On Generalized Signal Waveforms for Satellite Navigation (797942), страница 71
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We can furthersimplify this expression if we assume that all the power of the desired satellite fits into thebandwidth of the receiver, simplifying the effective Cd N 0 to:⎛ Cd ⎞⎟⎟ =⎜⎜⎝ N 0 ⎠eff 1 + CiN0CdN0∫βr / 2−βr / 2(H.2)Gi ( f ) Gd ( f ) dfWe can classify RF interference sources into narrowband interference and widebandinterference. For the case of narrowband interference, the power spectral density of theinterfering signal can be approximated as rectangular, such that:Gi ( f ) =12( f u − f l )fl ≤ f ≤ fu(H.3)Furthermore, if we assume that the whole interfering narrowband signal is in-band, then theeffective Cd N 0 will be:CdCd⎛ Cd ⎞N0N0⎜⎜⎟⎟ =(H.4)=β /2⎝ N 0 ⎠eff 1 + Ci r Gi ( f ) Gd ( f ) df 1 + Ci ⎡G ( f u + f l ) + G (− f u + f l )⎤dd⎥⎦N 0 ∫− β r / 22 N 0 ⎢⎣22In addition, since the PSD is even,Gd (fu + flf + fl) = Gd (− u)22(H.5)and assuming that the narrowband interference has a low frequency relative to the chip rate,we can use the following approximation for the particular case of BPSK(fc):⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟1⎛ f + fl ⎞⎝ fc ⎠(H.6)GBPSK ( f c ) ⎜ u≈⎟ = fc2fc(πf )⎝ 2 ⎠f=fu + fl2335Equivalent C/N0 in presence of RF interferenceFinally, as shown in [P.
Ward, 1994], the effective Cd N 0 will be for the case of anarrowband interferer:⎛ Cd ⎞⎟⎟⎜⎜⎝ N 0 ⎠effCd1N0≈=CiC /C1+ i d1+N 0 f c Cd / N 0fc(H.7)On the other hand, for the case of a wideband interferer, the expression to apply is thefollowing:Cd β r / 2Gd ( f ) df⎛ Cd ⎞N 0 ∫ −βr / 2⎟⎟ =⎜⎜(H.8)βr / 2Ci βr / 2N⎝ 0 ⎠effG(f)df+G(f)G(f)dfid∫ −βr / 2 dN 0 ∫ −βr / 2where we can recognize the spectral separation coefficient (SSC) in the denominator asdefined in chapter 5.
In fact, the lower the value the SSC adopts, the more robust will be thesignal against wideband and narrowband interferers as shown in the expressions above.Using again the example of a BPSK signal, the multiple access interference will adopt thefollowing form:I ma ( f ) = Ci ∫βr / 2−βr / 2Gi ( f ) Gd ( f ) df(H.9)where for the particular case of the intra-system interference or for the case of an interferermatching the spectrum of the desired signal, BPSK in our example, we have:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟⎝ fc ⎠(H.10)Gi ( f ) = Gd ( f ) = f c(πf )2As we can recognize, this term appears in the denominator of (H.8). Moreover, if we assume alarge processing gain, the multiple access interference will only be significant around zero aswe have shown in chapter 5 simplifying the interference to the following [J.J.
Spilker, 1997a]:2⎤⎡2 ⎛ πf ⎞⎢ sin ⎜⎜ ⎟⎟ ⎥∞⎝ f c ⎠ ⎥ df = 2 Ci(H.11)I ma (0) ≈ Ci ∫ ⎢ f c0 ⎢3 fc(πf )2 ⎥⎥⎢⎥⎦⎢⎣As we expected, if we express now 2 3 f c in dB for a chip rate of 1.023 MHz, we obtain thefamous figure of -61.8597 for the C/A Code Self SSC that we obtained in the simulations ofchapter 5.336Power Spectral Density of the AltBOC ModulationIAppendix. PSD of the AltBOCModulationIn the next lines the power spectral density of the AltBOC modulation will be derived. Wewill analyze the most general expressions that apply when data and pilot are considered, usingthus four different codes.
First the expressions with no constant envelope are obtained andthen the modified constant envelope version is studied and compared with that derived in[E. Rebeyrol et al., 2005].As we have done in previous chapters and Appendixes, we will assume that the AltBOCsignal is stationary in wide sense and that the PRN codes are ideal.
