Adaptive Sinusoidal Modelingspcc.csd.uoc.gr/_docs/Lectures2016/SPCC16_StylianouI.pdf · Adaptive...
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Adaptive Sinusoidal Modeling
Yannis Stylianou
University of Crete, Computer Science Dept., Greece,
SPCC 2016
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
1 Sinusoidal Modeling
2 Advanced Modeling
3 QHMValidation
4 aQHMDescriptionValidationAM-FM decomposition
5 eaQHM
6 ApplicationsSpeech technologies
7 Key papers
8 References
[email protected] Adaptive Sinusoidal Modeling 2/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Sinusoidal Models
Sinusoidal Model:
x(t) =K∑
k=−Kγke
j2πfk t
Special case, Harmonic Model:
x(t) =K∑
k=−Kγke
j2πkf0t
Estimation of parameters (Linear approach):
γ = Ffk s
Model Evaluation, Mean-Squared Error (MSE):
ε =
∫w|s(t)− x(t)|2 dt
[email protected] Adaptive Sinusoidal Modeling 3/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Sinusoidal Models
Sinusoidal Model:
x(t) =K∑
k=−Kγke
j2πfk t
Special case, Harmonic Model:
x(t) =K∑
k=−Kγke
j2πkf0t
Estimation of parameters (Linear approach):
γ = Ffk s
Model Evaluation, Mean-Squared Error (MSE):
ε =
∫w|s(t)− x(t)|2 dt
[email protected] Adaptive Sinusoidal Modeling 3/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Sinusoidal Models
Sinusoidal Model:
x(t) =K∑
k=−Kγke
j2πfk t
Special case, Harmonic Model:
x(t) =K∑
k=−Kγke
j2πkf0t
Estimation of parameters (Linear approach):
γ = Ffk s
Model Evaluation, Mean-Squared Error (MSE):
ε =
∫w|s(t)− x(t)|2 dt
[email protected] Adaptive Sinusoidal Modeling 3/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Sinusoidal Models
Sinusoidal Model:
x(t) =K∑
k=−Kγke
j2πfk t
Special case, Harmonic Model:
x(t) =K∑
k=−Kγke
j2πkf0t
Estimation of parameters (Linear approach):
γ = Ffk s
Model Evaluation, Mean-Squared Error (MSE):
ε =
∫w|s(t)− x(t)|2 dt
[email protected] Adaptive Sinusoidal Modeling 3/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
iQHM, aQHM, eaQHM, aHM
Towards Adaptive Sinusoidal Models
[email protected] Adaptive Sinusoidal Modeling 4/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Estimating Sinusoidal parameters
Sinusoidal representation for a speech/signal frame:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
Methods:
FFT-based methods (i.e., QIFFT [Abe et al., 2004-05, [1] [2]])Subspace methodsLeast Squares (LS) method
Frequency mismatch:
fk = fk + ηk
[email protected] Adaptive Sinusoidal Modeling 5/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Quasi-Harmonic Model, QHM [5]
Frequency mismatch:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
QHM (de Prony 1795, Laroche [3] (1989), Stylianou 1993,Pantazis [4] (2008, 2011) ):
x(t) =
(K∑
k=−K(ak + tbk)e j2πfk t
)w(t)
[email protected] Adaptive Sinusoidal Modeling 6/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Quasi-Harmonic Model, QHM [5]
Frequency mismatch:
x(t) =
(K∑
k=−Kake
j2πfk t
)w(t)
QHM (de Prony 1795, Laroche [3] (1989), Stylianou 1993,Pantazis [4] (2008, 2011) ):
x(t) =
(K∑
k=−K(ak + tbk)e j2πfk t
)w(t)
[email protected] Adaptive Sinusoidal Modeling 6/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
A few details of QHM
QHM in the frequency domain:
Xk(f ) = akW (f − fk) + jbk2π
W ′(f − fk)
Decomposition of bk : bk = ρ1,kak + ρ2,k jakThen
Xk(f ) = ak
[W (f − fk)−
ρ2,k
2πW ′(f − fk) + j
ρ1,k
2πW ′(f − fk)
]and taking into account:
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
Approximation of the kth component of QHM
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
][email protected] Adaptive Sinusoidal Modeling 7/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
A few details of QHM
QHM in the frequency domain:
Xk(f ) = akW (f − fk) + jbk2π
W ′(f − fk)
Decomposition of bk : bk = ρ1,kak + ρ2,k jakThen
Xk(f ) = ak
[W (f − fk)−
ρ2,k
2πW ′(f − fk) + j
ρ1,k
2πW ′(f − fk)
]and taking into account:
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
Approximation of the kth component of QHM
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
][email protected] Adaptive Sinusoidal Modeling 7/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
A few details of QHM
