Generalized spiral points

[narahahn]

The so-called generalized spiral points are generated according to Rakhmanov 1991.

• The azimuth and elevation angles are given in closed form
• The computation requires the number of microphones. This causes the same inconsistency problem we had before

[trettberg]

Currently we use only quadrature grids: The grid determines ISHT matrix Y. The SHT matrix is then given by Y^H D, where diagonal matrix D contains the quadrature weights.

This generalized spiral is not a quadrature grid. The SHT matrix must be computed via Y^+.

This makes me wonder:

1. Should the toolbox contain non-quadrature grids? Since Y^+ can be computed almost surely, these grids should be "particularly good" in some sense.

2. Is the generalized spiral a "particular good" grid?

(IMO merging should wait if this is not clear yet.)

[narahahn]

Good questions. Unfortunately, I don't have a satisfying answer. The advantages I see so far are just that it has a closed form, and it can be computed for any given number. Clearly, the usage of the generalized spiral grid as a microphone distribution needs to be further studied. Quoting Saff and Kuijlaars,

... one can introduce a parameter C in place of 3.6 and adjust its value appropriately for the application at hand

so I will test the grid for different values of C in the context of sound field analysis. I'll come back when I find out something new.