# Physically-Based Parametric Sound Synthesis and Control prc/ Physically-Based Parametric

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Course #2 Sound - 1

Perry R. Cook

Princeton Computer Science(also Music)

Physically-Based ParametricSound Synthesis and Control

Course Introduction

Parametric Synthesis and Control ofReal-World Sounds for

virtual reality

games

production

auditory display

interactive art

interaction design

Course #2 Sound - 2

Schedule0:00 Welcome, Overview0:05 Views of Sound0:15 Spectra, Spectral Models0:30 Subtractive and Modal Models1:00 Physical Models: Waveguides and variants1:20 Particle Models1:40 Friction and Turbulence1:45 Control Demos, Animation Examples1:55 Wrap Up

Views of Sound

Sound is a recorded waveform PCM playback is all we need for interactions, movies, games, etc.

(Not true!!)

Time Domain x( t ) (from physics)

Frequency Domain X( f ) (from math)

Production what caused it

Perception our image of it

Course #2 Sound - 3

Views of Sound

Time Domain is most closely related to

Production

Frequency Domain is most closely related to

Perception

we will see that many hybrids abound

Views of Sound: Time Domain

Sound is produced/modeled by physics,described by quantities of

Force force = mass * acceleration

Position x(t) actually < x(t), y(t), z(t) >

Velocity Rate of change of position dx/dt

Acceleration Rate of change of velocity dv/dt

Examples: Mass+Spring+Damper Wave Equation

Course #2 Sound - 4

Mass/Spring/Damper

F = ma = - ky - rv - mg

F = ma = - ky - rv

(if gravity negligible)

022

=++ ym

k

dt

dy

m

r

dt

yd

( )D Dr m k m2 0+ + =/ /

2nd Order Linear Diff Eq. Solution

1) Underdamped:

y(t) = Y0 e-t/ cos( t )

exp. * oscillation

2) Critically damped:

fast exponential decay

3) Overdamped:

slow exponential decay

Course #2 Sound - 5

Wave Equation

dfy = (T sin) x+dx - (Tsin)x f(x+dx) = f(x) + f/x dx + (Taylors series) sin = (for small ) F = ma = dx d2y/dt2 (for each dx of string)

The wave equation: (c2 = T / )) 2

2

22

2 1

dt

yd

cdx

yd =

Views of Sound: ProductionThroughout most of history, somephysical mechanism was responsiblefor sound production.

From our experience, certain gesturesproduce certain audible results

Examples:Hit harder --> louder AND brighterCant move instantaneouslyCant do exactly the same thing twice

Course #2 Sound - 6

Sound Views: Frequency Domain

Frequency Domain:

Many physical systems have modes (damped oscillations)

Wave equation (2nd order) orBar equation (4th order) need 2 or 4

boundary conditions for solution

Once boundary conditions are setsolutions are sums of exponentially damped sines

the sinusoids are Modes

The (discrete) Fourier Series

A time waveform is a sum of sinusoids

(A is complex)x n Aj nm

Nmn

N

( ) exp( )==

20

1

=

=

+=

+=

1

0

1

0

)2

cos(

)2

cos()2

sin(

N

nmm

N

nmm

N

nmD

N

nmC

N

nmB

Course #2 Sound - 7

The (discrete) Fourier Transform

A m X SRATE m N x njnm

Nn

N

( ) ( * / ) ( ) exp( )= =

=

20

1

sinusoidal ASpectrum is a decomposition

of a signal

This transform is unique and invertible

(non-parametric representation like sampling)

Spectra: Magnitude and PhaseOften only magnitude is plotted

Human perception is most sensitive to magnitude

Environment corrupts and changes phase

2 (pseudo-3) dimensional plots easy to view

Phase is important, however

Especially for transients (attacks, consonants, etc.)

If we know instantaneous amplitude and frequency, we can derive phase

Course #2 Sound - 8

Common Types of Spectra

Harmonic

sines at integer

multiple freqs.

Inharmonic

sines (modes),

but not integer

multiples

Common Types of Spectra

Noise

random

amplitudes

and phases

Mixtures

(most real-

world sounds)

Course #2 Sound - 9

Views of Sound: PerceptionHuman sound perception:

Auditory cortex:further refinetime & frequencyinformation

Cochlea:convert tofrequencydependentnerve firings

Ear:receive1-Dwaves

Brain:Higher levelcognition,objectformation,interpretation

Perception: Spectral Shape

Formants(resonances)are peaks inspectrum.

Human ear issensitive tothese peaks.

