Mathematical Tools for Neural and Cognitive Science

Fall semester, 2019

Section 3a: Early auditory system (an extended LSI example)

4000 Hz400 Hz

Critical Bands

• Loudness summation

• Critical band masking

“Designing” Filters

x(t)

y(t)inputx(t)

outputy(t)

RC

I= (x - y)/R

I = C dy/dt

y(t ) =VC = 1C

I dt or I =C dVCdt∫

x(t )− y(t ) =VR = IR

x − y =RC dydt

= τ dydt

For x = 0, solutions have the form y = Ae−t /τ

“Designing” Filtersx(t)

y(t)

10−1 100 101−20

−15

−10

−5

0

Frequency ω, radians per second

Frequency response of RC low−pass filter: log−log Bode plot

Gai

n m

agni

tude

, dB

3 dB corner

asymptoticslope−6 dB/octave

magnitude

−90

−45

0

Phas

e, d

egre

es

phase

H(ω ) = 1iτω +1

Cascaded or Gamma Filters

y ∝ t n−1e−t /τ

H(ω ) = 1iωτ +1

⎛⎝⎜

⎞⎠⎟n

…

0 10 20 30 40 500

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 = 3N = 4

Gamma distributions of different orders, fixed β = 0.2

x

P(x)

Gammatone Filters

y ∝ t n−1e−t /τ cos ωt −φ( )

0 10 20 30 40 50 60−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

gamma−distribution envelope

γ = 0.2ωR = 1

N = 4

Cosine and sine gammatone impulse responses

time

Res

pons

e am

plitu

de

Speech Spectrogram

Cochleogram

time (samples at 22 kHz)

ch

an

ne

l n

um

be

r

200 400 600 800 1000 1200 1400 1600

10

20

30

40

50

60

70

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

NextPrev

Low-pass or Bass Filter

High-pass or Treble Filter

Miles Davis “Half Nelson”

Bass only:

Treble only:

Time

Signal Frequency content

Filter

NextPrev

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

NextPrev

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

NextPrev

Example – Bass/Treble filters

Fourier spectrum representation of soundPeriod

This works for all “musical sounds”

0 0.5 1 1.5 2 2.5 0 400 800 1200 1600 2000

0 0.5 1 1.5 2 2.5 0 400 800 1200 1600 2000

Frequency (Hz)

Frequency (Hz)

Fundamental frequency and harmonics

Standing waves in a vibrating string

Flute (open pipe) harmonics

Other notes (shorten the pipe)

Reinforcing a harmonic (forcing a “node”)

Flute (open pipe) harmonics