Normalised coprime factorisation uncertainty: nu-gap metric and its geometric interpretation (simplified, real, SISO) -2-

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 14:16

Summary:

This video is a continuation of the video [gapm1EN], in which a number $p$ was associated with an infinitude of fractions $p=n∕d$ where $(d,n)$ laid on a straight line with slope $p$. Normalised representation was the point of the line where $d^2+n^2=1$. Also, the minimum-norm disturbance to $p_1$ so it rendered it equivalent to $p_2$ was the sine of the angle difference between the lines associated to $p_1$ and $p_2$.

In this video, we show two formulae to obtain the numerical value of the $nu$-gap, either from the normalised description, or from the original (unnormalised) one.

Also, we illustrate a second interpretation as a chordal distance on a stereographic projection, which is the original interpretation proposed in the seminal works of Vinnicombe.

Of course, the full matrix-valued complex case is developed in the cited works, but it is not considered here as the focus is simplicity and intuitive understanding. A glimpse of the underlying ideas appears in video [gapm2EN], continuation of this one.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.