Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability
Separation of High Order Impulse Responses in Methods based on the Exponential Swept-Sine
Robustness
Examples Configuration HOIR Separation
S. Tassart, A. Grand
Transfer Function Recovery
Conclusion
ST-Ericsson STS - Paris
AES #132, Budapest, 26–29 April 2012
Summary Extended ESS S. Tassart
1
Swept-Sine Analysis State of the Art Constraints
2
Principles Intermodulation Laws Separation Stability Robustness
3
Examples Configuration HOIR Separation Transfer Function Recovery
4
Conclusion
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Summary Extended ESS S. Tassart
1
Swept-Sine Analysis State of the Art Constraints
2
Principles Intermodulation Laws Separation Stability Robustness
3
Examples Configuration HOIR Separation Transfer Function Recovery
4
Conclusion
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009 S. Müller et al. (JAES): 2001 T. Kite (AES): 2004
Stability Robustness
Examples Configuration
A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010 C.L. Bennett (AES): 2010
HOIR Separation Transfer Function Recovery
M. Rébillat et al. (Journal of Sound and Vibr.): 2011
Conclusion
A. Farina, “Simultaneous measurement of impulse response and distortion with a swept-sine technique,” in 108th AES Convention, Feb. 2000, pp. 18–22.
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009 S. Müller et al. (JAES): 2001 T. Kite (AES): 2004
Stability Robustness
Examples Configuration
A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010 C.L. Bennett (AES): 2010
HOIR Separation Transfer Function Recovery
M. Rébillat et al. (Journal of Sound and Vibr.): 2011
Conclusion
S. Müller and P. Massarani, “Transfer-function measurement with sweeps,” Journal of the AES, vol. 49, no. 6, pp. 443–471, Jun. 2001.
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009 S. Müller et al. (JAES): 2001 T. Kite (AES): 2004
Stability Robustness
Examples Configuration
A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010 C.L. Bennett (AES): 2010
HOIR Separation Transfer Function Recovery
M. Rébillat et al. (Journal of Sound and Vibr.): 2011
Conclusion
T. Kite, “Measurement of audio equipment with log-swept sine chirps,” in 117th AES Convention, Oct. 2004, Convention Paper no. 6269.
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009
Constraints
Principles Intermodulation Laws
S. Müller et al. (JAES): 2001 T. Kite (AES): 2004
Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010 C.L. Bennett (AES): 2010 M. Rébillat et al. (Journal of Sound and Vibr.): 2011 A. Novák, L. Simon, F. Kadlec, and P. Lotton, “Nonlinear system identification using exponential swept-sine signal,” IEEE Trans. On Instrumentation and Measurement, vol. 59, no. 8, pp. 2220–2229, Aug. 2009.
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009
Constraints
Principles Intermodulation Laws
S. Müller et al. (JAES): 2001 T. Kite (AES): 2004
Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010 C.L. Bennett (AES): 2010 M. Rébillat et al. (Journal of Sound and Vibr.): 2011 C. L. Bennet, D. W. Harris, A. S. Tankanow and R. M. Twilley, “Effect of oversamlpling on SNR Swept-Sine Analysis,” in 129th AES Convention, Nov. 2010, Convention Paper no. 8232.
Swept-Sine Analysis Bibliography Extended ESS S. Tassart ESS Analysis State of the Art Constraints
State of the Art A. Farina (AES): 2000, 2001, 2007, 2009 S. Müller et al. (JAES): 2001
Principles Intermodulation Laws Separation Stability
T. Kite (AES): 2004 A. Novák et al.(IEEE, Trans. on Inst. and Meas.): 2010
Robustness
Examples Configuration HOIR Separation
C.L. Bennett (AES): 2010 M. Rébillat et al. (Journal of Sound and Vibr.): 2011
Transfer Function Recovery
Conclusion
M. Rébillat, R. Hennequin, E. Corteel, and B. F. G. Katz, “Identification of cascade of Hammerstein models for the description of non-linearities in vibrating devices,” Journal of Sound and Vibration, vol. 330, no. 5, pp. 1018–1038, Feb. 2011.
