Head and Ear Thurs. March 29, 2018 1 Impulse function at = 0. , , - - PowerPoint PPT Presentation

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COMP 546 Lecture 20 Head and Ear Thurs. March 29, 2018 1 Impulse function at = 0. , , , = ( 0 , 0 , 0 , ) To define an impulse function properly in a continuous space

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COMP 546

Lecture 20

Head and Ear

  • Thurs. March 29, 2018
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Impulse function at 𝑒 = 0.

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𝐽 π‘Œ, 𝑍, π‘Ž, 𝑒 = πœ€(π‘Œ βˆ’ π‘Œ0, 𝑍 βˆ’ 𝑍

0, π‘Ž βˆ’ π‘Ž0, 𝑒)

Sound obeys the wave equation. So, how is this function defined 𝑒 β‰  0 ?

To define an impulse function properly in a continuous space requires more math. Let’s not spend our time doing that, since we just want qualitative behavior here.

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Impulse becomes expanding sphere

𝑠 = 𝑀 𝑒

One can show that this follows from the wave equation.

𝑒 = 4 Ξ” 𝑒 𝑒 = 3 Ξ” 𝑒 𝑒 = 2 Ξ” 𝑒 𝑒 = Ξ” 𝑒

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𝑠 = 𝑀 𝑒

Impulse sound energy is spread over a thin sphere of fixed thickness and of area 4𝜌 𝑠2 where 𝑠2 = (π‘Œ βˆ’ π‘Œ0)2 + (𝑍 βˆ’ 𝑍

0)2 + (π‘Ž βˆ’ π‘Ž0)2 .

𝐽2 ~

1 𝑠2

𝐽 ~

1 𝑠

So, SPL

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𝐽 π‘Œ, 𝑍, π‘Ž, 𝑒

𝐽𝑑𝑠𝑑 πœ€(π‘Œ βˆ’ π‘Œ0, 𝑍 βˆ’ 𝑍

0, π‘Ž βˆ’ π‘Ž0), when 𝑒 = 0 𝐽𝑑𝑠𝑑 𝑠

πœ€ 𝑠 βˆ’ 𝑀 𝑒 , when 𝑒 > 0 and 𝐽𝑑𝑠𝑑 is constant (~energy in impulse)

𝑠 = (π‘Œ βˆ’ π‘Œ0)2 + (𝑍 βˆ’ 𝑍

0)2 + (π‘Ž βˆ’ π‘Ž0)2

=

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𝐽𝑑𝑠𝑑 𝑒 =

𝑒′ =0 π‘ˆβˆ’1

πœ€ 𝑒 βˆ’ 𝑒′ 𝐽𝑑𝑠𝑑(𝑒′)

We can write a general sound source a sum of impulse functions:

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Far from the source, where r is large, the wavefront is approximately locally planar.

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Binaural hearing

(preview of next lecture)

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If the sound arrives from the left (assuming planar wavefronts), what is the interaural delay?

𝑒 = 17 cm

𝑒 = 𝑒

𝑀 = .17 340 β‰ˆ .5 𝑛𝑑

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NaΓ―ve model: cone of confusion

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All incoming directions on a cone define the same delay & shadow effect.

Model head, shoulders, ears as a sphere.

Exercise: use time delay 𝜐 to estimate cone angle 𝜚

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Interaural differences

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How can the auditory system estimate the delay and shadowing ? Here is a simple model:

π½π‘š (𝑒) = 𝛽 𝐽𝑠(𝑒 βˆ’ 𝜐) + π‘œ(𝑒)

shadow (attenuation) noise delay

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Maximum likelihood: find the 𝛽 and 𝜐 that minimize where 𝜐 < 0.5 𝑛𝑑.

𝑒=1 π‘ˆ

{ π½π‘š (𝑒) βˆ’ 𝛽 𝐽𝑠(𝑒 βˆ’ 𝜐) }2

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To find the 𝛽 and 𝜐 that minimize we first find the 𝜐 that maximizes

𝑒

π½π‘š (𝑒) 𝐽𝑠(𝑒 βˆ’ 𝜐) .

𝑒=1 π‘ˆ

{π½π‘š (𝑒)2 βˆ’ 𝛽 π½π‘š (𝑒) 𝐽𝑠(𝑒 βˆ’ 𝜐) + 𝐽𝑠(𝑒 βˆ’ 𝜐)2} This ignores the small dependence of the 3rd term above on 𝜐.

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𝛽2 = 𝑒=1

π‘ˆ

π½π‘š (𝑒)2 𝑒=1

π‘ˆ

𝐽𝑠 (𝑒 βˆ’ 𝜐)2

Then estimate 𝛽 (shadowing): Note that this gives two cues which we can combine.

