I created a discrete signal sampled at 44.1kHz that is 330Hz for 1 second, 360Hz for 0.5s, and 390Hz for 0.25s. I did an FFT transformation of this, and got the magnitude graph below:
I don't understand how this magnitude information (combined with the phase information not shown) can reconstitute the original signal. I know it does -- I'm not asking for a proof. I'm hoping someone can supply the intuition for how that strange jagged FFT image ends up reconstructing the original signal.