Digital Audio Signal Processing Lecture-2: Microphone Array Processing - Fixed Beamforming - Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven homes.esat.kuleuven.be/~moonen/

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 2 Overview Introduction & beamforming basics –Data model & definitions Fixed beamforming –Filter-and-sum beamformer design –Matched filtering White noise gain maximization Ex: Delay-and-sum beamforming –Superdirective beamforming Directivity maximization –Directional microphones (delay-and-subtract) Lecture-3: Adaptive Beamforming

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 3 Introduction Directivity pattern of a microphone –A microphone is characterized by a `directivity pattern’ which specifies the gain (& phase shift) that the microphone gives to a signal coming from a certain direction (i.e. `angle-of-arrival’) –In general the directivity pattern is also a function of frequency (ω) –In a 3D scenario `angle-of-arrival’ is azimuth + elevation angle –Will consider only 2D scenarios for simplicity, with one angle-of arrival (θ), hence directivity pattern is H(ω,θ) –Directivity pattern is fixed and defined by physical microphone design |H(ω,θ)| for 1 frequency

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 4 Introduction By weighting or filtering (=freq.dependent weighting) and then summing signals from different microphones, a (software controlled) `virtual’ directivity pattern (=weigthed sum of individual patterns) can be produced This assumes all microphones receive the same signals (so are all in the same position). However… + :

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 5 Introduction However, an additional aspect is that in a microphone array different microphones are in different positions/locations, hence also receive different signals Example : uniform linear array i.e. microphones placed on a line & uniform inter-micr. distances (d) & ideal micr. characteristics (p.8) For a far-field source signal (plane waveforms), each microphone receives the same signal, up to an angle-dependent delay fs=sampling rate c=propagation speed + :

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 6 Introduction Beamforming = `spatial filtering’ based on microphone characteristics (directivity patterns) AND microphone array configuration (`spatial sampling’) + :

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 7 Introduction Background/history: ideas borrowed from antenna array design/processing for RADAR & (later) wireless communications. Microphone array processing considerably more difficult than antenna array processing: –narrowband radio signals versus broadband audio signals –far-field (plane wavefronts) versus near-field (spherical wavefronts) –pure-delay environment versus multi-path environment Classification: –fixed beamforming: data-independent, fixed filters F m e.g. delay-and-sum, filter-and-sum –adaptive beamforming: data-dependent adaptive filters F m e.g. LCMV-beamformer, Generalized Sidelobe canceler Applications: voice controlled systems (e.g. Xbox Kinect), speech communication systems, hearing aids,…

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 8 Beamforming basics Data model: source signal in far-field (see p.12 for near-field) Microphone signals are filtered versions of source signal S(  ) at angle  Stack all microphone signals in a vector d is `steering vector’ Output signal after `filter-and-sum’ is H instead of T for convenience (**)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 9 Data Model: source signal in far-field If all microphones have the same directivity pattern Ho(ω,θ), steering vector can be factored as… Will often consider arrays with ideal omni-directional microphones : Ho(ω,θ)=1 Example : uniform linear array, see p.5 Beamforming basics microphone-1 is used as a reference (=arbitrary)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 10 Beamforming basics Definitions (1) : In a linear array (p.5) :  =90 o =broadside direction  = 0 o =end-fire direction Array directivity pattern (compare to p.3) = `transfer function’ for source at angle  ( -π<  < π ) Steering direction = angle  with maximum amplification (for 1 freq.) Beamwidth (BW) = region around  max with (e.g.) amplification > -3dB (for 1 freq.)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 11 Beamforming basics Data model: source signal + noise Microphone signals are corrupted by additive noise Define noise correlation matrix as Will assume noise field is homogeneous, i.e. all diagonal elements of noise correlation matrix are equal : Then noise coherence matrix is

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 12 Beamforming basics Definitions (2) : Array Gain =improvement in SNR for source at angle  ( -π<  < π ) White Noise Gain =array gain for spatially uncorrelated noise (=white) (e.g. sensor noise) ps: often used as a measure for robustness Directivity =array gain for diffuse noise (=coming from all directions) DI and WNG evaluated at  max is often used as a performance criterion (ignore this formula) |signal transfer function|^2 |noise transfer function|^2

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 13 Far-field assumptions not valid for sources close to microphone array –spherical wavefronts instead of planar waveforms –include attenuation of signals –2 coordinates ,r (=position q) instead of 1 coordinate  (in 2D case) Different steering vector (e.g. with Hm(ω,θ)=1 m=1..M) : PS: Near-field beamforming with q position of source p ref position of reference microphone p m position of m th microphone e e=1 (3D)…2 (2D)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 14 PS: Multipath propagation In a multipath scenario, acoustic waves are reflected against walls, objects, etc.. Every reflection may be treated as a separate source (near-field or far-field) A more practical data model is: with q position of source and Hm(ω,q), complete transfer function from source position to m-the microphone (incl. micr. characteristic, position, and multipath propagation) `Beamforming’ aspect vanishes here, see also Lecture-3 (`multi-channel noise reduction’)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 15 Overview

