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Sinusoidal Binary Offset Carrier (SOC) Signals

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Title Sinusoidal Binary Offset Carrier (SOC) Signals
Author(s) J.A Ávila Rodríguez, University FAF Munich, Germany.
Level Advanced
Year of Publication 2011

To derive the spectrum of the SOC signals, the most convenient is to use the convolution theorem. According to it, the Fourier transform of the chip waveform can be expressed in terms of a convolution between the modulating carrier and the code bit. The problem reduces then to calculating the Fourier transforms for each signal as shown in [J. Winkel, 2002] [1]. In fact:

SOC Eq 1.png

which can be further simplified as follows, assuming that the code is ideal:

SOC Eq 2.png

Therefore, the power spectral density adopts the following form:

SOC Eq 3.png

It is interesting to note also that the same distinction between even and odd SOCs can also be made here as with the rectangular signals that we have already studied. Furthermore, the maximum of the spectrum is not located at [math]\displaystyle{ f = n f_c }[/math] as one might expect, but somewhere close to that point as shown in [J. Winkel, 2002] [1]. Finally, the autocorrelation function of the SOC signal for the sine-phased case is shown to be [J. Winkel, 2002] [1]:

SOC Eq 4.png

where [math]\displaystyle{ \Lambda \left ( \tau \right ) }[/math] is the triangular function and [math]\displaystyle{ \Theta \left ( \tau \right ) }[/math] represents the Heaviside step function. As we know the triangular function is defined as follows:

SOC Eq 5.png

and the Heaviside step function is equally shown to be defined as:

SOC Eq 6.png


  1. ^ a b c [J. Winkel, 2002] J. Winkel, Modeling and Simulating Generic GNSS Signal Structures and Receiver in a Multipath Environment, Ph.D. Thesis, University FAF Munich, Neubiberg, Germany.


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)