I have some signal 𝑠(𝑡) which is real data i.e. finite.

The time runs from −𝑇 to +𝑇. The signal amplitude is large at 𝑡=0 and small (→0) at the ±𝑇 limits.

I can do a finite (discrete) Fourier transform of the signal within the interval −𝑇/2 to + 𝑇/2. We will call this case A.

I can also do a finite FT of the signal within a smaller interval −𝑇/4 to +𝑇/4. We will call this case B.

Intuitively, I sort of expect the value of 𝑠(𝑓) - the FT of 𝑠(𝑡) - at a particular shared frequency to be greater in case A than in case B, since case A has more `contributions' to the total s(f).

However, when I code this up in Python, I can see that it is not necessarily the case. It is possible (depending on the form of 𝑠(𝑡)) for case B, the smaller time series, to have a greater value of 𝑠(𝑓).

Can someone explain why this is so? Thanks


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