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Transcript of Charla Colombia
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ADAPTIVE BLIND SIGNALPROCESSING:
Blind Source Separation
Guillermo Bedoya
UNIVERSITATPOLITCNICA DE
CATALUNYAUPC INPGrenoble
Grup d Arquitectures
Hardware Avanades[AHA]Laboratoire des Images et
des Signaux [LIS]
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1. Introduction to Blind Signal Processing: Problems andapplications
2. Blind Source Separation (BSS) : Statistical principles
3. Application of Information theory to BSS
Coffee Break
4. Adaptive Learning Algorithms forBSS
5. BSS of Nonlinear mixing models
6. Hardware considerations
OUTLINE
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1. INTRODUCTION TO BLINDSIGNAL PROCESSING
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Objective:
To discuss the basic principles of Blind Source Separation
in the context of Blind Signal Processing.
INTRODUCTION
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DEFINITION
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
For many authors the so-called Blind signal processing techniques
include:
BLIND SOURCE SEPARATION (BSS)
Blind Signal Extraction (BSE)
Blind Source Deconvolution (BSD)
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DEFINITION
Blind Source Separation (BSS) and Independent Component
Analysis (ICA) are emerging techniques ofarray processing
and data analysis that aim to recoverunobservedsignals orsources from observed mixtures (typically, the output of anarray of sensors), exploiting only the assumption of MutualIndependence between the signals.
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
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DEFINITION II
BSS / ICA
Algorithm
Environment
Sensor Array
Signal processing
s1
s2
s3
s1
s2
s3
x1
x2
x3
Speakers Microphones
s y = x
A B
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
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APPLICATIONS
Radiating Sources Estimating: Applications to Airport Surveillance.
Blind Beamforming.
Blind Separation of multiple co-channel BPSK signals arriving to antenna arrays.
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
Processing of Communication Signals:
Biomedical Signal Processing:
ECG & EEG.
Monitoring:
Multitag Contactless Identification Systems. Power Plant monitoring.
Environmental and Bio-medical Chemical Detection and Identification.
Alternative to Principal Components Analysis
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PRINCIPLES
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
The simplest BSS Model: NoiselessLinear Instantaneous Mixing.
We assume the existence ofj independent signals s1(t),,sj(t) and the
same number of observations (or observed mixtures) x1(t),,xn(t),
where number of sensors = number of sources, expressed as:
xi[t] a11s1[t]++a1jsj[t]
xn[t] ai1sj[t]++aijsj[t] For each i = 1, n.
and i=j
s1 x1
A =
Mixing Matrix
(sensor array)
s2
sj
x2
xn
x[t]=As[t]
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PRINCIPLES II
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
s1
sn
y1
yn
= s y =x
A B
Unobserved signals Observations Estimated source signals
=
x[t]=(x1
[t]xn[t])=As[t] y[t]=(y
1
[t]yn[t])=Bx[t]
B = A-1
y=BAs, C=BA, C=PD
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PRINCIPLES III
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
The BSS problem exists in recovering the sources vectors(t) using only:
The observed data x(t).
The assumption of mutual independence between the entries of
the input vectors(t).
Some prior information about the probability distribution of the
inputs.
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ECG
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
APPLICATIONS (ECG)
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CDMA System with N-users:
La seal recibida x es lasuperposicin de las seales
ensanchadas de los N usuarios msruido aditivo gaussiano.
b1
b2
bN
.
.
.
code1
code2
codeN
L chips
P1
P2
PN
NOISE
L chips
1L1NNL1L nbx
NLH
BSS permite recuperar b a partir de x sin conocer H independenciaestadsitica
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
APPLICATIONS (CDMA)
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APPLICATIONS (Image processing)
Image
decomposition
ICA
B
Components
processing
Image
Composition
I xt yt
X=
I=n
m/kx n/k
kx k
m
xt
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
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APPLICATIONS (Image processing)
1. INTRODUCTION TO BLIND SIGNAL PROCESSING
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2. BLIND SOURCE SEPARATION:STATISTICAL PRINCIPLES
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STATISTICAL MODEL
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
BSS exploits:
SPATIAL DIVERSITY: BSS looks for structure across the sensors, not across
the time.
SAMPLES DISTRIBUTION.
