On Generalized Signal Waveforms for Satellite Navigation (797942), страница 25

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W. Betz and D. B. Goldstein, 2002]. Moreover, Intra-system interference isexacerbated by the short C/A codes. The relatively slow 1.023 MHz spreading code rate of theBPSK-R modulation offers limited channel capacity, restricting the number of simultaneoussignals as well as the tolerable power differential between signals. Additionally, the datamessage modulated on the C/A code signal is inefficient and inflexible.An interesting aspect of the BOC signal regarding its complexity is that it can be consideredas two BPSKs shifted to –fs and +fs by the sub-carrier signal.

Indeed many receiverimplementations will make use of this principle to receive the future BOC signals. Side-lobeprocessing is thus a promising solution to treat BOC signals using the old BPSK architectureif we realize that a BOC signal is qualitatively similar to two BPSK signals with each half thepower [J.W. Betz et al., 2005]. This idea is also of interest to process AltBOC.4.3.3Generic BCS SignalsIn the previous lines we have studied two particular cases of the BCS modulation, namely theBPSK and BOC modulations. Nonetheless, the general expression derived at the beginning ofthe chapter is valid for any BCS vector.

For exemplification, we show in the next lines howthe PSD of a generic BCS could be derived.Let us assume a BCS signal with vector s = [+1, +1, -1]. We have thus s1 = +1 , s2 = +1 ,s 3 = −1 . The modulating term of the PSD can be easily calculated as:⎞⎛⎞⎛⎞⎛BCS([+1, +1, −1], f c )( f ) = 3 + 2 s1s2 cos⎜⎜ 2πf ⎟⎟ + 2s1s3 cos⎜⎜ 2 2πf ⎟⎟ + 2s2 s3 cos⎜⎜ 2πf ⎟⎟GMod⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 fc ⎠which can be further simplified to:⎛⎞⎛⎞⎛⎞⎛BCS([+1, +1, −1], f c )( f ) = 3 + 2 cos⎜⎜ 2πf ⎟⎟ − 2 cos⎜⎜ 2 2πf ⎟⎟ − 2 cos⎜⎜ 2πf ⎟⎟ = 3 − 2 cos⎜⎜ 2 2πfGMod⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 fcwhile(4.67)⎞⎟⎟⎠(4.68)100GNSS Signal Structure⎛ πf ⎞⎟sin 2 ⎜⎜3 f c ⎟⎠BPSK (3 f c )⎝( f ) = fcGpulse(πf )2(4.69)Thus the spectrum of this BCS sequence would adopt the following form:⎛ πf ⎞⎟sin 2 ⎜⎜3 f c ⎟⎠ ⎡⎛ 2πf⎝GBCS([+1, +1, −1], f c ) ( f ) = f c⎢3 − 2 cos⎜⎜ 22(πf ) ⎣⎝ 3 fc⎞⎤⎟⎟⎥⎠⎦(4.70)In general, in order to understand how the spectrum will look like for a given sequence, wehave to be able to understand how every term of the sum above contributes to the total PSD.4.4Sinusoidal Multilevel Coded Symbol (SMCS)SignalsIn the previous chapter we have examined signal waveforms with rectangular pulse shape.This is indeed the most typical case in most of the applications.

Now we will go one stepfurther and we will discuss a family of signals that results from modulating each subchip ofthe generation vector s with a sinusoidal function. Such signals receive the name ofSinusoidal Offset Carrier signals or SOC for short if the sinusoidal function is modulated by abinary code. The alternative use of Linear Offset Carrier or LOC is also offen observed in theliterature. As we can recognize, SMCS can be interpreted as a particular MCS that usessubchip pulses with sinusoidal shape. Accordingly, (4.28) could be applied. The original ideato use this signal for satellite navigation was presented in [J. W.

Betz, 1999] and has beenfurther developed in [J. Winkel, 2002]. The SOC signal can be defined as follows:sSOC (t ) = 2 ck sin (2 π n f c t )(4.71)where n corresponds to the number of periods of the sine wave that are contained in each codebit and the factor 2 was introduced to normalize the power to 1. Furthermore, c k refers tothe subchips modulating the chip waveform. It is important to note that the chip waveform isdefined by the sequence of subchips that forms it according to the generation vector s asdefined in chapter 4.2.While this definition applies only to the case of Offset Carrier Chips ([1, -1, 1, -1, …]), onecan imagine a generalized version for Binary Coded Symbols.

We will define these signalsthus in general as Sinusoidal Binary Coded Symbols or SBCS for short. Next figure shows anexample of SBCS with vector [1,-1,1].101GNSS Signal StructureFigure 4.19. Sinusoidal Binary Coded Symbol signal with generation vector [+1,-1,+1]As we can clearly recognize, there exists the same relationship between SBCS signals andSOCs as there was between BCS and BOC. In fact, the SOC signal that we defined above is aparticular case of the SBCS modulation that we have just described. According to this, if wetalk about SBCS([1,-1],1) and SOC(1,1) we are indeed referring to the same signal.In addition, it is important to mention that since, as we know from theory, the square-wavecontains tones at odd frequencies multiple of the elemental frequency, the SOC signal can beinterpreted qualitatively as a BOC signal that is filtered to have only the first tone.To summarize, we can conclude that this idea can be understood as a particular case ofMultilevel Coded Symbols (MCS) modulation with a pulse waveform of sinusoidal form.Next figure shows another example.Figure 4.20.

