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Blind Source Separation: Finding Needles in Haystacks
Scott C. Douglas
Department of Electrical Engineering
Southern Methodist University
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Signal Mixtures are Everywhere
• Cell Phones• Radio Astronomy• Brain Activity• Speech/Music
How do we make
sense of it all?
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Example: Speech Enhancement
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Example: Wireless Signal Separation
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Example: Wireless Signal Separation
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Example: Wireless Signal Separation
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Example: Wireless Signal Separation
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Outline of Talk
• Blind Source Separation General concepts and approaches
• Convolutive Blind Source Separation Application to multi-microphone speech
recordings
• Complex Blind Source Separation What differentiates the complex-valued case
• Conclusions
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Blind Source Separation (BSS) -A Simple Math Example
• Let s1(k), s2(k),…, sm(k) be signals of interest• Measurements: For 1 ≤ i ≤ m,
xi(k) = ai1 s1(k) + ai2 s2(k) + … + aim sm(k)• Sensor noise is neglected• Dispersion (echo/reverberation) is absent
A Bs(k) x(k) y(k)
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Blind Source Separation Example (continued)
A Bs(k) x(k) y(k)
• Can Show: The si(k)’s can be recovered as
yi(k) = bi1 x1(k) + bi2 x2(k) + … + bim xm(k)
up to permutation and scaling factors (the
matrix B “is like” the inverse of matrix A)
Problem: How do you find the demixing bij’s
when you don’t know the mixing aij’s or sj(k)’s?
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Why Blind Source Separation?(Why not Traditional Beamforming?)
• BSS requires no knowledge of sensor geometry. The system can be uncalibrated, with unmatched sensors.
• BSS does not need knowledge of source positions relative to the sensor array.
• BSS requires little to no knowledge of signal types - can push decisions/ detections to the end of the processing chain.
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What Properties Are Necessary for BSS to Work?
Separation can be achieved when (# sensors) ≥ (# of sources) • The talker signals {sj(t)} are statistically-independent
of each other and are non-Gaussian in amplitude
OR have spectra that differ from each other
OR are non-stationary
• Statistical independence is the critical assumption.
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Entropy is the Key to Source SeparationEntropy: A measure of regularity
In BSS, separated signals are demixed and, have “more order” as a group.
First used in 1996 for speech separation.
- In physics, entropy increases (less order)
- In biology, entropy decreases (more order)
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Convolutive Blind Source Separation
• Mixing system is dispersive:
• Separation System B(z) is a multichannel filter
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Goal of Convolutive BSS
• Key idea: For convolutive BSS, sources are arbitrarily filtered and arbitrarily shuffled
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Non-Gaussian-Based Blind Source Separation
• Basic Goal: Make the output signals look non-Gaussian, because mixtures look “more Gaussian” (from the Central Limit Theorem)
• Criteria Based On This Goal: Density Modeling Contrast Functions Property Restoral [e.g. (Non-)Constant Modulus
Algorithm]
• Implications: Separating capability of the criteria will be similar Implementation details (e.g. optimization strategy)
will yield performance differences
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BSS for Convolutive Mixtures• Idea: Translate separation task into
frequency domain and apply multiple independent instantaneous BSS procedures Does not work due to permutation problems
• A Better Idea: Reformulate separation tasks in the context of multichannel filtering Separation criterion “stays” in the time
domain – no implied permutation problem Can still employ fast convolution methods
for efficient implementation
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Natural Gradient Convolutive BSS Alg. [Amari/Douglas/Cichocki/Yang 1997]
where f(y) is a simple vector-valued nonlinearity.Criterion: Density-based (Maximum Likelihood)Complexity: about four multiply/adds per tap
=
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Blind Source Separation Toolbox
• A MATLAB toolbox of robust source separation algorithms for noisy convolutive mixtures (developed under govt. contract)
• Allows us to evaluate relationships and tradeoffs between different approaches easily and rapidly
• Used to determine when a particular algorithm or approach is appropriate for a particular (acoustic) measurement scenario
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Speech Enhancement Methods
• Classic (frequency selective) linear filtering Only useful for the simplest of situations
• Single-microphone spectral subtraction: Only useful if the signal is reasonably well-
separated to begin with ( > 5dB SINR ) Tends to introduce “musical” artifacts
• Research Focus: How to leverage multiple microphones to achieve robust signal enhancement with minimal knowledge.
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Novel Techniques for Speech Enhancement
• Blind Source Separation: Find all the talker signals in the room - loud and soft, high and low-pitched, near and far away … without knowledge of any of these characteristics.
• Multi-Microphone Signal Enhancement: Using only the knowledge of “target present” or “target absent” labels on the data, pull out the target signal from the noisy background.
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SMU Multimedia Systems LabAcoustic Facility
•Room (Nominal Configuration)Acoustically-treatedRT = 300 msNon-parallel walls to prevent flutter echo
•SourcesLoudspeakers playing Recordings as well as “live” talkers.Distance to mics: 50 cmAngles: -30
o, 0
o, 27.5
o
•SensorsOmnidirectional Micro- phones (AT803b)Linear array (4cm spacing)
• Data collection and processing entirely within MATLAB. • Allows for careful characterization, fast evaluation, and experimentation with artificial and human talkers.
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Performance improvement: Between 10 dB and 15 dB for “equal-level” mixtures, and even higher for
unequal-level ones.
