# Minimum Audible Angle in 3 rd-Order Ambisonics in Horizontal Plane for Different Ambsonic Decoders
## 4 Selected Ambsonic Decoders
### Listening Test Procedure  
Prior to the presented research, preliminary listening experiments were conducted. This process helped identify methodological optimisations, particularly regarding session duration based on the adaptive method in the context of the MAA [30]. This study implemented a non-parametric adaptive up-down method, which estimates thresholds without explicitly characterising the psychometric function's shape beyond monotonicity assumptions. Although threshold estimation typically involves averaging stimulus intensities from final reversals [31], this research used a hybrid-adaptive approach [32] that allows data to be  
\begin{table}
\begin{tabular}{c c c} \hline
**ID** & **AZIMUTH** & **ELEVATION** \\ \hline
1 & 0\({}^{\circ}\) & 0\({}^{\circ}\) \\
2 & \(-45^{\circ}\) & 0\({}^{\circ}\) \\
3 & \(-90^{\circ}\) & 0\({}^{\circ}\) \\
4 & \(-135^{\circ}\) & 0\({}^{\circ}\) \\
5 & \(180  
matched to an assumed psychometric function. Various transformed up-down procedures are used in psychoacoustics. The Mean Group Length (MGL) parameter characterises procedural complexity--larger values necessitate more reversals before test termination. The optimal measurement point on the psychometric function (the \"sweet point\") for minimising threshold estimate variance is generally considered to fall within the X80-X94 confidence range [33]. The research from [PERSON] [31] determines the X71 rule or X75 rule as the mid-point. The X71 rule (MGL = 1.71) executes approximately 50% faster than the X75 rule and was therefore chosen for the purposes of the presented research. Based on the aforementioned studies, the adaptive procedure implemented the following parameters:  
* Decrementing/incrementing step sequence: \([60^{\circ},30^{\circ},15^{\circ},8^{\circ},4^{\circ},2^{\circ},1^{\circ},0.5^{ \circ},0.25^{\circ}]\). * Initial angular displacement: \(30^{\circ}\) from reference position (100% recognition accuracy baseline). * A 2-up/1-down procedure with 7 reversal limits. The first reversal was excluded from the calculations. * A 75% recognition threshold determined using the Bayesian psignif algorithm. * The frontal plane (\(0^{\circ}\)-\(90^{\circ}\)) and rear plane (\(90^{\circ}\)-\(180^{\circ}\)) were examined separately to prevent systematic errors. The experimental framework maintained consistency across all tests regarding the interface design, psychoacoustic method, loudspeaker configuration, and ambisonic order. The testing environment used MATLAB 2020b connected to REAPER (digital audio workstation) through the Open Sound Control protocol. Open-source plugins were used for decoding: IEM AllRAD for the AllRAD and Aalto Sparta Decoder [34] for both the SAD and EPAD. The implementation utilised Schmidt semi-normalisation (SN3D) with max-\(\mathbf{r_{E}}\) weighting, as recommended in the literature [35; 36]. The stimulus used in the experiment was a white Gaussian noise burst of 500 ms, with 20 ms fade-in and fade-out and a frequency range 20 Hz-20 kHz. This range was selected to match both the effective operating frequency range of the ambisonic decoder's energy vector and the human auditory system's use of spatial localisation cues. Frequencies above approximately 200 Hz are known to contribute significantly to directional hearing, with low frequencies supporting Iteraural Time Difference (ITD)-based localisation and higher frequencies supporting ILD-based localisation mechanisms. A MATLAB-based interface facilitated the execution of the test. Participants were placed in the sweet spot and instructed to maintain visual focus on a screen behind the central loudspeaker to minimise head movements. The test sequences began with a reference sample played from predetermined azimuthal positions: \(15^{\circ},30^{\circ},45^{\circ},60^{\circ},75^{\circ}\), and \(90^{\circ}\) (frontal plane); \(105^{\circ},120^{\circ},135^{\circ},150^{\circ}\), and \(165^{\circ}\) (rear plane). Symmetry was assumed between the left and right hemispheres, while the \(0^{\circ}\) and \(180^{\circ}\) positions were specifically tested.