On Generalized Signal Waveforms for Satellite Navigation (797942), страница 41
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Therefore 0 ≤ α ≤ 1 . This can be clearly seen in the following figure fordifferent roll-off factors:178GNSS Signal StructureFigure 4.72. Raised Cosine Filter for different roll-off factorsWe show next the time representation hRC (t ) of the Raised Cosine pulses of the previousfigure for different roll-off factors:Figure 4.73. Raised Cosine pulses for different roll-off factorsAs we can recognize, the raised cosine pulse waveform with the optimum spectrumoccupation ( α = 0 ) is the pulse that also presents more oscillations in the time domain, whatis a non desired characteristic in principle. In fact, low values of α allow for a more efficientuse of the spectrum but increase the ISI.
Moreover, we have to keep in mind that none of thefilters of the figures above could correspond to a real implementation since they are not causalas h(t ) ≠ 0 for t < 0 . To avoid this, a minimum delay should be added for the pulse to gaincausality. Indeed, real implementations already add a delay of some chips from the momentthe signal enters the filter and leaves it. This is observed in the Raised Cosine filters too.179GNSS Signal StructureIt is important to note that a band-limited Nyquist pulse cannot avoid by itself the ISI unlessthe channel is ideal. This means that the RC pulses have got to be implemented together withan equalizer at the receiver for the correct identification of the symbols at the sampling time.We can express this in the following expression:H RC ( f ) = H TX ( f ) H C ( f ) H RX ( f ) H E ( f )(4.220)where H TX ( f ) is the transmission filter, H C ( f ) is the channel frequency response, H RX ( f )is the receiver filter and H E ( f ) is the equalizer.
The usual approach is to design thetransmitter and receiver filters such thatH RC ( f ) = H TX ( f ) H RX ( f )H TX ( f ) = H RX ( f ) = H RC ( f ) = H SRRC ( f )(4.221)and leave the equalizer filter to take care of the imperfections and ISI caused by the channel:HE ( f ) =1HC ( f )(4.222)According to this, the square-root raised cosine (SRRC) pulses are Nyquist pulses of finitebandwidth with power spectral density given by:1−α⎧Tcf ≤⎪2Tc⎪⎡π T ⎛⎪T ⎧⎪1 − α ⎞⎤ ⎫⎪ 1 − α1+α⎟⎟⎥ ⎬≤ f ≤GSRRC (α , f ) = ⎨ c ⎨1 + cos ⎢ c ⎜⎜ f −(4.223)2Tc ⎠⎦ ⎪⎭ Tc2Tc⎣ α ⎝⎪ 2 ⎪⎩⎪1+α0f >⎪2Tc⎩Moreover, it can be shown that∫∞−∞GSRRC (α , f )df = 1(4.224)where we can recognize that the bilateral bandwidth is finite and of value (1 + α ) Tc .
In thesame manner, the time representation of such SRRC pulses is shown to adopt the followingform [E.A. Lee and D.B. Messerschmitt, 1994]:⎡ (1 + α ) ⎤ Tc⎡ (1 − α ) ⎤cos ⎢π t⎥ +sin ⎢π t⎥4 α t ⎣ TcTc4α⎣⎦⎦s (t ) =2π Tc⎛ 4α t ⎞⎟⎟1 − ⎜⎜⎝ Tc ⎠(4.225)which is indeed a pulse shape with infinite support as we expected, since bandlimited signalsextend to infinity in the time-domain. The interesting aspect of this waveform is that itsatisfies the Nyquist condition for zero Inter-Symbol interference (ISI), so that the bit-error180GNSS Signal Structureprobability is identical to that of BPSK with Non Return to Zero (NRZ) pulses if the receiversamples at zero-ISI locations.In spite of its interesting properties, the Raised Cosine Signals proposed by[R. De Gaudenzi et al., 2000] for Galileo presented a series of major problems that made itnot recommendable for satellite navigation applications:•••••One of the most important disadvantages is the fact that the RC signal is handicappedfrom the beginning regarding its potential improvement of performance.
We have seenthat the SRRC modulation makes a very efficient use of the assigned spectrum. Thisremains true. However, the signal is by definition band-limited to a very narrowbandwidth so that the performance could never be as good as that of other signalssharing the band with wider bandwidths. A SRRC would have been maybe the best fora narrowband receiver of around 3 MHz as we commented in chapter 3 but Galileowould have lost the race in competitiveness as soon as other signals would have madeuse of wider bandwidths.Another consequence of the fact that the SRRC modulation is bandlimited is that itsauto-correlation function has a very rounded peak. As we have seen at the beginningof this chapter, the quality of a signal improves as the slope of the ACF becomessteeper around the main peak. In the case of the Raised Cosine Signal no matter howwide the receiver bandwidth would be, we would not be able to do anything toimprove the quality of our measurements.
If there is something that technology showsus permanently, that is the fact that we cannot design systems thinking of today’slimitations but we must challenge our potentials.As shown in [R. De Gaudenzi et al., 2000] the receiver complexity could have profitedfrom simplified receivers with lower complexity. While this might be true, an inherentdegradation would be introduced in the system per definition since the replica signalsin the receivers would be band-limited and thus handicapped. Furthermore, techniqueslike the narrow correlator would have brought no improvement due to the bandlimited property of the SRRC pulse.The Raised Cosine solutions that were proposed in [R. De Gaudenzi et al., 2000]presented another very serious inherent problem: namely a worse antijammingprotection compared with other signals, due to its spectrum concentrated in arelatively narrow bandwidth.
