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Blind Source Separation: Finding Needles in Haystacks
Scott C. Douglas
Department of Electrical Engineering
Southern Methodist University
Signal Mixtures are Everywhere
• Cell Phones• Radio Astronomy• Brain Activity• Speech/Music
How do we make
sense of it all?
Example: Speech Enhancement
Example: Wireless Signal Separation
Example: Wireless Signal Separation
Example: Wireless Signal Separation
Example: Wireless Signal Separation
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
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)
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?
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.
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.
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)
Convolutive Blind Source Separation
• Mixing system is dispersive:
• Separation System B(z) is a multichannel filter
Goal of Convolutive BSS
• Key idea: For convolutive BSS, sources are arbitrarily filtered and arbitrarily shuffled
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
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
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
=
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
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.
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.
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.
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)
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
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
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.
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
28
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
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:
Effect of Noise Segment Length on Overall Performance
31
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.
32
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.
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…
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
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.
Complex Fixed Point Algorithm [Douglas 2007]
NOTE: The MATLAB code involves both transposes and Hermitian transposes… and no, those aren’t mistakes!
Performance Comparisons
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
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.
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