QHM in the frequency domain:
Xk(f ) = akW (f − fk) + jbk2π
W ′(f − fk)
Decomposition of bk : bk = ρ1,kak + ρ2,k jakThen
Xk(f ) = ak
[W (f − fk)−
ρ2,k
2πW ′(f − fk) + j
ρ1,k
2πW ′(f − fk)
]and taking into account:
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
Approximation of the kth component of QHM
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
][email protected] Adaptive Sinusoidal Modeling 7/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
A few details of QHM
QHM in the frequency domain:
Xk(f ) = akW (f − fk) + jbk2π
W ′(f − fk)
Decomposition of bk : bk = ρ1,kak + ρ2,k jakThen
Xk(f ) = ak
[W (f − fk)−
ρ2,k
2πW ′(f − fk) + j
ρ1,k
2πW ′(f − fk)
]and taking into account:
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
Approximation of the kth component of QHM
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
][email protected] Adaptive Sinusoidal Modeling 7/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
A few details of QHM
QHM in the frequency domain:
Xk(f ) = akW (f − fk) + jbk2π
W ′(f − fk)
Decomposition of bk : bk = ρ1,kak + ρ2,k jakThen
Xk(f ) = ak
[W (f − fk)−
ρ2,k
2πW ′(f − fk) + j
ρ1,k
2πW ′(f − fk)
]and taking into account:
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
Approximation of the kth component of QHM
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
][email protected] Adaptive Sinusoidal Modeling 7/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Approximations in QHM
we found previously:
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
]Back to the time-domain:
xk(t) ≈ ak
[e j(2πfk+ρ2,k )t + ρ1,kte
j2πfk t]w(t)
Initially, we assumed:
xk(t) = ak
[e j(2π(fk+ηk ))t
]w(t)
in other words, it is suggested:
ηk = ρ2,k/2π
[email protected] Adaptive Sinusoidal Modeling 8/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Approximations in QHM
we found previously:
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
]Back to the time-domain:
xk(t) ≈ ak
[e j(2πfk+ρ2,k )t + ρ1,kte
j2πfk t]w(t)
Initially, we assumed:
xk(t) = ak
[e j(2π(fk+ηk ))t
]w(t)
in other words, it is suggested:
ηk = ρ2,k/2π
[email protected] Adaptive Sinusoidal Modeling 8/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Approximations in QHM
we found previously:
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
]Back to the time-domain:
xk(t) ≈ ak
[e j(2πfk+ρ2,k )t + ρ1,kte
j2πfk t]w(t)
Initially, we assumed:
xk(t) = ak
[e j(2π(fk+ηk ))t
]w(t)
in other words, it is suggested:
ηk = ρ2,k/2π
[email protected] Adaptive Sinusoidal Modeling 8/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Approximations in QHM
we found previously:
Xk(f ) ≈ ak
[W (f − fk −
ρ2,k
2π) + j
ρ1,k
2πW ′(f − fk)
]Back to the time-domain:
xk(t) ≈ ak
[e j(2πfk+ρ2,k )t + ρ1,kte
j2πfk t]w(t)
Initially, we assumed:
xk(t) = ak
[e j(2π(fk+ηk ))t
]w(t)
in other words, it is suggested:
ηk = ρ2,k/2π
[email protected] Adaptive Sinusoidal Modeling 8/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Constraints
W (f − fk −ρ2,k
2π) = W (f − fk)−
ρ2,k
2πW ′(f − fk)+
O(ρ22,kW
′′(f − fk))
[email protected] Adaptive Sinusoidal Modeling 9/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of the Window
We would like small value for W ′′(f ) at fk
W ′′(f ) is influenced by the length, T , of the analysis window,i.e., for rectangular window: W ′′(f ) ∝ T 3
So ... we would like a short analysis window
But ... long analysis window provides robustness
Length of the window � bandwidth
[email protected] Adaptive Sinusoidal Modeling 10/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of the Window
We would like small value for W ′′(f ) at fk
W ′′(f ) is influenced by the length, T , of the analysis window,i.e., for rectangular window: W ′′(f ) ∝ T 3
So ... we would like a short analysis window
But ... long analysis window provides robustness
Length of the window � bandwidth
[email protected] Adaptive Sinusoidal Modeling 10/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of the Window
We would like small value for W ′′(f ) at fk
W ′′(f ) is influenced by the length, T , of the analysis window,i.e., for rectangular window: W ′′(f ) ∝ T 3
