On Generalized Signal Waveforms for Satellite Navigation (797942), страница 24
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It must be noted that the variable n refers to the number of subchipsand not to the number of times that the sub-carrier contains the code rate as usually done inthe literature. Once we have obtained the modulating term of the power spectral density, thegeneral form of the PSD for any sine-phased BOC modulation can be expressed as:⎛ πf ⎞⎟⎟sin 2 ⎜⎜n −1nfBPSK ( nf c )BOC ( nf c 2 , f c )⎝ c ⎠ ⎧n + 2 (− 1)i (n − i )cos⎛⎜ i 2πf ⎞⎟⎫()()==GGfGff⎨∑cpulseModnf2⎛⎞⎜ nf ⎟⎬BOC sin ⎜⎜ f s = c , f c ⎟⎟()πfi =1c ⎠⎭⎝⎩2⎝⎠(4.46)which can be explicitly simplified as shown in Appendix B:92GNSS Signal StructureGnf⎛⎞BOCsin ⎜ f s = c , f c ⎟2⎝⎠⎛ πfsin 2 ⎜⎜⎝ fc= fc(πf )2⎞⎟⎟⎠ tan 2 ⎛⎜ πf ⎞⎟⎜ nf ⎟⎝ c⎠(4.47)As a conclusion, the BOC signal in sine phase can be considered as a BCS signal whosesequence is formed by concatenating [+1,-1] a number of times f s f c .
Thus the length n willbe 2 f s f c and (4.47) can also be expressed as:⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎞⎟⎟ ⎥⎟⎟⎢ sin ⎜⎜ ⎟⎟ sin⎜⎜f2f⎛⎞fπcs⎠⎥⎠ tan 2 ⎜⎢ ⎝ ⎠ ⎝⎜ 2 f ⎟⎟ = f c ⎢⎥⎞⎛πf⎝ s⎠⎟⎟ ⎥⎢ πf cos⎜⎜⎢⎣⎝ 2 f s ⎠ ⎥⎦which is the well known form we find in the literature [J.W. Betz, 2001].⎛ πfsin ⎜⎜⎝ fcGBOCsin ( f s , f c ) = f c(πf )2224.3.2.2(4.48)Binary Offset Carrier with cosine phasing: BOCcos(fs , fc)Following the same approach of the previous chapter, we will derive next the well knownexpression for the power spectral density of the BOC modulation with sub-carrier in cosinephasing.
Taking as an example the sine-phased BOC signal of the lines above, we will derivenow also a general expression by induction over n. Let us begin with BOCcos(fc, fc):BOCcos(fs, fc)= BOCcos(fc, fc) or BCS([1,-1,-1,1],fc) with fc=1.023 MHz(4.49)The corresponding definition matrix under these assumptions is shown to be:⎛ s1s1{0} s1s2 {1} s1s3 {2}⎜s2 s2 {0} s2 s3 {1}⎜M 4 ( [1,−1,−1,1] ) = ⎜s3 s3 {0}⎜⎜⎝s1s4 {3} ⎞ ⎛1{0} − 1{1} − 1{2} 1{3} ⎞⎟ ⎜⎟s2 s4 {2}⎟ ⎜1{0} 1{1} − 1{2}⎟=s3 s4 {1} ⎟ ⎜1{0} − 1{1} ⎟⎟ ⎜⎟s4 s4 {0}⎟⎠ ⎜⎝1{0} ⎟⎠(4.50)Thus, the modulating function adopts the following form:⎡⎛BOCcos ( f c , f c )( f ) = 4 + 2 ⎢− cos⎜⎜ 2πfGMod⎝ 4 fc⎣⎞⎛ 2πf⎟⎟ − 2 cos⎜⎜ 2⎠⎝ 4 fc⎞⎛ 2πf⎟⎟ + cos⎜⎜ 3⎠⎝ 4 fc⎞⎤⎟⎟⎥⎠⎦(4.51)If we repeat now the calculation