COMP 546 Lecture 20 Head and Ear Thurs. March 29, 2018 1
Impulse function at π’ = 0. π½ π, π, π, π’ = π(π β π 0 , π β π 0 , π β π 0 , π’) To define an impulse function properly in a continuous space requires more math. Letβs not spend our time doing that, since we just want qualitative behavior here. Sound obeys the wave equation. So, how is this function defined π’ β 0 ? 2
Impulse becomes expanding sphere One can show that this follows from the wave equation. π’ = 4 Ξ π’ π = π€ π’ π’ = 3 Ξ π’ π’ = 2 Ξ π’ π’ = Ξ π’ 3
Impulse sound energy is spread over a thin sphere of fixed thickness and of area 4π π 2 where π 2 = (π β π 0 ) 2 + (π β π 0 ) 2 + (π β π 0 ) 2 . π = π€ π’ 1 1 π½ 2 ~ π½ ~ So, SPL π 2 π 4
π½ π, π, π, π’ π½ π‘π π π(π β π 0 , π β π 0 , π β π 0 ) , when π’ = 0 = π½ π‘π π π π β π€ π’ , when π’ > 0 and π π = (π β π 0 ) 2 + (π β π 0 ) 2 + (π β π 0 ) 2 π½ π‘π π is constant (~energy in impulse) 5
We can write a general sound source a sum of impulse functions: πβ1 π π’ β π’ β² π½ π‘π π (π’ β² ) π½ π‘π π π’ = π’ β² =0 6
Far from the source, where r is large, the wavefront is approximately locally planar. 7
Binaural hearing (preview of next lecture) If the sound arrives from the left (assuming planar wavefronts), what is the interaural delay? π’ = π .17 π€ = 340 β .5 ππ‘ π = 17 cm 8
NaΓ―ve model: cone of confusion Model head, shoulders, ears as a sphere. All incoming directions on a cone define the same delay & shadow effect. Exercise: use time delay π to estimate cone angle π 9
Interaural differences How can the auditory system estimate the delay and shadowing ? Here is a simple model: π½ π ( π’ ) = π½ π½ π ( π’ β π ) + π(π’) noise shadow delay (attenuation) 10
Maximum likelihood: find the π½ and π that minimize π { π½ π ( π’ ) β π½ π½ π ( π’ β π ) } 2 π’=1 where π < 0.5 ππ‘ . 11
To find the π½ and π that minimize π {π½ π ( π’ ) 2 β π½ π½ π ( π’ ) π½ π ( π’ β π ) + π½ π ( π’ β π ) 2 } π’=1 we first find the π that maximizes π½ π ( π’ ) π½ π ( π’ β π ) . π’ This ignores the small dependence of the 3 rd term above on π. 12
Then estimate π½ (shadowing): π½ π ( π’ ) 2 π π’=1 π½ 2 = π½ π ( π’ β π ) 2 π π’=1 Note that this gives two cues which we can combine. 13
The Human Ear 14
Outer Ear Next ten slides: How do head and outer ear transform the sound that arrives at the ear from various directions ? 15
Head related impulse response (HRIR) Suppose sound is from direction ( π, π ). 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 π is: π½ π’ = β π (π’; π, π ) β π π β π€π’ left or right 16
Sound source π½ π‘π π π’; π, π transformed Suppose sound is from direction ( π, π ) and emits π½ π‘π π π’; π, π . Then the sound measured at the ear drum of ear π is: π½ π’ = β π (π’; π, π ) β π½ π‘π π π’; π, π (Ignoring time delay from source to ear.) 17
KEMAR mannequin In following slides, I will show HRIR measurements β π (π’; π, π ) . azimuth π elevation π 18
Azimuth π (Elevation π = 0) Suppose sound is measured at right ear drum. 19
0.7 ms HRIR Source direction (azimuth) 20
Arrival time differences are not as significant when azimuth = 0 and elevation is varied. HRIR Source direction (elevation) 21
If head is symmetric about the medial plane (left/right), then : β ππππ’ (π’; π, π ) = β π ππβπ’ (π’; π, βπ ) azimuth π elevation π 22
π½ π ππβπ’ π’; π, π = β π ππβπ’ (π’; π, π ) β π½ π‘π π π’; π, π HRIR For each incoming sound direction ( π, π ), what is the Fourier transform with respect to variable t ? 23
π½ π ππβπ’ π’; π, π = β π ππβπ’ (π’; π, π ) β π½ π‘π π π’; π, π HRIR For each incoming sound direction ( π, π ), what is the Fourier transform with respect to t ? = π½ π ππβπ’ π; π, π β π ππβπ’ (π; π, π ) π½ π‘π π π; π, π Head Related βTransfer Functionβ (HRTF) 24
HRTF β π ππβπ’ (π; π, π = 0) (plot for fixed elevation π = 0) Shadowing effect dominates: HRTF for each frequency π has a max at 90 degrees (right ear) and min at 270 degrees (left ear). π 25 π
HRTF β π ππβπ’ (π; π = 0, π) (plot for fixed azimuth π = 0. ) (medial plane) Curves shifted for visualization Valley is β pinnal notchβ (it distinguishes elevations) 26
Middle Ear Ossicles (bones) βEar drumβ 27
Ossicles act as a lever, transferring vibrations from ear drum to fluid in cochlea pinna cochlea auditory canal outer middle inner 28
Inner ear Vestibular apparatus Cochlea 29
Cochlea (unrolled) TOP VIEW SIDE VIEW 30
Cochlea (unrolled) TOP VIEW SIDE VIEW 31
π Recall vibrating string π = π Both π and π vary on fibres on basilar membrane. long (large L) short (small L) & loose (small c) & tense (large c) 20,000 Hz 20 Hz 32
Basilar Membrane (BM) http://auditoryneuroscience.com/topics/basilar-membrane-motion-0-frequency-modulated-tone 33 http://auditoryneuroscience.com/ear/bm_motion_2
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