Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Multi-Microphone Speech Dereverberation Based on Eigen-Decomposition - A Study Sharon Gannot
School of Electrical Engineering, Bar-Ilan University 42nd Asilomar Conference on Signals, Systems & Computers, October 27th, 2008
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
The Reverberation Phenomenon
4
2 0.6 1.5 0.5
Frequency [kHz]
0.7
0.9
3
0.8
2.5
1
3.5
0.9
3 Frequency [kHz]
4
1
3.5
0.8
2.5
0.7
2 0.6 1.5 0.5
1
1 0.4
0.5
0
0.1
0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
5
(a) Clean signal
Sharon Gannot
0
0.3
0 Amplitude
Amplitude
0.2
0
0.4 0.5
0.3
0
0.2 0
0
0.1
0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
(b) Reverberant signal (T60 = 0.4s)
Speech Dereverberation using EVD
5
0
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
The Room impulse Response (RIR) 0.07 direct path colouration tail
0.06
0.05
Amplitude
0.04
0.03
0.02
0.01
0
−0.01 0
0.05
0.1
0.15 Time [Sec]
0.2
0.25
0.3
3 Parts: Direct path. Colouration (early arrivals). Reverberation tail (late arrivals). Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Dereverberation Families of Algorithms: Reverberation supersession. Reverberation cancellation (usually multi-channel system identification)
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Dereverberation Families of Algorithms: Reverberation supersession. Reverberation cancellation (usually multi-channel system identification) The talk is based on: S. Gannot and M. Moonen, “Subspace methods for multi-microphone speech dereverberation,” EURASIP J. Appl. Signal Process., vol. 2003, no. 1, pp. 1074-1090, 2003. S. Gannot, “Multi-Microphone Speech Dereverberation using Eigen-decomposition”, to appear in “Speech Dereverberation”, P.A. Naylor and N.D. Gaubich (Eds.), Springer, 2008.
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
4
Extensions of the Basic Algorithm
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
4
Extensions of the Basic Algorithm
5
RIR Estimation in Subbands
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
4
Extensions of the Basic Algorithm
5
RIR Estimation in Subbands
6
Signal Reconstruction
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
4
Extensions of the Basic Algorithm
5
RIR Estimation in Subbands
6
Signal Reconstruction
7
Experimental Study
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Outline 1
Problem Formulation
2
Preliminaries
3
RIR Estimation - Algorithm Derivation
4
Extensions of the Basic Algorithm
5
RIR Estimation in Subbands
6
Signal Reconstruction
7
Experimental Study
8
Summary and Conclusions Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Problem Formulation ν1 (n) y1 (n)
H1 (z)
x1 (n)
ν2 (n) y2 (n)
H2 (z)
0 1 0 1 0 1
s(n)
νM (n) yM (tn
HM (z)
xm (n) = ym (n) + νm (n) =
x2 (n)
nh X
xM (n)
hm (k)s(n − k) + νm (n)
k=0
Hm (z) =
nh X
hm (k)z −k ;
m = 1, 2, . . . , M.
k=0
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Goal ˆ 1 (z) H
x1 (n) x2 (n)
ˆ 2 (z) H
RIR Estimation
ˆ M (z) H
xM (n)
Use a Two Stage Approach
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Goal ˆ 1 (z) H
x1 (n) x2 (n)
ˆ 2 (z) H
RIR Estimation
ˆ M (z) H
xM (n)
Use a Two Stage Approach Estimate the Acoustic Transfer Function (ATFs) Hm (z).
