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Just to add, when we talk about phase, it's always with respect to a reference "zero-degrees" sinusoid which is arbitrarily defined by you. As long as you conform to the same definition for all signals, things should work out.
In a Fourier Series (which applies to periodic signals), a number of sinusoids ( of integer multiples of the period of signal to be transformed) are summed up. Each of these sinusoides has a phase angle such that when we add up all of the individual sinusoids we recover the orignal waveform.
The Fourier Transform is a continuous version of the Fourier Series and is a generalization of the Fourier Series.
the phase signifies how the phase is changing with frequency.This can be understood if you take the example of an amplifier as a system and then if the phase plot of fourie transform of output for a sinusoidal input is plotted then it shows how the phase is changing with frequency.The phase change here is with respect to input signal to the amplifier.
Phasor is something like a vector which is rotating (changing its angle instantaneously) so the angle it substends at present with respect to a reference phasor is the phase angle.
Now The importance of Fourier Transform:
[this is my own predication if wrong pls correct me]
First v have to know what eigen vector says
e^(jwt) = coswt+jsinwt
here the eigen vector is a rotating phasor of unit magnitude now on multipling this vector with another signal it will rotate the signal by an angle wt. fourier uses this fact and build the freq domain coeff from time domain sampled values by rotating it with appropriate angle at that instant (wt) F(w) = ∫f(t)e^(jwt)dt so at each and every instant the signal is rotated to an appropriated angle if it have that w freq and on summing the coeff will become some non zero value if the freq is present in the signal and becomes zero if there is no such freq in the signal
The phase of the Fourier transform arises when you want to express the Fourier transform of a function in terms of a magnitude and phase, which are functions of frequency. Now, the meaning of the phase depends on the application you are working, mathematically it is the angle of a complex number. The phase can take values greater than 2*pi, then care should be taken when the fft is applied.
I guess somebody was referring to the Euler formula instead of "Eigen vector" in a previous post. The Fourier transform of cos(wo*t + c) = exp(i*c)*pi*Dirac(w-wo)+exp(-i*c)*pi*Dirac(w+wo), you can find it expanding cos(wo*t+c)=cos(wo*t)cos(c)-sin(wo*t)sin(c). Now, if we are talking about magnitude and phase, the magnitude introduced by exp(i*c) = 1, but it introduces a phase offset = arctg(c). It should be noted that the first term has a positive offset and the second is negative.