Section 3a: Early auditory system (an extended LSI example) 2 3 4 - PDF document

1 Mathematical Tools for Neural and Cognitive Science Fall semester, 2018 Section 3a: Early auditory system (an extended LSI example) 2 3

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16 Critical Bands • Loudness summation • Critical band masking 17 “Designing” Filters x(t) I = ( x - y )/ R input output y(t) x ( t ) y ( t ) I = C dy / dt R C y ( t ) = V C = 1 I = C dV C ∫ I dt or C dt x ( t ) − y ( t ) = V R = IR x − y = RC dy dt = τ dy dt For x = 0, solutions have the form y = Ae − t / τ 18 “Designing” Filters x(t) y(t) Frequency response of RC low−pass filter: log−log Bode plot 0 0 3 dB corner 1 H ( ω ) = Gain magnitude, dB −5 i τω + 1 Phase, degrees −45 asymptotic slope −10 phase −6 dB/octave −90 −15 magnitude −20 −1 0 1 10 10 10 Frequency ω , radians per second

19 Cascaded or Gamma Filters … Gamma distributions of different orders, fixed β = 0.2 0.2 0.18 y ∝ t n − 1 e − t / τ 0.16 0.14 0.12 N = 1 P(x) 0.1 n ⎛ ⎞ 1 N = 2 H ( ω ) = 0.08 ⎜ ⎟ N = 3 ⎝ ⎠ i ωτ + 1 N = 4 0.06 0.04 0.02 0 0 10 20 30 40 50 x 20 Gammatone Filters y ∝ t n − 1 e − t / τ cos ω t − φ ( ) Cosine and sine gammatone impulse responses 0.05 0.04 gamma−distribution envelope 0.03 γ = 0.2 ω R = 1 0.02 Response amplitude N = 4 0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 0 10 20 30 40 50 60 time 21 Speech Spectrogram

22 Cochleogram 10 20 channel number 30 40 50 60 70 200 400 600 800 1000 1200 1400 1600 time (samples at 22 kHz) 23 Example – Bass/Treble filters Frequency content Signal Miles Davis “Half Nelson” Filter Bass only: Low-pass or Bass Filter Treble only: High-pass or Treble Filter Prev Time Next 24 Example – Bass/Treble filters Frequency content Signal Miles Davis “Half Nelson” Filter Bass only: Low-pass or Bass Filter Treble only: High-pass or Treble Filter Prev Time Next

25 Example – Bass/Treble filters Frequency content Signal Miles Davis “Half Nelson” Filter Bass only: Low-pass or Bass Filter Treble only: High-pass or Treble Filter Prev Time Next 26 Example – Bass/Treble filters Frequency content Signal Miles Davis “Half Nelson” Filter Bass only: Low-pass or Bass Filter Treble only: High-pass or Treble Filter Prev Time Next 27 Fourier spectrum representation of sound Period

28 This works for all “musical sounds” 0 0.5 1 1.5 2 2.5 0 400 800 1200 1600 2000 Frequency (Hz) 0 0.5 1 1.5 2 2.5 0 400 800 1200 1600 2000 Frequency (Hz) 29 Fundamental frequency and harmonics 30 Standing waves in a vibrating string

31 Flute (open pipe) harmonics 32 Flute (open pipe) harmonics Other notes (shorten the pipe) Reinforcing a harmonic (forcing a “node”)