A Linear-Phase Filter-Bank Approach to Process Rigid Spherical Microphone Array Recordings Franz Zotter
Abstract—Rigid spherical microphone arrays offer the technology to capture immersive 3D audio environments in higher-order Ambisonics. The processing of their signals elegantly splits into a frequency-independent spherical harmonic decomposition part converting microphone signals into coefficient signals and an analytic filtering part super-directionally processing the coefficient signals for either higher-order Ambisonic playback or beamforming. This paper proposes to improve robustness by a linear-phase FIR filter bank to modify the analytic filters: A suitable design of cross-over frequencies and filter slopes limits boosts of self-noise and array imperfection by a controlled increase of directional resolution over frequency. Useful for 3D audio applications, the proposed sub-band signal treatment moreover yields diffuse-field equalization and side-lobe suppression. Index Terms—spherical microphone arrays, Ambisonics, robust array processing
ρ0 (!) ρ1 (!) ρ1 (!)
YNy
ρ1 (!) .. .
.. .
ρN (!) p(t)
spherical harmonics encoder
N (t)
radial focusing filters
χN (t)
Fig. 1. Front end for rigid spherical microphone array signal processing: sound pressure samples p(t) of microphones are spherical-harmonics decomposed by the frequency-independent matrix YN† , and the resulting coefficient signals ψN (t) are converted to Ambisonic signals χN (t) by filters ρn (ω).
I. I NTRODUCTION In the recent years, works by Meyer and Elko [1]–[3] made spherical microphone arrays popular, in particular with sound pressure sensors mounted on a rigid sphere. There is well-written literature to study them, e.g. [4]–[11], of which Rafaely’s book [12] often serves as a comprehensive tutorial. While basics and most of the required aspects of signal processing are already found in literature, e.g. in recent contribution of Alon [13] for beamforming, or Song [11] for 3D audio playback, our aim is to pick on a discussion of how to obtain model-based filters for processing as in L¨osler [14]. There are some advantages they might more easily offer than measurement-and-optimization-based alternatives as, e.g., in Zaunschirm [15] or Politis [16]. Common to all approaches as in Fig. 1, the received signals are decomposed into spherical harmonic coefficients, and the resulting coefficient signals are filtered to compensate for the attenuation (Fig. 3) that high-order spherical harmonic (Ambisonic) signals undergo in sound propagation and diffraction. The specific question that bothers us here is how to obtain the filters for compact rigid-sphere array signals with the specific target of an optimal Ambisonic 3D audio playback on loudspeakers. Measurement-and-optimization-based solutions undoubtedly have their benefits, in particular when introducing constraints on a limited white noise gain. However, the formulation of constraints for directional sidelobe suppression and diffuse-field equalization as in [14], [17] is challenging, therefore model-based filters are revisited, here. Franz Zotter is with the University of Music and Performing Arts Graz, Institute of Electronic Music And Acoustics, Inffeldgasse 10/III, 8010 Graz, Austria, (e-mail:
[email protected]).
It is not acceptable in the target 3D audio playback that (i) the perceived audio bandwidth gets limited or modified, (ii) directional sidelobes distort localization at off-center seats, (iii) group delay is distorted by phase-equalized IIR filter bank, (iv) the noise gets boosted too much when the signals are quiet. The above-mentioned measurement-and-optimization-based approaches in [13], [15], [16] tend to violate (i)-(ii), while the model-based approaches in [14], [17] tend to violate (iii), and the requirement (iv) would need level-adaptive filtering. To come up with an improved model-based design, this contribution reviews the theory of the acoustical model, its inverse formulation, regularization and signal-dependent white-noisegain limitation, as well as sidelobe suppression. The proposed filter design employs a linear-phase filter bank similar to the one presented for beamforming with the icosahedral loudspeaker [18], and the resulting energy-normalized, sidelobesuppressing, noise-boost limiting and regular encoding filters are best implemented using FIR block convolution. II. S PHERICAL HARMONICS , P LANE WAVES , S CATTERING According to literature, e.g. [6], compact rigid spheres offer an ideal measurement surface. This is because despite the gradual attenuation, every of the incident field modes is theoretically represented in the spherical surface sound pressure, without exception. Moreover, spherical harmonic decomposition separates off a diagonal frequency-dependent part that simplifies signal processing. This would not be the case for rigid scatterers of cylindrical or elliptical shape [19]. With the sound pressure p(t) expanded into spherical harmonics Ynm (θ) and coefficients ψnm (t) to express the
(a)
