multi-channel speech enhancement (1).ppt

3/24/2006 Lecture notes for Speech Communications

Multi-channel speech enhancement

Chunjian LiDICOM, Aalborg University


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)


Dual-channel spectral subtraction

- Hanson and Wong, ICASSP84.


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).


Revisiting the phase issue

a

a

a

a

fjaaa

fNfNfS

fNfS

fNfS

efNfNfSfS

1

22

)(

1

)(ˆ)cos()()(

2)()(

1)()(ˆ

)(ˆ)()()(ˆ

To see the dependency of magnitude on phase:

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.


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.


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.


Signal model


Signal estimation

)(ˆ)()(ˆ 1 ndnyns

1

021 )()(ˆ)(ˆ

M

i

indihnd

The optimization criterion:

The estimated signal:

21

02 )()(ˆ)(minargˆ

M

ihindihnyh


Signal estimation

The minimization can be solved by applying the orthogonality principle:

0)()(ˆ)(22

0

irihr d

M

iyd

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


LMS

ghh kkk 1ˆˆ

-It is a sequential, gradient descent minimization method,

- The estimate of the weights is updated each time a new sample is available:

Where the element of the gradient vector:

1

0

)( )()(ˆ)(2)(ˆ 22

M

idyd irihr

hg


LMS

222ˆ Hd ddR

)ˆ(222hRrg dyd

The most important trick is, in this sequential implementation, to approximate the correlation matrix and cross-correlation vector byThe instantaneous estimates.

Or, in matrix form:

)(22nyyd dr


LMS

max

10

max

The step size is often chosen empirically, as long as the following condition is satisfied for stability reason:

where is the largest eigenvalue of the matrix2d

R

The larger the step-size, the faster the convergence, but also the larger estimation variance.


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.


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.


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.


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.


A simple array model

Planar wave Uniform linear array Sensors responses are identical and

LTI Sensors are omni directional One parameter to estimate: DOA


ULA


ULA

)()()()( ttst eay

Tjj mcc ee ...1)( 2ac

The signal model:

where the array transfer vector :

Where is the delay with reference to the first sensor, and is the center frequency of the signal. By defining the spatial frequency as:

m

cd

cs sin

we can write the array transfer vector as:

Tmjj ss ee )1(...1)( a


ULA

A direct analogy between frequency analysis and spatial analysis using the spatial frequency.

To avoid spatial aliasing:

All frequency analysis techniques can be applied to the DOA estimation problem.

2/d


Spatial filtering

Analogy between spatial filter and temporal filter


Spatial filtering

The spatially filtered signal: Objective: find the filter that passes

undistorted the signals with a given DOA; and attenuates all the other DOAs as much as possible.

1)(min ** ahhhh

tosubject

)()()( * tstx ah


The beam pattern


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


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


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.


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


ICA model

)()()()()()()()()(

)()()(

333232131

323222112

313212111

3

2

1

tsatsatsatsatsatsatsatsatsa

txtxtx

Cocktail party problem. Three sources, three sensors:

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.

Asx

Or, in matrix form


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:

sxA 1

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.


Maximizing non-Gaussianity

Non-Gaussian is independent Measuring non-Gaussianity

- by kurtosis- by negentropy


ICA methods

ICA by maximizing non-Gaussianity ICA by Maximum Likelihood ICA by minimizing mutual information ICA by nonlinear decorrelation


Extensions to ICA

Noisy ICA ICA with non-square mixing matrix Independent Factor Analysis Convolutive mixture Methods using time structure


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


BSE

D. Mandic and A. Cichocki, An Online Algorithm For Blind Extraction Of Sources With Different Dynamical Structures.

multi-channel speech enhancement (1).ppt

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