So ... we would like a short analysis window
But ... long analysis window provides robustness
Length of the window � bandwidth
[email protected] Adaptive Sinusoidal Modeling 10/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of the Window
We would like small value for W ′′(f ) at fk
W ′′(f ) is influenced by the length, T , of the analysis window,i.e., for rectangular window: W ′′(f ) ∝ T 3
So ... we would like a short analysis window
But ... long analysis window provides robustness
Length of the window � bandwidth
[email protected] Adaptive Sinusoidal Modeling 10/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of the Window
We would like small value for W ′′(f ) at fk
W ′′(f ) is influenced by the length, T , of the analysis window,i.e., for rectangular window: W ′′(f ) ∝ T 3
So ... we would like a short analysis window
But ... long analysis window provides robustness
Length of the window � bandwidth
[email protected] Adaptive Sinusoidal Modeling 10/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of frequency mismatch
Assumey(t) = αe j(2πf t+ηt)
modeled by QHM as
x(t) = (a + tb)e j(2πf t) − T ≤ t ≤ T
LS solution provides:
a = αsin(ηT )
ηT
b = α 3j(sin(ηT )
η2T 3− cos(ηT )
ηT 2)
Frequency mismatch estimate:
η = 3
(1
ηT 2− cot(ηT )
T
)[email protected] Adaptive Sinusoidal Modeling 11/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of frequency mismatch
Assumey(t) = αe j(2πf t+ηt)
modeled by QHM as
x(t) = (a + tb)e j(2πf t) − T ≤ t ≤ T
LS solution provides:
a = αsin(ηT )
ηT
b = α 3j(sin(ηT )
η2T 3− cos(ηT )
ηT 2)
Frequency mismatch estimate:
η = 3
(1
ηT 2− cot(ηT )
T
)[email protected] Adaptive Sinusoidal Modeling 11/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of frequency mismatch
Assumey(t) = αe j(2πf t+ηt)
modeled by QHM as
x(t) = (a + tb)e j(2πf t) − T ≤ t ≤ T
LS solution provides:
a = αsin(ηT )
ηT
b = α 3j(sin(ηT )
η2T 3− cos(ηT )
ηT 2)
Frequency mismatch estimate:
η = 3
(1
ηT 2− cot(ηT )
T
)[email protected] Adaptive Sinusoidal Modeling 11/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Influence of frequency mismatch
Assumey(t) = αe j(2πf t+ηt)
modeled by QHM as
x(t) = (a + tb)e j(2πf t) − T ≤ t ≤ T
LS solution provides:
a = αsin(ηT )
ηT
b = α 3j(sin(ηT )
η2T 3− cos(ηT )
ηT 2)
Frequency mismatch estimate:
η = 3
(1
ηT 2− cot(ηT )
T
)[email protected] Adaptive Sinusoidal Modeling 11/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Combining Influences
−400 −300 −200 −100 0 100 200 300 400−400
−200
0
200
400
η1 (Hz)
(b)
er(η
1) (
Hz)
−400 −300 −200 −100 0 100 200 300 400−400
−200
0
200
400
η1 (Hz)
(a)
er(η
1) (
Hz)
RectWinHamming
no iter2 iter
• Iteratively, the bias can be removed when |η| < B/3, where B isthe bandwidth of the squared analysis [email protected] Adaptive Sinusoidal Modeling 12/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Robustness against additive noise
Signal contaminated by noise:
y(t) =4∑
k=1
akej2πfk + v(t)
Mean Squared Error (MSE):
MSE{fk} =1
M
M∑i=1
|fk(i)− fk |2
MSE{ak} =1
M
M∑i=1
|ak(i)− ak |2
Comparison with Cramer-Rao Bounds (CRB) and QIFFT(Abe et al. 2004)
10000 Monte Carlo simulations
[email protected] Adaptive Sinusoidal Modeling 13/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
MSE of frequencies as a function of SNR.
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(f
1)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(f
2)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(f
3)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(f
4)
CRBno iter3 iterFFT
[email protected] Adaptive Sinusoidal Modeling 14/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
MSE of amplitudes as a function of SNR.
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(a
1)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(a
2)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(a
3)
CRBno iter3 iterFFT
0 10 20 30 40 50 60 70 8010
−10
10−5
100
SNR (dB)
MS
E(a
4)
CRBno iter3 iterFFT
[email protected] Adaptive Sinusoidal Modeling 15/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Notes on QHM
The approximation made in QHM is valid provided that thefrequency mismatch lies in a specific interval.