Swept-Sine Analysis Cascade of Hammerstein Models Extended ESS S. Tassart
x(t)
y (t) H1 (z)
ESS Analysis
y1 (t)
State of the Art Constraints
Principles
H2 (z)
Intermodulation Laws
x 2 (t)
y2 (t)
Separation Stability Robustness
P
Examples Configuration HOIR Separation Transfer Function Recovery
HM (z)
x M (t)
Conclusion
Y (jω) =
X k
Yk (jω) =
yM (t)
X k
Hk (jω)X (k ) (jω)
Swept-Sine Analysis Response to a Sine Extended ESS S. Tassart
Pure Sine X (jω)
ESS Analysis State of the Art Constraints
Harmonic PPartials Y (jω) = k Yk (jω)
Spectrogram of a sine
Principles Intermodulation Laws
Spectrogram of the harmonic partials of a sine
20000
20000
Separation
Examples Configuration HOIR Separation
Frequency (Hz)
Robustness
Frequency (Hz)
Stability 15000
10000
5000
15000
10000
5000
Transfer Function Recovery 0
Conclusion
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
Time (s)
δ(ω − ω0 )
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
P
k
Hk (jk ω0 )δ(ω − k ω0 )
Swept-Sine Analysis Exponential swept Extended ESS
Exponential Swept E(jω) or C(jω)
S. Tassart ESS Analysis
Harmonic Partials U(jω)
Spectrogram of a exponential swept-sine...
... and its harmonics
State of the Art Constraints 20000
20000
Separation Stability
Frequency (Hz)
Intermodulation Laws
Frequency (Hz)
Principles 15000
10000
15000
10000
Robustness 5000
5000
Examples Configuration HOIR Separation
0
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
Time (s)
0.02
0.03
0.04
0.05
0.06
Time (s)
Transfer Function Recovery
Conclusion
E(jω): complex valued stimulus C(jω): real valued stimulus, i.e. 2C(jω) = E(jω) + E(−jω)
0.07
0.08
Swept-Sine Analysis ESS Principles Extended ESS S. Tassart
`¯3 (t − τ3 )
`¯4 (t − τ4 )
`¯2 (t − τ2 )
State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
log frequency (rad/sample)
ESS Analysis
`¯1 (t)
harmonic ↔ delay with
Conclusion
time (sample)
τ (3 ω)
τ (2 ω)
τ (ω) −τ2
−τ3
− τk ∝ log2 (k )
Swept-Sine Analysis Deconvolution Extended ESS
swept-sine response ¯ L1 (jω) = L1 (jω)C(jω)
S. Tassart ESS Analysis
impulse response ˜ ¯1 (jω)C(jω) L1 (jω) = L
Signal in the natural domain
State of the Art
Signal in the natural domain
4
2.5
Constraints
Separation Stability
3
2
2
1.5
1
Gain (linear)
Intermodulation Laws
Gain (linear)
Principles
0
-1
Robustness
1 0.5 0 -0.5
-2
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
-1 -3 -1.5 -4 500
1000
1500
2000
2500
500
1000
1500
Time (sample)
Time (sample)
`¯1 (t)
`1 (t)
with
˜ C(jω)C(jω) =1
2000
2500
Swept-Sine Analysis HOIRs separation Extended ESS
P
P
C(jω)
S. Tassart ESS Analysis
k
Lk (jω)e−jωτk
˜ C(jω)
State of the Art Constraints
Principles
Components separation in the deconvolution domain
Intermodulation Laws
`1 (t)
1
Separation Stability
0.8
Robustness
Configuration HOIR Separation Transfer Function Recovery
Conclusion
Amplitude (linear)
0.6
Examples
0.4 0.2 0 -0.2 -0.4 -0.6 3000
3200
3400
3600
3800
4000
Time (sample)
`3 (t − τ3 )
`2 (t − τ2 )
4200
Swept-Sine Analysis Caveats (partially solved) Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
1
time separation of the HOIRs
2
real valued stimuli (i.e. not complex valued)
Separation
C(jω) 6= E(jω)
Stability Robustness
⇒
Hk (jω) 6= Lk (jω)
Examples Configuration
3
discretization of the stimuli / measurements
4
artifacts at start and end of the stimulus (band limited issues)
HOIR Separation Transfer Function Recovery
Conclusion
Swept-Sine Analysis Caveats (partially solved) Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
1
time separation of the HOIRs
2
real valued stimuli (i.e. not complex valued)
Separation
C(jω) 6= E(jω)
Stability Robustness
⇒
Hk (jω) 6= Lk (jω)
Examples Configuration
3
discretization of the stimuli / measurements
4
artifacts at start and end of the stimulus (band limited issues)
HOIR Separation Transfer Function Recovery
Conclusion
Swept-Sine Analysis Caveats (partially solved) Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
