HOA Encoding

This page shows how a monophonic audio signal is rendered to Ambisonics by providing angular direction. This procedure is the standard approach for creating virtual sound sources in object-based spatialisation. This example assumes a plane-wave (far-field) source model. -----

Some Conventions

  • Cartesian coordinates:

    • \(x=\text{front->back}\)

    • \(y=\text{left->right}\)

    • \(z=\text{up->down}\)

  • Angles:

    • azimuth \(\varphi \in (-\pi, \pi]\) (CCW from +x toward +y),

    • elevation \(\theta \in [-\pi/2, \pi/2]\) (up from horizontal plane).

  • Normalisation/order:

    • AmbiX (ACN channel order, SN3D normalisation), unless noted.

    • ACN index \(n=\ell(\ell+1)+m\).

    • For FOA (order \(\ell=1\)), the mapping is \([n]=[0,1,2,3] \leftrightarrow [W,Y,Z,X]\).

First-Order Ambisonics for a Single Point Source

A monophonic source \(s(t)\) at direction \((\varphi,\theta)\) encodes to the FOA vector \(\mathbf a(t)=\begin{bmatrix}W&Y&Z&X\end{bmatrix}^{\mathsf T}\) (AmbiX ordering) as:

\begin{equation*} \begin{aligned} W(t) &= s(t)\,Y_0^0(\theta,\varphi),\\ Y(t) &= s(t)\,Y_1^{-1}(\theta,\varphi),\\ Z(t) &= s(t)\,Y_1^{0}(\theta,\varphi),\\ X(t) &= s(t)\,Y_1^{1}(\theta,\varphi), \end{aligned} \end{equation*}

with the real SN3D first-order spherical harmonics:

\begin{equation*} \begin{aligned} Y_0^0(\theta,\varphi) &= 1,\\ Y_1^{1}(\theta,\varphi) &= \cos\theta\,\cos\varphi,\\ Y_1^{-1}(\theta,\varphi) &= \cos\theta\,\sin\varphi,\\ Y_1^{0}(\theta,\varphi) &= \sin\theta. \end{aligned} \end{equation*}

Thus, explicitly:

\begin{equation*} \begin{aligned} W(t) &= s(t),\\ X(t) &= s(t)\,\cos\theta\,\cos\varphi,\\ Y(t) &= s(t)\,\cos\theta\,\sin\varphi,\\ Z(t) &= s(t)\,\sin\theta. \end{aligned} \end{equation*}

FOA — Multiple Point Sources (Object-Based)

For \(N\) sources \(s_i(t)\) at \((\varphi_i,\theta_i)\), FOA channels are a linear sum:

\begin{equation*} \mathbf a(t) = \sum_{i=1}^{N} s_i(t)\, \begin{bmatrix} 1\\[2pt] \cos\theta_i\,\sin\varphi_i\\[2pt] \sin\theta_i\\[2pt] \cos\theta_i\,\cos\varphi_i \end{bmatrix} \quad \text{(AmbiX/ACN order } [W,Y,Z,X]\text{).} \end{equation*}

Higher-Order Ambisonics (General Order \(L\))

Let \(Y_{\ell}^{m}(\theta,\varphi)\) be the real SN3D spherical harmonics with \(\ell=0..L\) and \(m=-\ell..\ell\). For a single source:

\begin{equation*} a_{\ell m}(t) = s(t)\,Y_{\ell}^{m}(\theta,\varphi),\qquad \ell=0..L,\; m=-\ell..\ell. \end{equation*}

For \(N\) sources:

\begin{equation*} a_{\ell m}(t) = \sum_{i=1}^{N} s_i(t)\,Y_{\ell}^{m}\!\bigl(\theta_i,\varphi_i\bigr). \end{equation*}

References

2019

  • Franz Zotter and Matthias Frank. Ambisonics: A Practical 3D Audio Theory for Recording, Studio Production, Sound Reinforcement, and Virtual Reality. Springer, 2019.
    [details] [BibTeX▼]

2015

  • Matthias Frank, Franz Zotter, and Alois Sontacchi. Producing 3d audio in ambisonics. In Audio Engineering Society Conference: 57th International Conference: The Future of Audio Entertainment Technology–Cinema, Television and the Internet. Audio Engineering Society, 2015.
    [details] [BibTeX▼]

2009

  • Frank Melchior, Andreas Gräfe, and Andreas Partzsch. Spatial audio authoring for ambisonics reproduction. In Proc. of the Ambisonics Symposium. 2009.
    [details] [BibTeX▼]

1973

  • Michael A. Gerzon. Periphony: With-Height Sound Reproduction. Journal of the Audio Engineering Society, 21(1):2–10, 1973.
    [details] [BibTeX▼]