# Why doesn't the magnitude of Fourier Transform change when signal is shifted (i.e when a time shift is introduced)?

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?

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.