Digital Auio Signal Processing DASP Lecture-3: Noise Reuction-II Fixe Beamforming arc oonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@kuleuven.be homes.esat.kuleuven.be/~moonen/ Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34
Introuction Directivity pattern of a microphone A microphone (* is characterize by a `irectivity pattern which specifies the gain & phase shift that the microphone gives to a signal coming from a certain irection (i.e. `angle-of-arrival In general the irectivity pattern is a function of frequency ( In a 3D scenario `angle-of-arrival ( for frequency is azimuth + elevation angle Will consier only D scenarios for simplicity, with one angle-of arrival (, hence irectivity pattern is ( Directivity pattern is fixe an efine by physical microphone esign (* We o igital signal prcessing, so this inclues front-en filtering/a-to-d/.. Digital Auio Signal Processing Version 7-8 3 / 34 Lecture-3: Fixe Beamforming Introuction Virtual irectivity pattern By weighting or filtering (freq. epenent weighting an then summing signals from ifferent microphones, a (software controlle virtual irectivity pattern (weigthe sum of iniviual patterns can be prouce F (ω z[k] + F (ω : F (ω y[k] y [k] y [k] N Fm (ω fm,n.e jω n n.5 Fr e que ncy ( z 3 45 9 35 8 Angle (eg virtual (ω,θ Fm (ω. m (ω,θ m This assumes all microphones receive the same signals (so are all in the same position. owever Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 4 / 34
Introuction owever, in a microphone array ifferent microphones are in ifferent positions/locations, hence also receive ifferent signals Example : uniform linear array i.e. microphones place on a line & uniform inter-micr. istances ( & ieal micr. characteristics (p. For a far-fiel source signal (i.e. plane wave, each microphone receives the same signal, up to an angle-epenent elay fssampling rate cpropagation spee F ( F ( : F ( Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 5 / 34 z[k] y + m[ m k] y [ k + τ ] m cosθ τ m( θ fs c y [k] virtual ( F m (.e jωτ m (θ m m ( m θ cosθ Introuction Beamforming `spatial filtering base on microphone characteristics (irectivity patterns AND microphone array configuration (`spatial sampling Classification: z[k] + F ( F ( : F ( y [k] θ cosθ Fixe beamforming: ata-inepenent, fixe filters F m e.g. elay-an-sum, filter-an-sum This lecture Aaptive beamforming: ata-epenent filters F m e.g. LCV-beamformer, generalize sielobe canceler Next lecture Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 6 / 34 3
Introuction Backgroun/history: ieas borrowe from antenna array esign an processing for raar & (later wireless communications icrophone array processing consierably more ifficult than antenna array processing: narrowban raio signals versus broaban auio signals far-fiel (plane waves versus near-fiel (spherical waves pure-elay environment versus multi-path environment Applications: voice controlle systems (e.g. Xbox Kinect, speech communication systems, hearing ais, Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 7 / 34 Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 8 / 34 4
Data moel & efinitions /5 Data moel: source signal in far-fiel (see p.4 for near-fiel icrophone signals are filtere versions of source signal S( at angle θ Y Stack all microphone signals (m.. in a vector m ir. pattern $!#!" ( (. jωτ ( θ jωτ [ ] T ( (. e... ( θ e θ (. is `steering vector Output signal after `filter-an-sum is Z( θ F m * m m Y (. S( (. Y m $!#!" jωτm ( θ e pos.-ep. phase shift. S( ( θ F (. Y( θ { F instea of T for convenience (** (. θ }. S( Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 9 / 34 Data moel & efinitions /5 Data moel: source signal in far-fiel Y (. S( If all microphones have the same irectivity pattern o(, steering vector can be factore as jωτ ( θ jωτ ( θ [ e e ] (.... $!#!" $!!!! #!!!!! " ir.pattern spatial positions microphone- is use as a reference (arbitrary Will often consier arrays with ieal omni-irectional microphones : o( Example : uniform linear array, see p.5 Will use microphone- as reference (e.g. efining input SNR: T ( Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34 5
Data moel & efinitions 3/5 Definitions: ( In a linear array (p.5 : θ 9 o broasie irection θ o en-fire irection Array irectivity pattern (compare to p.3 `transfer function for source at angle θ ( -π<θ< π Z( ( F (. S( Steering irection angle θ with maximum amplification (for freq. θ max ( arg max Beamwith (BW θ ( region aroun θ max with amplification > (max.amplif - 3B (for freq. Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34 Data moel & efinitions 4/5 Data moel: source signal + icrophone signals are corrupte by aitive Y (. S( + N( N ω [ N ( N (... ( ] T ( N ω Define correlation matrix as Φ ( E{ N(. N( Will assume fiel is homogeneous, i.e. all iagonal elements of correlation matrix are equal : Φ ( Φ (, i Then coherence matrix is } φ ( Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming..../ 34 ii.. Γ (. Φ (..!.... 6
