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 24�

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