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Constant Directivity Loudspeaker Beamforming

Yuancheng Luo

Abstract

Loudspeaker array beamforming is a common signal processing technique for acoustic directivity control and robust audio reproduction. Unlike their microphone counterpart, loudspeaker constraints are often heterogeneous due to arrayed transducers with varying operating ranges in frequency, acoustic-electrical sensitivity, efficiency, and directivity. This work proposes a frequency-regularization method for generalized Rayleigh quotient directivity specifications and two novel beamformer designs that optimize for maximum efficiency constant directivity (MECD) and maximum sensitivity constant directivity (MSCD). We derive fast converging and analytic solutions from their quadratic equality constrained quadratic program formulations. Experiments optimize generalized directivity index constrained beamformer designs for a full-band heterogeneous array.

Constant Directivity Loudspeaker Beamforming

Abstract

Loudspeaker array beamforming is a common signal processing technique for acoustic directivity control and robust audio reproduction. Unlike their microphone counterpart, loudspeaker constraints are often heterogeneous due to arrayed transducers with varying operating ranges in frequency, acoustic-electrical sensitivity, efficiency, and directivity. This work proposes a frequency-regularization method for generalized Rayleigh quotient directivity specifications and two novel beamformer designs that optimize for maximum efficiency constant directivity (MECD) and maximum sensitivity constant directivity (MSCD). We derive fast converging and analytic solutions from their quadratic equality constrained quadratic program formulations. Experiments optimize generalized directivity index constrained beamformer designs for a full-band heterogeneous array.
Paper Structure (7 sections, 37 equations, 5 figures, 1 algorithm)

This paper contains 7 sections, 37 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Loudspeaker Directivity Optimization
  3. Maximum Efficiency Constant Directivity Beamformer
  4. Maximum Sensitivity Constant Directivity Beamformer
  5. Quadratic Secular Equation Root-Finding
  6. Experiments
  7. Conclusion

Figures (5)

  • Figure 1: Sample accept $f_A(\boldsymbol{r})$ and reject $f_R(\boldsymbol{r})$ density functions define forward-facing listening and side-facing reflection windows respectively.
  • Figure 2: Max GRQ upper-bounds max GRPQ for a real sample mid-range, full-range and tweeter array. The maximizer $\boldsymbol{w}_*$ attenuates for frequencies outside each transducer's operating range curve specified by $\boldsymbol{\Lambda}$.
  • Figure 3: $S(\lambda)$ has positive pole $B_{\texttt{+}} = \left \{ {1} \right \}$, negative poles $B_{\texttt{-}} = \left \{ {-2, -1} \right \}$, and two roots. The monotonic increasing $S(b_{\texttt{-}} < \lambda < b_{\texttt{+}})$ bounds the root $\lambda_*$ nearest to $0$ and restricts other roots to outside $\lambda_*$ reflected across $b_{\texttt{-}}$, $b_{\texttt{+}}$.
  • Figure 4: MECD solver convergence rates for efficiency $G(\boldsymbol{C}, \boldsymbol{I}, \boldsymbol{w})$ objective and constant GDI $G(\boldsymbol{A}, \boldsymbol{R}, \boldsymbol{w}) = 6 \textrm{ dB}$ constraint are ranked PA $(\alpha=1, \boldsymbol{w}_0 = \boldsymbol{1})$$>$ SDP $>$ IPM $>$ DM ($\alpha_{\boldsymbol{w}} = 1e-2$, $\alpha_{\lambda} = 1e-3$ step-sizes).
  • Figure 5: Contour plots compare GRQ to GRPQ beam patterns constrained to each transducer's operating ranges across frequency on the horizontal plane.