentioned above, we areinterested in having autocorrelation functions as sharp as possible around the main peak,while at the same time a broad spectrum with minimum overlapping with other signals wouldminimize mutual interference with other existing signals in the band.One final but important comment is that normally it is assumed that the transmitted GNSSinterfering signals are band limited at the satellite transmitter.
Thus, if we assume an idealtransmit filter of rectangular form (also referred to as brick-wall filter in this thesis) withbandwidth βT, the normalized power spectral density of unit power within the satellitetransmission bandwidth should be expressed as follows:⎧ G (f )⎪ βT / 2⎪G ( f ) = ⎨ ∫ G ( f ) df⎪ − βT / 2⎪0⎩f ≤ βT / 2(4.13)f > βT / 281GNSS Signal Structure4.2Multilevel Coded Spreading Symbols (MCS)Generalizing the definition of [J.W.
Betz, 2003], [C.J. Hegarty, 2003] and[A.R. Pratt and J.I.R. Owen, 2003b] to non-binary signals, Multilevel Coded SpreadingSymbols can be seen as generalizations of BPSK and BOC. Each spreading symbol (which isphase modulated by a spreading code value) is divided into a number of equal-lengthsegments, each of which is assigned a deterministic value. As we can infere from thisdefinition, each of the segments, also called subchips in this thesis, can in principle adopt anyvalue.
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