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Presentation Outline Introduction Principals

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0 Signal Subspace Speech Enhancement
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1 Presentation Outline Introduction Principals
Orthogonal Transforms (KLT Overview) Papers Review Page 1 of 47

2 Introduction Two major classes of speech enhancement
By modeling of noise/speech: like HMM Highly dependent on speech signal syntax and noise characteristics Based on transformation: Spectral Subtraction Musical noise Signal Subspace belongs to the second class (nonparametric) Page 2 of 47

3 Schematic Diagram Orthogonal Inverse Noisy signal Transform Transform
Modifying Coefficients Orthogonal Transform Inverse Transform Noisy signal (time domain) Estimated Clean Signal Page 3 of 47

4 Schematic Diagram Noisy signal (time domain) Framing Orthogonal
Signal+Noise subspace Estimating Clean signal from Signal+Noise subspace Framing overlapping Orthogonal Transform Estimating Dimensions of Subspaces Gs Inverse Transform Clean Signal Gn Producing two orthogonal subspaces Noise subspace Page 4 of 47

5 Principals Procedure Estimate the dimension of the signal+noise subspace in each frame Estimate clean signal from (S+N) subspace by considering some criteria (main part) energy of the residual noise energy of the signal distortion Nulling the coefficients related to the noise subspace Page 5 of 47

6 Principals Assumptions Noise & speech are uncorrelated
Noise is additive & white (whitened) Covariance matrix of the noise in each frame is positive definite and close to a Toeplitz matrix Signal is more statistically structured than noise process Page 6 of 47

7 Principals Key Factor in Signal Subspace method
Covariance matrices of the clean signal have some zero eigenvalues. The improvement in SNR is proportional to the number of those zeros. Nullifying the coefficients of the noise subspace corresponds to that of weak spectral components in spectral subtraction. Page 7 of 47

8 Orthogonal Transforms
Signal Subspace decomposition can be achieved by applying: KLT via Eigenvalue Decomposition (ED) of signal covariance matrix via Singular Value Decomposition (SVD) of data matrix SVD approximation by recursive methods DCT as a good approximation to the KLT Walsh, Haar, Sine, Fourier,… Page 8 of 47

9 Orthogonal Transforms: Karhunen-Loeve Transform (KLT)
Also known as “Hotelling”, “Principal Component” or “Eigenvector" Transform Decorrelates the input vector perfectly Processing of one component has no effect on the others Applications Compression, Pattern Recognition, Classification, Image Restoration, Speech Recognition, Speaker Recognition,… Page 9 of 47

10 KLT Overview Let R be the correlation matrix of a random
complex sequence then Where E is the expectation operator and R is Hermitian matrix. Page 10 of 47

11 KLT Overview is called the KLT matrix.
Let be unitary matrix which diagonalizes R are the eigenvalues of R. is called the KLT matrix. Page 11 of 47

12 KLT Overview Property of : Consider the following transform:
sequence y is uncorrelated because : y has no cross-correlation Page 12 of 47

13 KLT Overview What is ? where and `s are ith column of
Thus are eigenvectors corresponding to Page 13 of 47

14 KLT Overview Comments The arrangement of y auto-correlations is the same as that of KLT can be based on Covariance matrix Using largest eigenvalues to reconstruct sequence with negligible error KLT is optimal Page 14 of 47

15 KLT Overview  Difficulties
Computational Complexity (no fast algorithm) Dependency on the statistics of the current frame Make uncorrelated not independent Utilize KLT as a Benchmark in evaluating the performance of the other transforms. Page 15 of 47

16 Papers Review A Signal Subspace Approach for S.E. [Ephraim 95]
On S.E. Algorithms based on Signal Subspace Methods [Hansen] Extension of the Signal Subspace S.E. Approach to Colored Noise [Ephraim] An Adaptive KLT Approach for S.E. [Gazor] Incorporating the Human Hearing Properties in Signal Subspace Approach for S.E. [Jabloun] An Energy-Constrained Signal Subspace Method for S.E. [Huang] S.E. Based on the Subspace Method [Asano] Page 16 of 47

17 A Signal Subspace Approach for S.E. [Ephraim 95]
Principal Decompose the input vector of the noisy signal into a signal+noise subspace and a noise subspace by applying KLT Enhancement Procedure Removing the noise subspace Estimating the clean signal from S+N subspace Two linear estimators by considering: Signal distortion Residual noise energy Page 17 of 47