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Goal ˆ 1 (z) H
x1 (n) x2 (n)
ˆ 2 (z) H
RIR Estimation
ˆ M (z) H
xM (n)
Use a Two Stage Approach Estimate the Acoustic Transfer Function (ATFs) Hm (z). ˆ m (z); m = 1, . . . , M to extract s(n). Use H Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone, Noiseless Case y1 (n) = h1 (n) ∗ s(n) y2 (n) = h2 (n) ∗ s(n) ATFs H1 (z)
Nullifying filters y1 (n)
H2 (z)
El (z)
s(n)
0
H2 (z)
y2 (n)
−H1 (z)
El (z)
Nullifying Filters [y2 (n) ∗ h1 (n) − y1 (n) ∗ h2 (n)] ∗ e` (n) = 0 h˜m,` (n) = hm (n) ∗ e` (n); m = 1, 2 Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Data Matrix
ym (0)
0
ym (1) ym (0) .. . ym (1) .. nh − 1) . ym (ˆ y (ˆ n ) y (ˆ n m h m h − 1) YT = . m .. ym (ˆ nh ) . y (N) .. m 0 ym (N) . . . 0 0 ··· Sharon Gannot
··· ..
.
..
.
..
.
..
.
..
.
..
.
0
0 .. .
0 ym (0) ym (1) .. . ym (ˆ nh − 1) ym (ˆ nh ) .. . ym (N)
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Filtered Room Impulse Responses (RIRs) Define: ˜T ˜ ˜ ˜ h nh ) ; m = 1, 2 m,` = hm,` (0) hm,` (1) . . . hm,` (ˆ
Concatenate:
˜ 1,` h ˜ h` = ˜ ; h2,`
Y2 Y= −Y1
Nullifying Filters: ˜ ` = 0; ∀`. YT h Therefore:
T˜ ˜T ˜T ˆ ˜ h ` YY h` = 0 ⇒ h` Ry h` = 0; ∀` Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Null Subspace
Eigenvalue (or Singular Value) Decomposition λ` = 0 ` = 0, 1, . . . , nˆh − nh λ` > 0 otherwise
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Null Subspace
Eigenvalue (or Singular Value) Decomposition λ` = 0 ` = 0, 1, . . . , nˆh − nh λ` > 0 otherwise Null Subspace Vectors
V = v0 v1 · · · vnˆh −nh
Sharon Gannot
˜ h = ˜ 1,0 h2,0
˜ 1,1 · · · h ˜ 2,1 · · · h
˜ 1,ˆn −n h h h ˜ 2,ˆn −n h h h
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Over-Estimated Room Impulse Responses Acoustical Transfer Functions For ` = 0, 1, . . . , nˆh − nh , m = 1, 2: ˜` ⇔ H ˜ m,` (z) h ˜ m,` (z) = Hm (z)E` (z) H
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Over-Estimated Room Impulse Responses Acoustical Transfer Functions For ` = 0, 1, . . . , nˆh − nh , m = 1, 2: ˜` ⇔ H ˜ m,` (z) h ˜ m,` (z) = Hm (z)E` (z) H Fundamental Lemma
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Over-Estimated Room Impulse Responses Acoustical Transfer Functions For ` = 0, 1, . . . , nˆh − nh , m = 1, 2: ˜` ⇔ H ˜ m,` (z) h ˜ m,` (z) = Hm (z)E` (z) H Fundamental Lemma For m = 1, 2, . . . , M: ˜ m,` (z) have nˆh − nh common roots ⇒ E` (z). H
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Over-Estimated Room Impulse Responses Acoustical Transfer Functions For ` = 0, 1, . . . , nˆh − nh , m = 1, 2: ˜` ⇔ H ˜ m,` (z) h ˜ m,` (z) = Hm (z)E` (z) H Fundamental Lemma For m = 1, 2, . . . , M: ˜ m,` (z) have nˆh − nh common roots ⇒ E` (z). H For ` = 0, 1, . . . , nˆh − nh : ˜ m,` (z) have nh common roots ⇒ Hm (z). H Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
RIR Estimation - Algorithm Derivation Filtering (Silvester) Matrix:
hm (0) 0 0 hm (1) hm (0) 0 .. . . hm (1) . . .. .. . . hm (nh ) Hm = hm (nh ) 0 . .. 0 .. . 0 0 ··· {z |
n ˆh −nh +1
Sharon Gannot
··· ···
0 0 .. .
.. . 0 .. . hm (0) hm (1) .. .. . . 0 hm (nh ) }
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Over-Estimated Room Impulse Responses Matrix Form
Define:
nh − nh ) eT ` = e` (0) e` (1) . . . e` (ˆ
Extraneous Filters:
E = e0 e1 · · · enˆh −nh .
