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Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 1 / 53
Simulating Hearing Loss Using a
Transmission Line Cochlea ModelDelft University of Technology
Leo Koop
October 28, 2015
Outline1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 2 / 53
Simulating Hearing Loss
• Applied Mathematics Department at TU Delft
• Supervisor: Kees Vuik
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 3 / 53
Simulating Hearing Loss
• An incas3 project
• Supervisor: Peter van Hengel
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 4 / 53
Motivation - Hearing Loss, Becoming morePrevalent
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 5 / 53
Motivation - Uses of a Hearing Loss Simulator
• Education
• Hearing loss prevention
• Simplifying hearing aid development and testing
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 6 / 53
Motivation - Uses of a Hearing Loss Simulator
• Education
• Hearing loss prevention
• Simplifying hearing aid development and testing
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 6 / 53
Motivation - Uses of a Hearing Loss Simulator
• Education
• Hearing loss prevention
• Simplifying hearing aid development and testing
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 6 / 53
The Ear
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 7 / 53
The Cochlea, a Cross Section
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 8 / 53
The Cochlea
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 9 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 10 / 53
Modeling the Cochlea
The Main Equation
∂2p
∂x2(x , t) − 2ρ∂2y
h∂t2(x , t) = 0, 0 ≤ x ≤ L, t ≥ 0 (1)
• y(x , t): The excitation of the oscillator
• p(x , t): Pressure on the cochlear partition
• ρ: Density of the cochlear fluid
• h: Height of the scala
• L: Length of the cochlea
• t: Time
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 11 / 53
Modeling the Cochlea
The Main Equation
∂2p
∂x2(x , t) − 2ρ∂2y
h∂t2(x , t) = 0, 0 ≤ x ≤ L, t ≥ 0 (1)
• y(x , t): The excitation of the oscillator
• p(x , t): Pressure on the cochlear partition
• ρ: Density of the cochlear fluid
• h: Height of the scala
• L: Length of the cochlea
• t: Time
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 11 / 53
Modeling the Cochlea
The Main Equation
∂2p
∂x2(x , t) − 2ρ∂2y
h∂t2(x , t) = 0, 0 ≤ x ≤ L, t ≥ 0 (1)
• y(x , t): The excitation of the oscillator
• p(x , t): Pressure on the cochlear partition
• ρ: Density of the cochlear fluid
• h: Height of the scala
• L: Length of the cochlea
• t: Time
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 11 / 53
Modeling the Cochlea
The Main Equation
∂2p
∂x2(x , t) − 2ρ∂2y
h∂t2(x , t) = 0, 0 ≤ x ≤ L, t ≥ 0 (1)
• y(x , t): The excitation of the oscillator
• p(x , t): Pressure on the cochlear partition
• ρ: Density of the cochlear fluid
• h: Height of the scala
• L: Length of the cochlea
• t: Time
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 11 / 53
Modeling the Cochlea
The Main Equation
∂2p
∂x2(x , t) − 2ρ∂2y
h∂t2(x , t) = 0, 0 ≤ x ≤ L, t ≥ 0 (1)
• y(x , t): The excitation of the oscillator
• p(x , t): Pressure on the cochlear partition
• ρ: Density of the cochlear fluid
• h: Height of the scala
• L: Length of the cochlea
• t: Time
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 11 / 53
Modeling the Cochlea
The pressure term
p(x , t) = my(x , t) + d(x)y(x , t) + s(x)y(x , t) (2)
• m: Mass of the membrane
• d(x): Position dependent damping
• s(x): Position dependent stiffness
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 12 / 53
Modeling the Cochlea
The pressure term
p(x , t) = my(x , t) + d(x)y(x , t) + s(x)y(x , t) (2)
• m: Mass of the membrane
• d(x): Position dependent damping
• s(x): Position dependent stiffness
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 12 / 53
The Stiffness Term
Linear stiffness term (theoretical)
s(x) = s0e−λx
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 13 / 53
The Stiffness Term
Linear stiffness term (theoretical)
s(x) = s0e−λx
Scaled length of the cochlea0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Stiffnessvalue
×104
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
s(x)
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 13 / 53
The Damping Term
Linear damping term (theoretical)
d(x) = ε√m s(x)
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 14 / 53
The Damping Term
Linear damping term (theoretical)
d(x) = ε√m s(x)
Scaled length of the cochlea0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Dam
pingvalue
0
1
2
3
4
5
6
7
8
d(x)
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 14 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 15 / 53
Model Output
Oscillators in scaled cochlea length0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1O
scillatordisplacement/velocity