This interval is a function of the bandwidth of the analysiswindow.
Robustness of QHM against noise was tested and verified.
It can be shown that iterative QHM is equivalent to (anapproximate) Gauss-Newton method.
There are also relations between QHM and ReassignedSpectrum
[email protected] Adaptive Sinusoidal Modeling 16/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Notes on QHM
The approximation made in QHM is valid provided that thefrequency mismatch lies in a specific interval.
This interval is a function of the bandwidth of the analysiswindow.
Robustness of QHM against noise was tested and verified.
It can be shown that iterative QHM is equivalent to (anapproximate) Gauss-Newton method.
There are also relations between QHM and ReassignedSpectrum
[email protected] Adaptive Sinusoidal Modeling 16/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Notes on QHM
The approximation made in QHM is valid provided that thefrequency mismatch lies in a specific interval.
This interval is a function of the bandwidth of the analysiswindow.
Robustness of QHM against noise was tested and verified.
It can be shown that iterative QHM is equivalent to (anapproximate) Gauss-Newton method.
There are also relations between QHM and ReassignedSpectrum
[email protected] Adaptive Sinusoidal Modeling 16/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Notes on QHM
The approximation made in QHM is valid provided that thefrequency mismatch lies in a specific interval.
This interval is a function of the bandwidth of the analysiswindow.
Robustness of QHM against noise was tested and verified.
It can be shown that iterative QHM is equivalent to (anapproximate) Gauss-Newton method.
There are also relations between QHM and ReassignedSpectrum
[email protected] Adaptive Sinusoidal Modeling 16/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Notes on QHM
The approximation made in QHM is valid provided that thefrequency mismatch lies in a specific interval.
This interval is a function of the bandwidth of the analysiswindow.
Robustness of QHM against noise was tested and verified.
It can be shown that iterative QHM is equivalent to (anapproximate) Gauss-Newton method.
There are also relations between QHM and ReassignedSpectrum
[email protected] Adaptive Sinusoidal Modeling 16/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
QHM and Gauss-Newton method
Assuming we have N samples of x(t) = ce jωtw(t)
Approximate Gauss-Newton solution for frequency
ω(1) = ω(0) −R
{∑Nt=−N tw2(t)x(t)e−jω
(0)t
c(0)∑N
t=−N t2w2(t)
}QHM
x(t) = (c + tb)e jωtw(t)
withω(1) = ω(0) + ρ2
= ω(0) −R
{jb(0)
c(0)
}where
b(0) =∑N
t=−N tw2(t)x(t)e−jω(0)t∑Nt=−N t2w2(t)
[email protected] Adaptive Sinusoidal Modeling 17/48
Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
QHM and Gauss-Newton method
Assuming we have N samples of x(t) = ce jωtw(t)
Approximate Gauss-Newton solution for frequency
ω(1) = ω(0) −R
{∑Nt=−N tw2(t)x(t)e−jω
(0)t
c(0)∑N
t=−N t2w2(t)
}QHM
x(t) = (c + tb)e jωtw(t)
withω(1) = ω(0) + ρ2
= ω(0) −R
{jb(0)
c(0)
}where
b(0) =∑N
t=−N tw2(t)x(t)e−jω(0)t∑Nt=−N t2w2(t)
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QHM and Gauss-Newton method
Assuming we have N samples of x(t) = ce jωtw(t)
Approximate Gauss-Newton solution for frequency
ω(1) = ω(0) −R
{∑Nt=−N tw2(t)x(t)e−jω
(0)t
c(0)∑N
t=−N t2w2(t)
}QHM
x(t) = (c + tb)e jωtw(t)
withω(1) = ω(0) + ρ2
= ω(0) −R
{jb(0)
c(0)
}where
b(0) =∑N
