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Power Spectral Density of the AltBOC Modulation

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FundamentalsFundamentals
Title Power Spectral Density of the AltBOC Modulation
Author(s) J.A Ávila Rodríguez, University FAF Munich, Germany.
Level Advanced
Year of Publication 2011

In the next lines the power spectral density of the AltBOC modulation will be derived. We will analyze the most general expressions that apply when data and pilot are considered, using thus four different codes. First, the expressions with no constant envelope are obtained and then the modified constant envelope version is studied and compared with that derived in [E. Rebeyrol et al., 2005][1]

It is assumed that the AltBOC signal is stationary in wide sense and that the PRN codes are ideal. A slight modification of the codes is necessary to consider that the sub-carrier is included in the chip waveform when the ratio (see Binary Offset Carrier (BOC)). Since Galileo will be transmitting AltBOC(15,10) in E5 and thus the ratio is odd, we will pay special attention to this particular 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][1]:

PSD ABOC Eq 1.png


where is the lower data code, is the lower pilot code, the upper data code and the upper pilot code. can also be expressed as follows:

PSD ABOC Eq 2.png


where

PSD ABOC Eq 3.png

and

PSD ABOC Eq 4.png


Using now the expressions above, the autocorrelation of AltBOC yields:

PSD ABOC Eq 5.png


If we assume now again that the crosscorrelation between the different codes is equal to zero and that the complex crosscorrelations cancel each other out, the power spectral density of the AltBOC modulation is then shown to be:

PSD ABOC Eq 6.png


where and are the Fourier Transforms of and . This means in other words that and are the power spectral densities of the cosine-phased and sine-phased BOC modulations that we have derived in Appendixes B and C. As we know, the expression of and depends on whether the ratio is even or odd and thus to have the general expression of the AltBOC signal we also have to distinguish between these two cases.


The normalized power spectral density of the cosine-phased BOC modulation for the ratio even is:

PSD ABOC Eq 7.png


and for odd

PSD ABOC Eq 8.png


The normalized power spectral density of the sine-phased BOC modulation for even is:

PSD ABOC Eq 9.png


and for odd

PSD ABOC Eq 10.png


Using these results, the normalized power spectral density of the AltBOC modulation is then shown to adopt the following expression for even:

PSD ABOC Eq 11.png

which can be further simplified as follows:

PSD ABOC Eq 12.png

which coincides perfectly with the expression derived in [E. Rebeyrol et al., 2005][1]. If we do the same now for the odd case, we obtain the following expression:

PSD ABOC Eq 13.png

or equivalently,

PSD ABOC Eq 14.png


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 to distinguish it from the general form, we will write the superindex c for constant envelope. The modified AltBOC signal waveform with constant envelope can be expressed as follows:

PSD ABOC Eq 15.png


where the superindex indicates constant envelope. Furthermore, is the in inverse of the sub-carrier frequency. According to this expression, can be further expanded as

PSD ABOC Eq 16.png


where

PSD ABOC Eq 17.png


and the data and pilot sub-carriers can be expressed as:

PSD ABOC Eq 18.png
PSD ABOC Eq 19.png


To facilitate now the analyses in the next lines let us rename the following signals in line with the approach followed in [E. Rebeyrol et al., 2005][1]. Thus:

PSD ABOC Eq 20.png


If we calculate now the autocorrelation function of , we can clearly recognize that most of the terms cancel out since we have assumed that the codes are ideally orthogonal with each other. Additionally, the cross-correlation between the data and pilot sub-carriers, namely and , will also be zero. In addition, the cross-correlations of each data and pilot sub-carrier correlates to zero with a delayed version of themselves by (and actually also with ). As a result, will simplify as follows:

PSD ABOC Eq 21.png


Since all the codes are of the same length and they do ideally correlate as expected from ideal random codes, the power spectral density of the modified constant envelope AltBOC modulation can be expressed as shown next:

PSD ABOC Eq 22.png


or also

PSD ABOC Eq 23.png

where , , and are the Fourier Transforms of ,, and respectively and consequently ,, and are the respective power spectral densities. We recall that the subindexes d and p indicate whether we refer to the data or pilot carrier and the super-index whether we work with the prompt or the delayed version. To make progress in our derivations it is necessary to calculate first the Fourier Transforms of the data and pilot sub-carriers. Let us begin now with the calculation of the Fourier Transform of over . To facilitate the understanding, we show again the shape of the AltBOC sub-carriers in the next figures:


Figure 1: Shapes of data and pilot sub-carriers.


We can define now the prompt data sub-carrier piecewise as follows:

PSD ABOC Eq 24.png


where

PSD ABOC Eq 25.png


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 envelope AltBOC modulation or the odd version. Indeed, we can employ here again the figure to make this distinction, where can be seen as the number of half periods within a chip.

To obtain the Fourier Transform of the prompt data sub-carrier, we derive first the Fourier Transform of the auxiliary signal . This is shown to be:

PSD ABOC Eq 26.png


which can be expanded as follows:

PSD ABOC Eq 27.png


Fortunately this complex expression can be further simplified and adopts the following form:

PSD ABOC Eq 28.png


As a result, the Fourier Transform of the prompt data sub-carrier can be expressed in terms of the equation above as:

PSD ABOC Eq 29.png


As we can immediately recognize, the sum term will be different depending on whether is even or odd. In order to have first the most general expression of the constant envelope AltBOC modulation we will not develop this expression any further until all the terms contributing to the computation of AltBOC are obtained.