(nm/m
s) ×10-5
-5
-4
-3
-2
-1
0
1
2
3
4
5progress 151.00 steps, t tilde = 0.00
osc. velocity
osc. displacement
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 16 / 53
Model Output
Oscillators in scaled cochlea length0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1O
scillatordisplacement/velocity
(nm/m
s) ×10-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
progress 1101.00 steps, t tilde = 0.00
osc. velocity
osc. displacement
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 17 / 53
Model Output
Oscillators in scaled cochlea length0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1O
scillatordisplacement/velocity
(nm/m
s) ×10-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
progress 4101.00 steps, t tilde = 0.01
osc. velocity
osc. displacement
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 18 / 53
Model Input and Output
Time (ms)80 90 100 110 120 130 140 150 160
Amplitude
-0.5
0
0.5
Singing sound
Time (ms)80 90 100 110 120 130 140 150 160
Amplitude(nm)
×10-5
-4
-2
0
2
4
Singing sound single oscillator trace
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 19 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 20 / 53
Simulation Approach
• We have a model of the normal cochlea
• Can we find the Cause given the Effect?
Idea
Find a method to get back the original sound from the model of ahealthy ear. Use this method on a model of a damaged ear tosimulate hearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 21 / 53
Simulation Approach
• We have a model of the normal cochlea
• Can we find the Cause given the Effect?
Idea
Find a method to get back the original sound from the model of ahealthy ear. Use this method on a model of a damaged ear tosimulate hearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 21 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 22 / 53
Inverse Filter Approach for a Single Oscillator
If the original sound is a and a result from the model is c then anoscillator in the model is b
c(t) = a(t) ∗ b(t)
This in the frequency domain is:
C (f ) = A(f ) · B(f )
A change in basic operation from convolution to multiplication
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 23 / 53
Finding the Inverse Filter
• b(t) is the impulse response of an oscillator
• B(f ) is the frequency response of an oscillator
In the frequency domain B−1(f ) is easily found:
B−1(f ) =1
B(f )
Given C (f ) and B(f ), A(f ) is:
A(f ) = C (f ) · B−1(f )
An inverse Fourier transform yields a(t)
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 24 / 53
The Idea
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 25 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 26 / 53
The Cochlea
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 27 / 53
A Good Impulse Response
Time in ms-2 0 2 4 6 8 10 12
Amplitude(nm)
×10-5
-2
-1
0
1
2
Clean Impulse Response
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 28 / 53
Non-Finite Impulse Response
Time in seconds0 0.05 0.1 0.15
Trace
ofoscillator410of600
×10-6
-4
-2
0
2
4
Impulse response with no update
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 29 / 53
Energy Build-up in the Model
Time (ms)0 500 1000 1500 2000
Amplitude(nm)
-0.05
0
0.05
Unstable CM output
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 30 / 53
The Solution
Scaled length of the cochlea0 0.2 0.4 0.6 0.8 1
Dam
pingvalue
0
5
10
15
20
25
d(x) updated
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 31 / 53
A Better Impulse Response
Time in seconds0 0.05 0.1 0.15
Trace
ofoscillator410of600
×10-6
-4
-2
0
2
4
Impulse response with no update
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 32 / 53
A Better Impulse Response
Time in seconds0 0.05 0.1 0.15
Trace
ofoscillator410of600
×10-6
-4
-2
0
2
4
Impulse response with update
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 33 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 34 / 53
Tuning Curves
frequency in kHz10
-110
010
1
Amplitude(dB)
-200
-180
-160
-140
-120
-100
-80
-60
-40
B(f ) - tuning curves
osc.#: 5
osc.#: 30
osc.#: 55
osc.#: 80
osc.#: 105
osc.#: 130
osc.#: 155
osc.#: 180
osc.#: 205
osc.#: 230
osc.#: 255
osc.#: 280
osc.#: 305
osc.#: 330
osc.#: 355
osc.#: 380
osc.#: 405
osc.#: 430
osc.#: 455
osc.#: 480
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 35 / 53
The Inverse Filters
frequency in kHz10
-110
010
1
Amplitude(dB)
40
60
80
100
120
140
160
180
200
B−1(f ) - filter functions
osc.#: 5
osc.#: 30
osc.#: 55
osc.#: 80
osc.#: 105
osc.#: 130
osc.#: 155
osc.#: 180
osc.#: 205
osc.#: 230
osc.#: 255
osc.#: 280
osc.#: 305
osc.#: 330
osc.#: 355
osc.#: 380
osc.#: 405
osc.#: 430
osc.#: 455
osc.#: 480
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 36 / 53
Error Amplification
frequency in kHz
10-1
100
101
Amplitude
0
500
1000
1500
2000
Frequency Content of Result
Sound
inv. Filt.