1
time separation of the HOIRs
2
real valued stimuli (i.e. not complex valued)
Separation
C(jω) 6= E(jω)
Stability Robustness
⇒
Hk (jω) 6= Lk (jω)
Examples Configuration
3
discretization of the stimuli / measurements
4
artifacts at start and end of the stimulus (band limited issues)
HOIR Separation Transfer Function Recovery
Conclusion
Swept-Sine Analysis Caveats (partially solved) Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
1
time separation of the HOIRs
2
real valued stimuli (i.e. not complex valued)
Separation
C(jω) 6= E(jω)
Stability Robustness
⇒
Hk (jω) 6= Lk (jω)
Examples Configuration
3
discretization of the stimuli / measurements
4
artifacts at start and end of the stimulus (band limited issues)
HOIR Separation Transfer Function Recovery
Conclusion
Swept-Sine Analysis Specific usecase Extended ESS S. Tassart ESS Analysis State of the Art
Long stimulus not suitable for the measurement: many “impulsive” noise superposed to the measurement
Constraints
Principles
replaced by the repeatition of short stimuli
Intermodulation Laws Separation Stability Robustness
Pros I/O synchronization
Examples Configuration HOIR Separation Transfer Function Recovery
Design in the frequency domain via IDFT average the measurements and improve SNR
Conclusion
Cons sensitive to sound speed variations difficulty to separate HOIRs
Swept-Sine Analysis Specific usecase Extended ESS S. Tassart ESS Analysis State of the Art
Long stimulus not suitable for the measurement: many “impulsive” noise superposed to the measurement
Constraints
Principles
replaced by the repeatition of short stimuli
Intermodulation Laws Separation Stability Robustness
Pros I/O synchronization
Examples Configuration HOIR Separation Transfer Function Recovery
Design in the frequency domain via IDFT average the measurements and improve SNR
Conclusion
Cons sensitive to sound speed variations difficulty to separate HOIRs
Swept-Sine Analysis Specific usecase Extended ESS S. Tassart ESS Analysis State of the Art
Long stimulus not suitable for the measurement: many “impulsive” noise superposed to the measurement
Constraints
Principles
replaced by the repeatition of short stimuli
Intermodulation Laws Separation Stability Robustness
Pros I/O synchronization
Examples Configuration HOIR Separation Transfer Function Recovery
Design in the frequency domain via IDFT average the measurements and improve SNR
Conclusion
Cons sensitive to sound speed variations difficulty to separate HOIRs
Swept-Sine Analysis Specific usecase Extended ESS S. Tassart ESS Analysis State of the Art
Long stimulus not suitable for the measurement: many “impulsive” noise superposed to the measurement
Constraints
Principles
replaced by the repeatition of short stimuli
Intermodulation Laws Separation Stability Robustness
Pros I/O synchronization
Examples Configuration HOIR Separation Transfer Function Recovery
Design in the frequency domain via IDFT average the measurements and improve SNR
Conclusion
Cons sensitive to sound speed variations difficulty to separate HOIRs
Summary Extended ESS S. Tassart
1
Swept-Sine Analysis State of the Art Constraints
2
Principles Intermodulation Laws Separation Stability Robustness
3
Examples Configuration HOIR Separation Transfer Function Recovery
4
Conclusion
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
New Principles Intermodulation Laws Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Analysis relying on a family of atoms, (ek )k : separable, i.e.
Principles Intermodulation Laws Separation
αk
recoverable from: y (t) =
Stability
Configuration
αk ek (t)
k
Robustness
Examples
X
stable by intermodulation, i.e.
HOIR Separation Transfer Function Recovery
Conclusion
∀i, j, robust to noise
∃k ,
∀t,
ei (t).ej (t) = ek (t)
New Principles Intermodulation Laws Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Analysis relying on a family of atoms, (ek )k : separable, i.e.