Statistical Model:
MIXING MATRIX: Its columns are assumed to be linearly independent (so that
it is invertible)
SOURCE DISTRIBUTION: the distribution of each source is a Nuisance
parameter, i.e., we are not primarily interested in it...
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LINEAR MIXING
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
If each source is assumed to have apdf denoted qi(.), the joint pdfq(s) of the
source vectors is:
q(s)=q1(s1) xx qn(sn)= qi(si)
i=1,n
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LINEAR MIXING
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
s1
s2
x1
x2
y1
y2z2
z1
x = As
p1(s1) p2(s2)
z = Wx
1.Whiteningy=Uz
2. Rotation
s1
s2
Joint PDF
p(s1, s2)
TE zz I
marginal PDF
General approach
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CONSTRAINTS
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
whitening
Rotation
Joint PDF of different mixing
matrices for two signals with uniform
distribution
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CONSTRAINTS
Joint PDF of different mixing matrices for twosignals with Gaussian distribution
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
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INDEPENDENCE & DECORRELATION
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
Intuitivo Las variables y1 e y2son independiente si el valor de y1no aporta
ninguna informacin acerca del valor de y2.
Matemtico: funciones de densidad de probabilidad (pdf) Seap(y1,y2) la funcin de densidad de probabilidad conjunta de y1 e y2.
Seap1(y1) la pdf marginal de y1
y de forma similar para y2
Entonces, y1 e y2 son independientes si y slo si:
1 1 1 2 2( ) ( , )p y p y y d y
1 2 1 1 2 2( , ) ( ) ( ).p y y p y p y
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INDEPENDENCE & DECORRELATION
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
Concept of Independence
Math: Ify1 and y2 are independents
Uncorrelatedness: h(), g() are equal to the Identity
1 2 1 2( ), ( ) ( ) ( ) .E g y h y E g y E h y
1 2 1 2, 0E y y E y E y
Uncorrelatedness Independence
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INDEPENDENCE
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
Statistical measurements of Independence and Gaussianity:
Kullback-Leibler Divergence (Signal distribution)
Kurtosis or fourth order cumulant (Signal Gaussianity)
Negentropy
( )
| ( ) log ( )
f
K f g f dg
y
y yy
24 2kurt( ) 3y E y E y
4kurt( ) 3y E y
Mutual
Information (I) &
Entropy (H)
;
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OBJECTIVE (CONTRAST) FUNCTIONS
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
BSS ALGORITHM:
Contrast Function Optimization Method+
SOME CONTRAST FUNTIONS:
Maximum Likelihood
Mutual Information or Entropy Maximization
Orthogonal Contrast
INFOMAX
High-Order approximations
Non linear cross correlations
Minimize or maximize
the measurement
Measurement
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OBJECTIVE (CONTRAST) FUNCTIONS
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
Properties of the BSS/ICA method depend on both of the
elements (Contrast function and optimization method). In
particular:
The statistical properties (e.g., consistency, asymptotic variance,robustness) of the ICA method depend on the choice of the objective
function.
The algorithmic properties (e.g., convergence speed, memoryrequirements, numerical stability) depend on the optimizationalgorithm.
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ENTROPY MAXIMIZATION CONTRAST
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
MUTUAL INFORMATION OR ENTROPY MAXIMIZATION
Where y is an independent components vector
Considering the whitening constraint:
The entropy of y, H[y], is invariant under rotations.
Separating algorithm.
|ME K y y y
cte.
1 1
N No
ME i i
i i
H y H H y
y y
min MEB
Bx
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MAXIMUM LIKELIHOOD CONTRAST
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
Maximum Likelihood:
Where the PDF of = A-1x is known.
The Maximum Likelihood contrast, maximizes the
independence of the output components and minimizes
the DISTANCE to the PDF of the data.
1 1
( )| ( ) log
( ) ( )ML
N N
pK p d
q s q s
yy y s y y
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KURTOSIS
Non gaussianity measurement (Central Limit Theorem)
Kurtosis (fourth-order cumulant)
Ify is unit variance
kurtosis = 0 for random Gaussian variables.
