# Minimum Audible Angle in 3 rd-Order Ambisonics in Horizontal Plane for Different Ambsonic Decoders
## 4 Selected Ambsonic Decoders  
Ambisonic decoding plays a crucial role in spatial audio reproduction, influencing sound localisation and perceived immersion [24]. Various decoding approaches exist, each with different trade-offs in terms of energy distribution, spatial accuracy, and computational complexity. This section examines three selected ambisonic decoders: the SAD, EPAD, and AllRAD. Their principles, performance characteristics, and practical implications for playback system configurations are analysed in detail. The **Sampling Ambisonic Decoder** (SAD) represents the most fundamental algorithm among those investigated in this study. Its core principle involves sampling virtual panning functions at the directions of \(L\) loudspeakers [6]. For spherical reproduction systems, the decoder matrix is simply  
\[D=\sqrt{\frac{4\pi}{L}}Y_{N}^{\mathsf{T}}. \tag{1}\]  
The factor \(\sqrt{\frac{4\pi}{L}}\) reflects that each loudspeaker contributes a fraction of the energy (\(E\) measure) distributed across the surrounding directional unit sphere. \(Y^{\mathsf{T}}\) is a transposed vector of the sampled spherical harmonics at the loudspeaker directions. This method provides an even distribution of objective metrics for uniform loudspeaker layouts. However, in non-uniform configurations, energy mismatches result in attenuation in varying degrees [6]. Figure 3 illustrates the distributions of amplitude, velocity, and energy for the investigated setup. Even distribution of all the measures can be observed on the horizontal plane, with fluctuations occurring above and below it. They reveal a strong dependence between the placement of the speaker and the energy measures. The uniformity of the distributions can be improved with mode-matching design (MMD), the primary objective of which is to align the spherical harmonic modes corresponding to loudspeaker signals with those of the sound field decomposed in Ambisonics. However, both the SAD and MMD share a significant limitation: when the playback system is not a \(t\)-design, that is, when there is some non-uniformity in the loudspeaker arrangement [6], the objective criteria of energy and the energy vector are not preserved (see Section 2). This limitation inspired researchers [25] to develop a more robust decoding approach. The **Energy-Preserving Ambsonic Decoder** (EPAD) maintains the MMD approach but eliminates energy scaling factors. This design is accurate for frequencies above 200 Hz and enables panning-invariant loudness. This preservation occurs for any loudspeaker configuration, provided that the number of spherical harmonics remains adequate [25]. Figure 4 demonstrates the uniform energy distribution alongside other numerical measures, showing significant improvement compared to the SAD algorithm: equalised energy on the whole surface and reduced \(\mathbf{r}_{\mathsf{V}}\) angular error. It is worth noting that if  
Figure 3: Distribution of the objective measures for the combination of the investigated playback system and the SAD. Red dots mark the loudspeaker positions. Please note that the scales for \(\mathbf{r}_{\mathsf{V}}\) angular error and energy are not equal to better visualise the differences in distributions. the loudspeaker arrangement was perfectly uniform, both the SAD and EPAD would be numerically equivalent [6]. The **All-Round Ambsonic Decoder** (AllRAD) represents a hybrid approach that integrates VBAP with SAD principles [26]. This methodology implements a sequential signal processing chain in which sound sources undergo \(t\)-design panning before transmission to loudspeakers on a virtual \(t\)-design layout and subsequent routing to physical loudspeakers via VBAP transformation matrices. AllRAD implementation requires computational discretisation through implementing a triangular grid between each of the three loudspeakers [6].