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Let a wave is represented by an equation

$$y=f(t)=10\sin(\frac{2\pi f_1t}{T} + \pi/6)+5\cos(\frac{2\pi f_2t}{T} +\pi/3)$$.

Here, let us take $f_1=10 ,f_2=5 ,T=100$

Then, from the Fourier transform we can compute it's phase spectrum(say $p(w)$) and magnitude spectrum(say $m(w)$).

So which information do we get from the phase spectrum about the wave ?

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  • $\begingroup$ What's a phase spectrum? $\endgroup$ – Nick P May 3 '16 at 17:47
  • $\begingroup$ @NickP - It's like a sliding FFT but you calculate the complex phase of the signal for each time window at a given number of frequencies. I am not sure if that is what the OP was implying but that is what the phase spectrum is (at least from what I have learned from signal processing and time series analysis). $\endgroup$ – honeste_vivere May 3 '16 at 23:40
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I believe that the question is asking you to perform the Fourier transform on the given function and instead of plotting the resulting complex function in 3D, to convert the values from the Fourier transform into the magnitude and phase.

That is, take the complex numbers you get from the transform $\mathbf{F}(y)=a(\omega)+ib(\omega)$, convert to $A(\omega)e^{i\theta(\omega)}$, and plot $A$ as a function of $\omega$ as the magnitude spectrum and $\theta$ as a function of $\omega$ as the phase spectrum. Unfortunately, these look like they will turn out to be $\delta$-function spikes at the given $\omega$'s.

Note: $a, b, A, \theta$ are all functions of $\omega$.

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