If you wish to contribute or participate in the discussions about articles you are invited to join Navipedia as a registered user

Power Spectral Density of Cosine-phased BOC signals

From Navipedia
Jump to: navigation, search


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

The Power Spectral Density of the even cosine-phased BOC is:

PSD COSBOC Eq 1.png

or equivalently,

PSD COSBOC Eq 2.png

where . In order to use the results obtained in the previous Appendixes, we will expand the modulation term in the brackets using the Euler´s formula:

PSD COSBOC Eq 3.png
PSD COSBOC Eq 4.png

According to this, can be expressed as follows:

PSD COSBOC Eq 5.png

or equivalently,

PSD COSBOC Eq 6.png

what can also be expressed as:

PSD COSBOC Eq 7.png

Decomposing the different terms of the sum, we have:

PSD COSBOC Eq 8.png


where,

PSD COSBOC Eq 9.png


As we can observe, remaining this identity true also for the other two summands and . Furthermore, if we look in detail at (9), we can see that it can be simplified again using the methodology of previous Appendixes. Indeed,

PSD COSBOC Eq 10.png


what can also be expressed as follows:

PSD COSBOC Eq 11.png


It must be noted, that according to the definition of the BOC modulation in cosine phasing as a BCS signal, and the term can be further simplified since will always be odd. In the same manner:

PSD COSBOC Eq 12.png


and consequently,

PSD COSBOC Eq 13.png


Furthermore, is shown to simplify to:

PSD COSBOC Eq 14.png


or equivalently,

PSD COSBOC Eq 15.png

For the third sum term, namely we have to solve first the following intermediate problem:

PSD COSBOC Eq 16.png

To do so, we define the following auxiliary function:

PSD COSBOC Eq 17.png

Since n/2 is even, we can further simplify the expression above as follows:

PSD COSBOC Eq 18.png


being the derivative of the function as shown next:

PSD COSBOC Eq 19.png


In an analogue way, substituting A by -A in (19) we can see that

PSD COSBOC Eq 20.png


and therefore,

PSD COSBOC Eq 21.png

what can be further simplified according to:

PSD COSBOC Eq 22.png


Now that we have calculated all the sum terms , and , we can have a simplified expression for the modulating term of the power spectral density of the cosine-phased BOC modulation. In fact,

PSD COSBOC Eq 23.png


If we further develop it, we obtain:

PSD COSBOC Eq 24.png


or equivalently,

PSD COSBOC Eq 25.png

Finally, since A=jB, we can simplify this expression as follows:

PSD COSBOC Eq 26.png

so that the power spectral density of is shown to be:

PSD COSBOC Eq 27.png


Finally since , it is trivial to see that the expression of the Power Spectral Density of an arbitrary cosine-phased BOC reduces in the even case to:

PSD COSBOC Eq 28.png

Once we have obtained the expression for the even BOC modulation in cosine phasing, we calculate its odd counterpart next.

For the case of the odd BOC modulation in cosine phasing, we have to derive a general expression for any n. As done in previous chapters, we will generalize over n. As in (2), the general expression for the odd case will be:

PSD COSBOC Eq 29.png

We begin with , where can also be expressed as BCS([+1,-1,-1,+1,+1,-1], ), being the generation matrix as follows:

PSD COSBOC Eq 30.png


In this case, the modulating term will adopt the following form,

PSD COSBOC Eq 31.png


while

PSD COSBOC Eq 32.png

In the same manner, for , what can also be defined in the general form as BCS([+1,-1,-1,+1,+1,-1,-1,+1,+1,-1], ). Thus

PSD COSBOC Eq 33.png


if we continue by induction we can see that the expression for any n adopts the form:

PSD COSBOC Eq 34.png


with and . Again, the modulating factor can be expressed as:

PSD COSBOC Eq 35.png


with

PSD COSBOC Eq 36.png


which is indeed the same expression we obtained in (7). However, since n is now twice an odd number, the results will vary slightly. Indeed it can be shown that:

PSD COSBOC Eq 37.png


since and will always be even. Thus we can simplify (37) as follows:

PSD COSBOC Eq 38.png

Therefore:

PSD COSBOC Eq 39.png


We can proceed in a similar way with and . To do so, we will use the already derived expressions for the even case and take into account that this time . According to this,

PSD COSBOC Eq 40.png


which can be further simplified to:


PSD COSBOC Eq 41.png

Similarly, to calculate now we will make use of the function defined above. Nevertheless, since now n/2 is always odd, the expression simplifies as follows:

PSD COSBOC Eq 42.png

and thus, for the odd case is shown to be:

PSD COSBOC Eq 43.png

Once we have calculated all the sum terms, it is time to obtain the expression for the modulating term of the power spectral density of the odd cosine-phased BOC modulation:

PSD COSBOC Eq 44.png


or equivalently:

PSD COSBOC Eq 45.png


The modulation term can also be expressed as follows:

PSD COSBOC Eq 46.png


or

PSD COSBOC Eq 47.png

In addition, since A=jB, we can simplify this expression as follows:

PSD COSBOC Eq 48.png


Thus the power spectral density of is shown to be in the odd case:

PSD COSBOC Eq 49.png


and since , we can also express it as follows:

PSD COSBOC Eq 50.png


As a conclusion, the normalized power spectral density of the cosine-phased BOC modulation is shown to be for n even:

PSD COSBOC Eq 51.png


and for n odd,

PSD COSBOC Eq 52.png

Now that we have derived the expressions of the power spectral density of the sine and cosine-phased BOC modulations, it is interesting to note the following relationship:

PSD COSBOC Eq 53.png


that allows us to go from the sine-phased expression to the other one. We show in the next figure the sine-phased BOC modulation together with its cosine-phased counterpart and the inverse tangent term of the expression above that relates both. For simplicity a sub-carrier frequency fs of 1.023 MHz and a carrier frequency fc of 1.023 MHz were assumed.


Figure 1: Power Spectral Density of Sine-phased, Cosine-phased and Inverse Tangent Function of BOC(10,5).


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)