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Digital Audio Signal Processing Lecture-2: Microphone Array Processing

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Title: 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
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