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An Introduction to Blind Source Separation

An Introduction to Blind Source Separation. Kenny Hild Sept. 19, 2001. Problem Statement. Communication System Transmitter Medium Receiver Data sent is unknown Transfer function of medium may be unknown Interference. Possible Solutions. Beamforming Uses geometric information

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An Introduction to Blind Source Separation

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  1. An Introduction to BlindSource Separation Kenny Hild Sept. 19, 2001

  2. Problem Statement • Communication System • Transmitter • Medium • Receiver • Data sent is unknown • Transfer function of medium may be unknown • Interference

  3. Possible Solutions • Beamforming • Uses geometric information • Steer antenna array to a desired angle of arrival • Filtering • Separate based on frequency information • Blind Source Separation, BSS • Statistical beamforming • Steer antenna array to directions based on statistics

  4. Beamforming s1(n) • Suppose • Direction of arrival is 00 azimuth • s1(n) = s2(n) = cos(wn) • Transfer functions are pure delays • Then • y(n) = x1(n) + x2(n-a), a = 0 • y(n) = cos(wn) + cos(wn) + cos(wn + f1) + cos(wn + f2) • y(n) = 2cos(wn) + 2cos((f1-f2)/2)cos(wn + (f1+f2)/2) s2(n) x1(n) x2(n)

  5. Filtering • Suppose • Signal is low pass, noise is white • Signals are bandpass • Then • Design LPF to remove high frequencies signal noise signal #2 signal #1 Frequency (normalized rad/s)

  6. Assumptions • Signals • Overlap in time • Angle of arrival is unknown – prevents beamforming • Overlap in frequency – prevents filtering • Blind Source Separation • Does not assume knowledge of DOA • Does not require signals to be separable in frequency domain

  7. Applications • Early diagnosis of pathology in fetus • Each EKG sensor contains a mixture of signals • Desire is to separate out fetus’ heartbeat • Hearing aids • Speech discrimination difficult with multiple speakers • The observations are the signals at each ear • Cellular communications • CDMA signals utilize overlapping frequency ranges • Additional signals, multi-path deteriorate performance

  8. Types of Mixtures • Memory • Instantaneous • Convolutive • Noise • Linearity • Over/under-determined

  9. Components of Adaptive Filter • Topology • Instantaneous • Convolutive • Criterion • Optimization method • Gradient descent • Fixed point

  10. Topology • Over-determined, linear mixture • N > M • H, W are matrices of ARMA filters • Types of topologies • Frequency-domain • Time-domain • Feedforward • Feedback • Lattice

  11. Topology • For Instantaneous Mixtures • H, W are matrices of constants • Often W is broken down into 2-3 operations • Dimension reduction, (N x M) matrix D • Spatial whitening, (N x N) matrix W • Rotations, (N x N) matrix R W = RWD (N x M) x = Hs (M x 1) y = Wx = WHs (N x 1)

  12. Topology • Spatial whitening • Makes outputs uncorrelated • This is insufficient • For separation • 4 possible rotations s2(n) y2(n) y1(n) s1(n)

  13. Criterion • Spatial whitening • x = Wx • E[xxT] = IN • W = FxLx-1/2 • J = SiSj (Rx(i,j) – IN(i,j))2 Rx=

  14. Criterion • Indeterminacies • Gain • Permutation • Rotations • Find characteristic of sources that is not true for any mixture

  15. Criterion • Nullify correlations • Between nonlinear functions of the outputs • Nonlinearity can be most any odd function • Cubic • Hyperbolic tangent • Requires source pdf’s to be even-symmetric • Non-linear PCA • If data is sphered, stable points are ICA solution • Minimizes joint entropy of nonlinear functions of outputs

  16. Criterion • Cancellation of HOS • 4th-order (kurtosis) is most common • If y1, y2, y3, y4 can be separated into 2 groups that are mutually independent, 4th-order cumulant is zero • Must check all 4th-order cumulants • Statistical properties of cumulant estimators are poor • Central limit theorem • Sum of independent, non-Gaussian sources approaches Gaussian • Maximize (K-L) distance between marginal pdf and Gaussian • Must know/estimate the kurtosis for each source

  17. Criterion • Maximum Likelihood • Must know/assume source distributions • Minimize K-L divergence between output pdf’s and known/assumed source pdf’s • Sensitive to outliers, model mismatch • Maximize the information flow • Maximize joint entropy of outputs (of the nonlinearities) • Nonlinearities should be source cdf’s • Equivalent to maximum likelihood

  18. Criterion • Mutual statistical independence • Oftentimes sources are independent • Uncorrelatedness does not imply independence • Canonical criterion • Difficult to estimate • Solution includes an infinite-limit integral • Marginal pdf’s estimated by truncated expansion about Gaussian

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