Data moel & efinitions 5/5 Definitions: ( Array Gain improvement in SNR for source at angle θ ( -π<θ< π SNR F (. output G ( SNR F (. Γ (. input White Noise Gain array gain for spatially uncorrelate F (. WNG ( θ F (. (e.g. sensor ps: often use as a measure for robustness Directivity array gain for iffuse (coming from all irections F (. θ DI( θ iffuse F (. Γ. white Γ Γ iffuse ij ω f ( sinc skip this formula ( j i c DI an WNG evaluate at θ max is often use as a performance criterion Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 3 / 34 I signal transfer function ^ (with micr- use as reference: transfer function ^ s PS: ω is ra/sample ( -Π ω Π ω fs is ra/sec PS: Near-fiel beamforming Far-fiel assumptions not vali for sources close to microphone array spherical waves instea of plane waves inclue attenuation of signals coorinates θ,r (position q instea of coorinate θ (in D case Different steering vector (e.g. with m( m.. : θ e (3D (D jωτ j j [ ] T ( q ωτ ( q ωτ ( q a e a e a e q e q pref q pref q pm a m τ m( q fs q pm c with q position of source p ref position of reference microphone p m position of m th microphone Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 4 / 34 7
PS: ultipath propagation In a multipath scenario, acoustic waves are reflecte against walls, objects, etc.. Every reflection may be treate as a separate source (near-fiel or far-fiel A more realistic ata moel is then.. Y ( q q. S( + N( [ ( q ( q... ( ] T ( ω, q q with q position of source an m(q, complete transfer function from source position to m-the microphone (incl. micr. characteristic, position, an multipath propagation `Beamforming aspect vanishes here, see also Lecture-5 (`multi-channel reuction Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 5 / 34 Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 6 / 34 8
Filter-an-sum beamformer esign Basic: proceure base on page Z( ( F (. S( N jnω fm, n. e T [ ] F (... F (, Fm ( n Array irectivity pattern to be matche to given (esire pattern over frequency/angle range of interest Non-linear optimization for FIR filter esign (ignore phase response min Quaratic optimization for FIR filter esign (co-esign phase response min fm m n N ( ( ( ω θ, n,..,.. fm θ ω m n N (, n,..,.. θ ω ( ω θ ( Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 7 / 34 Filter-an-sum beamformer esign Quaratic optimization for FIR filter esign (continue min f m fm With m n N (, n,..,.. θ ω T T T T [ f... f ], f [ f... f ] m, ( θ F f m, N (. θ optimal solution is [ F (... F ( ]. ( ω θ θ f. ( ω θ, p optimal Q. p, Q θ ω θ ω T Nx '!!! &!!! %. jω e.( I x (. θ ( N. jω e $!!!! #!!!! ". Kronecker prouct θ * ( ω θ Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 8 / 34 9
Filter-an-sum beamformer esign Design example 8 Logarithmic array N5 f s 8 kz.5 3 Frequency (z 45 9 35 Angle (eg 8 Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 9 / 34 Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34
atche filtering: WNG maximization Basic: proceure base on page 3 aximize White Noise Gain (WNG for given steering angle ψ F F ( arg{max WNG ( } arg{max F F (. (. } A priori knowlege/assumptions: angle-of-arrival ψ of esire signal + corresponing steering vector scenario white Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34 atche filtering: WNG maximization aximization in F (. F F ( arg{max } F (. is equivalent to minimization of output power (uner white input, subject to unit response for steering angle (** min F (., s.t. F (. Optimal solution (`matche filter is F F ( [FIR approximation]. F min f m n N ω ω m, n,..,.. F ( ω ω Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming / 34
atche filtering example: Delay-an-sum Basic: icrophone signals are elaye an then summe together ψ z[ k]. ( m y m [ k + Δ ] m Σ Δ Δ Δ m ( m cosψ jωδ e Fm ( m Fractional elays implemente with truncate interpolation filters (FIR Consier array with ieal omni-irectional micr s Then array can be steere to angle ψ : " ψ e jωτ ( #$... e jωτ ( T % &' Δ m τ m ( F ( ence (for ieal omni-ir. micr. s this is matche filter solution Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 3 / 34 atche filtering example: Delay-an-sum ieal omni-ir. micr. s Array irectivity pattern (: ( θ (. θ ( θ ( θ ( ψ θ max White gain : estructive interference constructive interference F (. WNG( θ.. F (. (inepenent of For ieal omni-ir. micr. array, elay-an-sum beamformer provies WNG equal to for all freqs (in the irection of steering angle ψ. Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 4 / 34
