Editing Short-time Fourier transform (section)

=== Explanation ===
It can also be explained with reference to the sampling and [[Nyquist frequency]].

Take a window of ''N''  samples from an arbitrary real-valued signal at sampling rate ''f''<sub>s</sub> .  Taking the Fourier transform produces ''N'' complex coefficients.  Of these coefficients only half are useful (the last ''N/2'' being the [[complex conjugate]] of the first ''N/2'' in reverse order, as this is a real valued signal).

These ''N/2'' coefficients represent the frequencies 0 to ''f''<sub>s</sub>/2 (Nyquist) and two consecutive coefficients are spaced apart by
''f''<sub>s</sub>/''N'' Hz.

To increase the frequency resolution of the window the frequency spacing of the coefficients needs to be reduced.  There are only two variables, but decreasing ''f''<sub>s</sub> (and keeping ''N'' constant) will cause the window size to increase — since there are now fewer samples per unit time.  The other alternative is to increase ''N'', but this again causes the window size to increase.  So any attempt to increase the frequency resolution causes a larger window size and therefore a reduction in time resolution—and vice versa.