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Power Spectral Density of Sine-phased BOC signals

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FundamentalsFundamentals
Title Power Spectral Density of Sine-phased BOC signals
Author(s) J.A Ávila Rodríguez, University FAF Munich, Germany.
Level Advanced
Year of Publication 2011

The power spectral density of any BOC in sine phasing can be expressed using the theory on Multilevel Coded Spreading Symbols (MCS) signals as follows:

PSD BOC Eq 1.png

(B.1) where the superindex e indicates the even case. Moreover, the modulating factor for the even case presents the following form:

PSD BOC Eq 2.png


(B.2) As we can see, the problem to calculate can be reduced into an easier one by means of the following auxiliary function :

PSD BOC Eq 3.png


and also with the auxiliary function , defined as follows:

PSD BOC Eq 4.png


where we have made the change

PSD BOC Eq 5.png

The interesting property about the above defined function is shown in the next relationship:

PSD BOC Eq 6.png


In fact, taking (4)we can see that

PSD BOC Eq 7.png


Thus (3) can be rewritten as follows

PSD BOC Eq 8.png


Combining now (4) and (6) according to (8) we obtain the following expression:

PSD BOC Eq 9.png


And the modulating function simplifies thus to:

PSD BOC Eq 10.png


where we have also taken into account that according to the definition of the BOC modulation in terms of a BCS vector, n is even in the even version. Additionally, since A can also be expressed as A=jB the expression above simplifies to

PSD BOC Eq 11.png


Once a simplified form has been derived for the BCS modulating factor of BOC(fs, fc), we substitute in (1) yielding the well-known expression for the Power Spectral Density (see Binary Offset Carrier (BOC)):

PSD BOC Eq 12.png


Since , the equation (12) can also be expressed as follows:

PSD BOC Eq 13.png

This is the well known expression that we find everywhere in the literature. Now that we have solved the case of the even BOC modulation in sine phasing, we calculate next its odd counterpart. For the case of the odd BOC modulation in sine phasing, we have to derive first a general expression for any odd n. We will proceed by generalizing over n.

For n = 3, can also be expressed as BCS([+1,-1,+1], fc), such that the generation matrix will adopt the following form:

PSD BOC Eq 14.png


Thus, the odd modulating term yields this time:

PSD BOC Eq 15.png

where o indicates the odd case, and

PSD BOC Eq 16.png


In the same manner, for , what can also be defined as in the general form BCS([+1,-1,+1,-1,+1], fc) with generation matrix given by:

PSD BOC Eq 17.png


Thus, for the case of , we will have:

PSD BOC Eq 18.png


If we continue by induction we can find the expression for any odd n:

PSD BOC Eq 19.png

As we can recognize, (19) is equal to (2) except that n is odd now with Moreover, it can be shown that for the odd is still valid. For simplicity, we express the modulating factor above using its exponential equivalent expression:

PSD BOC Eq 20.png

Using now the expressions derived above for the sum term , it can be shown that:

PSD BOC Eq 21.png

Again, this expression is similar to that obtained for the even case, but with a slight difference. Indeed, since n is odd, the second summand in the numerator has a changed sign with respect to (9). The modulating function is thus shown to present the following form:

PSD BOC Eq 22.png


Additionally, since A can also be expressed as A=jB, (22) simplifies to

PSD BOC Eq 23.png


Once we have the BCS modulating factor of an arbitrary odd it can be shown that the power spectral density is (see Binary Offset Carrier (BOC)):

PSD BOC Eq 24.png


which coincides perfectly with the expressions found in the literature [J. W. Betz, 1999][1], [A.R. Pratt and J.I.R. Owen, 2003][2] and [E. Rebeyrol et al., 2005][3].

Furthermore, since , the previous expression can also be shown as follows:

PSD BOC Eq 25.png


References

  1. ^ [J. W. Betz, 1999] J. W. Betz, The offset carrier modulation for GPS modernization, in Proceedings of the National Technical Meeting of the Institute of Navigation, ION-NTM 1999, pp. 639–648, January 1999, San Diego, California, USA.
  2. ^ [A.R. Pratt and J.I.R. Owen, 2003] Anthony R. Pratt & John I.R. Owen, BOC Modulations Waveform, Proceedings of the International Technical Meeting of the Institute of Navigation, ION-GNSS 2003, 9-12 September 2003, Portland, Oregon, USA.
  3. ^ [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)