modal decomposition using orthogonal basis

Hi,

I decomposed a real signal f(θ) into complex weights (including amplitude and phase) using an orthogonal basis given by exp(1i*m*θ), where m being an integer and denoting the order of modal decomposition. The decomposition results are given below with m = 1,2,3,4.

For the amplitude, the number of spikes is proportional to the order m. It seems that the high order one is the derivative of the low order one. Actually, I know it isn't. For the phase, the spikes becomes narrower with the order m and the sidelobe changes.

I referred to some literatures, which say that the high order decomposition corresponds to the high frequency conponent of the signal. But I wonder if there is some mathematic relation to describe such changes of the amplitude and the phase at the different orders.