Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Separate signal into periodic (deterministic) and aperiodic (stochastic) components: s ( t ) = s d ( t ) + s s ( t ) Highlight: In voiced frames , f k = kf 0 : deterministic − → harmonic K a k e j 2 π k ˆ � f 0 t w ( t ) s d ( t ) = k = − K Least Squares method for finding amplitudes and phases ( f 0 is considered as known) Window length in HNM < Window length in SM Reconstruction:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Separate signal into periodic (deterministic) and aperiodic (stochastic) components: s ( t ) = s d ( t ) + s s ( t ) Highlight: In voiced frames , f k = kf 0 : deterministic − → harmonic K a k e j 2 π k ˆ � f 0 t w ( t ) s d ( t ) = k = − K Least Squares method for finding amplitudes and phases ( f 0 is considered as known) Window length in HNM < Window length in SM Reconstruction: OLA or PI for harmonic part
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Separate signal into periodic (deterministic) and aperiodic (stochastic) components: s ( t ) = s d ( t ) + s s ( t ) Highlight: In voiced frames , f k = kf 0 : deterministic − → harmonic K a k e j 2 π k ˆ � f 0 t w ( t ) s d ( t ) = k = − K Least Squares method for finding amplitudes and phases ( f 0 is considered as known) Window length in HNM < Window length in SM Reconstruction: OLA or PI for harmonic part OLA for stochastic part
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications Cons:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications Cons: Local stationarity assumption holds
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications Cons: Local stationarity assumption holds Depends on good f 0 estimation
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications Cons: Local stationarity assumption holds Depends on good f 0 estimation Speech is not purely harmonic ( f k ≈ kf 0 )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Harmonic + Noise model - Stylianou, 1993-1996 Pros: Pitch synchronous analysis Smaller window lengths Convenient for modifications Cons: Local stationarity assumption holds Depends on good f 0 estimation Speech is not purely harmonic ( f k ≈ kf 0 ) ...this last observation is the motivation for the following model...
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Outline Parametric Techniques 1 QHM 2 iQHM 3 adaptive QHM 4 extended aQHM 5 References 6
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers usually f k = kf 0 , where f 0 is considered as known
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers usually f k = kf 0 , where f 0 is considered as known w ( t ) is the analysis window
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers usually f k = kf 0 , where f 0 is considered as known w ( t ) is the analysis window Again: Least Squares method for finding amplitudes
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers usually f k = kf 0 , where f 0 is considered as known w ( t ) is the analysis window Again: Least Squares method for finding amplitudes Window length ≈ 3 pitch periods
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model - Pantazis, Stylianou, Rosec, 2007-2010 K ( a k + tb k ) e j 2 π ˆ � f k t w ( t ) s d ( t ) = k = − K a k , b k are complex numbers usually f k = kf 0 , where f 0 is considered as known w ( t ) is the analysis window Again: Least Squares method for finding amplitudes Window length ≈ 3 pitch periods Reconstruction: K b k ) e j 2 π ˆ � a k + t ˆ f k t ˆ s d ( t ) = (ˆ (1) k = − K
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model HM versus QHM in frequency estimation - pure tone @ 100 Hz
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model HM versus QHM in frequency estimation - pure tone @ 100 Hz given frequency for both models: 90 Hz 0.7 Original Harmonic 0.6 Quasi−Harmonic 0.5 Magnitude 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 Frequency (Hz)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model Time domain properties: Inst. amplitude: � k ) 2 + ( a I ( a R k + tb R k + tb I k ) 2 M k ( t ) = | a k + tb k | =
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model Time domain properties: Inst. amplitude: � k ) 2 + ( a I ( a R k + tb R k + tb I k ) 2 M k ( t ) = | a k + tb k | = f k t + tan − 1 a I k + tb I Inst. phase: Φ k ( t ) = 2 π ˆ k a R k + tb R k
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model Time domain properties: Inst. amplitude: � k ) 2 + ( a I ( a R k + tb R k + tb I k ) 2 M k ( t ) = | a k + tb k | = f k t + tan − 1 a I k + tb I Inst. phase: Φ k ( t ) = 2 π ˆ k a R k + tb R k a R k b I k − a I k b R Inst. frequency: F k ( t ) = 1 f k + 1 2 π Φ ′ ( t ) = ˆ k M 2 2 π k ( t )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model Time domain properties: Inst. amplitude: � k ) 2 + ( a I ( a R k + tb R k + tb I k ) 2 M k ( t ) = | a k + tb k | = f k t + tan − 1 a I k + tb I Inst. phase: Φ k ( t ) = 2 π ˆ k a R k + tb R k a R k b I k − a I k b R Inst. frequency: F k ( t ) = 1 f k + 1 2 π Φ ′ ( t ) = ˆ k M 2 2 π k ( t ) where x R , x I denote the real and imaginary part of x
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model HM vs QHM on frequency tracks - pure tone @ 100 Hz: 105 True Freq. Estimated Freq. QHM Inst. Freq. 100 Frequency (Hz) 95 90 85 −20 −15 −10 −5 0 5 10 15 20 Time (ms) Highlight: frequency correction mechanism
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Quasi-Harmonic model HM vs QHM on frequency tracks - pure tone @ 100 Hz: 105 True Freq. Estimated Freq. QHM Inst. Freq. 100 Frequency (Hz) 95 90 85 −20 −15 −10 −5 0 5 10 15 20 Time (ms) Highlight: frequency correction mechanism Let’s discuss a bit on that...