Principles Intermodulation Laws Separation
αk
recoverable from: y (t) =
Stability
Configuration
αk ek (t)
k
Robustness
Examples
X
stable by intermodulation, i.e.
HOIR Separation Transfer Function Recovery
Conclusion
∀i, j, robust to noise
∃k ,
∀t,
ei (t).ej (t) = ek (t)
New Principles Intermodulation Laws Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Analysis relying on a family of atoms, (ek )k : separable, i.e.
Principles Intermodulation Laws Separation
αk
recoverable from: y (t) =
Stability
Configuration
αk ek (t)
k
Robustness
Examples
X
stable by intermodulation, i.e.
HOIR Separation Transfer Function Recovery
Conclusion
∀i, j, robust to noise
∃k ,
∀t,
ei (t).ej (t) = ek (t)
Separation Extended ESS S. Tassart ESS Analysis
in the time domain: ESS is suitable by combination of the measurements U (m)
m∈[0,M)
State of the Art Constraints
If the test vectors are phase shifted:
Principles Intermodulation Laws Separation
∀p ∈ [1, M],
p−1 E (p) (jω) = ωM E(jω)
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
then HOIRs can be reconvered by IDFT: H1 (jω)E (1) (jω) U (0) (jω) H2 (jω)E (2) (jω) U (1) (jω) = IDFT .. .. . . (M) (M−1) HM (jω)E (jω) U (jω)
A bit more complex with real signal, but same principles. . .
Separation Extended ESS S. Tassart ESS Analysis
in the time domain: ESS is suitable by combination of the measurements U (m)
m∈[0,M)
State of the Art Constraints
If the test vectors are phase shifted:
Principles Intermodulation Laws Separation
∀p ∈ [1, M],
p−1 E (p) (jω) = ωM E(jω)
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
then HOIRs can be reconvered by IDFT: H1 (jω)E (1) (jω) U (0) (jω) H2 (jω)E (2) (jω) U (1) (jω) = IDFT .. .. . . (M) (M−1) HM (jω)E (jω) U (jω)
A bit more complex with real signal, but same principles. . .
Separation Extended ESS S. Tassart ESS Analysis
in the time domain: ESS is suitable by combination of the measurements U (m)
m∈[0,M)
State of the Art Constraints
If the test vectors are phase shifted:
Principles Intermodulation Laws Separation
∀p ∈ [1, M],
p−1 E (p) (jω) = ωM E(jω)
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
then HOIRs can be reconvered by IDFT: H1 (jω)E (1) (jω) U (0) (jω) H2 (jω)E (2) (jω) U (1) (jω) = IDFT .. .. . . (M) (M−1) HM (jω)E (jω) U (jω)
A bit more complex with real signal, but same principles. . .
Separation Extended ESS S. Tassart ESS Analysis
in the time domain: ESS is suitable by combination of the measurements U (m)
m∈[0,M)
State of the Art Constraints
If the test vectors are phase shifted:
Principles Intermodulation Laws Separation
∀p ∈ [1, M],
p−1 E (p) (jω) = ωM E(jω)
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
then HOIRs can be reconvered by IDFT: H1 (jω)E (1) (jω) U (0) (jω) H2 (jω)E (2) (jω) U (1) (jω) = IDFT .. .. . . (M) (M−1) HM (jω)E (jω) U (jω)
A bit more complex with real signal, but same principles. . .
Stability Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Consider the atoms (designed in the Fourier domain): 1 ω Em,p (e jω ) = ω − 2 W m exp (−jϕ(ω) − jτp ω) p
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration
ϕ(ω) : designed algorithmically for ESS: N ω 0 ϕ (ω) = log2 + τ0 K ω0
HOIR Separation Transfer Function Recovery
Conclusion
N : length of one period K : width (in octave) covered by E in one period N/K : integer ω0 , τ0 , : reference values W (ω) : spectral weight
Stability Problem Extended ESS S. Tassart ESS Analysis
Design of the spectral weight W (if existent) so that the family of atoms Em,p is stable by intermodulation, i.e.:
State of the Art Constraints
Principles Intermodulation Laws
Em1 ,p1 ⊗ Em2 ,p2 Em1 ,p1 ⊗
Separation
∗ Em 2 ,p2
= α (m1 , p1 , m2 , p2 ) Em1 +m2 ,p1 +p2 = α (m1 , p1 , m2 , −p2 ) Em1 +m2 ,|p1 −p2 |
Stability Robustness
where α are parameters depending only on the shape of W .
