# Which information do we get from the phase spectrum about the wave?

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 ?

• What's a phase spectrum? – Nick P May 3 '16 at 17:47
• @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). – honeste_vivere May 3 '16 at 23:40

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 $\omega$.