signal processing - Can anybody give proof about features of phase and magnitude spectrum?

I have read that from Fourier transform we obtain magnitude and phase spectrum. The magnitude spectrum tells you how strong are the harmonics in a image and the phase spectrum tells where this harmonic lies in space.

But when I plotted the phase and magnitude spectrum of say an image, the spectrum were very weird and specifically the phase spectrum is very hard to understand. So I feel Extracting information from the phase and magnitude spectrum of a signal is very difficult .

So let us take any signal of your choice (as I am not concerned with any perticular signal) and plot it's phase and magnitude spectrum.

Now,can u extract information from Magnitude spectrum and Phase spectrum of the signal and justify sentences given in the 1st paragraph of question?

Can u give proof about these sentences?

Answer

The first image shows the addition of two harmonics (blue, orange), both are sine waves, the two add together to create the resultant red waveform.In the frequency spectrum for the resultant red waveform, there would be only two vertical lines at different frequencies.

The second image shows the addition of 5 harmonics, and the resultant (combined) waveform is again the red line, and you can see on this chart, the red line is closer to a square wave, it's flatter on top and the bottom, and the steep edge is steeper.

So you can see that the more sinewaves you add, so long as they are the correct multiples in frequency of the fundamental and correct amplitudes relative to each other, then the better the approximation to the squarewave.

Any repeating waveform can be constructed from sinewaves of different frequencies, amplitudes and phases.

The third image shows the spectrums and the time waveforms for two periodic waveforms. The first is a a square wave and above it is its spectrum showing the harmonics, what multiples in frequency they are of the fundamental and the relative amplitude of each.

The second waveform and spectrum isn't a standard waveform (square, sawtooth, ramp..), I don't know what it is, it might even be a section of waveform for a musical instrument. But it illustates that a periodic (repeating) waveform can be constructed from sinewaves (harmonics).