Features of phase and magnitude spectrum?

Question

I have read in many books that whether the signal is 1D or multidimensional ,

1. The magnitude spectrum tells you how strong are the harmonics in the signal

   and

2. The phase spectrum tells where this harmonic lies in time domain for 1 D signal (and in space domain in case of multidimensional)

But I didn't find any justification or explanation for the above sentences. I want to counter check (understand ) these sentences about phase spectrum and magnitude spectrum. So can anybody help for it ?

Answer 1

The phase tells you nothing about localization. Every sine and cosine is global. What phase tells us is a spatial offset to each wave. You will need a "short time" or "windowed" fourier transform to achieve temporal or spatial locality. Or you can use another transform like a Wavelet transform which gives a tradeoff between "frequency" and spatial/temporal information.

Answer 2

Ok, another attempt at explaining the basics of FFT phase and translation / time-shift.

For any (additive) part of the signal $f(x)$ which has fourier transform $$F\{f(x)\}(\omega)$$ is shifted to $f(x+x_0)$ will have Fourier transform : $$e^{-i x_0\omega}F\{f(x)\}(\omega)$$ So every coefficient $\omega$ has it's phase altered by the linear factor inside that exponential function.