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
⊗ means multiplication in the time domain ∗ Em,p
means conjugation in the time domain
Stability What does it give ? Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
Consider C(e jω ) and its modulation C (m) (e jω ): C e jω =E e jω + E ∗ e jω C (m+1) e jω =C (m) e jω ⊗ C e jω
Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
by virtue of the intermodulation laws: m W (ω)e−jτ1 ω W m ( ω2 )e−jτ2 ω (m) jω > C e = b:,m .. .
ω W m( m )e−jτm ω
C e jω
Stability So what ? Extended ESS S. Tassart ESS Analysis
The observed HOIRs Lk (e jω ) are given as:
State of the Art Constraints
Principles Intermodulation Laws Separation
k n X ω o Lk e jω = Hm e jω = bk ,: (ω)·H e jω bk ,m W m k m=1
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
band-limited approach of the Chebychev matrix version the matrix B(ω) is now frequency dependant The transfer functions H e jω can be evaluated once: Lk are properly separated B(ω) is inverted
Spectral Weight Design Heuristic Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
Good match is found with bell-shaped weights: log-normal: W (ω) = N log ωω¯ ; 0, σ 2 log-Hann: W (ω) = H σ1 log ωω¯
Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
with:
! 1 1 x −µ 2 N (x; µ, σ) = √ exp − 2 σ σ 2π cos2 πx , ∀x ∈ [−1, 1] 2 H(x) = 0 elsewhere.
Spectral Weight Design Heuristic Extended ESS S. Tassart
To find spectral weight W : evaluate actual vs. estimated C (m) for different W
ESS Analysis State of the Art
try the log-normal and log-Hann shapes
Constraints
Principles Intermodulation Laws
Spectral Shape
0
Separation
log-normal log-Hann
Stability -5
Robustness
Best choice for: log-Hann
-10
Examples Configuration
σ = 2,
-15
Transfer Function Recovery
Conclusion
Gain (dB)
HOIR Separation
ω ¯ = π/10,
-20 -25
N = 16384
-30
K = 16
-35
error = -27 dB
-40 -45 0.1
1
frequency (rad./sample)
Experimental Verification (m1 , p1 ) = (1, 2)
(m2 , p2 ) = (2, 1), log-normal shape
Extended ESS S. Tassart
Em1 ,p1 ⊗ Em2 ,p2
ESS Analysis State of the Art
gain
Constraints
Principles
phase difference Application of the intermodulation law for m1=1, m2=2, p1=2 and p2=1
Application of the intermodulation law for m1=1, m2=2, p1=2 and p2=1 2
Intermodulation Laws
EW ⊗ EW 1
0
EW ⊗ EW 1
2
2
Approximation 1.5
Separation Stability
-20
Group delay error (sample)
1
Examples Configuration HOIR Separation
gain (dB)
Robustness -40
-60
Transfer Function Recovery
0.5
0
-0.5
-1
-80
Conclusion
-1.5
-2
-100 10-1
100
Pulsation (radian/sample)
Frequency shift: +8.5%
10-1
100
Pulsation (radian/sample)
Experimental Verification (m1 , p1 ) = (1, 2)
(m2 , p2 ) = (2, 1), log-normal shape
Extended ESS S. Tassart
∗ Em1 ,p1 ⊗ Em 2 ,p2
ESS Analysis State of the Art
gain
Constraints
Principles
phase difference Application of the intermodulation law for m1=1, m2=2, p1=2 and p2=1
Application of the intermodulation law for m1=1, m2=2, p1=2 and p2=1 2
Intermodulation Laws
0
EW ⊗ conj(EW )
EW ⊗ conj(EW ) 1
1
2
2
Approximation 1.5
Separation Stability
-20
Group delay error (sample)
1
Examples Configuration HOIR Separation
gain (dB)
Robustness -40
-60
Transfer Function Recovery
0.5
0
-0.5
-1
-80
Conclusion
-1.5
-2
-100 10-1
100
Pulsation (radian/sample)
Frequency shift: +8.5%
10-1
100
Pulsation (radian/sample)
Robustness Time domain Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability
Time windowing in the deconvolution domain (when separating HOIRs) allows: the rejection of the uncorreleted noise the longest test vector, the more rejection we have Synchronized pattern repetition:
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
averaging patterns reduces the measurement noise, robust to impulse noise, not robust to models variations. It is probable that, at the end, only the length of the experiment is relevant with regard to the noise rejection performances.