t=−N tw2(t)x(t)e−jω(0)t∑Nt=−N t2w2(t)
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Evaluation: RMSE for frequency
B SNR = 0dB, N = 100 (left), N = 500 (right).CRB: Cramer-Rao lower bound for frequency estimation
2 4 6 8 10
10−4
10−3
10−2
Iterations(a)
RM
SE
GNiQHMCRB
2 4 6 8 10
10−4
10−3
10−2
RM
SE
Iterations(b)
GNiQHMCRB
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Evaluation: RMSE for amplitude
B SNR = 0dB, N = 100 (left), N = 500 (right).CRB: Cramer-Rao lower bound for amplitude estimation
2 4 6 8 10
10−1
100
Iterations(a)
RM
SE
2 4 6 8 10
10−1
100
Iterations(b)
RM
SE
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Reassigned Spectrum
Time relocation:
τ = −∂φ(τ, ω)
∂ω= τ −R
(XTw (τ, ω)
X (τ, ω)
)where
XTw (τ, ω) = −N∑
t=−Ntw(t)x(t + τ)e−jω(t+τ)
Frequency relocation:
ω = ω +∂φ(τ, ω)
∂τ= ω + I
(XDw (τ, ω)
X (τ, ω)
)where
XDw (τ, ω) = −N∑
t=−N
dw(t)
dtx(t + τ)e−jω(t+τ)
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Reassigned Spectrum and QHM
QHM: x(t) = (a + tb)e jωtw(t) with b = ρ1a + ρ2ja1
Then, it can be shown:
τ = τ +W2
W0ρ1
and (in case of using Gaussian windows):
ω = ω +W2
W0ρ2
where
Wk =N∑
t=−Ntkw(t)
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Reassigned Spectrum and QHM
QHM: x(t) = (a + tb)e jωtw(t) with b = ρ1a + ρ2ja1
Then, it can be shown:
τ = τ +W2
W0ρ1
and (in case of using Gaussian windows):
ω = ω +W2
W0ρ2
where
Wk =N∑
t=−Ntkw(t)
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Frequency mismatch estimation error
−600 −400 −200 0 200 400 600−2000
−1000
0
1000
2000
η1 (Hz)
er(
η1)
(H
z)
No Iterations
iQHM
−600 −400 −200 0 200 400 600−2000
−1000
0
1000
2000
η1 (Hz)
er(
η1)
(H
z)
RS−QHM
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From QHM to Adaptive QHM, aQHM [6]
QHM (stationarity assumption):
x(t) =
(K∑
k=−K(ak + tbk)e2πjfk t
)w(t)
Adaptive QHM (aQHM):
x(t) =
(K∑
k=−K(ak + tbk)e jφk (t)
)w(t)
where
φk(t) = 2π
∫ t
0fk(s)ds + ϕ, t ∈ [0,T ]
is the estimated instantaneous phase.
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From QHM to Adaptive QHM, aQHM [6]
QHM (stationarity assumption):
x(t) =
(K∑
k=−K(ak + tbk)e2πjfk t
)w(t)
Adaptive QHM (aQHM):
x(t) =
(K∑
k=−K(ak + tbk)e jφk (t)
)w(t)
where
φk(t) = 2π
∫ t
0fk(s)ds + ϕ, t ∈ [0,T ]
is the estimated instantaneous phase.
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From QHM to aQHM; Graphically
fk(t)
fk(t)
t
tl
tl+T
tl+Tt
l−T
tl−T
}ρ2,k
tl
t(b)
(a)
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Frame Rate in aQHM
One sample: no interpolation between estimations
Higher rates (i.e., 5ms, 10ms): Interpolation betweenestimates is required:
Amplitudes are linearly interpolatedFrequencies are interpolated with splinesPhases are interpolated by integration of instantaneousfrequency
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Example of estimation in aQHM: no iteration(QHM)
200 300 400 500 600 700 800 900 1000 1100900
920
940
960
980
1000
1020
1040
1060
1080
1100
Fre
quen
cy (
Hz)
Time (sample)
instantaneous frequency
estimated
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Example of estimation in aQHM: one iteration
200 300 400 500 600 700 800 900 1000 1100900
920
940
960
980
1000
1020
1040
1060
1080
1100F
requ
ency
(H
z)
Time (sample)
instantaneous frequency
estimated
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Example of estimation in aQHM: twoiterations
200 300 400 500 600 700 800 900 1000 1100900
920
940
960
980
1000
1020
1040
1060
1080