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model: Reminder: x k ( t ) = a k e j 2 π f k t w ( t ) + tb k e j 2 π f k t w ( t )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model: Reminder: x k ( t ) = a k e j 2 π f k t w ( t ) + tb k e j 2 π f k t w ( t ) f k ) + j b k X k ( f ) = a k W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (2)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model: Reminder: x k ( t ) = a k e j 2 π f k t w ( t ) + tb k e j 2 π f k t w ( t ) f k ) + j b k X k ( f ) = a k W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (2) where W ( f ) = FT { w ( t ) } and W ′ ( f ) = dW ( f ) / df .
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model: Reminder: x k ( t ) = a k e j 2 π f k t w ( t ) + tb k e j 2 π f k t w ( t ) f k ) + j b k X k ( f ) = a k W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (2) where W ( f ) = FT { w ( t ) } and W ′ ( f ) = dW ( f ) / df . Projecting b k to a k : b k = ρ 1 , k a k + ρ 2 , k ja k (3)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction Frequency domain view: Fourier Transform of the model: Reminder: x k ( t ) = a k e j 2 π f k t w ( t ) + tb k e j 2 π f k t w ( t ) f k ) + j b k X k ( f ) = a k W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (2) where W ( f ) = FT { w ( t ) } and W ′ ( f ) = dW ( f ) / df . Projecting b k to a k : b k = ρ 1 , k a k + ρ 2 , k ja k (3) Then, f k ) − ρ 2 , k � W ( f − ˆ 2 π W ′ ( f − ˆ X k ( f ) = a k f k ) (4) + j ρ 1 , k � 2 π W ′ ( f − ˆ f k )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction f k − ρ 2 , k Taylor series expansion of W ( f − ˆ 2 π ):
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction f k − ρ 2 , k Taylor series expansion of W ( f − ˆ 2 π ): f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ W ( f − ˆ 2 π W ′ ( f − ˆ 2 π ) = f k )+ (5) O ( ρ 2 2 , k W ′′ ( f − ˆ f k ))
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction f k − ρ 2 , k Taylor series expansion of W ( f − ˆ 2 π ): f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ W ( f − ˆ 2 π W ′ ( f − ˆ 2 π ) = f k )+ (5) O ( ρ 2 2 , k W ′′ ( f − ˆ f k )) If the value of term W ′′ ( f ) at f k is small, then for small values of ρ 2 , k , it is:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction f k − ρ 2 , k Taylor series expansion of W ( f − ˆ 2 π ): f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ W ( f − ˆ 2 π W ′ ( f − ˆ 2 π ) = f k )+ (5) O ( ρ 2 2 , k W ′′ ( f − ˆ f k )) If the value of term W ′′ ( f ) at f k is small, then for small values of ρ 2 , k , it is: f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ 2 π ) ≈ W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (6)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction f k − ρ 2 , k Taylor series expansion of W ( f − ˆ 2 π ): f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ W ( f − ˆ 2 π W ′ ( f − ˆ 2 π ) = f k )+ (5) O ( ρ 2 2 , k W ′′ ( f − ˆ f k )) If the value of term W ′′ ( f ) at f k is small, then for small values of ρ 2 , k , it is: f k − ρ 2 , k f k ) − ρ 2 , k W ( f − ˆ 2 π ) ≈ W ( f − ˆ 2 π W ′ ( f − ˆ f k ) (6) So, f k − ρ 2 , k 2 π ) + j ρ 1 , k � � W ( f − ˆ 2 π W ′ ( f − ˆ X k ( f ) ≈ a k f k ) (7)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction which goes back in time domain as...
