A fable …

1

THE SCOTTY WHO KNEW TOO MUCH JAMES THURBER

Several summers ago there was a Scotty who went to the country for a visit. He decided that all the farm dogs were cowards, because they were afraid of a certain animal that had a white stripe down its back.

“You are a pussycat and I can lick you,” the Scotty said to the farm dog who lived in the house where the Scotty was visiting.

“I can lick the little animal with the white stripe too. Show him to me.”

“Don’t you want to ask any questions?”

“Nah,” said the Scotty. “You ask the questions.”

So the farm dog took the Scotty into the woods and showed him the white-striped animal and the Scotty closed in on him, growling and slashing. It was all over in a moment and the Scotty lay on his back.

When he came to, the farm dog said, “What happened?”

“He threw vitriol,” * said the Scotty, “but he never laid a glove on me.”

A few days later the farm dog told the Scotty there was another animal all the farm dogs were afraid of.

* Vitriol is sulfuric acid, which also is used in car batteries.

2

“Lead me to him,” said the Scotty. “I can lick anything that doesn’t wear horseshoes.”

“Don’t you want to ask any questions about him” said the farm dog.

“Nah,” said the Scotty. “Just show me where he hangs out.”

So the farm dog led him to the place in the woods and pointed out the little animal when he came along.

“A clown,” said the Scotty, “a pushover,”

and he closed in, leading with his left and exhibiting some mighty fancy footwork. In less than a second Scotty was flat on his back, and when he woke up the farm dog was pulling quills out of him.

“What happened?” said the farm dog.

“He pulled a knife on me,” said the Scotty, “but at least I have learned how you fight out here in the country, and now I am going to beat you up.”

So he closed in on the farm dog, holding his nose with one front paw to ward off the vitriol and covering his eyes with the other front paw to keep out the knives. The Scotty couldn’t see his opponent and he couldn’t smell his opponent and he was so badly beaten that he had to be taken back to the city and put in a nursing home.

Moral: It is better to ask some of the questions than to know all the answers.

Topics: •  From simple filters to echo and reverb •  Variable delay single tap FIR filter •  Variable delay single tap IIR filter •  Plucked string filters •  Karplus - Strong plucked string models •  Waveguide modeling of wind musical instruments

3

Introduction to ���Audio and Music Engineering

Lecture 23

Revisit the simple FIR filter …

4

Y = X + anz −nX

n sample delay

an

+ input X output Y H (z) = 1 + anz

−n

H (ω ) = 1 + ane− jnω

n = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−15

−10

−5

0

5

Normalized Frequency (×π rad/sample)

Magnitude Response (dB)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Pole/Zero Plot

The FIR filter …

5

Y = X + anz −nX

n sample delay

an

+ input X output Y H (z) = 1 + anz

−n

H (ω ) = 1 + ane− jnω

n = 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−15

−10

−5

0

5

Normalized Frequency (×π rad/sample)

Magnitude Response (dB)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

6

Pole/Zero Plot

The FIR filter …

6

Y = X + anz −nX

n sample delay

an

+ input X output Y H (z) = 1 + anz

−n

H (ω ) = 1 + ane− jnω

n = 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−15

−10

−5

0

5

Normalized Frequency (×π rad/sample)

Magnitude Response (dB)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

100

Pole/Zero Plot

Revisit the simple IIR filter …

7

Y = X + bnz −nYH (z) = 1 + bnz

−n( )−1H (ω ) = 1 + bne − jnω( )−1

n = 1 Z-1 b1

+ input X output Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−6

−4

−2

0

2

4

6

8

10

12

14

Normalized Frequency (×π rad/sample)

Magnitude Response (dB)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Pole/Zero Plot

IIR filter …

8

Y = X + bnz −nYH (z) = 1 + bnz

−n( )−1H (ω ) = 1 + bne − jnω( )−1

n = 100 Z-1 b1

+ input X output Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−6

−4

−2

0

2

4

6

8

10

12

14

Normalized Frequency (×π rad/sample)

Magnitude Response (dB)

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Part

100

Pole/Zero Plot

9

Plucked string simulation

Karplus – Strong Model

Fine  tune  the  frequency  

Makes  high  harmonics  decay  

faster  

Makes  string  decay  

Delay  sets  the  frequency  

Pluck  the  string  

Musical Instrument Physical Modeling

Clarinet Physical Model

Digital Delay Line

Digital Delay Line

Cross-over network

Nonlinear “valve”

Blowing pressure

Bore Bell Reed

Output sound

(physical modeling is used widely in commercial synthesizers, e.g., Yamaha VL 70M)

Combine filters and delay lines, plus a model of the excitation mechanism, to generate musical instruments sounds by simulating the physics of the instrument.

10

Clarinet  Physics  

End View

Embouchure Force

P p

flow

Blowing P - internal p

“bias” region

Reed begins to close Greater

Embouchure Force

reed

P - p

Pressure Impulse

bell

Each time the pressure increases in the mouthpiece the reed opens and lets in more air – positive feedback.

11  

Clarinet Waveguide Model

Unit delay

Unit delay

Unit delay

Unit delay

Unit delay

Unit delay

p+(n)

p-(n)

+ p(n) Reflection Filter (LP)

Output Filter (HP)

Bore Bell

Nonlinear Scattering Junction

Blowing Pressure

Reed

Bi-directional delay line

12

13

60 80 100 120 140 160 180 200 220 240

4

5

6

7

8

9

10

11

12

Time (samples)

"a - re

d" "d -

blue"

Orig Sound - 10.6 Mb/min

Extracted Parameters - 0.1 Mb/min

Using Physical Models to encode musical performances

flow

Blowing P - internal p

Greater Embouchure

Force

Simple Waveguide Model���Maximum Likelihood Estimation ���Estimate parameters ~ 450/sec���Compress ~ 100x���

Lesser Embouchure

Force

14

Original  

Resynthesized  

15  

Even more compact … •  Employ measured acoustic

parameters of a clarinet to begin with a more accurate model.

•  1 time varying parameter –  20/sec (160 bits/sec) –  Compress ~ 7000 x

Original Resynthesized 16

Wav MP3

10X

Unco

mpr

esse

d Au

dio

Synthetic PM

100X

Empirical PM

7000X

Physical Model

Music Parameter Estimation

PM Parameters History

Physical Model Music

Physical Modeling Music Representation 7000 x smaller

Analysis Re-synthesis

Current Results

Continuing Work –  Refine models: include tonguing, vocal tract,

exciter (reed, lips) dynamics –  Extend to other wind, bowed, plucked

instruments –  Encode recordings of multiple instruments

•  Source separation

17