On Generalized Signal Waveforms for Satellite Navigation (797942), страница 22
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The shape of the subchip could be thus rectangular, sinusoidal or something different.The notation we will follow in this work to define an MCS signal as MCS([s], fc) where [s]represents the MCS sequence in one chip and fc is the chip rate. For the particular case that wework with binary signals we will have a BCS (Binary Coded Symbols) signal instead.As shown in [C.J. Hegarty et al., 2005] and [C.J. Hegarty et al., 2004], Binary CodedSymbols are already present in the literature since long time ago but it has not been untilrecent times that they have been considered as a serious alternative to the current BPSK andBOC modulations.
In fact, the BPSK modulation can also be denoted as BCS([1], fc) havingeach segment unit value all over the chip. In a similar way, BOC(1,1) has fs=1 and fc=1 andthus a BCS sequence with values [+1,-1] spread with a 1.023 MHz code would uniquelydefine it. According to our definition, BOC(1,1) can also be denoted as BCS([1,-1],1).
Thefollowing figure shows how a particular BCS signal could look like.Figure 4.3. BCS([1,-1,1,-1,1,-1,1,-1,1,1], fc) chip waveform in signal levels. Thiscorresponds to the BCS component proposed for Galileo E1 OS (CBCS proposal)We can go one step further in our definition of Binary Coded Symbols in line with[J.W. Betz, 2003] and define BCS([s1],[s2],…,[sn], fc) as the result of blending n differentbinary coded symbols at a chip rate fc. This can be achieved for example by time multiplexingthe different spreading symbols or by using different BCS sequences on In-phase andquadrature phase channels with independent spreading channels.
As we will see in thefollowing chapters, the MBOC signal relies on this blending concept. In the same mannerMCS([s1],[s2],…,[sn], fc) would be the result of blending n different Multilevel coded symbolsat a chip rate fc.82GNSS Signal StructureBinary Coded Symbols have gained in interest for the enormous flexibility that they couldoffer for future GNSS optimizations.In this chapter we will derive general expressions to calculate the power spectral density ofBCS and MCS sequences in general. Additionally, analytical expressions for calculating theSpectral Separation Coefficient (SSC) between two BCS signals will be derived in chapter 5.Finally, more complex signal waveforms that result from applying the theory on MCS andBCS sequences will be obtained and analyzed in terms of their potential use for navigation.MCS sequences are a promising field since well selected configurations offer clearperformance advantages as well as the possibility to control spectral properties in a moreefficient way.
This aspect has been of crucial importance during the design of Galileo in orderto be compatible and interoperable with GPS, and could show us the way to proceed in thefuture when new signals are planned to be placed in the already crowded RNSS bands.4.2.1MCS Power Spectral DensityThe power spectrum of any DSSS signal can be obtained by means of its correspondingautocorrelation function as we saw in (4.12):Gs ( f ) = ∫∞τ = −∞ℜ s (τ )e − j 2πfτ dτ(4.14)or by means of the Fourier Transform of the signal, as defined in (4.8).2P( f )22Gs ( f ) == f c P( f ) = f c S ( f )Tc(4.15)For the sake of convenience we will use the notation S ( f ) to refer to the chip waveformspectrum.
In the following lines, we will make use of this latter expression to derive the PSDof a generic MCS signal. Indeed, the Fourier transform of a generic signal MCS([s], fc) isshown to be:nS MCS ( jω ) = ∑kTcn∫skk =1 ( k −1)Tcne− jωt⎛ ωT ⎞dt = sin ⎜ c ⎟eω ⎝ 2n ⎠2jωTc2nn∑s ek =1k−jkωTcn(4.16)where n refers to the number of symbols in one chip and fc is the chip rate of the MCS signal.This can be further expressed in the frequency domain as follows:⎛ πf ⎞jπf sin ⎜⎜ nf ⎟⎟ n−nf c⎝ c ⎠ s e − j 2 πfk / nf cS MCS ( f ) = e(4.17)k(πf ) ∑k =1Once we have a general expression for the Fourier transform of an MCS signal, the powerspectral density can be derived according to (4.8) as follows:83GNSS Signal Structure⎛ πf ⎞⎟sin 2 ⎜⎜nf c ⎟⎠⎝GMCS([ s ], f c ) ( f ) = GMCS([ s ], f c ) ( f ) = f c(πf )2n∑s ek =1−jk2π f k 2nf c(4.18)We can go one step further in our derivation if we assume that the sequence consists of realcoefficients (not necessarily binary) such that s k = s k* .