Sinusoidal Multilevel Coded Symbol signal with generation vector [1,-2,2].In this example the amplitude was not normalized to have 1 W of powerFurthermore, it is important to note that unlike in the most straightforward definition of theSOC modulation, the factor accompanying the sine signal will not be in general 2 and willdepend on the particular symbol sequence. In fact, the factor has the mission to normalize thepower of the signal to unity.102GNSS Signal Structure4.4.1Sinusoidal Binary Offset Carrier (SOC) SignalsTo derive the spectrum of the SOC signals, the most convenient is to use the convolutiontheorem. According to it, the Fourier transform of the chip waveform can be expressed interms of a convolution between the modulating carrier and the code bit. The problem reducesthen to calculating the Fourier transforms for each signal as shown in [J.

Winkel, 2002]. Infact:SSOC ( f ) = FT { sSOC (t ) } = FT 2 sin (2 π n f c t ) ⊗ FT {ck (t )}(4.72){}which can be further simplified as follows, assuming that the code is ideal:SSOC ( f ) = j⎞⎤⎛ πf⎞⎛ πf2⎡⎢sinc⎜⎜ + nπ ⎟⎟ + sinc⎜⎜ − nπ ⎟⎟⎥fc ⎣⎠⎦⎝ fc⎠⎝ fc(4.73)Therefore, the power spectral density adopts the following form:2GSOC ( f ) =fc⎡⎞⎤⎛ πf⎞⎛ πf⎢sinc⎜⎜ + nπ ⎟⎟ + sinc⎜⎜ − nπ ⎟⎟⎥⎠⎦⎝ fc⎠⎝ fc⎣2(4.74)It is interesting to note also that the same distinction between even and odd SOCs can also bemade here as with the rectangular signals that we have already studied.

Furthermore, themaximum of the spectrum is not located at f = nf c as one might expect, but somewhere closeto that point as shown in [J. Winkel, 2002]. Finally, the autocorrelation function of the SOCsignal for the sine-phased case is shown to be [J. Winkel, 2002]:ℜSOC (τ ) =⎫⎛ nπ ⎞ sin (2nπτf )8 ⎧⎟⎟ −[θ (τ − Tc ) + θ (τ + Tc ) − 2θ (τ )]⎬⎨Λ(2τ f c ) cos⎜⎜π fc ⎩2nπ⎝ 2 fc ⎠⎭2(4.75)where Λ (τ ) is the triangular function and θ (τ ) represents the Heaviside step function. As weknow the triangular function is defined as follows:⎧1 − τΛ (τ ) = ⎨⎩ 0τ <1τ >1(4.76)and the Heaviside step function is equally shown to be defined as:⎧0 τ ≤ 0⎩1 τ > 0θ (τ ) = ⎨(4.77)103GNSS Signal Structure4.4.2Minimum Shift Keying (MSK)The MSK modulation is a constant envelope signal with continuous phase that results frommodulating the instantaneous frequency with rectangular pulses.

MSK is considered to be aspecial case of Offset QPSK (OQPSK) with half sinusoidal pulse weighting rather thanrectangular. Furthermore, MSK presents lower side lobes than QPSK and OQPSK as shownin [S. A. Gronemeyer and A. L. McBride, 1976] and [H. R. Mathwich et al., 1974]. As onecan imagine, this could be of great interest for those navigation bands where the Out of Bandemission constraints are stringent as in the case of the C-band between 5010 and 5030 MHz.Assuming that the PRN codes are ideal and making the same assumptions of previouschapters, the MSK modulation can be seen as a particular case of SMCS with sinusoidal pulsewaveform.

Moreover, since MSK is a particular case of MCS, all the expressions derived inprevious chapters can also be used for this particular case.In the MSK modulation the evolution of the phase over the time is linear. Indeed, recalling thegeneral expression of (4.2) and keeping in mind that MSK is a frequency modulation, it canbe shown that the evolution of the frequency over time adopts the following expression:f (t ) = ∑ ck (t ) p(t − kTc )(4.78)kwhere ck (t ) is the PRN code and p(t ) is the frequency pulse, defined for period n as follows:⎧ 1⎪p (t ) = ⎨ 2Tc⎪⎩ 0nTc ≤ t ≤ (n + 1)Tc(4.79)elsewhereAccordingly, the variation of the phase over the time will adopt the following form:tφ (t ) = 2πh ∫ f (τ )dτ(4.80)−∞with h = 1 2 .

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