Digital Audio Signal Processing Lecture-2: Microphone Array Processing

1
Digital Audio Signal Processing Lecture-2
Microphone Array Processing

• Marc Moonen Simon Doclo
• Dept. E.E./ESAT-STADIUS, KU Leuven
• marc.moonen_at_esat.kuleuven.be
• homes.esat.kuleuven.be/moonen/

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)
• Adaptive beamforming
• LCMV beamforming
• Frost beamforming
• Generalized sidelobe canceler

3
Introduction

• 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 (angle-of-arrival)
• Directivity pattern is a function of frequency
  (?)
• In a 3D scenario angle-of-arrival
• is azimuth elevation angle
• Will consider only 2D scenarios for
• simplicity, with one angle-of arrival (?),
• hence directivity pattern is H(?,?)
• Directivity pattern is fixed and defined
• by physical microphone design

H(?,?) for 1 frequency
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 positions).
  
  
  
• However...

N-tap FIR filters
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
• microphones placed on a line
• uniform inter-micr. distances (d)
• ideal micr. characteristics (see p.8)
• For a far-field source signal (plane
• waveforms), each microphone
• receives the same signal, up
• to an angle-dependent delay
• fssampling rate
• cpropagation speed
• Beamforming spatial filtering based on
  microphone characteristics (directivity patterns)
  AND micr. array configuration (spatial
  sampling).

6
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 Fm e.g. delay-and-sum,
  filter-and-sum
• adaptive beamforming data-dependent adaptive
  filters Fm e.g. LCMV-beamformer,
  Generalized Sidelobe canceler
• Applications voice controlled systems (e.g. Xbox
  Kinect), speech communication systems, hearing
  aids,

7
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

8
Beamforming basics

• Data Model source signal in far-field
• If all microphones have the same directivity
  pattern Ho(?,?), steering vector can be factored
  as
• Will often consider arrays with
• ideal omni-directional microphones
  Ho(?,?)1
• Example uniform linear array, see p.5

9
Beamforming basics

• Definitions (1)
• In a linear array (p.5) ? 90obroadside
  direction
• ? 0o
  end-fire direction
• Array directivity pattern (compare to p.3)
• transfer function for source at angle
  ? ( -plt?lt p )
• Steering direction
• angle ? with maximum amplification (for
  1 freq.)
• Beamwidth (BW)
• region around ?max with (e.g.)
  amplification gt -3dB (for 1 freq.)

10
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

11
Beamforming basics

• Definitions (2)
• Array Gain improvement in SNR for source at
  angle ? ( -plt?lt p )
• 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

signal transfer function2
noise transfer function2
(ignore this formula)
12
PS Near-field beamforming

• 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(?,?)1
  m1..M)

e
e1 (3D)2 (2D)
with q position of source pref
position of reference microphone pm
position of mth microphone
13
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(?,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)

14
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)
• Adaptive beamforming
• LCMV beamforming
• Frost beamforming
• Generalized sidelobe canceler

15
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)

16
Filter-and-sum beamformer design

• Quadratic optimization for FIR filter design
  (continued)
• With
• optimal solution is

Kronecker product
17
Filter-and-sum beamformer design

• Design example

M8 Logarithmic array N50 fs8 kHz
18
Matched filtering WNG maximization

• Basic procedure based on page 11
• Maximize White Noise Gain (WNG) for given
  steering angle ?
• A priori knowledge/assumptions
• angle-of-arrival ? of desired signal
  corresponding steering vector
• noise scenario white

19
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

20
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 micrs
• Then array can be steered to angle ?
• Hence (for ideal omni-dir. micr.s) this is
  matched filter solution

21
Matched filtering example Delay-and-sum
ideal omni-dir. micr.s

• Array directivity pattern H(?,?)
• 
  destructive interference
• 
  constructive interference
• White noise gain
• 
  (independent of ?)
• For ideal omni-dir. micr. array,
  delay-and-sum beamformer provides
• WNG equal to M for all freqs. (in the
  direction of steering angle ?).

22
Matched filtering example Delay-and-sum
ideal omni-dir. micr.s

• Array directivity pattern H(?,?) for uniform
  linear array
• H(?,?) has sinc-like shape and is
  frequency-dependent

M5 microphones d3 cm inter-microphone
distance ?60? steering angle fs16 kHz sampling
frequency
endfire
?60?
wavelength4cm
23
Matched filtering example Delay-and-sum
ideal omni-dir. micr.s

• 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

M5, ?60?, fs16 kHz, d8 cm
24
Matched filtering example Delay-and-sum
ideal omni-dir. micr.s

• 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)
25
Super-directive beamforming DI maximization

• Basic procedure based on page
  11
• Maximize Directivity (DI) for given steering
  angle ?
• A priori knowledge/assumptions
• angle-of-arrival ? of desired signal
  corresponding steering vector
• noise scenario diffuse

26
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

27
Super-directive beamforming DI maximization
ideal omni-dir. micr.s

• Directivity patterns for end-fire steering (?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) !