Null Subspace Vectors (Over-estimated RIRs): ˜ 1,0 h ˜ 1,1 · · · h ˜ 1,ˆn −n 4 h H1 h h V= ˜ ˜ ˜ 2,ˆn −n = H2 E = HE h2,0 h2,1 · · · h h h 4 Define Ei = inv(E) = ei0 ei1 · · · einˆh −nh Then: H = VEi Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
RIR Extraction Exploiting the Silvester Structure
V O . .. .. . .. . O |
O ··· V O . O .. .. . .. . O ···
ei0 ei1 0 .. 0 . =. i enˆ −n .. h h h1 0 | {z } h 0 · · · O V −S(ˆnh −nh ) | {z2 } {z } θ
··· ··· O ··· ··· O .. . . . . . .. . . . .. .. . . O
−S(0) −S(1) .. . .. . .. .
˜ V
O - all-zeros matrix S(`) - shift by ` matrix (` = 0, 1, . . . , nˆh − nh ) Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Algorithm Summary
RIR Estimation - Basic Case ˜θ=0 V
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Algorithm Summary
RIR Estimation - Basic Case ˜θ=0 V ˜ corresponding to eigenvalue 0 Find eigenvector of V
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Algorithm Summary
RIR Estimation - Basic Case ˜θ=0 V ˜ corresponding to eigenvalue 0 Find eigenvector of V Extract h1 , h2 from the eigenvector
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Extensions
Two Microphone Noisy Case
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Extensions
Two Microphone Noisy Case White Noise Case
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Extensions
Two Microphone Noisy Case White Noise Case Colored Noise Case
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Extensions
Two Microphone Noisy Case White Noise Case Colored Noise Case
Multi-Microphone Case (M > 2)
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Extensions
Two Microphone Noisy Case White Noise Case Colored Noise Case
Multi-Microphone Case (M > 2) Partial Knowledge of the Null Subspace
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Two Microphone Noisy Case
X = Y + Υ, X - noisy signal data matrix Υ - noise-only data matrix ˆx ≈ R ˆy + R ˆν R ˆx = R ˆν = R
XXT N+1 ΥΥT N+1
- noisy signal correlation matrix - noise-only signal correlation matrix
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
White2 Noise ˆ ν ≈ σν I R
RIR Estimation - White Noise V - eigenvectors corresponding to eigenvalue σν2 (remains intact)
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
White2 Noise ˆ ν ≈ σν I R
RIR Estimation - White Noise V - eigenvectors corresponding to eigenvalue σν2 (remains intact) ˜θ= V
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
White2 Noise ˆ ν ≈ σν I R
RIR Estimation - White Noise V - eigenvectors corresponding to eigenvalue σν2 (remains intact) ˜θ= V ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
White2 Noise ˆ ν ≈ σν I R
RIR Estimation - White Noise V - eigenvectors corresponding to eigenvalue σν2 (remains intact) ˜θ= V ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares Extract h1 , h2 from the eigenvector
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise ˆ x and R ˆν Calculate generalized EVD of R (or generalized SVD of X and Υ )
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise ˆ x and R ˆν Calculate generalized EVD of R (or generalized SVD of X and Υ ) V - generalized eigenvectors corresponding to generalized eigenvalue 1
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise ˆ x and R ˆν Calculate generalized EVD of R (or generalized SVD of X and Υ ) V - generalized eigenvectors corresponding to generalized eigenvalue 1 ˜ = Vθ