Robustness Time domain Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability
Time windowing in the deconvolution domain (when separating HOIRs) allows: the rejection of the uncorreleted noise the longest test vector, the more rejection we have Synchronized pattern repetition:
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
averaging patterns reduces the measurement noise, robust to impulse noise, not robust to models variations. It is probable that, at the end, only the length of the experiment is relevant with regard to the noise rejection performances.
Robustness Time domain Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability
Time windowing in the deconvolution domain (when separating HOIRs) allows: the rejection of the uncorreleted noise the longest test vector, the more rejection we have Synchronized pattern repetition:
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
averaging patterns reduces the measurement noise, robust to impulse noise, not robust to models variations. It is probable that, at the end, only the length of the experiment is relevant with regard to the noise rejection performances.
Robustness frequency domain Extended ESS S. Tassart ESS Analysis State of the Art Constraints
We have a relationship between observed HOIRs L e jω and actual high order transfer functions H e jω :
Principles Intermodulation Laws Separation
HOIR to TF relationship
Stability Robustness
Examples Configuration
L e jω = B(ω) · H e jω
⇒
H e jω = A(ω) · L e jω
HOIR Separation Transfer Function Recovery
Conclusion
Matrix inversion is sensitive. Different options to avoid 0/0: A = (B∗ B + I)−1 B∗ clamp the singular values from B−1 ...
Robustness frequency domain Extended ESS S. Tassart ESS Analysis State of the Art Constraints
We have a relationship between observed HOIRs L e jω and actual high order transfer functions H e jω :
Principles Intermodulation Laws Separation
HOIR to TF relationship
Stability Robustness
Examples Configuration
L e jω = B(ω) · H e jω
⇒
H e jω = A(ω) · L e jω
HOIR Separation Transfer Function Recovery
Conclusion
Matrix inversion is sensitive. Different options to avoid 0/0: A = (B∗ B + I)−1 B∗ clamp the singular values from B−1 ...
Summary Extended ESS S. Tassart
1
Swept-Sine Analysis State of the Art Constraints
2
Principles Intermodulation Laws Separation Stability Robustness
3
Examples Configuration HOIR Separation Transfer Function Recovery
4
Conclusion
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Validation Models description Extended ESS S. Tassart
Hammerstein models: M = 4 and Hm (jω) has a random phase and an exponential decay:
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples
m 1 2 3 4
decay 5 ms 2.5 ms 1.7 ms 1.25 ms
gain -20 dB -40 dB -60 dB -100 dB
Configuration HOIR Separation Transfer Function Recovery
Conclusion
Test vectors: Fs = 96 kHz, K = 16, N = 214 (170ms). Performance: estimated by H ejω − H jω ˆ e m m γm (ω) = 20 log10 Hm ejω
Transfer Function recovery Reliable Spectral Range Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws
Test vector C(jω): center frequency is 4.8 kHz bandwidth (at -40dB) is 5.4 octave
Separation Stability Robustness
m
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
1 2 3 4
lower bound 790 Hz 2.3 kHz 4.9 kHz 8.6 kHz
upper bound 29.2 kHz 39.5 kHz 45.6 kHz 47.5 kHz
octave range 5.2 4.1 3.2 2.4
HOIR Separation Noise Free Conditions Extended ESS S. Tassart
Separated Impulses
ESS Analysis
L1 L2 L3 L4
State of the Art
-60
Constraints
Principles Intermodulation Laws
-80
Separation
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Gain (dB)
Stability
-100 -120 -140 -160 -0.02
0
0.02
0.04 Time (s)
0.06
0.08
0.1
HOIR Separation Noisy conditions (-77 dB) Extended ESS Separated Impulses, P = 2M, SNRI = -77dB
S. Tassart
L1 L2 L3 L4
ESS Analysis State of the Art
-60
Constraints
Principles Intermodulation Laws
-80
Separation
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Gain (dB)
Stability
-100
-120
Conclusion -140
-160 -0.02
0
0.02
0.04 Time (s)
0.06
0.08
0.1
Transfer Function recovery Noise Free condition Extended ESS S. Tassart
Relative estimation error in clean conditions
20
ESS Analysis State of the Art Constraints
0
Principles
Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Performance (dB)
Intermodulation Laws
-20 -40 -60
Conclusion
-80
H1 H2 H3
-100 1000
10000 Frequency (Hz)
Transfer Function H1 recovery Noisy conditions (-77 dB, -57 dB, -37 dB) Extended ESS Evaluation of H1