Course #2 Sound - 10

Spectral Shape and Timbre

Quality of asound isdetermined bymany factors

Spectral shapeis one importantattribute

Spectra Vary in Time

Spectrogram (sonogram)amplitude as darkness (color) vs. frequency and time

Course #2 Sound - 11

Spectra in Time (cont.)

Waterfall Plotpseudo 3-d amplitude as heightvs. freq. and time

Each horizontal sliceis an amplitude vs.time magnitudespectrum

Additive Synthesis

( ) ( )[ ]=

=R

rrr ttAts

1

cos)(

The sinusoidal model:

R : number of sinewave components,Ar (t) : instantaneous amplitude,r (t) : instantaneous phase

Control the amplitudeand frequency of aset of oscillators

Course #2 Sound - 12

Sinusoidal Modeling

Vocoders Dudley 39, Many more since

Sinusoidal Models Macaulay and Quatieri 86

SANSY/SMS Sines + Stochastic Serra and Smith 87

Lemur Fitts and Hakken 92

FFT-1 Freed, Rodet and Depalle 96

Transients Verma, Meng 98

frequency of partials

magnitude of partials

Sinusoidal Analysis Tracks

Course #2 Sound - 13

Magnitude-only synthesis

original sound

magnitude-onlysynthesis

mS

AAAmA

lll )()(

11

+= m

Sm

lll )()(

11

+= )()1()( mlm rrr +=

( ) ( )[ ]=

=lR

r

lr

lr

l tmAms1

cos)(

Magnitude and Phase Synthesis

( ) rrt rr dt ++= )()( 00

original sound

synthesized soundwith phase matching

x(n)

s(n)

r(t) : instantaneous frequencyr(0) : initial phase valuer : fixed phase offset

Course #2 Sound - 14

Deterministic plus StochasticSynthesis (SMS)

[ ] )()(cos)()(1

tettAtsR

rrr +=

=

( ) dt t rr = 0)(when sinusoids are very stable, the instantaneous phasecan be calculated by:

otherwise:

model:

Ar(t), r(t): instantaneous amplitude and phase of rth sinusoid,e(t) : residual component.

( ) rrt

rr dt ++= )()( 00r(t) : instantaneous radian frequencyr(0) : initial phase value,r : fixed phase offset

Residual (stochastic component)

Resynthesis (with

phase) of sine

components

allows extraction

and modeling of

residual component

Course #2 Sound - 15

Basic SMS parameters Instantaneous frequency and amplitude of partials Instantaneous spectrum of residual

Instantaneous attributes Fundamental frequency Amplitude and spectral shape of sinusoidal components Amplitude and spectral shape of residual Degree of harmonicity Noisiness Spectral tilt Spectral centroid

Region Attributes

SMS High level attributes

Transients

Transients are vertical stripes in spectrogram

Use DCT to transform backto time domain, then dosinusoidal track analysis on that

Detection is the hard part

Course #2 Sound - 16

Sines + Noise + TransientsStrengths and WeaknessesStrengths: General signal model

(doesnt care what made it)

Closed form identity analysis/resynthesis

Perceptual motivations (somewhat, not all)

Weaknesses: No gestural parameterization

No physics without lots of extra work

No guaranteed compression or understanding

Subtractive Synthesis: LPC

=

=m

kk knxcnx

1

)()( LPC

=

=m

kk knxcnx

1

)()()()()( nxnxne =

=

=P

n

neP

E0

2)(1

Prediction

Error

Mean Squared Errorover block length P

Course #2 Sound - 17

LPC continued

LPC is wellsuited tospeech

Also wellsuited tosounds withresonances

LPC filter envelope (smooth line)fit to human vowel sound / i / (eee)

Subtractive Synthesis: Formants

Factor LPC into resonators

Eigenmodemotivations

Perceptual motivations

Vocal production motivations

Excite with pulse(s), noise, or residual

Course #2 Sound - 18

Modal Synthesis

Systems with resonances (eigenmodes of vibration)

Bars, plates, tubes, rooms, etc.

Practical and efficient, if few modes

Essentially a subtractive model in thatthere is some excitationand some filters to shape it.

Modal Synthesis: Strings

Strings are pinned at both ends

Generally harmonic relationship

Stiffness can cause minor stretching ofharmonic frequencies

Course #2 Sound - 19

Modal Synthesis: Bars

Modes of Bars: Free at each end

These would be harmonic, but stiffness ofrigid bars stretches frequencies.

Modes: 1.0, 2.765, 5.404, 8.933

Modal Synthesis: Tubes

Open or closed at each end, same asstrings and bars, b

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