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise ˆ x and R ˆν Calculate generalized EVD of R (or generalized SVD of X and Υ ) V - generalized eigenvectors corresponding to generalized eigenvalue 1 ˜ = Vθ ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Colored Noise RIR Estimation - Colored Noise ˆ x and R ˆν Calculate generalized EVD of R (or generalized SVD of X and Υ ) V - generalized eigenvectors corresponding to generalized eigenvalue 1 ˜ = Vθ ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares Extract h1 , h2 from the eigenvector
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Multi-Microphone Case (M > 2) Pairing
M×(M−1) 2
channels:
[yi (n) ∗ hj (n) − yj (n) ∗ hi (n)] ∗ el (n) = 0 i, j = 1, 2, . . . , M; l = 0, 1, . . . , nˆh − nh Construct an extended data matrix: X2 X3 · · · XM O · · · O · · · O −X1 O · · · X3 · · · XM O .. O −X1 −X O . 2 X= . . .. .. .. . O O .. .. . . O XM O O ··· −X1 · · · −X2 · · · −XM−1 Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone ˆ x and R ˆν Calculate generalized EVD of new R (or generalized SVD of new X and Υ )
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone ˆ x and R ˆν Calculate generalized EVD of new R (or generalized SVD of new X and Υ ) V - new null subspace
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone ˆ x and R ˆν Calculate generalized EVD of new R (or generalized SVD of new X and Υ ) V - new null subspace ˜ θ = , where: V i h T T . . . h h θ T = (ei0 )T (ei1 )T · · · (einˆh −nh )T hT 2 1 M
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone ˆ x and R ˆν Calculate generalized EVD of new R (or generalized SVD of new X and Υ ) V - new null subspace ˜ θ = , where: V i h T T . . . h h θ T = (ei0 )T (ei1 )T · · · (einˆh −nh )T hT 2 1 M ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Two Microphone Noisy Case Multi-Microphone Case (M > 2)
Algorithm RIR Estimation - Multi-Microphone ˆ x and R ˆν Calculate generalized EVD of new R (or generalized SVD of new X and Υ ) V - new null subspace ˜ θ = , where: V i h T T . . . h h θ T = (ei0 )T (ei1 )T · · · (einˆh −nh )T hT 2 1 M ˜ corresponding to the smallest Find eigenvector of V eigenvalue ⇒Total Least Squares Extract h1 , h2 , . . . , hM from the eigenvector Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Subband Filters 1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0 0
500
1000
1500
2000 2500 Frequency [Hz]
Sharon Gannot
3000
3500
4000
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
RIR Estimation in Subbands P0 (z)
↓L
P1 (z)
↓L
ˆ 0 (z) H 1
x0 1 (n)
ˆ 0 (z) H 2
x1 1 (n)
RIR
x1 (n)
↑L
Q0 (z)
↑L
Q0 (z)
EST PK−1 (z)
K−1 x (n) ↓L 1
P0 (z)
↓L
P1 (z)
↓L
ˆ 0 (z) H M ↑L
Q0 (z)
↑L
Q1 (z)
↑L
Q1 (z)
ˆ 1 (z) H 1
x0 2 (n)
ˆ 1 (z) H 2
1 (n) x2
RIR
x2 (n)
EST PK−1 (z)
↓L
P0 (z)
↓L
P1 (z)
↓L
K−1 x (n) 2
ˆ 1 (z) H M ↑L
ˆ K−1 (z) H 1 ↑L ˆ K−1 (z) H 2 ↑L
x0 M (tn x1 M (n)
EST PK−1 (z)
↓L
K−1 (n) M
Sharon Gannot
ˆ K−1 (z) H M ↑L
ˆ 2 (z) H
Q1 (z)
QK−1 (z) QK−1 (z)
RIR
xM (n)
x
ˆ 1 (z) H
QK−1 (z)
Speech Dereverberation using EVD
ˆ M (z) H
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Causal Equalizers Non-Causal Equalizers
Signal Reconstruction (general) gm (n); m = 1, 2, . . . , M - set of M equalizers. Estimated speech signal: ˆs (n) =
M X
gm (n) ∗ xm (n) =
m=1 M X
gm (n) ∗ hm (n) ∗ s(n) +
m=1
M X
gm (n) ∗ νm (n)
m=1
Equalization: M X
gm (n) ∗ hm (n) = δ(n) ⇔
m=1
M X
Gm (z)Hm (z) = 1
m=1 Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Causal Equalizers Non-Causal Equalizers
Multi-channel Inverse Filter Theorem (MINT) FIR Equalizers:
gT m = gm (0) gm (1) . . . gm (Lg )
Causal equalization:
H1 H2 · · · HM | {z } H
g1 g2 .. .