1100
Fre
quen
cy (
Hz)
Time (sample)
instantaneous frequency
estimated
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Validation
Let us consider as example:
x(t) = (11− 340t + 4000t2)e j2π(280t+19500t2)
and frequency mismatch: |fk − ζk | = 35Hz , using Hammingwindow
Consider simplified approaches: Sinusoidal-based analysis(SM)
Mean Absolute Error (MAE) [frame rate: one sample]
AM FM (Hz)
QHM, 10ms 0.46 2.15
aQHM (1) 0.008 0.006
SM, 10ms 0.44 3.08
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Validation
Let us consider as example:
x(t) = (11− 340t + 4000t2)e j2π(280t+19500t2)
and frequency mismatch: |fk − ζk | = 35Hz , using Hammingwindow
Consider simplified approaches: Sinusoidal-based analysis(SM)
Mean Absolute Error (MAE) [frame rate: one sample]
AM FM (Hz)
QHM, 10ms 0.46 2.15
aQHM (1) 0.008 0.006
SM, 10ms 0.44 3.08
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Validation
Let us consider as example:
x(t) = (11− 340t + 4000t2)e j2π(280t+19500t2)
and frequency mismatch: |fk − ζk | = 35Hz , using Hammingwindow
Consider simplified approaches: Sinusoidal-based analysis(SM)
Mean Absolute Error (MAE) [frame rate: one sample]
AM FM (Hz)
QHM, 10ms 0.46 2.15
aQHM (1) 0.008 0.006
SM, 10ms 0.44 3.08
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Validation
Let us consider as example:
x(t) = (11− 340t + 4000t2)e j2π(280t+19500t2)
and frequency mismatch: |fk − ζk | = 35Hz , using Hammingwindow
Consider simplified approaches: Sinusoidal-based analysis(SM)
Mean Absolute Error (MAE) [frame rate: one sample]
AM FM (Hz)
QHM, 10ms 0.46 2.15
aQHM (1) 0.008 0.006
SM, 10ms 0.44 3.08
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QHM reminder
QHM:
x(t) =
(K∑
k=−K(ak + tbk)e2πjfk t
)w(t)
Instantaneous parameters:
Ak(t) =√
(aRk + tbRk )2 + (aIk + tbIk)2
Φk(t) = 2πfkt + atanaIk+tbIkaRk +tbRk
Fk(t) = fk +1
2π
aRk bIk − aIkb
Rk
M2k (t)
= fk +1
2πρ2,k
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QHM reminder
QHM:
x(t) =
(K∑
k=−K(ak + tbk)e2πjfk t
)w(t)
Instantaneous parameters:
Ak(t) =√
(aRk + tbRk )2 + (aIk + tbIk)2
Φk(t) = 2πfkt + atanaIk+tbIkaRk +tbRk
Fk(t) = fk +1
2π
aRk bIk − aIkb
Rk
M2k (t)
= fk +1
2πρ2,k
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QHM and AM-FM decomposition
AM-FM signal
y(t) =
K(t)∑k=1
ak(t)cos(φk(t)),
Taylor series expansion of the instantaneous phase of kthcomponent:
φk(t) = 2πζkt +∞∑i=0
φk,it i
i !
Instantaneous frequency of the kth component at t = 0:
νk(0) = ζk +φk,12π
... and previously we had:
Fk(0) = fk +ρ2,k
2π
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QHM and AM-FM decomposition
AM-FM signal
y(t) =
K(t)∑k=1
ak(t)cos(φk(t)),
Taylor series expansion of the instantaneous phase of kthcomponent:
φk(t) = 2πζkt +∞∑i=0
φk,it i
i !
Instantaneous frequency of the kth component at t = 0:
νk(0) = ζk +φk,12π
... and previously we had:
Fk(0) = fk +ρ2,k
2π
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QHM and AM-FM decomposition
AM-FM signal
y(t) =
K(t)∑k=1
ak(t)cos(φk(t)),
Taylor series expansion of the instantaneous phase of kthcomponent:
φk(t) = 2πζkt +∞∑i=0
φk,it i
i !
Instantaneous frequency of the kth component at t = 0:
νk(0) = ζk +φk,12π
... and previously we had:
Fk(0) = fk +ρ2,k
2π
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QHM and AM-FM decomposition
AM-FM signal
y(t) =
K(t)∑k=1
ak(t)cos(φk(t)),
Taylor series expansion of the instantaneous phase of kthcomponent:
φk(t) = 2πζkt +∞∑i=0
φk,it i
i !