As we can see in Appendix M and in chapter 4.7.8, themore spread the frequency components of a signal are, the better the resistance againstnarrowband and wideband interference will be.Finally, the original proposal of [R. De Gaudenzi et al., 2000] did not contemplate thepossibility of having a military signal as the PRS. Such a signal would need widebandwidths per definition and using SRRC would have also implied important risksdue to its weakness against all sources of interference.181GNSS Signal Structure4.8.3Prolate Spheroidal Wave Functions (PSWF)Another family of waveforms that have gained in interest in the past months is that of the socalled Prolate Spheroidal Wave Functions (PSWF). The PSWF family offers an infinite baseof orthogonal functions {ψ 1 (t ),ψ 2 (t ),...,ψ i (t ),...} with associated Eigenvalues {λ1 , λ 2 ,..., λi ,...}that are real and positive.
Similar to the SRRC signal that we saw in the previous chapter,they also show ideal bandlimiting within [− β r 2 , β r 2] , making the PSWF signals veryinteresting to fulfil stringent demands on band-limitation.As the name well indicates, the Prolate Spheroidal Wave Functions are the result of solvingthe Helmholtz equation or wave equation in Prolate Spheroidal coordinates. Thiselectromagnetic identity is shown to adopt the following form:Δϕ ( x, y, z ) + k 2ϕ ( x, y, z ) = 0(4.226)where k defines the wave number in prolate spheroidal coordinates. The Prolate Spheroidalcoordinates can be expressed in Cartesian coordinates according to the followingtransformation [M. Abramovitz and I.A.
Stegun, 1965]:x =fy =z =f(1 − η )(ξ(1 − η )(ξ2222)− 1) sin φ− 1 cos φ(4.227)fηξwith− 1 ≤ η ≤ 1, 1 ≤ ξ ≤ ∞, 0 ≤ φ ≤ 2πwhere:• f• η• ξ• φ(4.228)is the semifocal distance,is the angular coordinate,is the radial coordinate, andis the azimuthal coordinate.Furthermore, the Laplace operator in the new coordinate system adopts the following form:Δh =1hξ hη hφ⎡∂⎢⎣⎢ ∂ξ⎛ hη hφ ∂ ⎞ ∂ ⎛ hξ hφ ∂ ⎞ ∂ ⎛ hξ hη ∂ ⎞⎤⎜⎟+⎜⎟+⎜⎟⎜ h ∂ξ ⎟ ∂η ⎜ h ∂η ⎟ ∂φ ⎜ h ∂φ ⎟⎥⎝ ξ⎠⎝ η⎠⎝ φ⎠⎥⎦(4.229)Thus the scalar wave equation can be written in prolate spheroidal coordinates as follows:Δψ + k 2ψ =∂∂ξ⎡ 2∂ψ ⎤ ∂ ⎡∂ 2ψξ 2 −η 22222 ∂ψ ⎤−+−+ξ11η⎢⎥ ∂η ⎢⎥ ξ 2 − 1 1 − η 2 ∂φ 2 + c ξ − η ψ = 0∂∂ξη⎣⎦⎣⎦()()()()()(4.230)being c = fk 2 per definition.
As it can be shown in [M. Abramovitz and I.A. Stegun, 1965],this partial differential equation of second order can be solved by variables separation as182GNSS Signal Structurefollows:⎧ cos(mφ ) ⎫⎬⎩ sin (mφ ) ⎭ψ = Rmn (c, ξ ) S mn (c,η ) ⎨(4.231)where Rmn (c, ξ ) represents the radial part and S mn (c, ξ ) depicts the angular component of thesolution. According to this, the radial solution is shown to comply with the following equation∂∂ξ⎡ 2⎤ ⎛m2 ⎞∂2 2⎜⎟ Rmn (c, ξ ) = 0()()ξ1Rc,ξλccξ−−−+mn⎢⎥ ⎜ mn∂ξξ 2 − 1 ⎟⎠⎣⎦ ⎝()(4.232)while the angular part needs to be a solution of the differential equation shown next⎤ ⎛m2 ⎞∂ ⎡∂22 2⎜⎟ S mn (c,η ) = 0 .()()1ηSc,ηλccη−+−−mn⎥ ⎜ mn∂η ⎢⎣∂η1 − η 2 ⎟⎠⎦ ⎝()(4.233)In the radial as well as in the angular differential equations shown above, the eigenvalues aredenoted as λmn (c ) .
In addition, since both differential equations are identical except for theirrespective definition supports, a transformation from the radial to the angular expression ispossible without great difficulties. Furthermore, if we particularize the previous equations forc = 0 , both the radial and the angular differential equations are shown to result in the sameexpression. The resulting homogeneous equation presents as solutions the so-called LegendreFunctions, which can be expressed as follows:μ⎛d ⎡μ2 ⎞ μ2 dPν ⎤⎜⎟ Pν = 0 ,()1z1νν−++−⎢⎥dz ⎣dz ⎦ ⎜⎝1 − z 2 ⎟⎠()(4.234)being Pνμ ( z ) the Legendre function of degree ν and order μ . In analogy to other similarproblems that can be found in the literature, the homogeneous solutions can be used to solvethe radial and angular differential equations.
In fact, the particular prolate angular solutionS mn (c,η ) can be expressed as an infinite series of Legendre functions in the following way:S mn (c,η ) =∞∑r = 0 ,1*d rmn (c )Pmm+ r (η )(4.235)where the symbol * indicates that the index r of the summation is even if n-m is even and thatthe index r is odd if n-m is odd. If we further introduce the homogeneous solution of theprolate angular function S mn (c,η ) into the general differential equation we obtain thefollowing recursive relationship [S.
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