I understand that discontinuities at the boundaries of a finite signal may introduce spectral leakage, that alter both the amplitude and phase spectra of the FFT, specially at low amplitudes. That is the explanation to several questions like (baffled by fft phase spectrum!"). In such questions, the examples are normally based on sine waves, where there is a dominant frequency, and the spectral leakage is noticeable in all the other frequencies.
Signals are often more complex, with a more continuous amplitude spectrum, but even in that case - specially if they are low pass-filtered, there is a point where there is a noticeable discontinuity in the amplitude and phase diagrams.
Example (in Octave):
N = 101; x = exp(-linspace(-5,5,N).^2); f = fftshift(fft(x)); subplot(211) semilogy(abs(f)) subplot(212) plot(unwrap(angle(f)))
Here the amplitude spectrum is bell-shaped as I expected at low frequencies. But when the amplitude goes below approximately
1e-11, the amplitude spectrum flattens.
At the frequencies where the amplitude spectrum is bell-shaped, the phase spectrum is a straight line with slope $\pi (N-1)/N$, which coincides with the effect of a time shift of $(N-1)/2$ from an even signal. At higher frequencies, it goes more or less linearly towards $\pi/2$ (although in this case the slope is not strictly constant).
Why there is such a harsh change in the spectra at that frequency? Is it a matter of numeric error in the computation of the FFT? The amplitude at the discontinuity is small (about
1e-11), but greater than machine error. And for other signals with narrower spectra, the amplitude at the discontinuity is higher. For instance, the same example but using a "wider" time signal:
x = exp(-0.5*linspace(-5,5*(N-1)/N).^2);