2
4 2kurt( ) 3y E y E y
4kurt( ) 3y E y
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
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KURTOSIS
Laplacian PDF
Pos. Kurtosis
leptokurtic
SuperGaussian
UNIFORM PDF
Negative Kurtosisplatykurtic
SubGaussian
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
21( )
2
yp y e
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3. APPLICATION OFINFORMATION THEORY TO BSS
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INTRODUCTION
3. APPLICATIONS OF INFORMATION THEORY TO BSS
Objective:
To discuss the INFOMAX criterion and apply it to the
problem of BSS
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INTRODUCTION TO INFORMATION THEORY
3. APPLICATIONS OF INFORMATION THEORY TO BSS
The binary Entropy function
The entropy of a random variableXis:
SupposeXis a binary random variable,
Then the entropy ofXis,
Since it depends onp, this is also writen sometimes as H(p)
H(X)=- p(x) logp(x)i=1
N
H(X)=- p logp - (1-p) log(1-p)
X=1 with probabilityp
0 with probability 1-p
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INTRODUCTION TO INFORMATION THEORY
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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INTRODUCTION TO INFORMATION THEORY
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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RELATIVE ENTROPY AND MUTUAL INFORMATION
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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H(.) and I(.) applied to BSS I
3. APPLICATIONS OF INFORMATION THEORY TO BSS
si(t) y(t)x(t)
A B
Let si (t), i=1,2,,n be a set of statistically independent signals,
x(t)=As(t)
We desire to determine matrix B so that: y(t)=Wx(t)=WAs(t)
recovers s(t) as fully as possible. We will take as a criterion the MutualInformation at the output H(y)
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H(.) and I(.) applied to BSS II
H(y)= H(yi) - I(y1,,yN)i=1
N
If we maximize H(y),we should :
1. Maximize each H(yi)
2. Minimize I(y1,,yN)
H(yi) are maximized when (and if) the otputs are uniformly distributed.
The mutual information is minimized when the are all independent!
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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393. APPLICATIONS OF INFORMATION THEORY TO BSS
H(.) and I(.) applied to BSS III
How to work with the outputs mutual information ?
We consider the case of adapting a processing function gwhich operates on
the scalarXusing a function Y = g(X) in order to maximize or minimize the
mutual information betweenXand Y.
But, achieving the MI miminimization and consecuently the
independence, requires that ghave the form of the Cumulative Density
Function (CDF) ofsi.
y(t)x(t) B g
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403. APPLICATIONS OF INFORMATION THEORY TO BSS
H(.) and I(.) applied to BSS IV
g(X) = g(X,b) g(x)=1/1+e-bx
We may not know the pdf of X, however, we can assume a particularfunction form, e.g.,Assuming thatp(si) is super-Gaussian :
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H(.) and I(.) applied to BSS V
Based on the last assumption (p(si) super-Gaussian), we can write:
H(yi)= -E[logp(yi)]
where we have
p(yi)=p(bxi) / yi/ bxi so that
H(yi)= -E[logp(bxi) / yi/ ui]and
H(y)= H(yi) - I(y)i=1
N
H(y)=- E[logp(bxi) / yi/ ui ] - I(y)i=1
N
;
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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H(.) and I(.) applied to BSS V
We want to determine B to maximize the joint entropy of the output H(y). So, an
adaptive scheme is to take
b
Hb =
In our specific case we have,
g(x) = y= 1/1+e-bx ; = by(1-y) ; = y(1-y)[1+by(1-2y)]
b
(ln y/ x) = yx
-1
b
yx
3. APPLICATIONS OF INFORMATION THEORY TO BSS
yx b
yx
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From,
b =
b = y(1-y)[1+by(1-2y)]
so,
b 1/b + x(1-2y) ; b = b-1 + (1-2y) x
And the weight update rule can be
b[k+1] = b[k] + b b
yx
-1
byx
yx
-1
Score function
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General form
H(y)=- E[logp(bxi) / yi/ ui ] - I(y)i=1
NFrom the term:
H(y)B = B -T - (u) xT
(u)= -
p(u)u
p(u)
Score function
3. APPLICATIONS OF INFORMATION THEORY TO BSS
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METHOD OVERVIEW
2. BLIND SOURCE SEPARATION: STATISTICAL PRINCIPLES
s yx
A B
1. Initialization of the separating matrix B (randomly).
2. Generation of the outputs y. y=Bx
3. Measurement of the outputs independence (contrast function).
For count = 1 to Number of iterations (nit):
Change the matrix B : ( ) and repeat until nit according to:
B[k+1] = B[k] + bB
y= B[k+1] x
4. End of algorithm.
H(y)B
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4. ADAPTIVE LEARNINGALGORITHMS FOR BSS
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ADAPTIVE LEARNING
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
BSS Framework:
En espacios eucldeos, la ley de aprendizaje basada en el gradiente
(estocstico) permite alcanzar una solucin:
El algoritmo es estable cuando el contraste es mnimo. Problema
el espacio de las matrices invertibles B no es eucldeo.