1 0 0 = .. .
gM | {z } g
d
−1 ˆ = argmin kHg − dk2 = HT H HT d g g
Sharon Gannot
0 | {z }
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Causal Equalizers Non-Causal Equalizers
Non-Causal Equalizers Matched Beamformer (MBF)
Gm (z) = PM
∗ (1/z ∗ ) Hm
∗ ∗ m=1 Hm (z)Hm (1/z )
Sharon Gannot
⇔ Gm (e jω ) = PM
∗ (e jω ) Hm
m=1 |Hm (e
Speech Dereverberation using EVD
jω )|2
.
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Causal Equalizers Non-Causal Equalizers
Non-Causal Equalizers Matched Beamformer (MBF)
Gm (z) = PM
∗ (1/z ∗ ) Hm
∗ ∗ m=1 Hm (z)Hm (1/z )
Inverse Filter
Gm (z) =
⇔ Gm (e jω ) = PM
∗ (e jω ) Hm
m=1 |Hm (e
1 1 ⇔ Gm (e jω ) = Hm (z) Hm (e jω )
Sharon Gannot
Speech Dereverberation using EVD
jω )|2
.
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Experimental Study Figures of Merit
Inspection of the estimated RIR and ATF
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Experimental Study Figures of Merit
Inspection of the estimated RIR and ATF Normalized Projection Misalignment (NPM) ˆ 2h ˆ (hT h) 1 kh − NPM [dB] = 20 log10 k2 ˆ 2 khk2 khk !2 Tˆ h h = 20 log10 1 − ˆ khkkhk
Sharon Gannot
Speech Dereverberation using EVD
!
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Experimental Study Figures of Merit
Inspection of the estimated RIR and ATF Normalized Projection Misalignment (NPM) ˆ 2h ˆ (hT h) 1 kh − NPM [dB] = 20 log10 k2 ˆ 2 khk2 khk !2 Tˆ h h = 20 log10 1 − ˆ khkkhk
!
Comparison of the input speech signal, the reverberant signal, and the processed signal Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results NPM vs. SNR
Scenario M = 2, nh = 16, nˆh = 21, Fs = 8000Hz, T = 0.5s, Discrete uniform distributed RIR coefficients, 50 “Monte Carlo” trials.
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results NPM vs. SNR
Scenario M = 2, nh = 16, nˆh = 21, Fs = 8000Hz, T = 0.5s, Discrete uniform distributed RIR coefficients, 50 “Monte Carlo” trials. White Noise Input SNR NPM
15 -3.5
20 -8.6
25 -16.5
30 -28.0
Sharon Gannot
35 -35.0
40 -44.0
45 -53
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results NPM vs. SNR
Scenario M = 2, nh = 16, nˆh = 21, Fs = 8000Hz, T = 0.5s, Discrete uniform distributed RIR coefficients, 50 “Monte Carlo” trials. White Noise Input SNR NPM
15 -3.5
20 -8.6
25 -16.5
30 -28.0
35 -35.0
40 -44.0
45 -53
Speech Input SNR NPM
35 0.0
40 0.0
45 -2.0
50 -10.0
Sharon Gannot
55 -11.0
60 -24.5
65 -38.0
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results NPM vs. filter order
Scenario M = 2, SNR=50dB, nˆh − nh = 5, Fs = 8000Hz, T = 0.5s, Gaussian distributed with decaying envelope RIR coefficients, 50 “Monte Carlo” trials.