As we saw in chapter4.3.2, a slight modification of the codes is necessary to consider that the sub-carrier isincluded in the chip waveform when the ratio Φ = 2 f s f c . Since Galileo will be transmittingAltBOC(15,10) in E5 and thus the ratio Φ is odd, we will pay special attention to thisparticular case deriving the expressions of AltBOC for both the even and odd cases.The AltBOC modulation can be defined as follows [E. Rebeyrol et al., 2005]:()()sAltBOC (t ) = cLD + j cLP cs (t ) + cUD + j cUP cs (t )*(I.1)where cLD is the lower data code, cLP is the lower pilot code, cUD the upper data code and cUPthe upper pilot code. sAltBOC (t ) can also be expressed as follows:sAltBOC (t ) = Cu (t ) m(t ) + Cl (t ) n(t )(I.2)whereCu (t ) = cUD (t ) + j cUP (t )andCl (t ) = cLD (t ) + j cLP (t )m(t ) = a (t ) + j b(t ) = sign [cos( 2 π f S t )] + j sign [sin( 2 π f S t )]n(t ) = a (t ) − j b(t ) = sign [cos(2 π f S t )] − j sign [sin( 2 π f S t )](I.3)(I.4)Using now the expressions above, the autocorrelation of AltBOC yields:⎧ℜ cUD (τ ) ℜ a (τ ) + ℜ cUD (τ ) ℜ b (τ ) +⎪⎪⎪ℜ cUP (τ ) ℜ a (τ ) + ℜ cUP (τ ) ℜ b (τ ) +ℜ sAltBOC (τ ) = ⎨⎪ℜ cLD (τ ) ℜ a (τ ) + ℜ cLD (τ ) ℜ b (τ ) +⎪ℜ (τ ) ℜ (τ ) + ℜ (τ ) ℜ (τ )⎪⎩ cLPabc LP(I.5)If we assume now again that the crosscorrelation between the different codes is equal to zeroand that the complex crosscorrelations cancel each other out, the power spectral density of theAltBOC modulation is then shown to be:GAltBOC ( f ) = 4 f c A( f ) + 4 f c B( f ) = 4[G A ( f ) + GB ( f )]22(I.6)337Power Spectral Density of the AltBOC Modulationwhere A( f ) and B ( f ) are the Fourier Transforms of sign[cos(2πf s t )] and sign[sin (2πf s t )]within a chip of length Tc.
This means in other words that G A ( f ) and G B ( f ) are the powerspectral densities of the cosine-phased and sine-phased BOC modulations that we havederived in Appendixes B and C. As we know, the expression of G A ( f ) and G B ( f ) dependson whether the ratio Φ is even or odd and thus to have the general expression of the AltBOCsignal we also have to distinguish between these two cases.If we recall now the results of (C.51) and (C.52) in Appendix C, for the ratio Φ even thenormalized power spectral density of the cosine-phased BOC modulation was shown to be:f c A( f )2and for Φ oddf c A( f )2⎡⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎟ ⎥⎢ 2 sin ⎜⎜ ⎟⎟ sin ⎜⎜f4fcs⎝⎠⎝⎠⎥= GBOC cos ( f s , f c ) = f c ⎢⎢⎥⎛ πf ⎞⎟⎟πf cos⎜⎜⎢⎥⎝ 2 fs ⎠⎣⎢⎦⎥2⎡⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎥⎢ 2 cos⎜⎜ ⎟⎟ sin ⎜⎜fc ⎠4 f s ⎟⎠ ⎥⎝⎝⎢= GBOC cos ( f s , f c ) = f c⎢⎥⎛ πf ⎞⎟⎟πf cos⎜⎜⎢⎥⎢⎣⎥⎦⎝ 2 fs ⎠(I.7)2(I.8)Equally, from Appendix B we know that for Φ even the normalized power spectral density ofthe sine-phased BOC modulation is shown to be:f c B( f )2and for Φ oddf c B( f )2⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟ ⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜f2f= GBOC ( f s , f c ) = f c ⎢ ⎝ c ⎠ ⎝ s ⎠ ⎥⎢⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎢⎣⎝ 2 f s ⎠ ⎥⎦2⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟ ⎥⎢ cos⎜⎜ ⎟⎟ sin ⎜⎜f2fcs⎝⎠⎝⎠⎥= GBOC (nf c , f c ) = f c ⎢⎢⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f c ⎠ ⎦⎥⎣⎢(I.9)2(I.10)Using these results, the normalized power spectral density of the AltBOC modulation is thenshown to adopt the following expression for Φ even:⎡⎛ πf ⎞ ⎤⎢ sin ⎜⎜ ⎟⎟ ⎥Φ even⎝ fc ⎠ ⎥( f ) = 4 GAΦ even ( f ) + GBΦ even ( f ) = 4 f c ⎢⎢GAltBOC⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢[]2⎡⎛ πf ⎞⎤4 ⎛ πf ⎞⎟⎟⎥ (I.11)⎟⎟ + sin 2 ⎜⎜⎢4 sin ⎜⎜4f2f⎝ s ⎠⎦⎝ s⎠⎣which can be further simplified as follows:338Power Spectral Density of the AltBOC Modulation2⎡⎛ πf ⎞ ⎤⎢ sin ⎜⎜ ⎟⎟ ⎥Φ even⎝ f c ⎠ ⎥ ⎧1 − cos⎛⎜ πf ⎞⎟⎫( f ) = 8 f c ⎢⎢GAltBOC⎨⎜ 2 f ⎟⎬⎛ πf ⎞ ⎥ ⎩⎝ s ⎠⎭⎟⎟ ⎥⎢ πf cos⎜⎜⎢⎣⎝ 2 f s ⎠ ⎥⎦(I.12)which coincides perfectly with the expression derived in [E.