1( )t t t B B B
s yx
A B
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GRADIENT TECHNIQUES
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
Gradiente convencional: espacio eucldeo
la transformacin infinitesimal de B se expresa como
Gradiente relativo:
la transformacin infinitesimal de B se expresa como
B I B B B
B B
B
B B
B
Espacio de las
matrices de separacin
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RELATIVE GRADIENT
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
Gradiente convencional: espacio eucldeo
Gradiente relativo:
( | () ( ) ( ) )o B B B
, 1
donde |N
T
ij ij
i j
Traza A B
A B A B y ( )ijB
B
( | () ( ) ( ) )o B B B B B ( | () ( ) ( ) )o B B B B
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RELATIVE GRADIENT
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
Comparacin:
( ) ( ) T B B B
BB B
B
Espacio de las
matrices de separacin
( ) B
( ) B
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NATURAL GRADIENT
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
Gradiente natural:
Planteamiento: encontrar dB que minimiza
Comparacin:
( |) ( ) ( )d d B B B B B2 2teniendo en cuenta que d B
BB B
B Espacio de las
matrices de separacin
( ) B( ) B
min ( )d
d B B BdB B
( ) B gradiente natural
( ) ( ) ( ) T B B B B B B
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LEARNING ALGORITHMS
4. ADAPTIVE LEARNING ALGORITHMS FOR BSS
Algoritmo basado en el Gradiente convencional:
En el caso de emplear ML:
Teniendo en cuenta que
Por tanto
y que
1 ( )t t t B B B
-1
1( )
TT
t t t B B y y I B
1( ) log ( ) ( )= = ( )( )
y k ki i
j i i jij i i
p q y d q yx y
B q y
y y B y
1 1
( )( ) | ( ) ( ) log
( ) ( )
y
ML ML y y
N N
pK p q p d
q s q s
yy Bx y s y y
.( ) ( ) log ( ) log det( )
cte
y yH p p d y y y y B
1( )T
H B y B
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LEARNING ALGORITHMS
Algoritmo basado en el Gradiente natural:
En el caso de emplear:
Algoritmo basado en el gradiente relativo
BB B
B Espacio de las
matrices de separacin
( ) B( ) B
min ( )d
d B
B BdB B
( ) B gradiente natural
1 ( )t t t B B B
1 ( )T
t t t B B y y I B
ML y
1, ,
( )siendo ( ) "score function"
( )T i i
i i i N
q y
q y
y
1( )T
t t
B B y y I
( ) ( ) ( ) T B B B B B B
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5. BSS OF NON LINEAR MIXINGMODELS
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THE POST NON-LINEAR MODEL
Given a set of sources s linearly mixed by means the matrix A, we have thelinear mixing part described by:
The invertible non-linear transfer function f(.) is represented by:
were e represents the observations at the output of the sensors system. The
sources s are recovered if the non-linear function gi(ei) and the de-mixing matrix
B are the inverse functions offi(xi) and A respectively.
5. BSS OF NON-LINEAR MIXING MODELS
sAx
)(xfe
Af1
f2
x1
x2
s1
s2
e1
e2
g1
g2
y1
y2
B
u1
u2
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THE POST NON-LINEAR MODEL
Consequently, the signals are related by the equation:
where,
The output is described by:
In order to separate the observations into statistically independent components,
we use a measure of the dependence degree of the components of u. When
this measure ofu reaches an absolute minimum, we ensure the independence
of the components.
5. BSS OF NON-LINEAR MIXING MODELS
)(egy)(( sAfgy
)(( sAfgWyWu
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THE POST NON-LINEAR MODEL
5. BSS OF NON-LINEAR MIXING MODELS
The mutual information I(.), can be employed as independence measure of the
components ofu. The mutual information (MI) of the output, can be expressed
via the signal entropy H:
where,
The MI is non-negative and zero when the components of u are statistically
independent from one other.
