Converter

This tool converts the Ambisonic signals from input convention conv_in to output convention conv_out. Indeed, depending on the chosen convention, the spherical harmonics norm and ordering change. In ambitools, the default convention used is ACN_N3D (see ymn.lib).

Available Conventions

ACN_N3D

The ACN_N3D convention stands for “Ambisonic Channel Number - Fully normalized”¹. The N3D norm is given by:

\begin{equation} N_{l,m}^\text{N3D} = \sqrt{(2- \delta_m) (2 l + 1)\frac{(l-|m|)!}{(l+|m|)!}}, \label{eq:n3d} \end{equation}

where $\delta_m$ is the Kronecker delta function. The ACN ordering is given by^[ambix]:

\[\begin{equation} \left\lbrace \begin{aligned} &\text{ACN}(l,m) = i = l^2 + l + m \\ &m = \lfloor \sqrt{i} \rfloor \\ &l = i - m^2 - m \end{aligned} \right. \label{eq:acn} \end{equation}\]

This results in a pyramidal shape for the spherical harmonics with increasing ACN $i$, as follows:

			0
		1	2	3
	4	5	6	7	8
9	10	11	12	13	14	15

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ACN_SN3D

The ACN_SN3D convention stands for “Ambisonic Channel Number - Schmidt semi-normalized”¹. The N3D norm is given by:

\begin{equation} N_{l,m}^\text{SN3D} = \sqrt{(2- \delta_m) \frac{(l-|m|)!}{(l+|m|)!}}, \label{eq:sn3d} \end{equation}

The convention uses ACN ordering (see Eq. $\eqref{eq:acn}$).

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FuMa

The FuMa convention stands for Furse and Malham¹. Its norm uses the MaxN (except for $l = 0$). This norm is such that:

\begin{equation} \max_{(\theta,\phi)}|Y_{l,m}(\theta,\phi)|=1 \label{eq:maxn} \end{equation}

This means that the maximum signal value is 1 for all components except the first component $(l = 0, m = 0)$ weighted with $ 1 / \sqrt{2}$. Finally, the FuMa norm is:

\[\begin{equation} N_{l,m}^\text{FuMa} = \left\lbrace \begin{aligned} 1/\sqrt{2} & & \text{for} & & l = 0 \\ \text{MaxN} & & \text{for} & & l > 0 \end{aligned} \right., \end{equation}\]

In FuMa convention, the signals are ordered in such a way that the corresponding ACN $i$ are sorted as follows:

			0
		2	3	1
	8	6	4	5	7
15	13	11	9	10	12	14

Note that the Furse and Malham convention was originally defined for $L \leq 3$. Therefore, for $L>3$ the inputs signals are killed in the current implementation.

Compilation parameter

• L: maximal Spherical Harmonics degree (i.e., Ambisonics order, $L > 0$)
• conv_in: Input signals convention.
  1. ACN_N3D
  2. ACN_SN3D
  3. FuMa
• conv_out: Ouput signals convention.
  1. ACN_N3D
  2. ACN_SN3D
  3. FuMa

Inputs / Outputs

• Inputs: $(L+1)^2$
• Outputs: $(L+1)^2$