Summarizing, the power spectral density of the prompt data sub-carrier can be expressed for the general case as follows:

PSD ABOC Eq 30.png


If we repeat now for the delayed data sub-carrier we have then:

PSD ABOC Eq 31.png


where in this case the function is defined as follows:

PSD ABOC Eq 32.png


so that,

PSD ABOC Eq 33.png


and therefore

PSD ABOC Eq 34.png


As we can see, the sum term of the expression above is similar to that of the prompt data sub-carrier adopting the power spectral density of the delayed data sub-carrier the following form in the general case:

PSD ABOC Eq 35.png


We repeat now for the prompt pilot sub-carrier in a similar way:

PSD ABOC Eq 36.png


with

PSD ABOC Eq 37.png


According to this, the Fourier Transform can be obtained from:

PSD ABOC Eq 38.png


yielding thus:

PSD ABOC Eq 39.png


As we can recognize, the sum term is fortunately again the same and thus the power spectral density will be:

PSD ABOC Eq 40.png


Finally, we calculate the Fourier Transform of the delayed pilot sub-carrier. This can be defined as follows:

PSD ABOC Eq 41.png


with

PSD ABOC Eq 42.png


being its Fourier Transform defined as follows:

PSD ABOC Eq 43.png


such that:

PSD ABOC Eq 44.png


Once again, the sum term is the same as that of the previous derivations and we can easily express the power spectral density for the general case as:

PSD ABOC Eq 45.png


Once we have obtained the individual elements that form the constant envelope AltBOC modulation, we can express the power spectral density for the general case as follows:

PSD ABOC Eq 46.png

or equivalently,

PSD ABOC Eq 47.png


what can be further simplified to:

PSD ABOC Eq 48.png


Until now, all the derived expressions are valid for both the even and odd case. However, as we can immediately recognize from the expression derived above, the common sum term will be different depending on whether is even or odd. In order to have a clearer view of the two cases to analyze, we present in the next figures how one could integrate the sub-carrier in the chip waveform for the two cases of .


Figure 2: Relationship between the sub-carrier frequency and the code frequency for the even AltBOC.


Figure 3: Relationship between the sub-carrier frequency and the code frequency for the odd AltBOC.


As we can see, for both cases the following relationship remains true:

PSD ABOC Eq 49.png


If we solve the sum term first for even we have:

PSD ABOC Eq 50.png


since for n even is always 1. We can further simplify the expression above if we recall the relationship between the sub-carrier and the code frequency. In fact, the square of the absolute value adopts the following form if is even:


PSD ABOC Eq 51.png


Equally, for the case of odd we have:

PSD ABOC Eq 52.png

since for n odd is always -1. Moreover, since the relationship is also valid when is odd, we can further simplify the previous expression as shown next:


PSD ABOC Eq 53.png

If we now put all the partial results of the lines above together, we can express the power spectral density of the modified even constant envelope AltBOC modulation as follows:


PSD ABOC Eq 54.png

while for the odd case we will have:


PSD ABOC Eq 55.png


which is the well-known expression of the normalized power spectral density of the constant envelope AltBOC modulation when is odd. In the next figure we show the two analyzed versions of the AltBOC modulation, namely the general expression and the constant envelope modified version with a sub-carrier frequency of 15.345 MHz and a code frequency of 10.23 MHz.

Figure 4: Power Spectral Density of the general AltBOC(15,10) and the modified constant envelope AltBOC(15,10).


Equally, for an AltBOC(10,10) – thus with even – the difference between the general AltBOC and the constant envelope solution is shown to be minimum:

Figure 5: Power Spectral Density of the general AltBOC(10,10) and the modified constant envelope AltBOC(10,10).


It is interesting to note that the power spectral densities of the general AltBOC modulation and the modified constant envelope AltBOC have similar shapes and only minor differences can be observed in the high order lobes. Indeed, the main and first side lobes are nearly identical being the differences lower than 1 dB. The only differences can be observed in the high frequency components and come from the extra terms that are needed to achieve constant envelope. Nonetheless in real implementations these would be filtered after amplification and thus for most of the bandwidths of interest we can state that there is no qualitative difference between both solutions regarding the spectrum. In terms of implementation, however, it is clear that the constant envelope solution is superior.


References

  1. ^ a b c d [E. Rebeyrol et al., 2005] E. Rebeyrol, C. Macabiau, L. Lestarquit, L. Ries, J-L. Issler, M.L. Boucheret, M. Bousquet , BOC Power Spectrum Densities, Proceedings of the National Technical Meeting of the Institute of Navigation, ION-NTM 2005, 24-26 January 2005, Long Beach, California, USA.


Credits

The information presented in this NAVIPEDIA’s article is an extract of the PhD work performed by Dr. Jose Ángel Ávila Rodríguez in the FAF University of Munich as part of his Doctoral Thesis “On Generalized Signal Waveforms for Satellite Navigation” presented in June 2008, Munich (Germany)