Recently, I came across a source which stated the model of a GPS L1 wave was:
My question is, how was this equation derived?
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That equation isn't derived, it describes the contents of a GPS signal that is created by the electronics in the transmitter in the GPS satellites.
The electronics create quadrature signals (I and Q) so that the available spectrum can be used more efficiently. Using a quadrature signal positive and negative frequencies (relative to the RF carrier) can be used separately.
So it is not that the GPS engineers have build a system and then derived the equation from that. It is the opposite, it was decided that the GPS system should use quadrature signals and that means this equation is what needs to be implemented in the system.