M 2
Maximum directivityM.M obtained for end-fire
steering and for frequency-gt0 (no proof)
28
Differential microphones Delay-and-subtract

• 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 0o
  (end-fire) and P(?max )1

?d/c ltlt?, ?? ltlt?
29
Differential microphones Delay-and-subtract

• Types dipole, cardioid, hypercardioid,
  supercardioid (HJ84)

Cardioid ?1 0.5 zero at 180o
DI 4.8 dB
Dipole ?1 0 (?0) zero at 90o
DI 4.8 dB
broadside
broadside
endfire
endfire
30
Differential microphones Delay-and-subtract
Hypercardioid ?1 0.25
zero at 109o highest
DI6.0 dB
Supercardioid ?1
zero at 125o, DI5.7 dB
highest front-to-back ratio
endfire
endfire
31
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)
• Adaptive beamforming
• LCMV beamforming
• Frost beamforming
• Generalized sidelobe canceler

32
LCMV-beamforming

• Adaptive filter-and-sum structure
• Aim is to minimize noise output power, while
  maintaining a chosen response in a given look
  direction (and/or other linear constraints, see
  below). compare to () p.1926
• I.e. similar to operation of a superdirective
  array (in diffuse noise), or delay-and-sum (in
  white noise), but now noise field is unknown !
• Implemented as adaptive FIR filter (cfr DSP-II)

33
LCMV-beamforming

• LCMV Linearly Constrained Minimum Variance
• f designed to minimize power (variance) of output
  zk
• To avoid desired signal cancellation, add (J)
  linear constraints
• Ex fix array response in look-direction ? for
  sample freqs wi, i1..J ()
• 
  
• 
  
• With () (for sufficiently large J) constrained
  output power minimization approximately
  corresponds to constrained noise power
  minimization (why?)
• Solution is (obtained using Lagrange-multipliers,
  etc..)

34
Frost Beamforming

• Frost-beamformer adaptive version of
  LCMV-beamformer
• If Ryy is known, a gradient-descent procedure for
  LCMV is
• in each iteration filters f are updated in
  the direction of the constrained gradient. The P
  and B are such that fk1 statisfies the
  constraints (verify!). The mu is a step size
  parameter (to be tuned)
• If Ryy is unknown, an instantaneous (stochastic)
  approximation may be substituted, leading to a
  constrained LMS-algorithm

35
Generalized Sidelobe Canceler (GSC)

• GSC alternative adaptive filter formulation
  of the LCMV-problem constrained optimisation
  is reformulated as a constraint pre-processing,
  followed by an unconstrained optimisation,
  leading to a simpler adaptation scheme
• LCMV-problem is
• Define blocking matrix Ca, ,with columns
  spanning the null-space of C
• 
  
• Parametrize all fs that satisfy constraints
  (verify!)
• I.e. filter f can be decomposed in a fixed
  part fq and a variable part Ca. fa
• Unconstrained optimization of fa
• (MN-J coefficients)

36
Generalized Sidelobe Canceler

• GSC (continued)
• Hence unconstrained optimization of fa can be
  implemented as an
• adaptive filter (adaptive linear combiner),
  with filter inputs (left-
• hand sides) equal to and desired
  filter output (right-hand
• side) equal to
• LMS algorithm

37
Generalized Sidelobe Canceler

• GSC then consists of three parts
• Fixed beamformer (cfr. fq ), satisfying
  constraints but not yet minimum variance),
  creating speech reference
• Blocking matrix (cfr. Ca), placing spatial nulls
  in the direction of the speech source (at
  sampling frequencies) (cfr. C.Ca0), creating
  noise references
• Multi-channel adaptive filter
• (linear combiner)
• your favourite one, e.g. LMS

38
Generalized Sidelobe Canceler

• A popular GSC realization is as follows
• Note that some reorganization has been done
  the blocking matrix now generates (typically)
  M-1 (instead of MN-J) noise references, the
  multichannel adaptive filter performs
  FIR-filtering on each noise reference (instead of
  merely scaling in the linear combiner).
  Philosophy is the same, mathematics are different
  (details on next slide).

39
Generalized Sidelobe Canceler

• Math details (for Deltas0)

select sparse blocking matrix such that
input to multi-channel adaptive filter
use this as blocking matrix now
40
Generalized Sidelobe Canceler

• Blocking matrix Ca
• Creating (M-1) independent noise references by
  placing spatial nulls in look-direction
• different possibilities (a la p.38)
• (broadside
  steering)

Griffiths-Jim

• Problems of GSC
• impossible to reduce noise from look-direction
• reverberation effects cause signal leakage in
  noise reference
• adaptive filter should only be updated when
  no speech is present !
• 
  

Walsh