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction which goes back in time domain as... f k + ρ 2 , k ) t + ρ 1 , k te j 2 π ˆ � e j (2 π ˆ f k t � x k ( t ) ≈ a k w ( t ) (8)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction which goes back in time domain as... f k + ρ 2 , k ) t + ρ 1 , k te j 2 π ˆ � e j (2 π ˆ f k t � x k ( t ) ≈ a k w ( t ) (8) Then, ρ 2 , k / 2 π can be an estimator of the frequency error η k : a R k b I k − a I k b R η k = ρ 2 , k / 2 π = 1 k ˆ (9) 2 π | a k | 2
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction which goes back in time domain as... f k + ρ 2 , k ) t + ρ 1 , k te j 2 π ˆ � e j (2 π ˆ f k t � x k ( t ) ≈ a k w ( t ) (8) Then, ρ 2 , k / 2 π can be an estimator of the frequency error η k : a R k b I k − a I k b R η k = ρ 2 , k / 2 π = 1 k ˆ (9) 2 π | a k | 2 In other words, QHM suggests a frequency correction to the input frequencies ˆ f k (or a frequency estimator). This suggestion is however conditional on the magnitude of ρ 2 , k and the value of term W ′′ ( f ) at f k
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Frequency mismatch correction which goes back in time domain as... f k + ρ 2 , k ) t + ρ 1 , k te j 2 π ˆ � e j (2 π ˆ f k t � x k ( t ) ≈ a k w ( t ) (8) Then, ρ 2 , k / 2 π can be an estimator of the frequency error η k : a R k b I k − a I k b R η k = ρ 2 , k / 2 π = 1 k ˆ (9) 2 π | a k | 2 In other words, QHM suggests a frequency correction to the input frequencies ˆ f k (or a frequency estimator). This suggestion is however conditional on the magnitude of ρ 2 , k and the value of term W ′′ ( f ) at f k Also, the correction term depends on the window mainlobe width
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Outline Parametric Techniques 1 QHM 2 iQHM 3 adaptive QHM 4 extended aQHM 5 References 6
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM - Pantazis, Stylianou, Rosec, 2007-2010 This frequency updating mechanism provides frequencies which can be used in the model iteratively and result in better parameter estimation ( a k , b k )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM - Pantazis, Stylianou, Rosec, 2007-2010 This frequency updating mechanism provides frequencies which can be used in the model iteratively and result in better parameter estimation ( a k , b k ) This iterative parameter estimation is referred to as the iterative QHM
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM HM versus iQHM in frequency estimation - speech signal: 0.2 0.1 Speech 0 −0.1 −0.2 0 50 100 150 200 250 300 350 400 450 Time (samples) Speech spectrum 20 kf 0 k(f 0 + ∆ f 0 ) Magnitude 0 −20 −40 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Noise robustness: 0 0 10 10 0 0 10 10 MSE(a 1 ) MSE(a 2 ) MSE(f 1 ) MSE(f 2 ) −5 −5 10 10 −5 −5 10 10 CRLB CRLB CRLB CRLB no iter no iter no iter no iter 3 iter 3 iter 3 iter 3 iter −10 −10 −10 −10 10 10 10 10 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 SNR (dB) SNR (dB) SNR (dB) SNR (dB) 0 0 10 10 0 0 10 10 MSE(a 3 ) MSE(a 4 ) MSE(f 3 ) MSE(f 4 ) −5 −5 10 10 −5 −5 10 10 CRLB CRLB CRLB CRLB no iter no iter no iter no iter 3 iter 3 iter 3 iter 3 iter −10 −10 −10 −10 10 10 10 10 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 SNR (dB) SNR (dB) SNR (dB) SNR (dB) (a) MSE for amplitudes (b) MSE for frequencies Figure: Noise Robustness
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons: Needs larger analysis window
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons: Needs larger analysis window And what about...
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons: Needs larger analysis window And what about... local stationarity??
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons: Needs larger analysis window And what about... local stationarity?? Speech is non stationary even in very short time intervals
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References iterative QHM Pros: Linear amplitude evolution Frequency mismatch correction: η k = f k − ˆ f k Cons: Needs larger analysis window And what about... local stationarity?? Speech is non stationary even in very short time intervals iQHM still holds the local stationary assumption
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Outline Parametric Techniques 1 QHM 2 iQHM 3 adaptive QHM 4 extended aQHM 5 References 6
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How?
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) !
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K complex amplitudes a k , b k : estimated via Least Squares
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K complex amplitudes a k , b k : estimated via Least Squares � ˆ a � = ( E H W H WE ) − 1 E H W H Ws (10) ˆ b
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K complex amplitudes a k , b k : estimated via Least Squares � ˆ a � = ( E H W H WE ) − 1 E H W H Ws (10) ˆ b where E = [ E 0 | E 1 ]:
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K complex amplitudes a k , b k : estimated via Least Squares � ˆ a � = ( E H W H WE ) − 1 E H W H Ws (10) ˆ b where E = [ E 0 | E 1 ]: ( E 0 ) n , k =) e j φ k ( t n )
Outline Parametric Techniques QHM iQHM adaptive QHM extended aQHM References Adaptive Quasi-Harmonic Model Adaptive models tackle the problem of local non stationarity[5] - How? By projecting the signal onto non-stationary basis functions e j φ k ( t ) ! K Model: � e j φ k ( t ) � � s d ( t ) = ( a k + tb k ) w ( t ) k = − K complex amplitudes a k , b k : estimated via Least Squares � ˆ a � = ( E H W H WE ) − 1 E H W H Ws (10) ˆ b where E = [ E 0 | E 1 ]: ( E 0 ) n , k =) e j φ k ( t n ) ( E 1 ) n , k = t n e j φ k ( t n ) = t n ( E 0 ) n , k
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