In this case,n∑ sk e−j 2 kπfnf ck =12j 2 kπf−⎛ n⎜= ∑ s k e nf c⎜ k =1⎝2πf−2 j⎛ − j 2nfπf⎜ s e c + s e nf c2⎜ 1⎝j 2 kπf−⎞⎛ n⎟⎜ s e nf ck⎟⎜ ∑⎠⎝ k =12πf− nj⎞⎛ j 2nfπfnf c ⎟⎜+ ... + s n ese c⎟⎜ 1⎠⎝For simplicity in the notation we call ejk2πfnf c*j 2 kπf−⎞ ⎛ n⎟ = ⎜ s e nf ck⎟ ⎜∑⎠ ⎝ k =1+ s2 e2j2πfnf c⎞⎛ n * j 2nfkπf⎟⎜ s e ck⎟⎜ ∑⎠⎝ k =12πfnj⎞+ ...
+ s n e nf c ⎟⎟⎠⎞⎟=⎟⎠(4.19)= {k } and we will express the product of sums of(4.19) by means of the following matrix representation:⇒ms1 s1 {0}s1 s 2 {1}⎛⎜s 2 s 2 {0}⎜ s 2 s1 {− 1}n⎜l ⇓ M ( [s ] ) = s 3 s1 {− 2}s 3 s 2 {− 1}⎜......⎜⎜ s s {1 − n} s s {2 − n}n 2⎝ n 1s1 s 3 {2}s 2 s 3 {1}s 3 s 3 {0}...s n s 3 {3 − n}... s1 s n {n − 1} ⎞⎟... s 2 s n {n − 2}⎟... s 3 s n {n − 3} ⎟⎟......⎟...s n s n {0}⎟⎠(4.20)where l denotes the row index and m the column index. Since the coefficients are assumed tobe real, s l s m* = sl* s m = sl s m and the sum of all the terms in the matrix above can be expressedas follows:n −1⎧n 2⎛ ωT ⎞+s2sl sl +1 cos⎜ c ⎟ +j 2 kπf∑∑l⎪n−⎧ n n −l +1ωTc ⎤ n 2⎝ n ⎠⎪ l =1⎡l =1nf c()==−se2sscosl1⎨ n−2⎨ ∑ ∑ l l + m −1∑k⎢⎥ − ∑ slnTω⎣⎦ l =1⎛⎞=1=1k =1lm⎩⎪+ 2 s s cos⎜ 2 c ⎟ + ...∑l l +2⎪⎩ l =1⎝ n ⎠(4.21)Furthermore, we can express the term is parentheses in a more simplified form if we apply thevariable change l+m-1=k’, yielding thus:2n∑ sk ek =1−j 2 kπfnf c2nnωT ⎤ n⎡= 2∑∑ sl sk ′ cos ⎢(k ′ − l ) c ⎥ − ∑ sl2n ⎦ l =1⎣l =1 k ′ =l(4.22)Observing the matrix of (4.20) above, we can clearly see that we only need to look at theterms of the right superior triangular part of the matrix to compute the PSD of the MCS signalfor any given real sequence.
