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Section 3a: Early auditory system (an extended LSI example) 2 3 4 - - PDF document
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 4 5 6 7 8 9 10 400 Hz 4000 Hz 11 12 13 14 15 16 Critical Bands Loudness summation
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4000 Hz 400 Hz
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Critical Bands
- Loudness summation
- Critical band masking
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“Designing” Filters
x(t) y(t) input x(t)
- utput
y(t) R C I= (x - y)/R I = C dy/dt
y(t) =VC = 1 C I dt
- r
I = C dVC dt
∫
x(t)− y(t) =VR = IR x − y = RC dy dt = τ dy dt For x = 0, solutions have the form y = Ae −t/τ
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“Designing” Filters
x(t) y(t)
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−1
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−20 −15 −10 −5 Frequency ω, radians per second Frequency response of RC low−pass filter: log−log Bode plot Gain magnitude, dB 3 dB corner asymptotic slope −6 dB/octave magnitude −90 −45 Phase, degrees phase
H(ω) = 1 iτω +1 18
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Cascaded or Gamma Filters
y ∝t n−1e −t/τ H(ω) = 1 iωτ +1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
n
…
10 20 30 40 50 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 N = 1 N = 2 N = 3 N = 4 Gamma distributions of different orders, fixed β = 0.2 x P(x)
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Gammatone Filters
y ∝t n−1e −t/τ cos ωt −φ
( )
10 20 30 40 50 60 −0.05 −0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04 0.05 gamma−distribution envelope γ = 0.2 ωR = 1 N = 4 Cosine and sine gammatone impulse responses time Response amplitude
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Speech Spectrogram
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Cochleogram
time (samples at 22 kHz) channel number 200 400 600 800 1000 1200 1400 1600 10 20 30 40 50 60 70
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Example – Bass/Treble filters
Low-pass or Bass Filter High-pass or Treble Filter Miles Davis “Half Nelson” Bass only: Treble only: Time Signal Frequency content Filter Next Prev
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Low-pass or Bass Filter High-pass or Treble Filter Miles Davis “Half Nelson” Bass only: Treble only: Time Signal Frequency content Filter Next Prev
Example – Bass/Treble filters
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Low-pass or Bass Filter High-pass or Treble Filter Miles Davis “Half Nelson” Bass only: Treble only: Time Signal Frequency content Filter Next Prev
Example – Bass/Treble filters
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Low-pass or Bass Filter High-pass or Treble Filter Miles Davis “Half Nelson” Bass only: Treble only: Time Signal Frequency content Filter Next Prev
Example – Bass/Treble filters
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Fourier spectrum representation of sound
Period
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This works for all “musical sounds”
0.5 1 1.5 2 2.5 400 800 1200 1600 2000 0.5 1 1.5 2 2.5 400 800 1200 1600 2000
Frequency (Hz) Frequency (Hz)
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Fundamental frequency and harmonics
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Standing waves in a vibrating string
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