for BOCcos(2fc, fc) in order to derive the generalizedexpression by induction over n, we have BOCcos(fs, fc) = BOCcos(2fc, fc) orBCS([1,-1,-1,1,1,-1,-1,1] , fc), so that⎡⎛ 2πf ⎞⎛ 2πf ⎞⎛ 2πf ⎞⎟⎟ − 6 cos⎜⎜ 2⎟⎟ + cos⎜⎜ 3⎟⎟ +⎢− cos⎜⎜f8f8f8ccc⎝⎠⎝⎠⎝⎠BOCcos ( 2 f c , f c )( f ) = 8 + 2 ⎢⎢GMod⎛⎞⎛⎞⎛⎞⎛⎢+ 4 cos⎜ 4 2πf ⎟ − cos⎜ 5 2πf ⎟ − 2 cos⎜ 6 2πf ⎟ + cos⎜ 7 2πf⎜⎟⎜⎟⎜⎟⎜ 8f⎢⎣c⎝ 8 fc ⎠⎝ 8 fc ⎠⎝ 8 fc ⎠⎝⎤⎥⎥⎞⎥⎟⎟⎥⎠⎥⎦(4.52)If we generalize now, we can see that the expression for any n will adopt the following form:93GNSS Signal StructureBOCcos ( nf cGMod4, f c )⎧n/2⎡⎤⎞⎫⎛( f ) = n + 2 ⎨∑ (− 1) i cos⎢(2 i − 1) 2πf ⎥ + ∑ 2(− 1) i (n / 2 − i ) cos ⎜⎜ 2i 2πf ⎟⎟⎬nf c ⎠⎭(4.53)where n ∈ { 4, 8,12,16...}.
Finally, once we have obtained the modulating term of the power⎣⎩ i =1nf c ⎦n / 2 −1⎝i =1spectral density for any n, we can express the power spectral density of any cosine-phaseBOC signal as follows:⎤⎫⎡ n/2⎡2πf ⎤⎛ πf ⎞ ⎧i+⎟⎟ ⎪sin 2 ⎜⎜⎥⎪⎢∑ (− 1) cos ⎢(2i − 1)⎥nf c ⎦nf c ⎠ ⎪i =1⎣⎥⎪⎢⎝(4.54)G= fc⎬⎨n + 2 ⎢ n / 2−1nf c2⎛⎞BOCcos ⎜ f s =, fc ⎟(πf ) ⎪⎛ 2πf ⎞⎥ ⎪i4⎝⎠⎥⎢+ ∑ 2(− 1) (n / 2 − i ) cos⎜ 2i⎜ nf ⎟⎟⎥ ⎪⎪⎢1i=c ⎠⎦ ⎭⎝⎣⎩As derived in Appendix C, after some math (4.54) can still be explicitly expressed for theeven case as follows:⎛ πf ⎞⎛ πf ⎞⎟⎟sin 2 ⎜⎜ ⎟⎟ sin 4 ⎜⎜fnfcc⎠⎝⎝ ⎠(4.55)G= 4 fcnf⎛⎞BOCcos ⎜ f s = c , f c ⎟⎞⎛f2π224⎝⎠(πf ) cos ⎜⎜ ⎟⎟⎝ nf c ⎠The BOC signal in cosine phase can be considered as a BCS signal whose sequence is formedby concatenating [1,-1,-1,1] a number of times f s f c . Thus the length n will be 4 f s f c and(4.55) simplifies to:⎡⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎛ πf ⎞⎛ πf ⎞⎟⎟ ⎥⎟⎟sin ⎜⎜ ⎟⎟ sin 4 ⎜⎜⎢ 2 sin ⎜⎜ ⎟⎟ sin ⎜⎜fc ⎠4 fs ⎠f4fcs⎠⎥⎝⎠⎝⎝⎝GBOCcos ( f s , f c ) = 4 f c= fc ⎢⎢⎥⎛ πf ⎞⎞⎛⎟⎟(πf )2 cos 2 ⎜⎜ πf ⎟⎟πf cos⎜⎜⎢⎥⎢⎣⎝ 2 fs ⎠⎝ 2 fs ⎠⎦⎥22(4.56)In the same manner, for the odd case cosine-phase we have according to Appendix C:⎞ 4 ⎛ πf⎟⎟ sin ⎜⎜⎝ 4 fs⎠⎞⎛(πf )2 cos 2 ⎜⎜ πf ⎟⎟⎝ 2 fs ⎠⎛ πfcos ⎜⎜⎝ fc= 4 fc2GBOCcos ( f s , f c )4.3.2.3⎡⎛ πf ⎞ 2 ⎛ πf⎞⎟⎟⎢ 2 cos⎜⎜ ⎟⎟ sin ⎜⎜⎝ 4 fs⎝ fc ⎠⎠= f ⎢c⎢⎛ πf ⎞⎟⎟⎢πf cos⎜⎜2f⎢⎣⎝ s⎠⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(4.57)Autocorrelation function of a generic BOC signalOne of the most interesting figures in the analysis of the signal structure is the autocorrelationfunction of the chip waveform as we saw at the beginning of this chapter.
