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2

Under the assumption that your words "extract the amplitude of those four components" actually mean "extract the spectral magnitudes of those four components", then the initial phases of the original four periodic signals do not matter. It's straightforward to prove this behavior by modeling your down-conversion process using software.


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A non-linear phase response is equivalent to group delay variation, and similar to multi-path fading can cause inter-symbol interference if it is severe enough, but either can be corrected with channel equalization. In this case it would be a static correction so would not be overly complicated to implement in the transmitter, receiver or both. Factoring the ...


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Do an fftshift (Rotate the input vector by n/2) before the FFT to place the peak of the Gaussian at index 0.


2

The group delay will be flat assuming the filter is linear phase (which means the phase is linear vs frequency). Group delay is the derivative of phase with respect to frequency. Either unwrap your phase and detrend it or use the group delay function that is part of scipy.signal to evaluate directly. Any FIR filter (not just the Gaussian) will be linear ...


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You've got a system of equations with 2 unknowns ($A$ and the phase offsets). You hence need two equations. That means two samples.


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This is a real signal, so the phase can only be $0$ to $\pi$.


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My attempt to get the phase, see figure below:


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