On Generalized Signal Waveforms for Satellite Navigation (797942), страница 72

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In addition, the cross-correlations of each data and pilotsub-carrier correlates to zero with a delayed version of themselves by Ts / 4 (and actually alsowith 3Ts / 4). As a result, ℜcAltBOC (τ ) will simplify as follows:⎧ℜ cLD scd* (τ ) + ℜ cLD scd** (τ ) + ℜ cLP scd* (τ ) + ℜ cLP scd** (τ ) +⎪⎪⎪ℜ cUD scd* (τ ) + ℜ cUD scd** (τ ) + ℜ cUP scd* (τ ) + ℜ cUP scd** (τ ) +cℜ AltBOC (τ ) = ⎨⎪ℜ cLD sc*p (τ ) + ℜ cLD sc**p (τ ) + ℜ cLP sc*p (τ ) + ℜ cLP sc**p (τ ) +⎪ℜ⎪⎩ cUD sc*p (τ ) + ℜ cUD sc**p (τ ) + ℜ cUP sc*p (τ ) + ℜ cUP sc**p (τ )(I.21)Since all the codes are of the same length and they do ideally correlate as expected from idealrandom codes, the power spectral density of the modified constant envelope AltBOCmodulation can be expressed as shown next:222cG AltBOC( f ) = 4 f c SC d* ( f ) + 4 f c SC d** ( f ) + 4 f c SC *p ( f ) + 4 f c SC *p* ( f )2(I.22)or also[cG AltBOC( f ) = 4 G d* ( f ) + G d** ( f ) + G *p ( f ) + G *p* ( f )](I.23)340Power Spectral Density of the AltBOC Modulationwhere SCd* ( f ) , SCd** ( f ) , SC *p ( f ) and SC *p* ( f ) are the Fourier Transforms of scd* , scd** , sc*pand sc*p* respectively and consequently Gd* ( f ) , Gd** ( f ) , G *p ( f ) and G *p* ( f ) are the respectivepower spectral densities.

We recall that the subindexes d and p indicate whether we refer tothe data or pilot carrier and the super-index whether we work with the prompt or the delayedversion. To make progress in our derivations it is necessary to calculate first the FourierTransforms of the data and pilot sub-carriers. Let us begin now with the calculation of theFourier Transform of scd* over [0, Tc]. To facilitate the understanding, we show again theshape of the AltBOC sub-carriers in the next figures:Figure I.1. Shapes of data and pilot sub-carriersWe can define now the prompt data sub-carrier piecewise as follows:wheren −1T ⎞⎛scd* (t ) = ∑ (−1) m μTs / 2 ⎜ t − m s ⎟2⎠⎝m =0(I.24)⎧⎪⎪⎪⎪⎪μTs / 2 (t ) = ⎨⎪⎪⎪⎪−⎪⎩(I.25)2 +1212122 +12⎡ Ts ⎤⎢0, 8 ⎥⎣⎦⎡ Ts Ts ⎤⎢8 , 4⎥⎣⎦⎡ Ts 3Ts ⎤⎢4 , 8 ⎥⎣⎦TT3⎡ s s⎤⎢ 8 , 2⎥⎣⎦It is important to remember that depending on whether the number of times that m fits in n(see previous Figure) is even or odd we will have the even version of the constant envelopeAltBOC modulation or the odd version.

Indeed, we can employ here again the figureΦ = 2 f s f c to make this distinction, where Φ can be seen as the number of half periodswithin a chip.341Power Spectral Density of the AltBOC ModulationTo obtain the Fourier Transform of the prompt data sub-carrier, we derive first the FourierTransform of the auxiliary signal μTs / 2 . This is shown to be:⎧T⎛FT ⎨μ Ts / 2 ⎜ t − m s2⎝⎩⎞⎫⎟⎬ =⎠⎭( m +1)∫Tm s2Ts2⎛μT / 2 ⎜ t − ms⎝Ts ⎞ − 2 jπftdt⎟e2⎠(I.26)which can be expanded as follows:T T⎧m T2s + T8sm s+ s2 4211 − 2 jπft+⎪− 2 jπftededt −t+∫⎪ ∫2TsTs Ts 2m +⎧t ⎞⎫ ⎪⎪ m 2⎛2 8FT ⎨μ Ts / 2 ⎜ t − m s ⎟⎬ = ⎨ T 3TTs 3Tsss2 ⎠⎭ ⎪ m +m +⎝⎩2 82 812 + 1 − 2 jπftjft2−π⎪−edt − ∫edt∫⎪ Ts Ts 22Ts Tsm +⎪⎩ m 2 + 42 4(I.27)Fortunately this complex expression can be further simplified and adopts the following form:− 2 jπfmTs2T− jπf s⎧t s ⎞⎫e⎛2FT ⎨μ Ts / 2 ⎜ t − m ⎟⎬ = −ejf22π⎝⎠⎩⎭⎡⎢−⎣()T ⎞T ⎞ ⎤⎛⎛2 + 1 cos⎜ π f s ⎟ + 2 cos⎜ π f s ⎟ + 1⎥2⎠4⎠ ⎦⎝⎝(I.28)As a result, the Fourier Transform of the prompt data sub-carrier can be expressed in terms ofthe equation above as:− jπfSC d* ( f ) = −Ts2⎡e−2 jπf ⎢⎣()T⎛2 + 1 cos⎜ π f s2⎝T⎛⎞⎟ + 2 cos⎜ π f s4⎝⎠Ts⎞ ⎤ n −1m − 2 jπfm 2()+1−1e(I.29)⎟ ⎥∑⎠ ⎦ m =0As we can immediately recognize, the sum term will be different depending on whether Φ iseven or odd.