In this chapter wewill derive general expressions to define the analytical shape of the autocorrelation function ageneric BOC signal with infinite bandwidth. This will help us understand the importance ofhaving a good autocorrelation function in order to have good ranging potential forpositioning. Additionally, analytical expressions will permit us establishing comparisonsbetween sine- and cosine-phased BOC modulations and investigate the effect that extra termsin the definition of the ACF can bring.94GNSS Signal StructureBefore that, we derive first the inverse Fourier Transform of some functions of interest.
Theimportance of these functions lies in the fact that since the Power Spectral Density of anyMCS signal can be developed as a series with them, the derivation of analytical expressionsfor the ACF will be then possible no matter how complex the shape of the signal is.As we know, the inverse Fourier transform of Tk (ω ) = cos(kω f c ) ω 2 can be defined asfollows:⎧τ⎪ 2⎪⎪ kTk (τ ) = ⎨−⎪ 2⎪− τ⎪⎩ 2τ < −k−k ≤τ ≤ k(4.58)τ >kwhere k fixes the height of the function at τ = 0 . Moreover, the function Tk (τ ) is expressed asa function of τ [chips].
Next the trapezoid function is shown graphically:Figure 4.11. Definition of the trapezoid function Tk(τ )Also, the Fourier inverse Transform of S k (ω ) =function that we will call S k (τ ) as follows:⎧ τ2⎪S k (τ ) = ⎨ 2kτ⎪− 2⎩ 2k1(kω )2can be expressed by means of aτ <0τ >0(4.59)Figure 4.12. Definition of the S k(τ ) function95GNSS Signal StructureFor the more general case, we will define the inverse Fourier Transform of⎛ ω⎞ ⎛ ω⎞cos⎜⎜ k1 ⎟⎟ cos⎜⎜ k2 ⎟⎟⎝ fc ⎠ ⎝ fc ⎠M k1 , k 2 (ω ) =2ω(4.60)as:k⎧− 2τ ≤ (k2 − k1 )⎪2⎪⎪ (τ + k + k )12(k2 − k1 ) ≤ τ ≤ (k2 + k1 )M k1 ,k2 (τ ) = ⎨−(4.61)4⎪τ⎪−τ ≥ (k2 + k1 )⎪⎩2where we assume that k 2 ≥ k1 without loss of generality. In addition, we can clearly see thatTk (τ ) is a particular case of M k1 ,k2 (τ ) since M (0, k 2 ) (τ ) = Tk 2 (τ ) .Next we compare the BOC signal in sine and cosine phasing for different chip rates.
