On Generalized Signal Waveforms for Satellite Navigation (797942), страница 67
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This thesis provides the analytical models and expressions required to supportfuture studies in the field. This work has shown that backward compatibility with otherexisting signals is possible if some constraints are introduced in the signal design. Thisconcept has proved to work as it has been recently demonstrated with the design of MBOC inE1/L1 and should be further exploited using the models of this thesis.While this work provides the fundamental models and theory required to design a signal planfrom the point of view of the signal waveforms.
Further research is found necessary in severalother fields of comparable relevance. Next lines describe those aspects of the signal designwhere further work is recommended in the future:••New frequency allocations for navigation services: The efforts invested by allSatellite Navigation Systems have mainly concentrated on placing new services in thevery few frequency bands that are already allocated to Radio-Navigation SatelliteServices (RNSS) today.
However, the present RNSS bands are scarce andovercrowded. Moreover, they provide very limited physical performance given therelatively narrow band they possess. Future efforts should be dedicated to finding newfrequency resources with the required protection to provide any potential service.RNSS C-band and S-band for navigation: In consonance with the previousparagraph and keeping in mind that these two bands are already accepted as RNSSeither on a global or a regional basis, they should be further explored in coming years:o The C-band range from 5000 MHz to 5030 MHz is an RNSS band in thewhole world and could provide improved navigation services given its highercarrier frequency. The main contributions to the error budget, namelyionospheric errors and multipath, are in this band considerably lower.
Thisband could give answer to those limitations that the L-band shows today.However, the technology in C-band poses important challenges that are still tobe well understood.306Conclusions and Recommendationso The S-band range from 2483.5 MHz to 2500 MHz is already a navigation bandin many Asian countries (ITU Region 3). In fact, The Chinese Beidou GEOsatellites use as carrier frequency 2491 MHz and the Japanese EngineeringTest Satellite Number 8 (ETS/8) too. In addition, the Indian IRNSS systemalso plans navigation signals in this band for the near future as we have seen inchapter 2.8.2. For this band to be used for a global satellite navigation servicefurther work is required in understanding the advantages and limitations of thisfrequency allocation. Given its proximity to communication services,important synergies are to be expected.
Further work could use this thesis asfundament to understand the potential of this band.••••New signal waveforms for navigation: While the BOC, AltBOC and MBOCmodulations have represented important innovations in the signal design, this thesishas shown that there are still other signal options of great interest, not necessarilybinary, that should be further investigated in the future. The current payloadlimitations to have binary signals will soon be overcome and future work is thusrequired to identify new signals. This thesis offers the mathematical framework underwhich these new signals could be investigated.Relative reduction in the importance of backward compatibility: In the design andmodernization of all current navigation systems, backward compatibility with legacysignals has played an outstanding role.
However, as software receivers gain inimportance, this constraint could lose part of its weight in future decisions. New andmore powerful signals could thus be designed. Future work should also be carried outusing the guidelines proposed in this thesis.Need of improved multiplexing techniques: As the number of signals increases, newtechniques will be required in the future to accommodate new services and targetedapplications. Today, the same navigation signal serves completely different users.However, as suggested in this thesis, several signals optimized to particular users ofinterest could be multiplexed using advanced multiplexing techniques already presenttoday in the literature. Further work is found to be required in coming years to give ananswer to the ever increasing needs of the different user communities. In particular forservices that require improved robustness, combinations of CDMA with frequencyhopping and Orthogonal Frequency Division Multiplexing (OFDM) could providepromising solutions.New code families: Significant progress has been made in the field of code theoryover the past years.
However, much of these developments have not yet been appliedto satellite navigation. While the modernized GPS and Galileo will make use of newcode concepts, it is clear that further research will still be required in the future. Inaddition, new encryption methods should be further explored to give an answer to theneeds of certain services.307Conclusions and Recommendations•••Modernized Message Structure: Message structure is one of the aspects in the signaldesign that has less evolved in the past years.
While GPS III will introduce some slightmodifications to the actual designs, it is clear that still further improvements can beachieved. Fundamental research should thus be realized in coming years to elaborateoptimized message designs for new satellite navigation signals. The use of dynamicmessages with data rates that change depending on the needs of the system is a fieldwhere future work is required. Some of these ideas could be well answered using thetheory presented in this thesis, exploiting the multiplexing characteristics.Compatibility and Interoperability: This thesis has underlined the importance of thesignal design to achieve compatibility.
Future work will be required in coming yearsto understand how the coexistence between all the different services of the variousGlobal and Regional Satellite Navigation Systems could be guaranteed as newsystems come into play.Interoperable Integrity: In line with the previous paragraph, interoperability in theintegrity concept of the different navigation systems should equally be furtherinvestigated. The user will only be capable of really profiting from the differentintegrity concepts that exist today if harmonized statistical models are developed.308Power Spectral Density of BPSK SignalsAAppendix.