In order to have first the most general expression of the constant envelopeAltBOC modulation we will not develop this expression any further until all the termscontributing to the computation of AltBOC are obtained.Summarizing, the power spectral density of the prompt data sub-carrier can be expressed forthe general case as follows:⎡G (f)=−2 2 ⎢4π f ⎣*dfc()⎛ T ⎞⎛ T ⎞ ⎤2 + 1 cos⎜ πf s ⎟ + 2 cos⎜ πf s ⎟ + 1⎥⎝ 2⎠⎝ 4⎠ ⎦2n −1∑ (− 1)me− 2 jπfmTs22(I.30)m =0It is interesting to note that a similar formulation is obtained using the MCS definitions ofchapter 4.2.1. If we repeat now for the delayed data sub-carrier sc d** (t ) we have then:where in this case the μTs / 2n −1T ⎞⎛sc d** (t ) = ∑ (−1) m μ Ts / 2 ⎜ t − m s ⎟2⎠⎝m =0function is defined as follows:(I.31)342Power Spectral Density of the AltBOC Modulation⎧ 1⎪ 2⎪⎪ 2 +1μ Ts / 2 (t ) = ⎨⎪ 2⎪ 1⎪ 2⎩⎡ Ts ⎞⎢0, 8 ⎟⎣⎠⎛ Ts 3Ts ⎞⎜ ,⎟⎝8 8 ⎠⎛ 3Ts Ts ⎤, ⎥⎜⎝ 8 2⎦(I.32)so that,⎧T ⎞⎫⎛FT ⎨μ Ts / 2 ⎜ t − m s ⎟⎬ =2 ⎠⎭⎝⎩( m +1)∫mTs2− 2 jπfmμtTs2s/2Ts2T ⎞e⎛e⎜ t − m s ⎟ e − 2 jπft dt = −2⎠2 jπf⎝T− jπf s2and therefore− jπf⎡⎛ Ts ⎞ ⎤⎢− j sin ⎜ πf 2 ⎟ ⎥⎝⎠ ⎥⎢⎢⎛ Ts ⎞⎥⎢− j 2 sin ⎜ πf ⎟⎥⎝ 4 ⎠⎦⎣(I.33)Ts2Ts⎡e⎛ Ts ⎞⎛ Ts ⎞⎤ n −1m − 2 jπfm 2()− j sin ⎜ πf ⎟ − j 2 sin ⎜ πf ⎟⎥ ∑ − 1 eSC ( f ) = −2 jπf ⎢⎣⎝ 2⎠⎝ 4 ⎠⎦ m =0**d(I.34)As we can see, the sum term of the expression above is similar to that of the prompt datasub-carrier adopting the power spectral density of the delayed data sub-carrier the followingform in the general case:⎡ ⎛ Ts ⎞⎛ T ⎞⎤sin⎜ πf ⎟ + 2 sin⎜ πf s ⎟⎥G (f)=2 2 ⎢4π f ⎣ ⎝ 2 ⎠⎝ 4 ⎠⎦**dfc2n −1∑ (− 1)me− 2 jπfmTs22(I.35))m =0We repeat now for the prompt pilot sub-carrier sc *p (t ) in a similar way:n −1T ⎞⎛sc *p (t ) = ∑ (−1) m μ Ts / 2 ⎜ t − m s ⎟2⎠⎝m =0(I.36)⎧⎪−⎪⎪⎪⎪μ Ts / 2 (t ) = ⎨⎪⎪⎪⎪⎪⎩(I.37)with2 −1212122 −12⎡ Ts ⎞⎢0, 8 ⎟⎣⎠TT⎛ s s⎞⎜ , ⎟⎝8 4⎠⎛ Ts 3Ts ⎞⎜ ,⎟⎝4 8 ⎠⎛ 3Ts Ts ⎤, ⎥⎜⎝ 8 2⎦According to this, the Fourier Transform can be obtained from:⎧T⎛FT ⎨μ Ts / 2 ⎜ t − m s2⎝⎩− 2 jπfmTsT()2− jπf s ⎡e⎞⎫⎛ T ⎞ ⎤⎛ T ⎞2e2 − 1 cos⎜ πf s ⎟ − 2 cos⎜ πf s ⎟ + 1⎥⎟⎬ = −⎢2 jπf2⎠4⎠ ⎦⎠⎭⎝⎝⎣(I.38)yielding thus:343Power Spectral Density of the AltBOC Modulation− jπfTs2()⎡e⎛ T ⎞⎛ T ⎞ ⎤2 − 1 cos⎜ πf s ⎟ − 2 cos⎜ πf s ⎟ + 1⎥SC ( f ) = −⎢2 jπf ⎣⎝ 2⎠⎝ 4⎠ ⎦*pn −1m∑ (− 1) e− 2 jπfmTs2(I.39)m =0As we can recognize, the sum term is fortunately again the same and thus the power spectraldensity will be:()⎡⎛ T ⎞⎛ T ⎞ ⎤2 − 1 cos⎜ πf s ⎟ − 2 cos⎜ πf s ⎟ + 1⎥G (f)=2 2 ⎢4π f ⎣⎝ 2⎠⎝ 4⎠ ⎦fc*p2n −1∑ (− 1)me− 2 jπfmTs22(I.40)m =0Finally, we calculate the Fourier Transform of the delayed pilot sub-carrier.