18 A Signal Subspace Approach for S.E. [Ephraim 95]
Notes Keeping the residual noise below some threshold to avoid producing musical noise Since DFT & KLT are related, SS is a particular case of this method if # of basis vectors (for linear combination of a vector) are less than the dim of the vector, then there are some zero eigenvalues for its correlation matrix Page 18 of 47

19 A Signal Subspace Approach for S.E. [Ephraim 95]
Basics speech signal : z=y+w , K-dimensional If M=K, representation is always possible. Else “damped complex sinusoid model” can be used. Span( V ): produces all vector y Are zero mean complex variables Page 19 of 47

20 A Signal Subspace Approach for S.E. [Ephraim 95]
When M<K, all vectors y lie in a subspace of spanned by the columns of V  SIGNAL+NOISE SUBSPACE Covariance matrix of clean signal y zero eigenvalues Page 20 of 47

21 A Signal Subspace Approach for S.E. [Ephraim 95]
Covariance matrix of noise w : (K-Dim) White noise vectors fill the entire Euclidean space RK Thus the noise exists in both S+N subspace and complementary subspace NOISE SUBSPACE RK n S n n Page 21 of 47

22 A Signal Subspace Approach for S.E. [Ephraim 95]
The discussion indicates that Euclidean space of the noisy signal is composed of a signal subspace and a complementary noise subspace This decomposition can be performed by applying KLT to the noisy signal : Let The covariance matrix of z is: Page 22 of 47

23 A Signal Subspace Approach for S.E. [Ephraim 95]
Noise is additive Let be the eigendecomposition of Rz Where are eigenvectors of Rz and Eigenvalues of Rw are Page 23 of 47

24 A Signal Subspace Approach for S.E. [Ephraim 95]
Estimating Dimensions of Signal Subspace M Because ,Hence is the orthogonal projector onto the S+N subspace Let : principal eigenvectors Page 24 of 47

25 A Signal Subspace Approach for S.E. [Ephraim 95]
Thus a vector z of noisy signal can be decomposed as is the Karhunen-Loeve Transform Matrix. The vector does not contain signal information and can be nulled when estimating the clean signal. However, M (dim of S+N subspace) must be calculated precisely Page 25 of 47

26 A Signal Subspace Approach for S.E. [Ephraim 95]
Linear Estimation of the clean signal Time Domain Constrained Estimator Minimize signal distortion while constraining the energy of residual noise in every frame below a given threshold Spectral Domain Constrained Estimator Minimize signal distortion while constraining the energy of residual noise in each spectral component below a given threshold Page 26 of 47

27 A Signal Subspace Approach for S.E. [Ephraim 95]
Time Domain Constrained Estimator Having z=y+w Let be a linear estimator of y where H is a K*K matrix The residual signal is Representing signal distortion and residual noise respectively Page 27 of 47

28 A Signal Subspace Approach for S.E. [Ephraim 95]
Defining Criterion Solving : Energy: Energy: Minimize signal distortion while constraining the energy of residual noise in the entire frame below a given threshold Page 28 of 47

29 A Signal Subspace Approach for S.E. [Ephraim 95]
After solving the Constrained minimization by ‘‘Kuhn-Tucker’’ necessary conditions we obtain Eigendecomposition of HTDC Where is the Lagrange multiplier that must satisfy Page 29 of 47

30 A Signal Subspace Approach for S.E. [Ephraim 95]
In order to null noisy components If then HTDC=I, which means minimum distortion and maximum noise Page 30 of 47

31 A Signal Subspace Approach for S.E. [Ephraim 95]
Spectral Domain Constrained Estimator Minimize signal distortion while constraining the energy of residual noise in each spectral component below a given threshold. Results: Page 31 of 47

32 A Signal Subspace Approach for S.E. [Ephraim 95]
Notes The most computational complexity is in Eigendecomposition of the estimated covariance. Eigendecomposition of Toeplitz covariance matrix of the noisy vector is used as an approximate to KLT Compromise between large T in estimating Rz ,and large K to satisfy M<K, while KT can not be too large Page 32 of 47

33 A Signal Subspace Approach for S.E. [Ephraim 95]
Implementation Results The improvement in SNR is proportional to K /M The SDC estimator is more powerful than the TDC estimator SNR improvements in Signal Subspace and SS are similar Subjective Test 83.9 preferred Signal Subspace over noisy signal 98.2 preferred Signal Subspace over SS Page 33 of 47

