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Design of fixed broadband beamformers robust against gain and phase deviations

Design of fixed broadband beamformers robust against gain and phase deviations. Simon Doclo, Marc Moonen

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Design of fixed broadband beamformers robust against gain and phase deviations

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  1. Design of fixed broadband beamformers robust against gain and phase deviations Simon Doclo, Marc Moonen Katholieke Universiteit LeuvenDept. of Electrical Engineering, ESAT-SISTA Kasteelpark Arenberg 10, B-3001 Leuven, Belgium{simon.doclo,marc.moonen}@esat.kuleuven.ac.beTel: +32-16-321899, Fax: +32-16-321970} May 15, 2003

  2. Overview • Robust multi-microphone signal enhancement • Beamforming basics • Broadband beamformer design procedures • Filter-and-sum beamforming • Cost functions • Weighted least-squares • Non-linear cost function • Eigenfilter-based • Robust broadband beamforming • Robustness against gain/phase/position errors • Mean cost function • Maximum cost function • Simulations

  3. Speech reference f[k] Delay-sum beamformer S S Noise references Blocking matrix Introduction • Multi-microphone noise reduction scheme which is robust to model imperfections (closely spaced microphones) • Generalised Sidelobe Canceller (GSC) structure: • Robustness • Fixed beamformer: broadband beamforming, robust against deviations in microphone gain, phase and position • Adaptive stage: quadratic constraint, leaky LMS, optimal filter • Measurement or calibration: time-consuming, expensive Take into account robustness in design procedure

  4. Beamforming basics • Microphone array configuration: N microphones, distance dn • Goal: compute FIR filters wn[k]such that beamformer provides desired (fixed) frequency and spatial discrimination Spatial directivity pattern: Design of broadband beamformer with arbitrary desired directivity pattern for a given arbitrary microphone array configuration, using FIR filter-and-sum structure

  5. Broadband beamformer design • Calculate wsuch that optimally fits desired spatial directivity pattern • Broadband problem: don’t split up problem for several distinct frequencies  design over total frequency-angle plane • No approximation of double integrals by finite Riemann-sum • No incorporation of microphone configuration in optimisation problem • Far-field assumption: plane waves + no attenuation • An(,) : microphone characteristics + mounting + deviations from nominal situation Minimisation of cost function (LS, ME, TLS, NL) [Kajala 99]

  6. Integrals have to be recalculated for each iteration Integrals only have to be calculated once Cost functions: Overview • Cost functions: • Weighted least squares  quadratic function • Non-linear design procedure  iterative optimisation • Conventional eigenfilter technique  GEVD, reference point • Eigenfilter based on TLS error  GEVD, best non-iterative technique

  7. Robust broadband beamforming • Small deviations from assumed characteristics (gain/phase/position) may lead to large deviations in spatial directivity pattern • In practice microphone characteristics are never exactly known • Instead of measuring/calibrating or limiting WNG, take all feasible charateristics into account and minimise: • Mean cost function using probability as weights • Requires probability density functions • Gain: higher-order moments, phase: knowledge about complete pdf • Maximum cost function for all feasible characteristics • Minimax criterion, optimise for worst-case scenario • (dense) grid of characteristics  high complexity Take into account stochastic deviations in design

  8. Simulations (non-linear design) • N = 3, positions: (-0.01 0 0.015) m, L = 20, fs=8 kHz • Passband = 0o-60o, 300-4000 Hz (endfire)Stopband = 80o-180o, 300-4000 Hz • Gain deviation = [0.9 1.1 1.05], Phase deviation = [5o -2o 5o] • Uniform gain pdf: 0.85-1.15Uniform phase pdf: -5o-10o

  9. dB dB dB dB Angle (deg) Angle (deg) Angle (deg) Angle (deg) Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Simulations (non-linear design)

  10. Simulations (non-linear design) 10

  11. Conclusion • Design of fixed beamformers with arbitrary spatial directivity pattern for arbitrary microphone configuration • FIR filter-and-sum beamformer: different cost functions • LS : amplitude and phase • NL : only amplitude • Design of robust beamformers by incorporating stochastic gain/phase/position errors  statistical information • Mean cost function • Maximum cost function • Simulations show performance improvement when deviations occur, certainly for small microphone arrays

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