Moreover, it must be noted again that this sequence has not to benecessarily binary, being BCS a particular case of MCS as defined in the lines above.Combining now (4.18) and (4.22) we have the general expression for the power spectraldensity of a generic MCS signal:84GNSS Signal Structure⎛ πf ⎞⎟sin 2 ⎜⎜n −1 nnf c ⎟⎠ ⎧ n 2ωT ⎤ ⎫⎡⎝+2GMCS([ s ], f c ) ( f ) = f cssl sk ′ cos ⎢(k ′ − l ) c ⎥ ⎬⎨∑ l∑∑2n ⎦⎭(πf ) ⎩ l =1⎣l =1 k ′ =l +1(4.23)or equivalently,⎛ πf ⎞⎟sin 2 ⎜⎜nf c ⎟⎠ ⎧ n nωTc ⎤ n 2 ⎫⎡ ′⎝()2sscoskl−− ∑ sl ⎬GMCS([ s ], f c ) ( f ) = f c⎨′∑∑lk⎢n ⎥⎦ l =1 ⎭(πf )2 ⎩ l =1 k ′=l⎣4.3(4.24)Binary Coded Symbols (BCS)Binary coded symbols are a particular case of MCS signal with a binary chip sequencesl ∈ {+ 1,−1}. The power spectral density of a generic BCS ( [s1 , s 2 , s3 ,..., s n ], f c ) can thus bederived from (4.23) or (4.24) and is shown to adopt the following simplified form:⎛ πf ⎞⎟sin 2 ⎜⎜nf c ⎟⎠ ⎧ n n⎡2πf ⎤ ⎫⎝GBCS([ s ],1) ( f ) = f c⎨∑∑ 2sl s k′ cos ⎢(k ′ − l )⎥ − n⎬2nf c ⎦ ⎭(πf ) ⎩ l =1 k ′=l⎣(4.25)Or equivalently:⎛ πf ⎞⎟sin 2 ⎜⎜nf c ⎟⎠ ⎧ 1 n −1 n⎡2πf ⎤ ⎫⎝GBCS([ s ],1) ( f ) = nf c⎨1 + ∑ ∑ 2sl sk ′ cos ⎢(k ′ − l )⎥⎬2nf c ⎦ ⎭(πf ) ⎩ n l =1 k ′=l +1⎣(4.26)where the first term of the product corresponds to the PSD of a BPSK with nfc MHz of chiprate and the second term can be represented by the matrix defined in (4.20) and is shown tohave a power of 1 in an infinite bandwidth.
We can thus represent the PSD as follows:GBCS([ s ],1) ( f ) = GBPSK (nf c ) ( f )GMod ( f )(4.27)According to this, the function that seems to bring more information about the BCS sequenceis the sum of the second factor. This term, namely GMod ( f ) , can be easily computed with thematrix we defined in (4.20) and we will call it modulation term for short. Furthermore, it isimportant to realize that the modulation term and the spreading symbol are used indistinctly inthe literature to indicate the same.Furthermore, it is important to note that (4.27) can be further generalized toGBCS([ s ],1) ( f ) = GSubchip pulse ( f )GMod ( f )(4.28)for the case that the sub-chip is modulated by a generic non-rectangular pulse.
This will beused in further chapters when we derive more general expressions and is also shown in theAppendixes of this thesis.85GNSS Signal StructureThe main conclusion that can be drawn from observing this definition matrix is that a subchipalone, or in other words an element of the sequence alone, does not directly affect the PSD ofthe whole signal. It must be understood in relationship with all the other elements of thesequence. This adds an important complexity.
In fact, it is not trivial to derive qualitativelythe shape of any generic BCS signal by only having a look at its generation sequence [s ] = s .We can clearly see this in the following example. Let us imagine different BCS signals oflength 21 consisting only of ones expect for one logical zero (or -1 at signal level) where thezero subchip is placed at different locations within the BCS vector.Figure 4.4.
Autocorrelation Function of different BCS sequencesFigure 4.5. First Derivative of the Autocorrelation of different BCS sequences86GNSS Signal StructureFigure 4.6. Multipath Envelopes of different BCSFor the previous figures, an E-L discriminator with 0.1 chips of spacing was employed. As wecan recognize, a single shift to the right in the BCS sequence alters the final multipathperformance significantly. This leads us thus to the conclusion that evaluating the multipathperformance of a signal with a quick look is not an easy task as it was with BPSK and BOCsignals, for example.
Even though we knew how the BCS sequence looks like, a simple shiftwould significantly modify the multipath properties of the signal. Indeed, the mathematicalideas and conclusions gathered above have driven many of the works that were carried out inthe past years to find the MBOC signal.4.3.1Binary Phase Shift Keying Modulation (BPSK)A very important and useful signal in satellite navigation is the BPSK modulation which wasin fact the first one to be used for Satellite Navigation. In spite of its simplicity, it is still usednowadays but could eventually be substituted by the BCS modulation or combinations withthis one in the medium-long term.We will derive now the power spectral density of a BPSK(fc) using the general definition ofBCS that we saw in chapter 4.3.
According to this, any BPSK(fc) signal can be described as aBCS sequence with vector s = [1 1 1 …1] whatever the length of the vector. We will derivethe expression of the PSD generalizing over n.First we build the M n ( [s ] ) matrix for any n, which is shown to be:⎛1{0} 1{1} 1{2}⎜1{0} 1{1}⎜nM ( [1,1,...,1] ) = ⎜1{0}⎜⎜⎜⎝... 1{n − 1} ⎞⎟...
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