Blind Separation of Acoustic Signals

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Blind Separation of Acoustic Signals

• Scott C. Douglas
• Southern Methodist University, Dallas TX, USA

Presented by - Praharshana Perera
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Cocktail Party Problem Convolutive Mixtures

x1(k)
s1(k)
xn(k)
sm(k)
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Convolutive Blind Source Separation Task
Unknown Mixing System
Separation System

• BLIND -
• The source signals are not observed
• No information is available about the mixture

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Unknown Mixing System

• Source signal vector sequence
• These signals pass through an (m x n) linear
  time-invariant system with matrix impulse
  response Ai ,
• The measured signal vector sequence

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Unknown Mixing System

• The mixing model is convolutive
• (
  convolution operator)
• Causality of the system model is assumed
• for l lt 0
• No broadband noise is present
• where

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Separation System

• Each source can be uniquely extracted from the
  sensor measurements
• The sequence of (m x n) matrices describe
  the separation system
• Contains the estimates of the individual sources
• Causality of the separation is assumed
• Number of sensors n gt m number of sources

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Goal of Convolutive BSS

• Goal Adjust the impulse response of the demixing
  system such that each output signal yi(k)contains
  one filtered version of each source signal
  sj(k)without replacement and without any loss of
  information
• where is one to one and
  arbitrary
• Assumption
• Each si(k) is statistically independent of each
  sj(l) for all i j all k and all l
• For Samples sisi(k), the joint PDF is of the
  form

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The Simple Cocktail Party Problem Instantaneous
Blind Signal Separation
Mixing matrix A
x1
s1
Observations
Sources
xn
sm
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Instantaneous Blind Signal Separation

• Mixing model
• Linear and Instantaneous Mixing
• No dispersive effects or time delays are present
• Instantaneous Separation
• Solution
• Adapt the separation matrix such that
• Where is an (m x m) permutation matrix with
  one unity entry in any row or column and D is a
  diagonal scaling matrix

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Multi Channel Blind Deconvolution

• Convolutive BSS
• Attempts to enforce spatial independence of
  output signals
• Blind Deconvolution
• Attempts to enforce both spatial and temporal
  independence
• Additional assumption
• All source signals are spatially and temporally
  independent
• Ideal Multichannel blind deconvolution
• Each yi(k) is a scaled time-shifted version of
  sj(k)

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Separating Criteria for BSS

• Efficiency of BSS depends on the separation
  criteria that is employed
• Convolutive BSS separation criteria
• Density modeling
• Contrast functions
• Correlation based criteria

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Density Modeling Criteria

• Based on Information Theory
• Characterize the shared information in a set of
  signals
• Separation when no shared information can be
  found in out put signals
• Method
• Adjust the separation system Bl(k) so that the
  joint PDF of y(k) , Py(y)is as close as possible
  to some model distribution
• Kullback-Leibler divergence measure
• The formula can be expressed using expectation
  operator

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• Choice of depends on the assumptions
  and priory knowledge of s(k).
• If all si(k) are identically distributed
• Leads to a ML estimate of the separation matrix
  for given signal statistics
• depends on the p.d.f of
  the extracted sources in y(k) ? depends on the
  impulse response Bl.
• A cost function for Bl
  can be developed that matches the above formula
  up to a constant

Density Modeling Criteria
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Contrast Functions

• Depends on a single extracted output
• Identifies when one output yi(k) contains
  elements of only one source signal sj(k)
• Do not require significant knowledge about the
  nature of the source PDF
• Define combine system matrix C BA
• ith extracted output signal
• Contrast function is a cost
  function of the distribution of yi(k)
• A local maximum over all elements of Cij, 0lt j lt
  m defines a separate solution
• The normalized kurtosis of the random variable y
  is a good candidate for a contrast function
• 
  

