I know that we can get the phasor representation(frequency domain) of a signal through Fourier transform, but the picture above gives a method that can get the phasor representation of a signal in a different way, in this method according to the writer, it numerically "demodulate" the signal to create a phasor representation of the signal, however, i can't understand that, anyone can help me? Thanks in advance!
The method described above is not different from a Fourier transform.
Fourier transform of a signal computes the integral of the signal multiplied by e-i.omega.t.
Remember that ei.x = cos(x) + i.sin(x)
By multiplying by sine and cosines, taking the average, and combining the results into a complex number, you are doing mathematically a very similar calculation, and practically the same operation.
Demodulation and fourier transform are very closely related. But in order to talk about demodulation, you need to define what the modulation is.
If the signal uses amplitude modulation, then the signal is demodulated by taking the length of the obtained phasor. If the signal uses phase modulation, you demodulate by taking the argument of the phasor. Frequency demodulation is a somewhat more complex task.