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Prof. , School of EE&T Term 3, 2022
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ELEC3104: Mini-Project – Cochlear Signal Processing
Prof. , School of EE&T Term 3, 2022
ELEC3104: Mini-Project – Cochlear Signal Processing
TLT – Level 2 (Pass Level): Implementation of a cascaded filter bank
model of the cochlea for analysis purposes.
TLT-Level 2 Project Implementation
✓ For this project, you should implement a digital model of the peripheral auditory system
comprising of a model of the outer ear and the middle ear (from TLT Level 1) and the transmission
line mode of the cochlea (TLT Level 2).
✓ The transmission line model of the cochlea can be implemented as a cascade of many band-pass
✓ You should understand how the characteristics of the model are related to the functioning of the
cochlea explained in TLT Level 1
✓ Validate that all parts of your model operate as desired in terms of impulse responses, frequency
responses for a variety of input signals.
Inner ear (Cochlea)
A longitudinal section of an uncoiled cochlea
Basilar membrane
0.05155 cm
TLT-Level 2 Project Implementation: Cochlear Modelling
The Cascade Model
• A simple electrical model of a section of the BM is shown below figure below.
Pressure and Displacement Transfer Functions
✓ The Voltage or Pressure transfer function of the isolated section can be obtained as follows:
✓ One can see that the displacement transfer function is contained in the pressure transfer
function and therefore a simple cascade arrangement is possible.
Low pass filter Resonant pole Resonant zero
Frequency Response – one section of the membrane
Pressure Transfer Function
Low pass filter Resonant pole Resonant zero
Frequency Response – one section of the membrane
Frequency Response
of Displacement
Transfer Function of
Displacement Transfer Function
Low pass filter Resonant pole
Digital filter model of the basilar membrane
✓ A digital filter model of the basilar membrane can be obtained by transforming the analogue filter equation
(given below) to an equivalent digital filter equation.
✓ The impulse response of the Basilar Membrane (BM) is an important property and it should be preserved in the
digital filter model of the BM. So use impulse invariant transformation and is given by:
; T – sampling period
✓ On applying the impulse invariant transformation, the pressure transfer function in digital domain can be
✓ Where 𝑎0, 𝑎1, 𝑎2, 𝑏1, 𝑎𝑛𝑑 𝑏2 are the digital filter coefficients;
Digital filter coefficients
−𝒑𝟏 𝑻cos(𝒒𝟏𝑻); 𝒂𝟐 = 𝒆
−𝟐𝒑𝟏 𝑻; 𝒑𝟏 =
; 𝒒𝟏 = 𝒑𝟏 𝟒𝑸𝒛
−𝒑𝟐 𝑻cos(𝒒𝟐𝑻); 𝒃𝟐 = 𝒆
−𝟐𝒑𝟐 𝑻; 𝒑𝟐 =
; 𝒒𝟐= 𝒑𝟐 𝟒𝑸𝒑
Sampling frequency (fs) = 48 kHz, T = 1/ fs
✓ In each section, pressure is converted into displacement of the basilar membrane and transmitted to the
following section. This leads to two transfer functions, one relating the output pressure, 𝑉𝑜 𝑛 , and the
input pressure, 𝑉𝑖 𝑛 , and the other relating the output displacement, 𝑉𝑚 𝑛 to the input pressure.
✓ Each section of the basilar membrane can be realised as a digital filter as follows:
One can see that the displacement transfer function is contained
in the pressure transfer function and hence a simple cascade
arrangement is sufficient to represent the cochlear model
𝒂𝟎 = (𝟐 − cos𝜽𝒄) − (𝟐 − cos𝜽𝒄)
𝜽𝒄 𝑖𝑠 𝑡ℎ𝑒 3dB cut-off frequency of the
low pass filter (choose 𝜽𝒄 = 1.4 * 𝝎𝒛)
Transmission Line Model of the Cochlea
✓ The basic model of the cochlea is a transmission line model in which the basilar membrane is modelled as a cascade
of 128 low pass filters, notch filters and resonators as shown below. Assume a sampling frequency of 48 kHz.
