I have $n$ signals that are all oscillatory. I've estimated instantaneous phases using the Hilbert Transform (discarding amplitude), leaving me with a set of angles forming points on the $n$-sphere. All angles are ranged $[-\pi,\pi]$
I now need to transform these spherical points to the (n+1)-dimensional Cartesian coordinate system and found this guide.
In short: \begin{align} x_1&=r\cos(\phi_1)\\ x_2&=r\sin(\phi_1)\cos(\phi_2)\\ x_3&=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\\ \vdots \\ x_{n}&=r\sin(\phi_1)\cdots\sin(\phi_{n-1})\cos(\phi_{n})\\ x_{n+1}&=r\sin(\phi_1)\cdots\sin(\phi_{n-1})\sin(\phi_{n})\\ \end{align}
In my case, $r=1$. But in the guide it also says that $\phi_1,\phi_2,\cdots,\phi_{n-1}$ range over $[0,\pi]$ radians, whereas $\phi_n$ ranges over $[0,\pi]$. Should I in my implementation take the absolute values of the first $(n-1)$ angles before transforming to Cartesian? E.g., $\phi_{10}=-\pi/2$ should be transformed to $\phi_{10}=\pi/2$ ?
I realize there's a bunch of symmetry going on that I don't understand, but hopefully one of you can help me with the implementation :-)
EDIT: By following the procedure above, the first few coordinates will carry large weight, whereas the rest will be nearly zero. I wish to model my n-dimensional signals using the spherical uniform distribution, but with these coordinates this doesn't make sense. Once again I believe I am missing something?