This can bedefined as follows:n −1T ⎞⎛(I.41)sc *p* (t ) = ∑ (−1) m μ Ts / 2 ⎜ t − m s ⎟2⎠⎝m =0with⎧ 1⎡ Ts ⎞⎪⎢0, 8 ⎟⎣⎠⎪ 22 − 1 ⎛ Ts 3Ts ⎞⎪(I.42)μ Ts / 2 (t ) = ⎨−⎜ ,⎟2⎝8 8 ⎠⎪⎪ 1⎛ 3Ts Ts ⎤, ⎥⎜⎪ 2⎝ 8 2⎦⎩being its Fourier Transform defined as follows:− 2 jπfm⎧T ⎞⎫e⎛FT ⎨μ Ts / 2 ⎜ t − m s ⎟⎬ = −2 ⎠⎭2 jπf⎝⎩Ts2e− jπfTs2⎡⎛ Ts ⎞⎛ Ts ⎞ ⎤⎢− j sin ⎜ πf 2 ⎟ + j 2 sin ⎜ πf 4 ⎟⎥⎝⎠⎝⎠⎦⎣(I.43)such that:− jπfTs2⎡e⎛ T ⎞⎤⎛ T ⎞SC ( f ) = −− j sin ⎜ πf s ⎟ + j 2 sin ⎜ πf s ⎟⎥⎢2 jπf ⎣⎝ 2⎠⎝ 4 ⎠⎦**pn −1∑ (− 1)me− 2 jπfmTs2(I.44)m =0Once again, the sum term is the same as that of the previous derivations and we can easilyexpress the power spectral density for the general case as:⎡⎛ πf− sin⎜⎜G (f)=2 2 ⎢4π f ⎣⎝ 2 fs**pfc⎞⎛ πf⎟⎟ + 2 sin⎜⎜⎠⎝ 4 fs⎞⎤⎟⎟⎥⎠⎦2n −1∑ (− 1)me− 2 jπfmTs22(I.45)m =0Once we have obtained the individual elements that form the constant envelope AltBOCmodulation, we can express the power spectral density for the general case as follows:[cG AltBOC( f )( f ) = 4 G d* ( f ) + G d** ( f ) + G *p ( f ) + G *p* ( f )](I.46)or equivalently,344Power Spectral Density of the AltBOC Modulation2⎧⎡⎛ πf ⎞⎛ πf ⎞ ⎤ ⎫⎪ ⎢− 2 + 1 cos⎜⎜⎟⎟ + 2 cos⎜⎜⎟⎟ + 1⎥ + ⎪⎪⎣⎝ 2 fs ⎠⎝ 4 fs ⎠ ⎦ ⎪⎪⎪2⎪⎪ ⎡ ⎛ πf ⎞⎤⎛ πf ⎞⎟⎟ + 2 sin ⎜⎜⎟⎟⎥ +⎪⎪+ ⎢sin ⎜⎜Ts⎝ 4 f s ⎠⎦fc ⎪ ⎣ ⎝ 2 f s ⎠⎪ n −1m − 2 jπfm 2cGAltBOC ( f ) = 2 2 ⎨⎬ ∑ (− 1) e2π f ⎪ ⎡⎛ πf ⎞⎛ πf ⎞ ⎤ ⎪ m = 0⎟⎟ − 2 cos⎜⎜⎟⎟ + 1⎥ + ⎪⎪+ ⎢ 2 − 1 cos⎜⎜24ffss⎝⎠⎝⎠ ⎦ ⎪⎪ ⎣⎪⎪2⎛ πf ⎞⎛ πf ⎞⎤⎪⎪ ⎡⎪⎪+ ⎢− sin ⎜⎜ 2 f ⎟⎟ + 2 sin ⎜⎜ 4 f ⎟⎟⎥⎝ s⎠⎝ s ⎠⎦⎭⎩ ⎣(())what can be further simplified to:cAltBOCG4(f)= 2 2π f Tc2⎡ 2 ⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎟ − cos⎜⎜⎟⎟ − 2 cos⎜⎜⎟⎟ cos⎜⎜⎟⎟ + 2⎥⎢cos ⎜⎜ffff2224ssss⎝⎠⎝⎠⎝⎠⎝⎠ ⎦⎣n −1∑ (− 1)me− 2 jπfm(I.47)Ts 22m=0(I.48)Until now, all the derived expressions are valid for both the even and odd case.

