I understand that a time shift in the time domain produces a corresponding phase change / phase shift in the frequency domain. But I don't understand why the magnitude is unchanged (I am referring to the time shift property here). I am thinking of the following example: Consider An Aperiodic signal, now we observe this signal over the time interval delta t. (and it is my understanding that the longer this interval is , the more accurate the peak of the obtained Fourier transform). But since the signal is aperiodic, this means by definition that it does not repeat itself. So , if we introduce a time shift , then within that time interval delta t that we are observing , a different part of the signal will appear. So how come the magnitude of the Fourier transform is still the same?
So how come the magnitude of the Fourier transform is still the same?
Because the Fourier Transform integrates from $-\infty$ to $+\infty$. There is no finite "observation window" and you always integrate over the entire signal. The time shift doesn't change that.
Once you consider a finite interval, you are windowing the signal and that does indeed change both magnitude and phase.