Forexemplification we will take a sub-carrier rate of 10.23 MHz and a code rate of 5.115 MHz.We recall that these modulations correspond to the GPS M-Code and the Galileo PRS (E6).Figure 4.13. Power Spectral Densities of BOCsin(10,5) and BOCcos(10,5)The difference between the sine-phased and cosine-phased BOC modulation is even moreobvious when we look at the BOCsin(15,2.5) and BOCcos(15,2.5) signals.
As we can see, whilethe sine-phased concentrates more power at inner frequencies, so does the cosine version atouter frequencies.96GNSS Signal StructureFigure 4.14. Power Spectral Densities of BOCsin(15,2.5) and BOCcos(15,2.5)Now that we have the tools to derive the generic form of the ACF of any MCS signal, wedevelop the power spectral density of BOCsin(15,2.5) as follows:⎡⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎥⎢ 2 sin ⎜⎜ ⎟⎟ sin ⎜⎜4 f s ⎟⎠ ⎥fc ⎠⎝⎝⎢GBOCsin ( f s =15, f c = 2.5 ) = f c⎢⎥⎛ πf ⎞⎟⎟πf cos⎜⎜⎢⎥⎝ 2 fs ⎠⎣⎢⎦⎥22f =ω / 2π=⎡⎛ ω ⎞ 2 ⎛ ωπ ⎞ ⎤⎟ sin ⎜⎜⎟⎥⎢ 4 sin ⎜⎜2 f c ⎟⎠8 f s ⎟⎠ ⎥⎝⎝⎢fc=⎢⎥⎛ ω ⎞⎟⎟ω cos⎜⎜⎢⎥⎝ 4 fs ⎠⎣⎢⎦⎥⎡⎛1 ω⎞⎛5 ω ⎞⎛3 ω ⎞⎛2 ω ⎞⎛1 ω ⎞ ⎤⎟⎟ + 16 cos⎜⎜⎟⎟ − 24 cos⎜⎜⎟⎟ + 32 cos⎜⎜⎟⎟ − 72 cos⎜⎜⎟⎟ + ⎥⎢− 88 cos⎜⎜⎝ 12 f c ⎠⎝ 6 fc ⎠⎝ 4 fc ⎠⎝ 3 fc ⎠⎝ 4 fc ⎠ ⎥⎢⎢⎛1 ω ⎞⎛1 ω ⎞⎛1 ω ⎞⎛7 ω⎞⎛5 ω⎞ ⎥fc ⎢⎥⎜⎟⎜⎟⎜⎟⎜⎟= 2 + 80 cos⎜⎟ + 64 cos⎜ 3 f ⎟ + 48 cos⎜ 2 f ⎟ − 40 cos⎜ 12 f ⎟ − 56 cos⎜⎜ 12 f ⎟⎟ − ⎥6ω ⎢fc ⎠c ⎠c ⎠⎝ c⎠⎝ c⎠⎝⎝⎝⎢⎥⎛ 11 ω ⎞⎛ω ⎞⎢⎥⎢− 8 cos⎜⎜ 12 f ⎟⎟ + 2 cos⎜⎜ f ⎟⎟ + 46⎥c ⎠⎝⎝ c⎠⎣⎦(4.62)And using the formulations derived in previous pages, we can express the ACF as follows:⎡⎤⎢− 88 T 1 (τ ) + 16 T5 (τ ) − 24 T3 (τ ) + 32 T2 (τ ) − 72 T1 (τ ) + ⎥126434⎢⎥⎢ACFBOCsin (15, 2.5) (τ ) = + 80 T1 (τ ) + 64 T1 (τ ) + 48 T1 (τ ) − 40 T 7 (τ ) − 56 T 5 (τ ) − ⎥⎢⎥6321212⎢⎥⎢− 8 T11 (τ ) + 2 T1 (τ ) + 46 S1 (τ )⎥12⎣⎦(4.63)which adopts graphically the following form:97GNSS Signal StructureFigure 4.15.