Instantaneous frequency of the kth component at t = 0:
νk(0) = ζk +φk,12π
... and previously we had:
Fk(0) = fk +ρ2,k
2π
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Reconstruction errors with QHM, aQHM, SM
0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.5
0
0.5
Time (s) (a)
0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.1
−0.05
0
0.05
0.1
Time (s) (b)
0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.1
−0.05
0
0.05
0.1
Time (s) (c)
0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.1
−0.05
0
0.05
0.1
Time (s) (d)
Original
QHM Recon. Error
aQHM Recon. Error
SM Recon. Error
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aQHM: AM-FM decomposition of VoicedSpeech
Signal Reconstruction
x(t) =K∑
k=1
Ak(t) cos (Φk(t))
Test on 5 minutes of 20 female and male voiced speech(TIMIT)
Average Signal-to-Error Reconstruction Ratio (in dB)
Male Female
QHM 23.9 29.1
aQHM (3) 29.1 34.1
SM 17.2 21.1
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aQHM: AM-FM decomposition of VoicedSpeech
Signal Reconstruction
x(t) =K∑
k=1
Ak(t) cos (Φk(t))
Test on 5 minutes of 20 female and male voiced speech(TIMIT)
Average Signal-to-Error Reconstruction Ratio (in dB)
Male Female
QHM 23.9 29.1
aQHM (3) 29.1 34.1
SM 17.2 21.1
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aQHM: AM-FM decomposition of VoicedSpeech
Signal Reconstruction
x(t) =K∑
k=1
Ak(t) cos (Φk(t))
Test on 5 minutes of 20 female and male voiced speech(TIMIT)
Average Signal-to-Error Reconstruction Ratio (in dB)
Male Female
QHM 23.9 29.1
aQHM (3) 29.1 34.1
SM 17.2 21.1
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Extended aQHM
Recall aQHM:
x(t) =K∑
k=−K(ak + tbk)e jφk (t)
Extended aQHM:
x(t) =K∑
k=−K(ak + tbk)α(t)e jφk (t)
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AM-FM modeling: aQHM
0 0.05 0.11
1.5
2
2.5
3
Time (s) (a)
AM
1
0 0.05 0.1600
650
700
750
800
Time (s) (b)
FM
1 (
Hz)
0 0.05 0.11
1.5
2
2.5
3
Time (s) (c)
AM
2
0 0.05 0.1900
950
1000
1050
1100
Time (s) (d)
FM
2 (
Hz)
TrueEstimated
TrueEstimated
TrueEstimated
TrueEstimated
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AM-FM modeling: extended aQHM
0 0.05 0.11
1.5
2
2.5
3
Time (s) (a)
AM
1
0 0.05 0.1600
650
700
750
800
Time (s) (b)
FM
1 (
Hz)
0 0.05 0.11
1.5
2
2.5
3
Time (s) (c)
AM
2
0 0.05 0.1900
950
1000
1050
1100
Time (s) (d)
FM
2 (
Hz)
TrueEstimated
TrueEstimated
TrueEstimated
TrueEstimated
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Comparing Adaptive Models
0 0.1 0.2 0.3 0.4−0.5
0
0.5
Time (s)
Orig
inal
Sig
nal
0 0.1 0.2 0.3 0.4−0.5
0
0.5
Time (s)
aQH
M R
ec. S
igna
l
0 0.1 0.2 0.3 0.4−0.5
0
0.5
Time (s)
eaQ
HM
Rec
. Sig
nal
0 0.1 0.2 0.3 0.4−0.01
0
0.01
Time (s)
aQH
M R
ec. E
rror
0 0.1 0.2 0.3 0.4−0.01
0
0.01
Time (s)
eaQ
HM
Rec
. Err
or
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Stop modeling
0.05 0.1 0.15−0.05
0
0.05
0.1
Time (s)
Original signal /t/
0.05 0.1 0.15−0.05
0
0.05
0.1
Time (s)
adaptive QHM reconstruction
0.05 0.1 0.15−0.05
0
0.05
0.1
Time (s)
extended adaptive QHM reconstruction
0.05 0.1 0.15−0.05
0
0.05
0.1
Time (s)
Sinusoidal reconstruction
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Stop modeling
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Large scale evaluation
Global Signal to Reconstruction Error Ratio (dB)
Model /p/ /t/ /k/ /b/ /d/ /g/
SM 12.7 12.8 12.4 16.6 14.9 15.3
aQHM 19.9 20.6 21.7 28.3 26.9 27.5
eaQHM 25.4 25.7 27.2 32.9 32.2 32.9
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Speech Modifications
High quality pitch scale modification
Within glottal cycle formant tracking and modificationstowards voice conversion
Free pre-echo effect in time scaling
Emotion detection and control
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Speech Modifications
High quality pitch scale modification
Within glottal cycle formant tracking and modificationstowards voice conversion
Free pre-echo effect in time scaling
Emotion detection and control
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Speech Modifications
High quality pitch scale modification
Within glottal cycle formant tracking and modificationstowards voice conversion
Free pre-echo effect in time scaling
Emotion detection and control
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Speech Modifications
High quality pitch scale modification
Within glottal cycle formant tracking and modificationstowards voice conversion
Free pre-echo effect in time scaling
Emotion detection and control
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Speech Synthesis
Unit selection based: fast synthesis, smooth trajectories,high quality prosodic modifications, voice conversion
Statistical modeling: naturally smooth trajectories,articulators tracking, adaptation and controllability,modulations
Sinusoidal models work nicely with DNNs
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Speech Synthesis
Unit selection based: fast synthesis, smooth trajectories,high quality prosodic modifications, voice conversion
Statistical modeling: naturally smooth trajectories,articulators tracking, adaptation and controllability,modulations
Sinusoidal models work nicely with DNNs
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Speech Synthesis
Unit selection based: fast synthesis, smooth trajectories,high quality prosodic modifications, voice conversion
Statistical modeling: naturally smooth trajectories,articulators tracking, adaptation and controllability,modulations
Sinusoidal models work nicely with DNNs
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Speech Recognition
Robust to noise filter bank estimation
Data driven dynamic information
High-resolution modulations
Fits well to the DNN framework.
