When estimating frequency, the uncertainty principle creates a well known limit on the bounds of what can be done.

What’s not so apparent is how far frequency resolution can be pushed, without violating the uncertainty principle.

Window performance table

Window performance plot in dB

I’m hoping to pursue this topic to work up to how to perform the operation necessary to achieve this frequency resolution. This is going to include:

  • An outline of the criteria for valid spectral analysis. (Foreshadow: the Wigner Distribution Function violates these criteria.)

  • A discussion outlining the criteria of a useful Energy Spectral Density function. This discussion will specifically focus on the conservation of energy, and how to achieve it.

  • A proof of showing that one spectral analysis method outperforms all others when it comes to achieving the minimum uncertainty.

  • A discussion developing how individual channels in this transform can be visualized

  • A description of how the FFT can be used to speed up the transform

  • A description of how the transform can be inverted

Reference(s): [1] [2] [3] [4] [5] [6]