PSD of BPSK signalsAs shown in chapter 4.3.1, BPSK is a particular case of MCS. Thus, the power spectraldensity of a generic BPSK(fc) can be described using the theory on MCS signals as follows:⎛ πf ⎞⎟sin 2 ⎜⎜⎡ n −1nf c ⎟⎠ ⎧⎪⎛ 2πf ⎞⎤ ⎫⎪BPSK ( nf c )BPSK ( f c )⎝⎟⎟⎥ ⎬( f ) = fc2GBPSK ( f c ) = GSubchip pulse ( f )GMod+n⎨⎢∑ (n − i )cos⎜⎜ i2nf(πf ) ⎪⎩1i=c⎝⎠⎦ ⎪⎭⎣(A.1)In this Appendix we will show that after some math this expression leads to the well knownform of the BPSK Power Spectral Density that can be found in the literature. To do so, wewill concentrate on the term in the brackets, namely the modulating factor. For simplicity inthe manipulations we will express the cosine as a function of complex exponentials using theEuler´s formula.
According to this, the modulating term to simplify is as follows:n −1⎡ n −1⎡⎤⎤BPSK ( f c )( f ) = n + 2⎢∑ (n − i )cos ⎢i 2πf ⎥ ⎥ = n + ∑ (n − i )(eiA + e− iA )GMod(A.2)2πfi =1⎣ nf c ⎦ ⎦⎣ i =1A= jnf cAs we can recognize from the expression above, the problem to solve reduces to calculatingthe following sum:n −1n −1n −1i =1i =1i =1∑ (n − i )eiA = n∑ eiA −∑ i eiA(A.3)Let us define the following function:n −1n −1( )f ( A) = ∑ e = ∑ eiAi =1A ii =1e An − e A= Ae −1(A.4)which shows the interesting property thatdf ( A) n −1 iA= ∑i edAi =1(A.5)From (A.4) we can also see that()()()df ( A) n e An − e A e A − 1 − e A e An − e A=2dAeA −1Using the previous results, we define now the function Φ ( A) as follows:(n −1n −1n −1i =1i =1i =1)Φ ( A) = ∑ (n − i ) e iA = n∑ e iA −∑ i e iA = n f ( A) −df ( A)dA(A.6)(A.7)Combining now (A.4) and (A.6) according to (A.7) yields then:n −1Φ ( A) = ∑ (n − i ) e =i =1iAe A (n − 1) + e A (n +1) − n e 2 A(eA)−12(A.8)We can further express the original expression of (A.2) as follows in terms of Φ ( A) :309Power Spectral Density of BPSK Signals⎡ n −1⎡BPSK ( f c )( f ) = n + 2⎢∑ (n − i )cos⎢i 2πfGMod⎣ nf c⎣ i =1=n+e (n − 1) + eA(eAA ( n +1 )− 1)2− ne2A+e−A⎤⎤=⎥ ⎥ = { n + Φ( A) + Φ (− A)}2πf⎦⎦A= j(n − 1) + e(e−Anf c− A ( n +1 )− 1)− ne(A.9)−2 A2This can be simplified after extracting the common factor e An and e − An respectively andforming squares, as shown next:[]⎧n (e − A − 1)(e A − 1) 2 + (4n − 2 )(e A + e − A ) −⎫⎪⎪⎪⎪AA 2⎨⎬− ⎞⎛An−2 A− An2A⎪− 6n + 4 − n (e + e ) + ⎜⎜ e 2 − e 2 ⎟⎟ (e + e )⎪⎪⎪⎭⎝⎠BPSK ( f c )(f )= ⎩GMod2(e− A − 1)(e A − 1)[(A.10)]Moreover, since A can also be expressed as A=jB, the modulating term simplifies to⎡⎤⎡4⎛ B ⎞2 ⎛ B ⎞⎤⎢16n sin ⎜ ⎟ + (8n − 4 ) ⎢1 − 2 sin ⎜ ⎟⎥ − 6n + 4 − ⎥⎝2⎠⎝ 2 ⎠⎦⎣⎢⎥⎢⎥⎡⎤BBB⎛ ⎞⎛ ⎞⎛ ⎞⎢− 2n ⎢1 − 8 sin 2 ⎜ ⎟ cos 2 ⎜ ⎟⎥ − 8 sin 2 ⎜ ⎟ cos(nB )⎥⎝2⎠⎝2⎠⎝ 2 ⎠⎦BPSK ( f c )⎦⎥( f ) = ⎣⎢ ⎣GMod⎛B⎞16 sin 4 ⎜ ⎟⎝2⎠B=2πfnf c(A.11)resulting in the next expression:⎡⎤2⎛ B ⎞⎢8 sin ⎜ 2 ⎟ [1 − cos(nB )]⎥⎝ ⎠BPSK ( f c )⎦(f )= ⎣GMod⎛B⎞16 sin 4 ⎜ ⎟⎝2⎠B=2πfnf c⎛ πf ⎞⎛ nB ⎞sin 2 ⎜⎜ ⎟⎟sin 2 ⎜ ⎟⎝ 2 ⎠⎝ fc ⎠==⎛B⎞⎛ πf ⎞sin 2 ⎜ ⎟⎟sin 2 ⎜⎜⎝ 2 ⎠ B = 2πfnf c ⎟⎠⎝nfc(A.12)Now that we have the BCS modulating factor of BPSK(fc), it can be shown that the powerspectral density is the well known expression we saw in chapter 4.3.1:⎛ πf⎛ πf ⎞⎛ πf ⎞⎟⎟ sin 2 ⎜⎜ ⎟⎟sin 2 ⎜⎜sin 2 ⎜⎜ffnfBPSK ( nf c )BPSK ( f c )( f ) = f c ⎝ 2 c ⎠ ⎝ c ⎠ = f c ⎝ 2cGBPSK ( f c ) = GSubchippulse ( f )GMod(πf ) sin 2 ⎛⎜ πf ⎞⎟(πf )⎜ nf ⎟⎝ c⎠⎞⎟⎟⎠ (A.13)310Power Spectral Density of sine-phased BOC signalsBAppendix.