Autocorrelation Function of BOCsin(15,2.5)In the same manner, the power spectral density of BOCcos(15,2.5) can be equally expressed interms of the functions defined above such that the ACF is shown to adopt the following form:⎡⎤⎢− 88 T 1 (τ ) + 16 T5 (τ ) − 24 T3 (τ ) + 32 T2 (τ ) − 72 T1 (τ ) +⎥126434⎢⎥⎢+ 80 T1 (τ ) + 64 T1 (τ ) + 48 T1 (τ ) − 40 T 7 (τ ) − 56 T 5 (τ ) −⎥⎢⎥6321212⎢⎥ACFBOCcos (15, 2.5) (τ ) = ⎢− 8 T11 (τ ) − 2 T1 (τ ) + 50 S1 (τ ) − 8 T 1 (τ ) + 8 T23 (τ ) + 80 T1 (τ ) + ⎥1224246⎢⎥⎢+ 8 T1 (τ ) − 8 T 5 (τ ) − 8 T13 (τ ) + 8 T5 (τ ) + 8 T11 (τ ) + 8 T 7 (τ ) − ⎥⎢⎥8242482424⎢⎥⎢− 8 T17 (τ ) + 8 T19 (τ ) − 8 T3 (τ ) − 8 T7 (τ )⎥242488⎣⎦(4.64)Figure 4.16. Autocorrelation Function of BOCcos(15,2.5)If we compare now (4.63) and (4.64) we see that we can express the ACF of thecosine-phase BOC as a function of the ACF of the sine-phased BOC in the following way:98GNSS Signal StructureACFBOCcos (15, 2.5) (τ ) = ACFBOCsin (15, 2.5) (τ ) + DBOC(15, 2.5) (τ )(4.65)where⎡− 4 T1 (τ ) + 4 S1 (τ ) − 8 T 1 (τ ) + 8 T23 (τ ) + 80 T1 (τ ) + 8 T1 (τ ) − 8 T 5 (τ ) − 8 T13 (τ ) + ⎤⎢⎥2424682424DBOC(15, 2.5) (τ ) = ⎢⎥+ 8 T5 (τ ) + 8 T11 (τ ) + 8 T 7 (τ ) − 8 T17 (τ ) + 8 T19 (τ ) − 8 T3 (τ ) − 8 T7 (τ )⎢⎣⎥⎦82424242488what can be graphically shown as follows:(4.66)Figure 4.17.
Difference DBOC(15,2.5)(τ) of ACF of BOCcos(15,2.5) and BOCsin(15,2.5)By looking at Figure 4.17 the following interesting properties can be observed:•We can distinguish 6 peaks on every side with an amplitude of 1 (2 ⋅ 6 ) . In fact, for thegeneral case we will have n peaks on every side with an amplitude 1 (2 ⋅ n ) wheren = f s fc .•This function shows the interesting property that we can easily convert the sine-phasedautocorrelation function of any BOC signal into its cosine-phased counterpart byadding the corresponding difference function shown above.Moreover, if we calculate now the Fourier transform of (4.66), we obtain the followingspectrum:Figure 4.18.
Power Spectral Density of the Difference Function DBOC(15,2.5)(τ )As we can recognize, this spectrum is the difference between the power spectral densities ofthe sine-phased and cosine-phased BOC(15,2.5) modulations.99GNSS Signal Structure4.3.2.4BOC signals vs. BPSK signalsThe BOC modulation was the first attempt to modernize the GNSS signals and has indeedopened a new field of research in navigation that has recently lead to the AltBOC and MBOCsolutions. These will be described in the following pages. As commented by[J.-A. Avila-Rodriguez et al., 2006d] and [G.W.
Hein et al., 2006a], while very goodperformance can be obtained with the C/A code signal, it has been recognized that betterperformance can be obtained using spreading modulations that provide more power at highfrequencies away from the centre frequency. In fact, this is the main idea behind the BOCmodulation where a sub-carrier signal shifts spectral components to outer parts of the. The oldBPSK modulation that is currently still used for GPS C/A code has limited capability forranging and requires high performance receivers to use very wide front-end bandwidths asshown in [J.
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