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results NPM vs. filter order
Scenario M = 2, SNR=50dB, nˆh − nh = 5, Fs = 8000Hz, T = 0.5s, Gaussian distributed with decaying envelope RIR coefficients, 50 “Monte Carlo” trials. White Noise Input nh NPM
16 -60.0
32 -49.5
64 -33.0
Sharon Gannot
128 -18.0
256 -0.5
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results Truncated Simulated RIR
Scenario M = 2, SNR=50dB, nˆh − nh = 5, Fs = 8000Hz, T = 0.5s, T60 = 0.7s, RIR truncated to nh = 600. NPM=-26dB. Real TLS−Fullband
1
0.8
Amplitude
0.6
0.4
0.2
0
−0.2
−0.4 0
0.01
0.02
0.03 0.04 Time [Sec]
Sharon Gannot
0.05
0.06
0.07
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Full-band Version - Results Sonograms 4
2 0.6 1.5 0.5
Frequency [kHz]
0.7
0.9
3
0.8
2.5
1
3.5
0.9
3 Frequency [kHz]
4
1
3.5
1
0.8
2.5
0.7
2 0.6 1.5 0.5 1
0.4
0.4
0.5
0.5
Amplitude
0.2
0
0.1
−1 0
0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
Amplitude
0.3
0 1
0
(a) Clean signal
0.1
0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
5
4
4
1
0.6 1.5 0.5
Frequency [kHz]
0.7
2
0.9
3
0.8
2.5
1
3.5
0.9
3
1
0.8
2.5
0.7
2 0.6 1.5 0.5 1
0.4
0.4 0.5
0.3
0 1
0.2
0
0.1
−1 0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
5
0
(a) Dereverberated signal (MINT) Sharon Gannot
Amplitude
Amplitude
0.5
0
0
(b) Reverberant signal (500 taps)
3.5
Frequency [kHz]
0.2
0 −1
0
5
0.3
0 1
0.3
0 1
0.2
0
0.1
−1 0
0.5
1
1.5
2
2.5 3 Time [Sec]
3.5
4
4.5
5
(b) Dereverberated signal (MBF) Speech Dereverberation using EVD
0
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Full-band Version - Results Subband Version - Results
Subband Version - Results Scenario M = 2, SNR=120dB, nh = 24, 6 bands, nˆhk − nhk = 2 per-band, T=4000, Gaussian distributed with decaying envelope RIR coefficients, white noise input, gain ambiguity compensated.
12
12
Real TLS−−Sub
Real TLS−−Sub
10
10
8
8
6
6
4
4
2
2
0 0
500
1000
1500
2000
2500
3000
3500
4000
Sharon Gannot
0 0
500
1000
1500
2000
2500
Speech Dereverberation using EVD
3000
3500
4000
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Filter-bank Design
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Filter-bank Design Band overlap
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Filter-bank Design Band overlap Band gaps
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Filter-bank Design Band overlap Band gaps
Gain Ambiguity Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Limitations of the Proposed Methods Noise Robustness Null Subspace MINT
Common Zeros Room Impulse Responses Extraneous zeros resulting in from the overestimation
The Demand for the Entire RIR Compensation nˆh ≥ nh
Filter-bank Design Band overlap Band gaps
Gain Ambiguity Subband method Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix The channel order overestimation and the additive colored noise are treated by employing TLS-based procedure
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix The channel order overestimation and the additive colored noise are treated by employing TLS-based procedure Both full-band and subband versions
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix The channel order overestimation and the additive colored noise are treated by employing TLS-based procedure Both full-band and subband versions Both variants demonstrate high sensitivity to SNR level and the RIR order
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix The channel order overestimation and the additive colored noise are treated by employing TLS-based procedure Both full-band and subband versions Both variants demonstrate high sensitivity to SNR level and the RIR order At the current stage, the proposed methods are incapable of solving the dereverberation problem
Sharon Gannot
Speech Dereverberation using EVD
Problem Formulation Preliminaries RIR Estimation - Algorithm Derivation Extensions of the Basic Algorithm RIR Estimation in Subbands Signal Reconstruction Experimental Study Summary and Conclusions
Limitations of the Proposed Methods Summary
Summary The reverberating filters are embedded in the null subspace of the multi-channel received data The null subspace is estimated using either the GSVD of the data matrix or the GEVD of the respective correlation matrix The channel order overestimation and the additive colored noise are treated by employing TLS-based procedure Both full-band and subband versions Both variants demonstrate high sensitivity to SNR level and the RIR order At the current stage, the proposed methods are incapable of solving the dereverberation problem Subband structures might be able to bring the prospective solution for the dereverberation problem Sharon Gannot
Speech Dereverberation using EVD