Transmission Line Model
✓ Each digital filter section in the model above represents a section of the basilar membrane (tuned to a specific
frequency) with 128 sections representing the entire basilar membrane.
✓ A stimuls representing pressure at the ear drum (after the outer ear model) is the input to the model shown in
the figure above. This stimulus then moves along the transmission line as a travelling wave corresponding to the
pressure in the cochlear fluid.
Selection of Frequency Scale
✓ The model considers the BM length to be 3.5 cm.
✓ If the membrane is simulated using 128 digital filters connected in cascade then the section length (∆𝑥) and the frequency
ratio between adjacent sections are constant throughout. That is ∆𝑥 = 3.5/128 = 0.0275 cm;
(20000 ) 10−0.667𝑥
(20000 ) 10−0.667(𝑥+∆𝑥)
= 100.667∆𝑥 = 1.0429
DISPLACEMENT
DISPLACEMENT
FILTER 128
DISPLACEMENT
OUTPUT 128
Filter number Distance (cm) Frequency(fp)
1 0 20 kHz
66 1.777 1304.79 Hz
128 3.4727 96.553 Hz
Implementation – Step 1
✓ Number of filters N = 128; Length of the BM = 3.5 cm. ∆𝑥 =
= 0.0273 𝑐𝑚
✓ 𝑥 = 0, ∆𝑥, 2∆𝑥, 3∆𝑥,……… . , 127∆𝑥; 𝑹𝒆𝒔𝒐𝒏𝒂𝒏𝒕 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚: 𝒇𝒑(𝒏) = (𝟐𝟎𝟎𝟎𝟎 ) 𝟏𝟎
𝒇𝒑 𝒏 : Resonant
Frequency (Hz)
𝒇𝒛 𝒏 : Resonant zero (Hz)
(Notch filer)
1 0 𝑓𝑝 1 = 20000
= 1.0429 𝑓𝑧 1 = 1.0429× 𝑓𝑝 1 = 20858
2 ∆𝑥 𝑓𝑝 2 = 19177
= 1.0429 𝑓𝑧 2 = 1.0429× 𝑓𝑝 2 = 20000
3 2∆𝑥 𝑓𝑝 3 = 18389
= 1.0429 𝑓𝑧 3 = 1.0429× 𝑓𝑝 3 = 19178
4 3∆𝑥 𝑓𝑝 4 = 17633 . .
128 127∆𝑥 𝑓𝑝 128 = 96.55 – 𝑓𝑧 128 = 1.0429× 𝑓𝑝 128 =100.70
Implementation – Step 2
✓ Calculate the quality factor values, 𝑄𝑝 and 𝑄𝑧; and bandwidths, 𝐵𝑊𝑝 and 𝐵𝑊𝑧.
✓ 𝑄𝑝 varies linearly from 10 (first filter) to 5.5 (128
th filter)
✓ 𝑄𝑧 varies linearly from 22 (first filter) to 12 (128
th filter)
✓ You can change these around and observe what happens but ensure 𝑄𝑧 > 𝑄𝑝.
and 𝐵𝑊𝑧 𝑛 =
Filter No (𝒏): 𝒇𝒑 𝒏
𝑸𝒑 𝒏 𝑩𝑾𝒑 𝒏 in
𝑸𝒛 𝒏 𝑩𝑾𝒛 𝒏
1 20000 10 2000 20858 22 948.1
2 19177 9.96 1925 20000 21.92 912.4
3 18389 9.93 1852 19178 21.84 878.0
128 96.55 5.5 17.56 100.70 12 8.4
Design Criteria
✓ In order to simulate the basilar membrane accurately with the transmission line model, it is critical that the complex
zero frequency is slightly higher than the resonant frequency of the preceding resonator (complex pole filter).