PSD of sine-phased BOC signalsAs we saw in chapter 4.3.2.1 the power spectral density of any BOC(fs , fc) in sine phasing canbe expressed using the theory on MCS signals as follows:⎛ πf ⎞⎟⎟sin 2 ⎜⎜n −1⎧nf⎡⎤⎫BPSK ( 2 f c )BOC sin ( f s , f c )ce( f )GMod,e( f ) = f c ⎝ 2 ⎠ ⎨n + 2∑ (− 1)i (n − i )cos ⎢i 2πf ⎥ ⎬GBOCsin ( f s , f c ) = Gpulse(πf ) ⎩i =1⎣ nf c ⎦ ⎭(B.1)where the superindex e indicates the even case. Moreover, the modulating factor for the evencase presents the following form:n −1⎡ n −1⎡ 2 πf ⎤ ⎤iBOC sin ( f s , f c )()()()(− 1)i (n − i ) eiA + e −iAGMod,f=n+2−1n−icosi=n+⎢∑∑e⎢⎥⎥i =1⎣ nf c ⎦ ⎦⎣ i =1()A= j2πfnf c(B.2)As we can see, the problem to calculate can be reduced into an easier one by means of thefollowing auxiliary function Φ ( A) :n −1n −1n −1Φ ( A) = ∑ (− 1) (n − i ) e = n∑ (− 1) e −∑ i (− 1) e iAiiAi =1iiAi =1i(B.3)i =1and also with the auxiliary function f ( A) , defined as follows:n −1n −1(f ( A) = ∑ (− 1) e = ∑ − eiiAi =1) = (− −e e) −+1en(− 1) e An + e A=A− e A −1Ai =1where we have made the changeA nA i(B.4)xn − xx =(B.5)∑x −1i =1The interesting property about the above defined function f ( A) is shown in the nextn −1irelationship:In fact, taking (B.4) we can see that[df ( A) n −1i= ∑ i (− 1) eiAdAi =1]((B.6)[))df ( A) − n(− 1) e An + e A e A + 1 + e A (− 1) e An + e A=2dAeA + 1Thus (B.3) can be rewritten as followsn −1n(n −1n −1n]Φ ( A) = ∑ (− 1) (n − i ) e = n∑ (− 1) e −∑ i (− 1) e iA = nf ( A) −iiAi =1i =1iiAii =1(B.7)df ( A)dA(B.8)Combining now (B.4) and (B.6) according to (B.8) we obtain the following expression:n −1Φ ( A) = ∑ (− 1) (n − i ) e iA =i =1ie A (1 − n ) − n e 2 A + (− 1)(eA+ 1)2n +1e A(n +1)(B.9)311Power Spectral Density of sine-phased BOC signalsAnd the modulating function simplifies thus to:⎡ n −1⎡BOC sin ( f s , f c )( f ) = n + 2⎢∑ (− 1)i (n − i )cos ⎢i 2πfGMod,e⎣ nf c⎣ i =1e A (1 − n ) + (− 1) e A(n +1) − n e 2 An +1=n+n+(e()A)+12+(− e An + e − An + 2 (1 − n ) − n e A + e − Ae A + e− A + 2()⎤⎤=⎥ ⎥ = { n + Φ( A) + Φ(− A)}2πf⎦⎦A= jnf ce − A (1 − n ) + (− 1) en +1 − A ( n +1)(e)−A)+12− n e− 2 A=An−⎛ An2⎞⎜e − e 2 ⎟⎜⎟− e An + e − An + 2⎝⎠=−=AA 2AA 2− ⎞− ⎞⎛ 2⎛ 2⎜e + e 2 ⎟⎜e + e 2 ⎟⎟⎟⎜⎜⎠⎠⎝⎝((B.10)2)where we have also taken into account that according to the definition of the BOC modulationin terms of a BCS vector, n is even in the even version.
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