The MI has a property that, if we perform invertible transformations on theindividual components ofu, resulting in zi= i(ui), the mutual information of the
components zi is equal to the mutual information of the components ui.
i
HiyHI )()()( uu
uuuu dppH )(log)()(
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REALISTIC EXAMPLE
5. BSS OF NON-LINEAR MIXING MODELS
Linear mixing
stage
Nonlinear
distortion
Nonlinear
compensation
Linear de-mixing
stage
A B
f1
f2
g1
g2
ai
aj
i
j
ID1
ID2
gID1
gID2
ID= a+ b*ln(ai + Kijaj ) ;zi/zj
gID = e(IDa)/ b
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1. Initialization B = I, gID = ID, a and b.
2. Loop
1. Compute outputs by: y =Bx;
2. Estimation of parameters
a[k+1]= a[k]+ stepsize a
b[k+1]= b[k]+ stepsize b
3. Normalization
4. Linear BSS algorithm: b[k+1]=b[k]+ stepsize b
5. Repeat until convergence
ALGORITHM
5. BSS OF NON-LINEAR MIXING MODELS
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6. HARDWARECONSIDERATIONS
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OVERVIEW
HARDWARE IMPLEMENTATION:
DSPbased implementation
FPGA
Fully Analog system Integration
HYBRID INTEGRATION
ASIC with a DSP core associated to the sensor array.
6. HARDWARE CONSIDERATIONS
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DSP IMPLEMENTATION
A higher flexibility at a low cost system can be obtained by using a DSP for
software fast algorithm implementation
Finite Precision errors
Fixed point - floating point
Number of bits
Analog to Digital conversion
Finite word-length used to store all internal algorithmic quantities
6. HARDWARE CONSIDERATIONS
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DSP ARCHITECTURE
6. HARDWARE CONSIDERATIONS
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ANALOG CMOS CIRCUIT IMPLEMENTATION
6. HARDWARE CONSIDERATIONS
The INFOMAX algorithm finds the un-mixingmatrix B by means of maximizing the joint
entropy H(y) of the outputs. Its learning rule
(using te natural gradient) is:
B BBT = [I-(u)uT]BH(y)
B
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MULTIPLIER CIRCUIT
The rule can be writen as:
u = BX; SUM=uT B
B = [B- (u)SUM]
6. HARDWARE CONSIDERATIONS
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WEIGHT UPDATE CIRCUIT
6. HARDWARE CONSIDERATIONS
Vout = outp-outm = [ (Vc1 - Vc2)/sC](vip - vim)
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PROPUESTA DE TRABAJO !
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PROPUESTA
IMPLEMENTACION HARDWARE DE UN ALGORITMO DE SEPARACION CIEGA
DE FUENTES APLICADO A PROCESADO DE VOZ.
REQUISITOS BASICOS:
2 Estudiantes (buen nivel de Ingls).
Conocimientos de MATLAB, C/C++ y Assembler.
Conocimientos de Estadstica Multivariable.
Tarjeta y plataforma de desarrollo para DSP.
DURACION Y FORMA DE TRABAJO:
5 Meses. El algoritmo ya esta diseado y Listo!
Cx via INTERNET (recoleccin bibliogrfica, algoritmia,
etc) y dirigido por docente CUTB.
IMPLEMENTACION HARDWARE DE UN ALGORITMO DE SEPARACION
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CIEGA DE FUENTES APLICADO A PROCESADO DE VOZ.
PLAN GENERAL DE TRABAJO:
Recoleccin Bibliogrfica y contextualizacin (- 20 dias).
Simulacin en MATLAB del algoritmo
Separar dos o ms seales de voz (- 10 dias).
Simulacin del algoritmo en C/C++
Separar dos o ms seales de voz (- 30 dias).
Desarrollo del algoritmo en la Tarjeta DSP (90 dias).
Seleccion y adquisicin de la tarjeta de desarrollo.
Definicin de los parmetros de diseo.
Generacin de reportes cada 20 dias (cada uno ser un
captulo del proyecto de final de carrera).
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GRACIAS !
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ADAPTIVE BLIND SIGNALPROCESSING:
Blind Source Separation
Guillermo Bedoya
UNIVERSITATPOLITCNICA DE
Grup d Arquitectures
Hardware Avanades [AHA]
Laboratoire des Images et
des Signaux [LIS]