The meaning of the angle of Fourier coefficients.

Question

I was wondering what is the meaning of the argument of fourier transform?

It is known that the amplitude space happens to be useful and sometimes people even refer to fourier transform as a transformation to amplitude space directly.

It is generally said, that the amplitude measures how much of a certain frequency is present in a signal, which is rather straightforward and intuitive.

I have not however found any explanation for the angle of fourier coefficients ( angle or argument ). It is interesting to know, what would it be?

Answer 1

Phase in a sense measures the alignment of two signals. For example, consider the two signals $u_1(t) = e^{2\pi it}$ and $u_2(t) = e^{i\delta}e^{2\pi it}$, where $\delta$ is some real number. Both signals have the same frequency and amplitude. However, depending on $\delta$, the signals might combine in different ways. Let $v = u_1+u_2$, then $v = (1+e^{i\delta})e^{2\pi i t}$.

Now notice the following:

• When $\delta=0$, $v = 2u_1$ and $|v|=2$. This is constructive interference.
• When $\delta = \pi$, $e^{i\delta} = -1$ and $v=0$. This is destructive interference.

Other values of $\delta$ are somewhere in between. This shows how the phase is supremely important in determining how two signals interact. Coincidentally, this the reason that quantum mechanics uses complex probability amplitudes: so that the probability of a combination of states can destructively interfere such as in the double slit experiment, for example.

Answer 2

The angle represents delay, or phase. The sum of sinusoids is not uniquely-defined unless you pick a phase offset between them.