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The Human Ear

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Outer Ear

Next ten slides: How do head and outer ear transform the sound that arrives at the ear from various directions ?

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Head related impulse response (HRIR)

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𝐽 𝑒 = β„Žπ‘— (𝑒; 𝜚, πœ„) βˆ— πœ€ 𝑠 βˆ’ 𝑀𝑒

Suppose sound is from direction (𝜚, πœ„). The wave is planar when it arrives at the head. If the source is an impulse then sound measured at the ear drum of ear 𝑗 is:

left or right

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Sound source 𝐽𝑑𝑠𝑑 𝑒; 𝜚, πœ„ transformed

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Suppose sound is from direction (𝜚, πœ„) and emits 𝐽𝑑𝑠𝑑 𝑒; 𝜚, πœ„ . Then the sound measured at the ear drum of ear 𝑗 is:

𝐽 𝑒 = β„Žπ‘— (𝑒; 𝜚, πœ„) βˆ— 𝐽𝑑𝑠𝑑 𝑒; 𝜚, πœ„

(Ignoring time delay from source to ear.)

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azimuth πœ„ elevation 𝜚

KEMAR mannequin

In following slides, I will show HRIR measurements β„Žπ‘— (𝑒; 𝜚, πœ„).

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Azimuth πœ„ (Elevation 𝜚 = 0)

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Suppose sound is measured at right ear drum.

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Source direction (azimuth)

HRIR

0.7 ms

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Arrival time differences are not as significant when azimuth = 0 and elevation is varied. Source direction (elevation)

HRIR

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If head is symmetric about the medial plane (left/right), then : β„Žπ‘šπ‘“π‘”π‘’ (𝑒; 𝜚, πœ„) = β„Žπ‘ π‘—π‘•β„Žπ‘’ (𝑒; 𝜚, βˆ’πœ„)

azimuth πœ„ elevation 𝜚

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For each incoming sound direction (𝜚, πœ„), what is the Fourier transform with respect to variable t ? π½π‘ π‘—π‘•β„Žπ‘’ 𝑒; 𝜚, πœ„ = β„Žπ‘ π‘—π‘•β„Žπ‘’ (𝑒; 𝜚, πœ„) βˆ— 𝐽𝑑𝑠𝑑 𝑒; 𝜚, πœ„

HRIR

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For each incoming sound direction (𝜚, πœ„), what is the Fourier transform with respect to t ? π½π‘ π‘—π‘•β„Žπ‘’ 𝑒; 𝜚, πœ„ = β„Žπ‘ π‘—π‘•β„Žπ‘’ (𝑒; 𝜚, πœ„) βˆ— 𝐽𝑑𝑠𝑑 𝑒; 𝜚, πœ„

HRIR

π½π‘ π‘—π‘•β„Žπ‘’ πœ•; 𝜚, πœ„ = β„Žπ‘ π‘—π‘•β„Žπ‘’ (πœ•; 𝜚, πœ„) 𝐽𝑑𝑠𝑑 πœ•; 𝜚, πœ„

Head Related β€œTransfer Function” (HRTF)

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Shadowing effect dominates: HRTF for each frequency πœ• has a max at 90 degrees (right ear) and min at 270 degrees (left ear).

HRTF β„Žπ‘ π‘—π‘•β„Žπ‘’(πœ•; πœ„, 𝜚 = 0)

(plot for fixed elevation 𝜚 = 0)

πœ„

πœ•

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Curves shifted for visualization Valley is β€œpinnal notch” (it distinguishes elevations) (medial plane)

HRTF β„Žπ‘ π‘—π‘•β„Žπ‘’ (πœ•; πœ„ = 0, 𝜚)

(plot for fixed azimuth πœ„ = 0. )

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Middle Ear

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β€œEar drum” Ossicles (bones)

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auditory canal pinna cochlea

  • uter middle inner

Ossicles act as a lever, transferring vibrations from ear drum to fluid in cochlea

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Inner ear

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Cochlea Vestibular apparatus

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Cochlea (unrolled)

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TOP VIEW SIDE VIEW

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Cochlea (unrolled)

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TOP VIEW SIDE VIEW

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short (small L) & tense (large c) long (large L) & loose (small c)

20,000 Hz 20 Hz

Recall vibrating string πœ• =

𝑑 𝑀

Both 𝑀 and 𝑑 vary on fibres on basilar membrane.

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Basilar Membrane (BM)

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http://auditoryneuroscience.com/topics/basilar-membrane-motion-0-frequency-modulated-tone

http://auditoryneuroscience.com/ear/bm_motion_2