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Correlation Based Criteria

• Assumes that the source signals are statistically
  independent but temporally correlated
• Such that the normalized cross-correlation matrix
• has m unique eigenvalues for some value of l
  0
• This condition yields a separating solution
• Normalized cross-correlation matrix of input
  signals can be simplified to
• By defining the eigenvalue decomposition of
• The demixing matrix B can be calculated as

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Structures and Algorithms for BSS

• Filter Structures
• Design issues
• Room reverberation
• Stability of the separation system
• Computational complexity
• Solution - Finite impulse response (FIR) filters
• The separation equation will be changed into
• L is the systems filter length
• Bl(k), 0 lt l ltL are FIR filter parameters

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Density Matching BSS using Natural Gradient
Adaptation

• The algorithm based on the cost function
• Calculate the gradient of the cost function
• Make differential updates for the filter
  parameters
• Problem- Difficult to calculate the gradient
• Solution - Transform the update into a natural
  gradient adaptation procedure
• Natural gradient method alters the true gradient
  search direction for more efficient adaptation

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Natural Gradient Adaptation

• The conditions on the output sequence y(k)
  corresponding to
• i.e a stationary point of the update.
• 
  1 lt i j lt m, all l
• Where
• This condition implies spatial statistical
  independence of the extracted output signals

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Contrast Based BSS under Prewhitening Constraints

• A contrast function identifies an independent
  source when it is extracted from a linear mixture
• To extract m sources in parallel
• such a procedure does not guarantee that yi(k)
  and yj(k) corresponds to different source signals
  
• To extract all the sources, parameter-dependant
  constraints can be used
• The separation system B(k) is factorized into two
  separation systems
• B(k)W(k)P(k)
• P(k) and W(k) are optimized separately

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x(k)
y(k)
S(k)
v(k)
A
P
W
m
m
n

• Two processing stages
• A prewhitening stage
• The goal of the prewhitening stage is to
  calculate a signal sequence v(k)P(k)x(k), whose
  covariance matrix
• outputs temporally and spatially uncorrelated
  with unit variance
• A separation stage
• in which (m x m) separation matrix W is used to
  extract the individual sources from v(k) , y(k)
  W(k)v(k)
• separation matrix W(k) can be constrained to be
  orthonormal
• This constraint guarantee that each output signal
  yi(k) corresponds to a different source signal,
  when contrast function optimization is performed

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Numerical Evaluations

• The real world signal mixtures used are
• A two channel recording of two male speakers in
  a conference room
• ( fs16kHz sampled,16 bit, t18.75s)
• A moderate level of reverberation
• Nominal level of fan noise
• A two channel recording of a male female a
  cappella duet taken from an audio cd
• ( fs44.1kHz sampled,16 bit, t139.3s)
• Close harmonies with significant spectral overlap
• Both natural and artificial reverberation effects
• Algorithm - density based convolutive BSS
• To evaluate the performance, a signal-to-interfere
  nce ratio (SIR) was calculated for the mixed and
  separated signals

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Speech-Speech Separation Example

• Signal-to-interference and signal-to-noise ratios
  for the mixed and separated signals in the
  speech-speech separation example
• The audio quality of the extracted outputs were
  natural and listenable

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a capella Duet Example

• Spectrograms of the left and the right channels
  before processing over the time interval
  113lttlt114.2 s
• Spectrograms of the outputs of the algorithm over
  the same time interval
• As can be seen the males voice is enhanced in
  the left output channel, whereas the females
  voice is enhanced in the right output channel

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BSS Audio Examples

• The algorithm is based on density modeling
• A speaker has been recorded in a normal office
  room with two distance talking microphones
  (sampling rate 16 khz) with loud music in the
  background
• The distance between the speaker,cassette player
  and the microphone is about 60 cm in a square
  ordering
• Microphone 1 Microphone 2
• Separated Source 1 Separated Source 2

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Conclusions and Open Issues

• Time-varying acoustical environments
• Changes in source number
• Robustness issues
• Separating mixtures containing more sources
  than sensors
• Statistical efficiency of separation methods