34 On S.E. Algorithms based on Signal Subspace Methods [Hansen]
The dimension of the signal subspace is chosen at a point with almost equal singular values Gain matrices for different estimators SDC TDC MV Lowest residual noise LS  G=I Lowest signal distortion and highest residual noise K /M improvement in SNR SDC improves the SNR in the range 0-20 db Less sensitive to errors in the noise estimation Musical noise Page 34 of 47

35 Extension of the Signal Subspace S. E
Extension of the Signal Subspace S.E. Approach to Colored Noise [Ephraim] Whitening approach is not desirable for SDC estimator. Obtaining gain matrix H for SDC estimator is not diagonal when the input noise is colored Whitening  Orthogonal Transformation U’  modify components by Page 35 of 47

36 An Adaptive KLT Approach for S.E. [Gazor]
Goal Enhancement of speech degraded by additive colored noise Novelty Adaptive tracking based algorithm for obtaining KLT components A VAD based on principle eigenvalues Page 36 of 47

37 An Adaptive KLT Approach for S.E. [Gazor]
Objective Minimize the distortion when residual noise power is limited to a specific level Type of colored noise Have a diagonal covariance matrix in KLT domain Replaced by Page 37 of 47

38 An Adaptive KLT Approach for S.E. [Gazor]
Adaptive KLT tracking algorithm named “projection approximation subspace tracking” reducing computational time Eigendecomposition is considered as a constrained optimization problem Solving the problem considering quasi-stationarity of speech Then a recursive algorithm is planned to find a close approximation of eigenvectors of the noisy signal Page 38 of 47

39 An Adaptive KLT Approach for S.E. [Gazor]
Voice activity detector When the current principle components’ energy is above 1/12 its past minimum and maximum Implementation Results SNR (dB) Non-Processed Ephraim’s Noise Type 10 85% 55% white 5 75% 69% 64% 89% 73% office 79% 68% Page 39 of 47

40 Incorporating the Human Hearing Properties in the Signal Subspace Approach for S.E. [Jabloun]
Goal Keep the residual noise as much as possible, in order to minimize signal distortion Novelty Transformation from Frequency to Eigendomain for modeling masking threshold. Many masking models were introduced in frequency domain; like Bark scale IFET Masking FET eigendomain Page 40 of 47

41 Incorporating the Human Hearing Properties in the Signal Subspace Approach for S.E. [Jabloun]
Use noise prewhitening to handle the colored noise Implementation results Input SNR Compared with noisy signal Compared with Signal Subspace 20 dB 92% 71% 10 dB 85% 78% 5 dB Page 41 of 47

42 An Energy-Constrained Signal Subspace Method for S.E. [Huang]
Novelty The colored noise is modelled by an AR process. Estimating energy of clean signal to adjust the speech enhancement Prewhitening filter is constructed based on the estimated AR parameters. Optimal AR coeffs is given by [Key 98] Page 42 of 47

43 An Energy-Constrained Signal Subspace Method for S.E. [Huang]
Implementation Results Word Recognition Accuracy for noisy digits Input SNR 0 dB 5 dB 10 dB 20 dB Baseline 40 % 70 % 90 % 100 % ECSS SNR improvement for isolated noisy digits Input SNR 0 dB 5 dB 10 dB 20 dB Improvement 7.6 6.4 5.2 2.9 Page 43 of 47

44 S.E. Based on the Subspace Method [Asano]—Microphone Array
The input spectrum observed at the mth microphone Vector notation for all microphones (spatial) correlation matrix for xk is Then Eigenvalue Decomposition is applied to Microphone array Ambient Noise Directional Sources Page 44 of 47

45 S.E. Based on the Subspace Method [Asano]—Microphone Array
Procedure Weighting the eigenvalues of spatial correlation matrix Energy of D directional sources is concentrated on D largest eigenvalues Ambient noise is reduced by weighting eigenvalues of the noise-dominant subspace discarding M-D smallest eigenvalues when direct-ambient ratio is high Using MV beamformer to extract directional component from modified spatial correlation matrix Page 45 of 47

46 S.E. Based on the Subspace Method [Asano]—Microphone Array
Implementation results Two directional speech signals + Ambient noise Recognition Rate: 87.2% 81.5% 86.6% 81.1% 10 dB 78% 72.3% 71.5% 66.9% 5 dB B1 A SNR MV-NSR MV Page 46 of 47

47 Thanks For Your Attention
The End Page 47 of 47


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