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Blind Source Separation: Finding Needles in Haystacks

Scott C. Douglas

Department of Electrical Engineering

Southern Methodist University

douglas@lyle.smu.edu

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