However, aswe can immediately recognize from the expression derived above, the common sum term willbe different depending on whether Φ is even or odd. In order to have a clearer view of the twocases to analyze, we present in the next figures how one could integrate the sub-carrier in thechip waveform for the two cases of Φ.Figure I.2. Relationship between the sub-carrier frequency and the code frequency forthe even AltBOC345Power Spectral Density of the AltBOC ModulationFigure I.3. Relationship between the sub-carrier frequency and the code frequency forthe odd AltBOCAs we can see, for both cases the following relationship remains true:n = 2 fs fc(I.49)If we solve the sum term first for Φ even we have:n −1∑ (− 1)meT− 2 jπfm s2=m =0(− 1)n−ee− jπfTs n− jπfTs−1= je−1T− j ( n −1)πf s2T ⎞⎛sin ⎜ nπf s ⎟2⎠⎝⎛ T ⎞cos⎜ πf s ⎟2⎠⎝(I.50)since (-1)n for n even is always 1.

We can further simplify the expression above if we recallthe relationship between the sub-carrier and the code frequency. In fact, the square of theabsolute value adopts the following form if Φ is even:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟Ts 2n −1− 2 jπfm⎝ fc ⎠2(I.51)(− 1)m e=∑⎞⎛πfm =0⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠Equally, for the case of Φ odd we have:Ts ⎞⎛⎟Ts cos⎜ nπfTsn − jπfTs nn −1− j ( n −1)πf2⎠(− 1) e−1m − 2 jπfm 2⎝2(I.52)=e=∑ (− 1) e− e − jπfTs − 1⎛ Ts ⎞m =0cos⎜ πf⎟2⎠⎝since (-1)n for n odd is always -1.

Moreover, since the relationship n = 2 f s f c is also validwhen Φ is odd, we can further simplify the previous expression as shown next:n −1∑ (− 1)m =0meT− 2 jπfm s22⎛ πf ⎞cos 2 ⎜⎜ ⎟⎟⎝ fc ⎠=⎛ πf ⎞⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠(I.53)346Power Spectral Density of the AltBOC ModulationIf we now put all the partial results of the lines above together, we can express the powerspectral density of the modified even constant envelope AltBOC modulation as follows:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟4fΦ even ,c⎝ f c ⎠ ⎡cos 2 ⎛⎜ πf ⎞⎟ − cos⎛⎜ πf ⎞⎟ − 2 cos⎛⎜ πf ⎞⎟ cos⎛⎜ πf ⎞⎟ + 2⎤( f ) = 2 c2GAltBOC⎢⎜2f ⎟⎜2f ⎟⎜2f ⎟ ⎜4f ⎟ ⎥π f2 ⎛ πf ⎞ ⎣s ⎠s ⎠⎝⎝⎝ s⎠ ⎝ s⎠ ⎦⎟⎟cos ⎜⎜⎝ 2 fs ⎠(I.54)while for the odd case we will have:⎛ πf ⎞cos 2 ⎜⎜ ⎟⎟4fΦ odd ,c⎝ f c ⎠ ⎡cos 2 ⎛⎜ πf ⎞⎟ − cos⎛⎜ πf ⎞⎟ − 2 cos⎛⎜ πf ⎞⎟ cos⎛⎜ πf ⎞⎟ + 2⎤( f ) = 2 c2GAltBOC⎢⎜2f ⎟⎜2f ⎟⎜2f ⎟ ⎜4f ⎟ ⎥π f2 ⎛ πf ⎞ ⎣s ⎠s ⎠⎝⎝⎝ s⎠ ⎝ s⎠ ⎦⎟⎜cos ⎜⎟⎝ 2 fs ⎠(I.55)which is the well known expression of the normalized power spectral density of the constantenvelope AltBOC modulation when Φ is odd.

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