Rebeyrol et al., 2005]. If we dothe same now for the odd case, we obtain the following expression:⎡⎛ πf ⎞ ⎤⎢ cos⎜⎜ ⎟⎟ ⎥Φ odd⎝ fc ⎠ ⎥( f ) = 4 GAΦ odd ( f ) + GBΦ odd ( f ) = 4 f c ⎢⎢GAltBOC⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜2f⎢⎣⎝ s ⎠ ⎥⎦or equivalently,[]⎡⎛ πf ⎞ ⎤⎢ cos⎜⎜ ⎟⎟ ⎥Φ odd⎝ fc ⎠ ⎥( f ) = 8 f c ⎢⎢GAltBOC⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢2⎡⎛ πf ⎞⎤4 ⎛ πf ⎞⎟⎟⎥ (I.13)⎟⎟ + sin 2 ⎜⎜⎢4 sin ⎜⎜4f2fss⎠⎦⎝⎠⎝⎣2⎧⎛ πf ⎞⎫⎟⎟⎬⎨1 − cos⎜⎜2f⎝ s ⎠⎭⎩(I.14)Once we have derived the general AltBOC expressions for the case of non-constant envelope,we concentrate now on the modified AltBOC modulation with constant envelope.
In order todistinguish it from the general form, we will write the superindex c for constant envelope. Aswe have shown in chapter 4.8.1 the modified AltBOC signal waveform with constantenvelope can be expressed as follows:⎧ D⎛ Ts ⎞⎤P ⎡⎪ cL + j cL ⎢ scd (t ) − j scd ⎜ t − ⎟⎥ +4 ⎠⎦⎝⎣⎪⎪ D⎡⎛ T ⎞⎤⎪ cU + j cUP ⎢ scd (t ) + j scd ⎜ t − s ⎟⎥ +4 ⎠⎦⎝⎪⎣csAltBOC(t ) = ⎨(I.15)⎪ c D + j c P ⎡ sc (t ) − j sc ⎛ t − Ts ⎞⎤ +⎟L ⎢pp⎜⎪ L4 ⎠⎥⎦⎝⎣⎪⎪ D⎛ Ts ⎞⎤P ⎡⎪ cU + j cU ⎢ sc p (t ) + j sc p ⎜ t − 4 ⎟⎥⎠⎦⎝⎣⎩()()()()where the superindex indicates constant envelope.
Furthermore, Ts is the in inverse of thesub-carrier frequency. According to this expression, sAltBOC (t ) can be further expanded as⎧ D⎪cL scd (t ) −⎪⎪ D⎪cU scd (t ) +⎪csAltBOC (t ) = ⎨⎪c D sc (t ) −⎪ L p⎪⎪c D sc (t ) +⎪⎩ U p⎛ T ⎞j cLD scd ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cUD scd ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cLD sc p ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cUD sc p ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cLP scd (t ) + cLP scd ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cUP scd (t ) − cUP scd ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cLP sc p (t ) + cLP sc p ⎜ t − s ⎟ +4⎠⎝⎛ T ⎞j cUP sc p (t ) − cUP sc p ⎜ t − s ⎟4⎠⎝(I.16)339Power Spectral Density of the AltBOC ModulationwherecLD = cUD cUP cLPcLP = cUD cUP cLDcUD = cLD cUP cLPcUP = cUD cLD cLP(I.17)and the data and pilot sub-carriers can be expressed as:⎧ 2⎡ ⎛⎡ ⎛2π ⎞⎤ 1π ⎞⎤sign ⎢cos⎜ 2πf S t − ⎟⎥ + sign[cos(2πf S t )] +sign ⎢cos⎜ 2πf S t + ⎟⎥sc d (t ) = ⎨4 ⎠⎦ 244 ⎠⎦⎣ ⎝⎣ ⎝⎩ 4(I.18)⎧⎡ ⎛⎡ ⎛22π ⎞⎤ 1π ⎞⎤sign ⎢cos⎜ 2πf S t − ⎟⎥ + sign[cos(2πf S t )] −sign ⎢cos⎜ 2πf S t + ⎟⎥sc p (t ) = ⎨−4 ⎠⎦ 244 ⎠⎦⎣ ⎝⎣ ⎝⎩ 4(I.19)To facilitate now the analyses in the next lines let us rename the following signals in line withthe approach followed in [E.
Rebeyrol et al., 2005]. Thus:scd* (t ) = scd (t )⎛ T ⎞scd** (t ) = scd ⎜ t − s ⎟4⎠⎝*sc p (t ) = sc p (t )(I.20)⎛ Ts ⎞sc*p* (t ) = sc p ⎜ t − ⎟4⎠⎝If we calculate now the autocorrelation function of ℜcAltBOC (τ ) , we can clearly recognize thatmost of the terms cancel out since we have assumed that the codes are ideally orthogonal witheach other. Additionally, the cross-correlation between the data and pilot sub-carriers, namelyscd and scp, will also be zero.
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