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Multi-channel speech enhancement Chunjian Li DICOM, Aalborg University 3/24/2006 Lecture notes for Speech Communications Methods & applied fields Dual-channel spectral subtraction - noise reduction in speech Adaptive Noise Canceling (ANC) - noise reduction and interference elimination - echo canceling - adaptive beamforming Blind Source Separation (BSS) Blind Source Extraction (BSE) 3/24/2006 Lecture notes for Speech Communications Dual-channel spectral subtraction - Hanson and Wong, ICASSP84. 3/24/2006 Lecture notes for Speech Communications The method The exponent is chosen to be a=1 based on listening test and spectral distortion measure. The noisy phase is used in the reconstruction of signal. The estimate of noise spectrum is either obtained from a reference channel or estimated from the noisy signal assuming the SNR is very low (about -12 dB). 3/24/2006 Lecture notes for Speech Communications Revisiting the phase issue To see the dependency of magnitude on phase: 1 a Sˆ ( f ) S ( f ) N ( f ) Nˆ ( f ) e j ( f ) a a S( f ) a ˆ S ( f ) N ( f ) 1 N( f ) 2 S( f ) 2 N( f ) a cos() Nˆ ( f ) a 2 1 a where is the phase difference between the two signals. It is clear that the estimate of signal magnitude spectrum depends on both the SNR and the phase difference. But phase is not estimated in this method because the enhanced quality is acceptable. 3/24/2006 Lecture notes for Speech Communications Comments The simplest (and a bit unrealistic) form of exploiting multi-channel. Aims at improving intelligibility. Significant intel. gains only at very low SNR (-12dB). Unvoiced speech is not processed. 3/24/2006 Lecture notes for Speech Communications Adaptive Noise Canceling First proposed by Widrow et al. [1] in 1975. It is adaptive because of the use of adaptive filter such as the LMS algorithm. The objective: estimate the noise in the primary channel using the noise recorded in the secondary channel, and subtract the estimate from the primary channel recordings. [1] B. Widrow, J. R. Grover, J. M. McCool et al. ”Adaptive noise canceling: Principles and applications,” Proceedings of the IEEE, vol.63, pp. 1692-1716, Dec. 1975. 3/24/2006 Lecture notes for Speech Communications Signal model 3/24/2006 Lecture notes for Speech Communications Signal estimation The estimated signal: sˆ(n) y(n) dˆ1 (n) M 1 dˆ1 (n) hˆ(i )d 2 (n i ) i 0 The optimization criterion: hˆ arg min y(n) hˆ(i)d 2 (n i) h i 0 M 1 3/24/2006 Lecture notes for Speech Communications 2 Signal estimation The minimization can be solved by applying the orthogonality principle: M ryd2 ( ) hˆ(i)rd 2 ( i) 0 i 0 This can be solved in the same way as solving the normal equations. But it is usually solved by sequential algorithms such as the LMS algorithm. The advantages of the LMS are: -No matrix inversion, low complexity -Fully adaptive, suitable to non-stationary signal and noise -Low delay 3/24/2006 Lecture notes for Speech Communications LMS -It is a sequential, gradient descent minimization method, - The estimate of the weights is updated each time a new sample is available: hˆ k hˆ k 1 k g Where the element of the gradient vector: g 3/24/2006 ( ) M 1 2ryd2 ( ) hˆ(i)rd2 ( i) hˆ( ) i 0 Lecture notes for Speech Communications LMS Or, in matrix form: g 2(ryd2 R d 2 hˆ ) The most important trick is, in this sequential implementation, to approximate the correlation matrix and cross-correlation vector by The instantaneous estimates. ˆ d dH 2 R d2 2 ryd2 d2 y(n) 3/24/2006 Lecture notes for Speech Communications LMS The step size is often chosen empirically, as long as the following condition is satisfied for stability reason: 0 1 max where max is the largest eigenvalue of the matrix R d 2 The larger the step-size, the faster the convergence, but also the larger estimation variance. 3/24/2006 Lecture notes for Speech Communications Comments The LMS belongs to the stochastic gradient algorithm. The algorithm is based on the instantaneous estimates of correlation function, which are of high variance. But the algorithm works well because of its iterative nature, which averages the estimate over time. Low complexity: O(M), where M is the filter order. Although the derivation is based on WSS assumption, the algorithm is applicable to stationary signals, due to the sequential implementation. 3/24/2006 Lecture notes for Speech Communications Implementation issues of ANC Microphones must be sufficiently separated in space or contain acoustic barriers. Typically 1500 taps are needed => large misadjustment => pronounced echo => must use small step-size => long convergence time. Different delays from the sources to the two microphones must be taken care of. Frequency domain LMS can reduces the number of taps needed. ANC can be generalizes to a multi-channel system, which can be seen as a generalized beamforming system. 