Normalised factorisation uncertainty: nu-gap metric and its geometric interpretation (simplified, real, SISO) -1-

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 10:50

*Enlace a Spanish version

Summary:

This video discusses the geometric interpretation of the normalised coprime factor uncertainty, dealt with with matlab commands ncfmargin, gapmetric, ncfsyn.

Specifically, we consider a simplified SISO case in which we have two numbers $p_1$ and $p_2$, real ones.

Each of them, say $p$, can be represented by an infinite number of pairs $(d,n)\in {ℝ}^2$ such that $n∕d=p$, so $p$ can be mapped to a straight line because $(d,n)\equiv (qd,qn)$ for any $q\ne 0$. Then, $p$ is the slope of such line. The normalised representation is the intersection of the line with the unit radius circumference.

In this learning object, it is shown that the $\nu$-gap (minimum coprime factor uncertainty) that transforms $p_1\equiv (d_1,n_1)$ onto $p_2\equiv (d_2,n_2)$ is the sine of the angle diference between the lines associated to each of the $p_i$.

Further details on numerical computations of the $\nu$-gap and an alternate geometric interpretation appear on the video [gapm1bEN], a continuation of the present video.

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