Electronic Music

Electronic Music



Dr Ian Drumm

Physical Modelling

Sound Synthesis

• Aims

– Introduce physical modelling as a sound

synthesis technique



• Learning outcomes

– Discrete time domain representation

– Digital Waveguides

– Tuning waveguides

– Realistic boundary conditions

What is physical modelling?

• Traditional techniques synthesise waveforms

usually with respect to analysis of real acoustic

waveforms.

• Physical modelling emulates the sound

production process of real instruments

• Technique suited for digital implementation

• Computationally expense becoming less of and

issue

– Computers become faster/cheaper

– Optimisations

Why physical modelling?

• Traditional techniques

– feedforward … source->filter

– control parameters don’t directly reflect players

experience





• Physical modelling

– takes account of two way (often non-linear)

interaction between excitation source and filter

(feedback)

– control parameters intuitive (e.g. soft/hard plucking,

blowing pressure).

– greater potential for expression

– more realistic emulations – and opportunities for

creative synthesis

• Characterise excitation source’s behaviour

over a range amplitudes

• Characterise resonator (usually does not

depend on amplitude of signal passing

through it)

Waveguides

 y 2

2  y

2

T

• Wave equation c c

L

t 2

x 2



• Da Lembert’s solution



y  f1 (ct  x)  f 2 (ct  x)

Signal in digital domain

• Discrete time interval T

• Time elapsed t n  nT

Waveguide in digital domain

• Discrete distance interval X  cT

• Distance travelled x m  mX

Digital Waveguide Theory

Putting solution in discrete time domain



y  f1 (ct n  x m )  f 2 (ct n  x m )

 y  f1 (cnT  mX )  f 2 (cnT  mX )



 y  f1 (cnT  mcT )  f 2 (cnT  mcT )



 y  f1 (cT n  m)  f 2 (cT n  m)

Knowing c and T are constant

 

y  y ( n  m)  y ( n  m)

• So y is a function of indexes to the digital signal

and waveguide hence we can model wave

propagation by shifting values along digital

waveguides in both forward and backward

direction.

Rigid Boundary Condition

• Assume rigid boundary x=0, y=0



y  f1 (ct  x)  f 2 (ct  x)

 f1 ct   f 2 ct   0

 f1 ct    f 2 ct 

Rigid Boundary Condition Digital

Domain

• Previous suggests discrete signal elements

multiplied by gain of -1 at boundary

How long should digital waveguide be?

• E.g. require A at 440Hz at sample rate of 20000Hz

• Sample interval

1 1

T   0.00005 s

f sample _ rate 20000

• It takes to travel along waveguide t n  nT

• Gives frequency

f 1

nT

1

n n=1/(440*0.00005) = 44.455~44 or 45 samples

fT

Linear Interpolation

• If we wanted to find y3 given

delay d we would look at other

points and interpolate

• Given the gradient is constant

(linear)





 y 2  y1   yd  y2   y  y1 

  

 t  t    t  t   t  t t d  t 2   y 2

  y d   2 

 2 1   d 2   2 1 

 y1  y 2 

 yd    t  t d  y 2

 Where d  t d  t 2

 1 2 

• Writing this in the discrete time

domain 

y[n  d ]  y[n  1]  y[n]d  yn

Tuning waveguide

• Add fractional delay d to waveguide



y[n  d ]   y[n  1]  y[n]d  yn

 y[n  d ]  dz 1  (1  d ) y[n]









• Linear interpolation work best with lower

frequencies

All pass interpolation

• An all pass filter can

be thought of as a

combination of low

and high pass filters

• No resulting change

in amplitude with

respect to frequency

• Does change phase

hence can add delay

to delay line

Tuning Waveguide with All Pass Filter

• Difference equation for all pass filter



y[n]  ax[n]  x[n  1]  ayn  1

• Magnitude response G(ω) and phase response Θ(ω)





Y ( z )  aX ( z )  z 1 X ( z )  az 1Y ( z )

jT

1 1

 Y ( z )  az Y ( z )  aX ( z )  z X ( z ) ze

Y ( z ) a  z 1

 H ( z)  

X ( z ) 1  az 1





G   H e  jT

  

   arg H e jT



Tuning Waveguide with All Pass filter









• The coefficient a can be given in terms of

delay Δ

1 

a

1 

More realistic boundary conditions

• Consider body response for real instruments

• Can measure with accelerometer / tuned hammer

• Could convolve body response with waveguide;

finite impulse response filter





y[n]  b0 xn  b1 xn  1  b2 xn  2....





• Large number of coefficients required hence

computationally expensive to apply each cycle

• For efficiency could employ Infinite Impulse

Response filter (e.g. Biquad)

• Difference equation for biquad filter



b0 xn  b1 xn  1  b2 xn  2  a1 yn  1  a 2 yn  2

y[n] 

a0



• Response



b0 X ( z )  b1 z 1 X ( z )  b2 z 2 X ( z )  a1 z 1Y ( z )  a 2 z 2Y ( z )

Y ( z) 

a0

 a0Y ( z )  a1 z 1Y ( z )  a 2 z  2Y ( z )  b0 X ( z )  b1 z 1 X ( z )  b2 z  2 X ( z )

Y ( z ) b0  b1 z 1  b2 z  2

 H ( z)  

X ( z ) a0  a1 z 1  a 2 z  2

E.g. Design For Low pass filter :

E.g. Acoustic Guitar

• Body of instrument has modes

– Bending mode (~60Hz) – guitar bends like

free beam

– Breathing mode (~100Hz) – guitar top and

bottom plates move out of phase sucking air

in and out

– Higher modes (~190Hz, 200Hz, etc, etc)

• Could model by finding coefficients of for

corresponding band pass filters