S. Tassart 20
-77dB -57dB -37dB
ESS Analysis State of the Art Constraints
0
Principles Intermodulation Laws
-20
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Performance (dB)
Separation
-40
-60
Conclusion -80
-100 1000
10000 Frequency (Hz)
Transfer Function H2 recovery Noisy conditions (-77 dB, -57 dB, -37 dB) Extended ESS Evaluation of H2
S. Tassart 20
-77dB -57dB -37dB
ESS Analysis State of the Art Constraints
0
Principles Intermodulation Laws
-20
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Performance (dB)
Separation
-40
-60
Conclusion -80
-100 1000
10000 Frequency (Hz)
Transfer Function H3 recovery Noisy conditions (-77 dB, -57 dB, -37 dB) Extended ESS Evaluation of H3
S. Tassart 20
-77dB -57dB -37dB
ESS Analysis State of the Art Constraints
0
Principles Intermodulation Laws
-20
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Performance (dB)
Separation
-40
-60
Conclusion -80
-100 1000
10000 Frequency (Hz)
Summary Extended ESS S. Tassart
1
Swept-Sine Analysis State of the Art Constraints
2
Principles Intermodulation Laws Separation Stability Robustness
3
Examples Configuration HOIR Separation Transfer Function Recovery
4
Conclusion
ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Conclusion Wrap-up Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability
generate the test vector C e jω and: ˜ e jω its band-limited inverse; C B(ω) band-limited intermodulation matrix derive multiple P test vectors, with phase shift ωp : w:,k =
ω1k
ω2k
· · · ωPk
>
Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
pseudo-inverse computation: V = (W∗ W)−1 W∗ HOIR (i.e. Lk ) separation, given the observations U: ˜ e jω = Lk e jω e−jτk ω vk ,: U e jω C Transfer functions (i.e. H) recovery, e.g.: ∗ jω ∗ (B (ω)B(ω) + (ω)IM ) H e = B (ω)L e jω
Conclusion Summary Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Extension of the ESS analysis framework separation of HOIR is enhanced with the help of phase shifted test vectors time overlap of HOIRs is no longer a limitation band-limitation of the test vectors is taken into account
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Limitations operating bandwidth of test vector still limited (start at 800 Hz for a sampling rate of 96 kHz) performance limited by the appromixation quality of the intermodulation laws
Conclusion Summary Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Extension of the ESS analysis framework separation of HOIR is enhanced with the help of phase shifted test vectors time overlap of HOIRs is no longer a limitation band-limitation of the test vectors is taken into account
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Limitations operating bandwidth of test vector still limited (start at 800 Hz for a sampling rate of 96 kHz) performance limited by the appromixation quality of the intermodulation laws
Conclusion Future Directions Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
to find new spectral weights W (ω) that reduce the residual error observed when applying the intermodulation laws
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
to integrate different amplitudes in the set of test vectors to improve the separation of HOIR (and rejection of uncorrelated noise) with exponential decayed windows
Conclusion Future Directions Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
to find new spectral weights W (ω) that reduce the residual error observed when applying the intermodulation laws
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
to integrate different amplitudes in the set of test vectors to improve the separation of HOIR (and rejection of uncorrelated noise) with exponential decayed windows
Conclusion Future Directions Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation
to find new spectral weights W (ω) that reduce the residual error observed when applying the intermodulation laws
Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
to integrate different amplitudes in the set of test vectors to improve the separation of HOIR (and rejection of uncorrelated noise) with exponential decayed windows
Conclusion Questions and Answers Extended ESS S. Tassart ESS Analysis State of the Art Constraints
Principles Intermodulation Laws Separation Stability Robustness
Examples Configuration HOIR Separation Transfer Function Recovery
Conclusion
Thank you for your attention. Question & Answers
