Advanced DSP - FFT windows: frequency resolution

An interesting measure of a window function's quality is the minimum separation needed between two frequency components of equal amplitude, that still allows them to be resolved. By 'resolved', we mean that there is a local minimum between the two peaks:

The usual 'rule of thumb' for being able to resolve two adjacent signal peaks is the width of the peak at the half power point (the '3 dB bandwidth'). This is because two frequency components of equal amplitude will show only a single peak if separated by less than their 3 dB bandwidth, and so cannot be resolved. But the calculation on which this 'rule of thumb' is based assumes that the signals add incoherently. This is an important problem with the Fourier Transform because the addition which is involved in the Fourier Transform is coherent, not incoherent.

The Fourier Transform is the addition of frequency components, weighted by the window function's kernel at each frequency. Because of the coherence, the 6 dB bandwidth determines resolution rather than the 3 dB bandwidth.