✓ Selection of frequency scale: The ratios of the resonant frequencies of two adjacent sections are always constant and
equal to 1.0429. i.e., the resonant frequency of the 𝑖𝑡ℎ section is 1.0429 times the resonant frequency of the 𝑖 + 1𝑡ℎ
✓ From experiments it is known that the Q values (quality factor) for the complex pole section, 𝑄𝑝, go from 10 (1
st filter)
down to 5.5 (128th filter). You can interpolate linearly for the intermediate filters.
✓ The Q values (quality factor) for the complex zeros section, 𝑄𝑧, go from 22 (1
st filter) down to 12 (128th filter). You can
interpolate linearly for intermediate filters.
✓ For each section, 𝑄𝑧 > 𝑄𝑝
✓ 𝐾 is chosen as the ratio of complex pole frequency to the complex zero frequency (K =
✓ The cut-off frequency of the low pass filter can be chosen as follows: 𝒇𝒄 = 1.4 * 𝒇𝒛
✓ Note that this model gives the basilar membrane displacement without taking into account fluid coupling. In order to
take fluid coupling into account you must apply the spatial differentiation (TLT Level1 slide 12).
✓ Use impulse invariant transformations (as outlined earlier) to design the digital filters.
Implementation – Step 3
Filter Number 𝒇𝒑 (Hz) 𝒇𝒛 (Hz) 𝑸𝒑 𝑸𝒛 𝑩𝑾𝒑(Hz) 𝑩𝑾𝒛 (Hz)
1 20000 20858 10 22 2000 948.1
2 19177 20000 9.96 21.92 1925 912.4
3 18389 19178 9.93 21.84 1852 878.0
128 96.55 100.70 5.5 12 17.56 8.4
Filter Number 𝑲 𝑮𝟎 𝒂𝟎 𝑮𝒑 𝒃𝟏 𝒃𝟐 𝑮𝒛 𝒂𝟏 𝒂𝟐
1 0.9589 0.8137 0.1863 3.2863 -1.5167 0.7697 0.2729 -1.7514 0.9130
2 0.9589 0.8199 0.1801 3.1975 -1.4202 0.7773 0.2798 -1.6574 0.9160
3 0.9589 0.8242 0.1758 3.0960 -1.3113 0.7847 0.2885 -1.5473 0.9189
128 0.9589 0.0183 0.9817 0.0002 1.9975 0.9977 5759 1.9985 0.9987
If your implementation is right, you should get these parameter values for the selected filters if you started with the same
assumptions
Implementation – Step 4
✓ Digital filtering in the time domain for following inputs (sampled at 48kHz). You can use filter() in MATLAB to implement
filtering.
✓ single sinusoid – Try initially with 1kHz, then others of your choice.
✓ Make sure you obtain all the 128 displacement outputs and 128 pressure outputs for each input signal. You can store
these as 2 matrices with 128 columns and as many rows as there are samples in the input signal.
✓ From the impulse responses of each section, obtain the magnitude response (take FFT of the impulse response) and
make sure it is what you expect.
✓ For the sinusoidal input, plot a row of the output displacement matrix to observe basilar membrane displacement at
that time.
✓ Note that spatial differentiation is carried out across columns (within each row).
Spatial Differentiation
✓ Spatial differentiation of the membrane displacement
represents coupling between the cilia of the inner hair
cells, through the fluid in the subtectorial space.
✓ Spatial differentiation refers to taking the derivative
with respect to the position (along the basilar
membrane). A discrete model is given by:
𝑑𝑚[n]=𝑠𝑚[n]- 𝑠𝑚+1[n]
{𝑒. 𝑔. 𝑑1[n]=𝑠1[n]- 𝑠2[n]}
✓ The second spatial differentiation is given by:
𝑒𝑚[n]=𝑑𝑚[n]- 𝑑𝑚+1[n]
{𝑒. 𝑔. 𝑒1[n]=𝑑1[n]- 𝑑2[n]}
Spatial Differentiation
Filter 1 Filter 2 Filter 3 Filter m Filter N
s1[n] s2[n] s3[n] sm[n] sN[n]
Spatial differentiation
d1[n] d2[n] d3[n] dm[n] dN[n]
Spatial differentiation
e1[n] e2[n] e3[n] em[n] eN[n]
Digital filter model of the basilar membrane
Are you on the right track?