atche filtering example: Delay-an-sum Array irectivity pattern ( for uniform linear array: ( θ ( has sinc-like shape an is frequency-epenent.8.6.4. e m jγ / e e (cos θ cosψ j( m ω fs c jγ / sin( γ / sin( γ / γ ( θ 5 microphones 3 cm inter-microphone istance ψ6 steering angle f s 6 kz sampling frequency Spatial irectivity pattern for f5 z 9 - - ieal omni-ir. micr. s 8 enfire 4 6 Frequency (z 45 8 9 35 Angle (eg 8 7 Digital Auio Signal Processing Version 7-8 ψ6 Lecture-3: Fixe Beamforming 5 / 34 wavelength4cm atche filtering example: Delay-an-sum For c f.( + cosψ an ambiguity, calle spatial aliasing, occurs. This is analogous to time-omain aliasing where now the spatial sampling ( is too large. c c λmin Aliasing oes not occur (for any ψ if f. f s max ieal omni-ir. micr. s.8.6.4. Frequency (z 4 6 5 8 Angle (eg 5 5, ψ6, f s 6 kz, 8 cm c f.( + cosψ Details... ( θ iff γ π.p for integer p γ for θ ψ (for all π c if ψ then γ π occurs for θ π an f...( + cosψ π c 3 if ψ then γ π occurs for θ an f...( cosψ Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 6 / 34 3
atche filtering example: Delay-an-sum Beamwith for a uniform linear array: ieal omni-ir. micr. s BW c 96( ν sec ψ ω with e.g. ν/sqrt( (-3 B hence large epenence on # microphones, istance (compare p.4 & 5 an frequency (e.g. BW infinitely large at DC Array topologies: Uniformly space arrays Neste (logarithmic arrays (small for high large for small D- (planar / 3D-arrays 4 Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 7 / 34 Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 8 / 34 4
Super-irective beamforming : DI maximization Basic: proceure base on page 3 aximize Directivity (DI for given steering angle ψ F SD ( arg{max DI( } arg{max F (. } iffuse F (. Γ. A priori knowlege/assumptions: angle-of-arrival ψ of esire signal + corresponing steering vector scenario iffuse Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 9 / 34 Super-irective beamforming : DI maximization aximization in F (. SD F ( arg{max } iffuse F (. Γ. is equivalent to minimization of output power (uner iffuse input, subject to unit response for steering angle (** iffuse min F (. Γ (., s.t. F (. Optimal solution is F SD ( [FIR approximation].{ Γ iffuse ( }..{ Γ iffuse SD min f m n N ω ω ω m, n,..,.. F ( ω ( }. Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 3 / 34 5
Super-irective beamforming : DI maximization Directivity patterns for en-fire steering (ψ: ieal omni-ir. micr. s Delay-an-sum beamformer (f3 z 9 - - 8 Superirective beamformer (f3 z 9 - - 8 5 3 cm f s 6 kz 7 7 DI 5 Directivity (linear Superirective beamformer has highest DI, but very poor WNG (at low frequencies, where iffuse coherence matrix becomes ill-conitione hence problems with robustness (e.g. sensor! 5 5 5 Superirective Delay-an-sum DIWNG5 4 6 8 Frequency (z White gain (B - - -3-4 WNG 5-5 Superirective Delay-an-sum -6 4 6 8 Frequency (z PS: iffuse white for high frequencies (cfr. ωè Π an c/fsλmin/ min(j-i in iffuse coherence matrix Digital aximum Auio Signal irectivity. Processing obtaine Version for 7-8 en-fire steering Lecture-3: an for Fixe frequency-> Beamforming (no proof 3 / 34 Overview Introuction & beamforming basics Data moel & efinitions Filter-an-sum beamformer esign atche filtering White gain maximization Ex: Delay-an-sum beamforming Superirective beamforming Directivity maximization Directional microphones (elay-an-subtract Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 3 / 34 6
Differential microphones : Delay-an-subtract First-orer ifferential microphone irectional microphone closely space microphones, where one microphone signal is elaye (harware an then subtracte from the other micropone signal Σ + _ τ θ cosθ jω ( τ + c ( θ e ω/c <<π, ωτ <<π Array irectivity pattern: ( jω.(τ + c cos!.(τ + c. high-pass jω angle epenence P(! τ α First-orer high-pass frequency epenence τ + / c P( freq.inepenent (! irectional response P( α + ( α α : P( is scale cosine, shifte up with α such that θ max o (en-fire an P(θ max cosθ Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 33 / 34 Differential microphones : Delay-an-subtract Types: ipole, carioi, hypercarioi, supercarioi (J84 broasie enfire Dipole: α (τ zero at 9 o DI4.8B ypercarioi: α.5 zero at 9 o highest DI6.B Supercarioi: α ( 3 /.35 zero at 5 o, DI5.7 B highest front-to-back ratio Carioi: α.5 zero at 8 o DI4.8B Digital Auio Signal Processing Version 7-8 Lecture-3: Fixe Beamforming 34 / 34 7