Blind Source Separation Example
Convolutive Mixing (Room)
Separation System (Code)
Talker 1
(MG)
Talker 2(SCD)
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Unequal Power Scenario ResultsUnequal Power Scenario Results
Time-domain CBSS Time-domain CBSS methods provide methods provide the greatest SIR the greatest SIR improvements for improvements for weak sources; no weak sources; no significant significant improvement in SIR improvement in SIR if the initial SIR is if the initial SIR is already largealready large
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Noise Source
Noise Source
Speech Source
Linear Processing
AdaptiveAlgorithm
Multi-Microphone Speech Enhancement
Contains most speech
Contains most noise
y1
y2
y3
yn
z1
z2
z3
zn
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Speech Enhancement via Iterative Multichannel Filtering
• System output at time k: a linear adaptive filter
• is a sequence of (n x n) matrices at iteration k.
• Goal: Adapt , over time such that the multichannel output contains signals with maximum speech energy in the first output.
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Multichannel Speech Enhancement Algorithm
• A novel* technique for enhancing target speech in noise using two or more microphones via joint decorrelation
• Requires rough target identifier (i.e. when talker speech is present)
• Is adaptive to changing noise characteristics• Knowledge of source locations, microphone
positions, other characteristics not needed.• Details in [Gupta and Douglas, IEEE Trans.
Audio, Speech, Lang. Proc., May 2009] *Patent
pending
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Performance Evaluations
• Room– Acoustically-treated, RT = 300 ms– Non-parallel walls to prevent flutter echo
• Sources– Loudspeakers playing BBC Recordings
(Fs = 8kHz), 1 male/1-2 noise sources– Distance to mics: 1.3 m– Angles: -30
o, 0
o, 27.5
o
• Sensors– Linear array adjustable (4cm spacing)
• Room– Ordinary conference room (RT=600ms)
• Sources– Loudspeakers playing BBC Recordings
(Fs = 8kHz), 1 male/1-2 noise sources– Angles: -15
o, 15
o, 30
o
• Sensors– Omnidirectional Microphones (AT803b)– Linear array adjustable (4cm nominal
spacing)
6 7
867
8
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Audio Examples
• Acoustic Lab: Initial SIR = -10dB, 3-Mic System
Before: After:• Acoustic Lab: Initial SIR = 0dB, 2-Mic System
Before: After:• Conference Room: Initial SIR = -10dB, 3-Mic System
Before: After:• Conference Room: Initial SIR = 5dB, 2-Mic System
Before: After:
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Effect of Noise Segment Length on Overall Performance
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Diffuse Noise Source Example
• Noise Source: SMU Campus-Wide Air Handling System
• Data was recorded using a simple two-channel portable M-Audio recorder (16-bit, 48kHz) with it associated “T”-shaped omnidirectional stereo array at arm’s length, then downsampled to 8kHz.
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Air Handler Data Processing
• Step 1: Spatio-Temporal GEVD Processing on a frame-by-frame basis with L = 256, where Rv(k) = Ry(k-1); that is, data was whitened to the previous frame.
• Step 2: Least-squares multichannel linear prediction was used to remove tones.
• Step 3: Log-STSA spectral subtraction was applied to the first output channel.
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Complex Blind Source Separation
A Bs(k) x(k) y(k)
• Signal Model: x(k) = A s(k)
• Both the si(k)’s in s(k) and the elements of A are complex-valued.
• Separating matrix B is complex-valued as well.
• It appears that there is little difference from the real-valued case…
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Complex Circular vs. Complex Non-Circular Sources
• (Second-Order) Circular Source: The energies of the real and imaginary parts of si(k) are the same.
• (Second-Order) Non-Circular Source: The energies of the real and imaginary parts of si(k) are not the same.
Non-CircularCircular Circular
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Why Complex Circularity Matters in Blind Source Separation
• Fact #1: It is possible to separate non-circular sources by decorrelation alone if their non-circularities differ [Eriksson and Koivunen, IEEE Trans. IT, 2006]
• Fact #2: The strong-uncorrelating transform is a unique linear transformation for identifying non-circular source subspaces using only covariance matrices.
• Fact #3: Knowledge of source non-circularity is required to obtain the best performance of a complex BSS procedure.
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Complex Fixed Point Algorithm [Douglas 2007]
NOTE: The MATLAB code involves both transposes and Hermitian transposes… and no, those aren’t mistakes!
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Performance Comparisons
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Complex BSS ExampleOriginal Sources
SensorSignals
16-elem ULA, /4Spacing 3000 Snapshots SINRs/elem: -17,-12,-5,-12,-12 (dB) . DOAs(o): -45,20,-15,49,35
CFPA1Outputs
Output SINRs (dB):7,24,18,15,23
Complexity: ~3500 FLOPSper output sample
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Conclusions• Blind Source Separation provides unique
capabilities for extracting useful signals from multiple sensor measurements corrupted by noise.
• Little to no knowledge of the sensor array geometry, the source positions, or the source statistics or characteristics is required.
• Algorithm design can be tricky. • Opportunities for applications in speech
enhancement, wireless communications, other areas.
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For Further Reading
My publications page at SMU:
http://lyle.smu.edu/~douglas/puball.html
• It has available for download • 82% of my published journal papers• 75% of my published conference papers