3/24/2006 Lecture notes for Speech Communications Eliminating cross-talk Cross-talk: If the signal is also captured in the reference channel, the ANC will suppress part of the signal. Cross-talk can be reduced by employing two adaptive filter within a feedback loop. 3/24/2006 Lecture notes for Speech Communications Beamforming Compared to ANC, beamforming is truly a spatial filtering technique. First, locate the source direction; then form a beam directing to the source. The source location problem is a analogy of the spectral analysis problem, with the frequency domain replaced by the spatial domain. 3/24/2006 Lecture notes for Speech Communications A simple array model Planar wave Uniform linear array Sensors responses are identical and LTI Sensors are omni directional One parameter to estimate: DOA 3/24/2006 Lecture notes for Speech Communications ULA 3/24/2006 Lecture notes for Speech Communications ULA The signal model: y(t ) a( )s(t ) e(t ) where the array transfer vector : a( ) 1 e jc 2 ... e jc m T Where m is the delay with reference to the first sensor, and c is the center frequency of the signal. By defining the spatial frequency as: s c d sin c we can write the array transfer vector as: a( ) 1 e 3/24/2006 js ... e j ( m 1)s T Lecture notes for Speech Communications ULA A direct analogy between frequency analysis and spatial analysis using the spatial frequency. To avoid spatial aliasing: d /2 All frequency analysis techniques can be applied to the DOA estimation problem. 3/24/2006 Lecture notes for Speech Communications Spatial filtering Analogy between spatial filter and temporal filter 3/24/2006 Lecture notes for Speech Communications Spatial filtering The spatially filtered signal: x(t ) h*a( )s(t ) Objective: find the filter that passes undistorted the signals with a given DOA; and attenuates all the other DOAs as much as possible. min h*h subject to h*a( ) 1 h 3/24/2006 Lecture notes for Speech Communications The beam pattern 3/24/2006 Lecture notes for Speech Communications Restrictions to beamforming Very sensitive to array geometry, need good calibration Has only directivity, no selectivity in range or other location parameters Frequency response is not flat Ambient noises are assumed to be spatially white Beam width (or selectivity) depends on the size of the array Spatial aliasing problem 3/24/2006 Lecture notes for Speech Communications Blind Source Separation (BSS) MIMO systems Spatial processing techniques with no knowledge of array geometry Invisible beam Arbitrarily high spatial resolution Do not depend on signal frequency Spatial noise is not assumed to be white Not a spatial sampling system 3/24/2006 Lecture notes for Speech Communications Solutions to BSS Independent Component Analysis (ICA) [2] Independent Factor Analysis (IFA) [3] [2] A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons, Inc. 2001 [3] H. Attias, “Independent factor analysis”, Neural Computation, 1999. 3/24/2006 Lecture notes for Speech Communications Independent component analysis (ICA) Instantaneous mixing The number of sensors is greater than or equal to the number of sources No system noise The sources (components) are independent of each other The sources are non-Gaussian processes 3/24/2006 Lecture notes for Speech Communications ICA model Cocktail party problem. Three sources, three sensors: x1 (t ) a11s1 (t ) a12 s2 (t ) a13 s3 (t ) x (t ) a s (t ) a s (t ) a s (t ) 22 2 23 3 2 12 1 x3 (t ) a31s1 (t ) a32 s2 (t ) a33 s3 (t ) Or, in matrix form x As Neither s nor A are known. Can not be solved by linear algebra. If the sources are independent non-Gaussian, the A matrix can be found by maximizing the non-Gaussianity of the sources. 3/24/2006 Lecture notes for Speech Communications Contrast function An iterative gradient method. First initialize the A matrix. If the mixing matrix A is square and non-singular, move it to the left: A 1x s Calculate the non-Gaussianity of s, and find the next estimate of A that gives a higher non-Gaussianity. Iterate until convergence. The contrast function is the objective function to maximize or minimize. 3/24/2006 Lecture notes for Speech Communications Maximizing non-Gaussianity Non-Gaussian is independent Measuring non-Gaussianity - by kurtosis - by negentropy 3/24/2006 Lecture notes for Speech Communications ICA methods ICA by maximizing non-Gaussianity ICA by Maximum Likelihood ICA by minimizing mutual information ICA by nonlinear decorrelation 3/24/2006 Lecture notes for Speech Communications Extensions to ICA Noisy ICA ICA with non-square mixing matrix Independent Factor Analysis Convolutive mixture Methods using time structure 3/24/2006 Lecture notes for Speech Communications Blind Source Extraction Only interested in one or a few sources out of many (feature extraction) Save computation Don’t know the exact number of sources 3/24/2006 Lecture notes for Speech Communications BSE D. Mandic and A. Cichocki, An Online Algorithm For Blind Extraction Of Sources With Different Dynamical Structures. 3/24/2006 Lecture notes for Speech Communications