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 37 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 38 / 53
Choosing Contribution Bounds per InverseFilter
frequency in kHz10
-110
010
1
Amplitude(dB)
60
80
100
120
140
160
Two inverse filters and f and f
tilde f f
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 39 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 40 / 53
Modifying the Damping Term
Oscillator number in the cochlea model0 100 200 300 400 500 600
Dampingvalue
0
5
10
15
20
25
Damping while simulating damage
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 41 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 42 / 53
The Resulting Sound
frequency in kHz10
-110
010
1
Amplitude
0
200
400
600
800
1000
1200
1400
1600
1800
Frequency Content of Result
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 43 / 53
The Resulting Sound
frequency in kHz
10-1
100
101
Amplitude
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ratio between damage and original signal
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 44 / 53
Next Subsection1 Introductions and Motivation2 Modeling the Cochlea
The General ModelModel Output
3 Inverse ModelThe IdeaInverse Filtering Method
4 Energy ReflectionThe Problem
5 Sound ResynthesisTuning Curves and Inverse FiltersFilter Contribution Regions
6 Simulating Hearing LossModeling a Damaged CochleaThe Resulting SoundComparison of Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 45 / 53
A Normal Cochleogram
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 46 / 53
Simulating Damage
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 47 / 53
Cochleogram in Response to Damaged Sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 48 / 53
Comparing the Cochleograms
Oscillator number in the cochlea model
0 100 200 300 400 500 600
Averageresponse
level
intime
-60
-55
-50
-45
-40
-35
-30
Average of cochleogram per oscillator in time
Original sound
While simulating damage
Response to resynthesized sound
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 49 / 53
Sound Samples
• “1, 2, 3” From the TIDigits database• “1, 2, 3 w/ noise” With background noise• “1, 2, 3 w/ noise” Resynthesized without simulating damage• “1, 2, 3 w/ noise” Approximate hearing loss of a 40 year old• “1, 2, 3 w/ noise” Approximate hearing loss of a 60 year old• “1, 2, 3 w/ noise” Approximate hearing loss of a 90 year old
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 50 / 53
Accomplishments
• Fixed an energy reflection problem in the model
• Found a way to combine contributions from different oscillators
• A transmission line cochlea model can now be used to simulatehearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 51 / 53
Accomplishments
• Fixed an energy reflection problem in the model
• Found a way to combine contributions from different oscillators
• A transmission line cochlea model can now be used to simulatehearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 51 / 53
Accomplishments
• Fixed an energy reflection problem in the model
• Found a way to combine contributions from different oscillators
• A transmission line cochlea model can now be used to simulatehearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 51 / 53
Recommendations
• Sound resynthesis from a nonlinear transmission line cochleamodel
• The best method to solve the energy reflection problem
• How to best modify the cochlea model to simulate various kindsof hearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 52 / 53
Recommendations
• Sound resynthesis from a nonlinear transmission line cochleamodel
• The best method to solve the energy reflection problem
• How to best modify the cochlea model to simulate various kindsof hearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 52 / 53
Recommendations
• Sound resynthesis from a nonlinear transmission line cochleamodel
• The best method to solve the energy reflection problem
• How to best modify the cochlea model to simulate various kindsof hearing loss
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 52 / 53
The End
“He who has an ear, let him hear...”
- Revelation 2:17
Leo Koop (TU Delft) Simulating Hearing Loss Using a Transmission Line Cochlea ModelOctober 28, 2015 53 / 53