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 16 Filter-and-sum beamformer design Basic: procedure based on page 9 Array directivity pattern to be matched to given (desired) pattern over frequency/angle range of interest Non-linear optimization for FIR filter design (=ignore phase response) Quadratic optimization for FIR filter design (=co-design phase response)

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 17 Filter-and-sum beamformer design Quadratic optimization for FIR filter design (continued) With optimal solution is Kronecker product

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 18 Filter-and-sum beamformer design Design example M=8 Logarithmic array N=50 f s =8 kHz

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 19 Overview

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 20 Matched filtering : WNG maximization Basic: procedure based on page 11 Maximize White Noise Gain (WNG) for given steering angle ψ A priori knowledge/assumptions: –angle-of-arrival ψ of desired signal + corresponding steering vector –noise scenario = white

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 21 Matched filtering : WNG maximization Maximization in is equivalent to minimization of noise output power (under white input noise), subject to unit response for steering angle (**) Optimal solution (`matched filter’) is [FIR approximation]

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 22 Matched filtering example: Delay-and-sum Basic : Microphone signals are delayed and then summed together Fractional delays implemented with truncated interpolation filters (=FIR) Consider array with ideal omni-directional micr’s Then array can be steered to angle  : Hence (for ideal omni-dir. micr.’s) this is matched filter solution

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 23 Matched filtering example: Delay-and-sum Array directivity pattern H(ω,θ): =destructive interference =constructive interference White noise gain : (independent of ω) For ideal omni-dir. micr. array, delay-and-sum beamformer provides WNG equal to M for all freqs. (in the direction of steering angle ψ). ideal omni-dir. micr.’s

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 24 Array directivity pattern H( ,  ) for uniform linear array: H( ,  ) has sinc-like shape and is frequency-dependent Matched filtering example: Delay-and-sum M=5 microphones d=3 cm inter-microphone distance  =60  steering angle f s =16 kHz sampling frequency =endfire  =60  wavelength=4cm ideal omni-dir. micr.’s

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 25 Matched filtering example: Delay-and-sum For an ambiguity, called spatial aliasing, occurs. This is analogous to time-domain aliasing where now the spatial sampling (=d) is too large. Aliasing does not occur (for any  ) if M=5,  =60 , f s =16 kHz, d=8 cm ideal omni-dir. micr.’s

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 26 Matched filtering example: Delay-and-sum Beamwidth for a uniform linear array: hence large dependence on # microphones, distance (compare p.22 & 23) and frequency (e.g. BW infinitely large at DC) Array topologies : –Uniformly spaced arrays –Nested (logarithmic) arrays (small d for high , large d for small  ) –2D- (planar) / 3D-arrays with e.g. =1/sqrt(2) (-3 dB) ideal omni-dir. micr.’s

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 27 Overview

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 28 Super-directive beamforming : DI maximization Basic: procedure based on page 11 Maximize Directivity (DI) for given steering angle ψ A priori knowledge/assumptions: –angle-of-arrival ψ of desired signal + corresponding steering vector –noise scenario = diffuse

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 29 Super-directive beamforming : DI maximization Maximization in is equivalent to minimization of noise output power (under diffuse input noise), subject to unit response for steering angle (**) Optimal solution is [FIR approximation]

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 30 Directivity patterns for end-fire steering ( ψ =0): Superdirective beamformer has highest DI, but very poor WNG (at low frequencies, where diffuse noise coherence matrix becomes ill-conditioned) hence problems with robustness (e.g. sensor noise) ! Super-directive beamforming : DI maximization M 2 M=5 d=3 cm f s =16 kHz Maximum directivity=M.M obtained for end-fire steering and for frequency->0 (no proof) ideal omni-dir. micr.’s WNG=5 PS: diffuse noise = white noise for high frequencies DI=5 DI=25

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 31 Overview

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 32 First-order differential microphone = directional microphone 2 closely spaced microphones, where one microphone is delayed (=hardware) and whose outputs are then subtracted from each other Differential microphones : Delay-and-subtract

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 33 First-order differential microphone = directional microphone 2 closely spaced microphones, where one microphone is delayed (=hardware) and whose outputs are then subtracted from each other Array directivity pattern: –First-order high-pass frequency dependence –P(  ) = freq.independent (!) directional response –0   1  1 : P(  ) is scaled cosine, shifted up with  1 such that  max = 0 o (=end-fire) and P(  max )=1  d/c << ,  <<  Differential microphones : Delay-and-subtract

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 34 Differential microphones : Delay-and-subtract Types: dipole, cardioid, hypercardioid, supercardioid (HJ84) Dipole:  1 = 0 (  =0) zero at 90 o DI = 4.8 dB Cardioid:  1 = 0.5 zero at 180 o DI = 4.8 dB =endfire =broadside

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 35 Differential microphones : Delay-and-subtract Hypercardioid:  1 = 0.25 zero at 109 o highest DI=6.0 dB Supercardioid:  1 = zero at 125 o, DI=5.7 dB highest front-to-back ratio =endfire

Digital Audio Signal Processing Version Lecture-2: Microphone Array Processing p. 36 Overview