✓ If your implementation is on the right track, you should observe the following impulse responses (roughly) at
different sections of the membrane. Look at the membrane displacements when giving an impulse input.
Without spatial differentiation With two spatial differentiations
Filter 30 (5917.2 Hz) Filter 30 (5917.2 Hz)
Filter 60 (1678.7 Hz)
Filter 60 (1678.7 Hz)
Filter 90 (476.2 Hz)
Filter 90 (476.2 Hz)
Displacement without spatial
differentiation
Filter 1 Filter 2 Filter 3 Filter m Filter N
s1[n] s2[n] s3[n] sm[n] sN[n]
Spatial differentiation
d1[n] d2[n] d3[n] dm[n] dN[n]
Spatial differentiation
e1[n] e2[n] e3[n] em[n] eN[n]
Digital filter model of the basilar membrane
Are you on the right track?
✓ If your implementation is on the right track, you should observe the following magnitude response (roughly) at the
appropriate section of the basilar membrane with and without spatial differentiation.
Filter 72 (1.01 kHz)
Filter 72 (1.01 kHz)
Frequency response of a particular filter can be
obtained by taking the FFT of the impulse
response of that filter.
Filter 1 Filter 2 Filter 3 Filter m Filter N
s1[n] s2[n] s3[n] sm[n] sN[n]
Spatial differentiation
d1[n] d2[n] d3[n] dm[n] dN[n]
Spatial differentiation
e1[n] e2[n] e3[n] em[n] eN[n]
Digital filter model of the basilar membrane
Are you on the right track?
✓If your implementation of spatial
differentiation is right, you should observe
the following (roughly) for a 1kHz
sinusoidal input.
✓ If your implementation is on the right track, given a sinusoidal input, you should observe the following displacement
(after two spatial differentiation) plotted against section number. As expected the membrane should exhibit activities up
to the appropriate resonant section (where it shows maximum displacement) and no appreciable activity thereafter.
After 4000 samples (83.3 ms)
After 4000 samples (83.3 ms)
Filter 1 Filter 2 Filter 3 Filter m Filter N
s1[n] s2[n] s3[n] sm[n] sN[n]
Spatial differentiation
d1[n] d2[n] d3[n] dm[n] dN[n]
Spatial differentiation
e1[n] e2[n] e3[n] em[n] eN[n]
Digital filter model of the basilar membrane
Are you on the right track?
✓ If you plot the pressure outputs of different sections at a
series of regular time steps (superimpose them on the same
plot), you should observe something similar to the following.
Observing the envelope, the pressure is high at the basal end
and decays down to zero at the resonant position.
TLT level 2 Final Validation:
1. Apply a signal which is a sum of three
sinusoids (Try a sum of a low frequency, a
mid frequency and a high frequency
sinusoid), 1000-2000 samples, of equal
amplitude and frequencies of your choice,
to the input of the cochlear model.
2. Plot membrane displacement (in one
i. Without spatial differentiation
ii. With one spatial differentiation
iii. With two spatial differentiation
3. Explain your results to your lab
demonstrator.
Pressure envelope due to a 1 kHz
(filter number 72) sine wave at the
Reflection
You should reflect on your project to see the following:
✓ What is the function of the basilar membrane and how does it respond to various input stimuli?
✓ What will happen if you include the outer and middle ear models at the input of the transmission line
model of the cochlea in terms of hair cell output?
✓ What is the effect of spatial differentiation on the basilar membrane displacement?
✓ As a low frequency wave travels along the basilar membrane, the fluid pressure decreases and becomes
zero. Can you explain this in terms of the travelling wave?
✓ When the input has multiple frequencies and one of the tones is removed after a period of time, how and
when would the hair cell response change?
✓ What is the effect of the Q factors of your filters on the overall model? And what is the function of the
complex zeros in your model?
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