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Speech Recognition
Robust to noise filter bank estimation
Data driven dynamic information
High-resolution modulations
Fits well to the DNN framework.
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Speech Recognition
Robust to noise filter bank estimation
Data driven dynamic information
High-resolution modulations
Fits well to the DNN framework.
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Speech Recognition
Robust to noise filter bank estimation
Data driven dynamic information
High-resolution modulations
Fits well to the DNN framework.
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Speech Pathology
Jitter and shimmer estimators
Vocal fatigue
Robust estimation in spasmodic dysphonia
Vocal tremor.
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Speech Pathology
Jitter and shimmer estimators
Vocal fatigue
Robust estimation in spasmodic dysphonia
Vocal tremor.
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Speech Pathology
Jitter and shimmer estimators
Vocal fatigue
Robust estimation in spasmodic dysphonia
Vocal tremor.
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Speech Pathology
Jitter and shimmer estimators
Vocal fatigue
Robust estimation in spasmodic dysphonia
Vocal tremor.
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Key papers to read
R. McAulay and T. Quatieri: Speech analysis/Synthesis based on a sinusoidal representation IEEE TASSP,Vol.34, No.4, Aug 1986, pp 744-754.
Y. Pantazis, O. Rosec, and Y. Stylianou: Adaptive AM-FM Signal Decomposition with Application toSpeech Analysis IEEE TASLP, Vol.19, No.2, Feb 2011, pp 290-300.
Y. Pantazis, O. Rosec, and Y. Stylianou: Iterative Estimation of Sinusoidal Signal Parameters. IEEE SPL,Vol.17, No.5, May 2010, pp. 461-464
P. Babu and P. Stoica: Comments on Iterative Estimation of Sinusoidal Signal Parameters. Comments onIEEE SPL, pp.1022-1023, Vol.17, No.12, Dec 2010
Y. Pantazis, O. Rosec, and Y. Stylianou: Reply to ”Comments on ’Iterative Estimation of Sinusoidal SignalParameters’ ” IEEE SPL, pp. 1024-1026, Vol.17, No.12, Dec 2010
G. Degottex and Y. Stylianou: Analysis and Synthesis of Speech using an Adaptive Full-band HarmonicModel IEEE TASLP, Vol.21, Issue 10, pp 2085-2095, 2013
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Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
M. Abe and J. S. III, “CQIFFT: Correcting Bias in a Sinusoidal Parameter Estimator based on Quadratic
Interpolation of FFT Magnitude Peaks,” Tech. Rep. STAN-M-117, Stanford University, California, Oct 2004.
M. Abe and J. S. III, “AM/FM Estimation for Time-varying Sinusoidal Modeling,” in Proc. IEEE Int. Conf.
Acoust., Speech, Signal Processing, (Philadelphia), pp. III 201–204, 2005.
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heavily damped percussive sounds.,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, (Glasgow,UK), pp. 2053–2056, May 1989.
Y. Pantazis, O. Rosec, and Y. Stylianou, “On the Properties of a Time-Varying Quasi-Harmonic Model of
Speech,” (Brisbane), Sep 2008.
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no. 5, pp. 461–464, 2010.
Y. Pantazis, O. Rosec, and Y. Stylianou, “Adaptive AMFM signal decomposition with application to speech
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Engewood Cliffs, NJ: Prentice Hall, 2002.
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Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
Acknowledgments
My colleagues: Yannis Pantazis, Olivier Rosec (Voxygen),Vassilis Tsiaras, Gilles Degottex
My ex-PhD Student: Giorgos Kafentzis
Tom Quatieri and Prentice Hall for gave me the permission touse figures from Tom’s book[7]
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Outline Sinusoidal Modeling Advanced Modeling QHM aQHM eaQHM Applications Key papers References
THANK YOU
for your attention on this part as well!
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