Observing the Effects of Waveguide Model Elements in Acoustic Tube Measurements
Tamara Smyth
Jonathan Abel
School of Computing Science, Simon Fraser University
Universal Audio Inc. and Stanford University (CCRMA)
Meeting of the Acoustical Society of America, Honolulu, Hawaii
December 2, 2006
Outline
• Digital waveguide theory• Technique for measuring an impulse
response from acoustic tube structures• Observing waveguide theory in
measured responses• Comparing model and measurement
A Digital Waveguide Section
()
• Both the plane waves of a cylinder and the spherical waves a cone can be modeled using a digital waveguide.
()Z-L
Z-L
• The effects of viscous drag and thermal conduction along the bore walls, lead to an attenuation in the propagating waves, determined by
() = 2 x 10-5 / a
Theoretical Wall Loss
• The round trip attenuation for a tube of L is given by,
2() = e-2L
valid for diameters seen in most musical instruments.
R2()R1()
Termination and ScatteringA change of impedance, such as a termination or connection to another waveguide section, will require additional filters to account for reflection and possibly transmission.
Termination Scattering
R() R()
T()
+
+T1()
T2()
Open End Reflection Filter
• The reflection filter for an open is given by
Rop() = ZL () / Z0 - 1
ZL () / Z0 +1
Z0 = cS
,
where
and ZL() is the complex terminating impedance at the open end of a cylinder, given by the expression by Levine and Schwinger.
is the wave impedance,
j + c/x
Z2 / Z1 +1
Theoretical Junction ReflectionThe reflection at the junction is given by
R() = Z2 / Z1 - 1 ,
*
The impedance for the spherical waves is given by
Zn = cS
j
This leads to a first-order, one-pole, one-zero, filter.
.
The impedance for plane waves is given by
Zy = cS
Cylinder
Cylicone
Cylinder Scattering Cone
Example Waveguide Models
Four Measured Tube StructuresCylinder, speaker-closed Cylinder, speaker-open
Cylicone, speaker-closed Cylicone, speaker-open
Obtaining an Impulse Response from an LTI System
The impulse is limited in amplitude and has poor noise rejection
Measurement noise
Measured responseTest
signalLTI system
h(t) +s(t) r(t)
n(t)
Impulse Response Using a Swept Sine
• The sine is swept over a frequency trajectory (t) effectively smearing the impulse over a longer period of time.
• Since higher frequencies go into the system at later times, they must be realigned to recover the impulse response.
Our Measurement System
Cylinder, Speaker-Closed
From the first measurement we observe:
• The speaker transfer function, ()• The speaker reflection, ()• The round trip wall losses for a cylinder, 2()
Arrival Responses for a Closed Cylinder
L1 = ()
L2 = ()2()(1+())
L3 = ()4() ()(1+())
Closed Cylinder Arrival Spectra
L1
L2
L3
Speaker Reflection Transfer Function• Given the arrival responses:
L1 = ()
L2 = ()2()(1+())
L3 = ()4()()(1+()),
() = ^ 1 -
= L1L3
(L2)2
()
1 + ()=
• We are able to estimate the speaker reflection transfer function
Cylinder Wall Loss Transfer Function
• Given the arrival responses:
L1 = ()
L2 = ()2()(1+())
L3 = ()4()()(1+()),
and the estimate for the speaker reflection, we are able to estimate the wall loss transfer function
2() =L3
() L2^
^
Estimated and Theoretical Propagation Losses
() ^
Cylinder, Speaker-Open
From this measurement we observethe reflection from an open end, Rop().
Arrival Responses for an Open Cylinder
Y1 = ()
Y2 = ()2()Rop()(1+())
Y3 = ()4()R2op() ()(1+())
Open Cylinder Arrival Spectra
Y1
Y2
Y3
Open End Reflection
• Given the second arrival for the closed tube:
Y2 = ()2()Rop()(1+()),
• We are able to estimate the reflection from an open end
Rop() = Y2
L2
L2 = ()2()(1+())
• and the second arrival for the open tube:
^
Cylinder Open End Reflection
Cylicone, Speaker-Closed
We consolidate this measurement with the theoretical reflection and transmission filters at the junction:
• the cylinder: Ry() and Ty(), • the cone: Rn() and Tn()
Arrival Responses for Closed Cylicone
A1 = ()
A2 = …
A3 = …
Second Arrival, Closed Cylicone
A(2,1) = ()y()Ry()(1+())2
A(2,2) = ()y() n()Ty()Tn()(1+())2 2
Measured and Modeled Closed Cone Arrival
Cylicone, Speaker-Open
From this measurement, we observe the behaviourof the reflection filter from the cone’s open end.
Arrival Responses for Open Cylicone
N1 = ()
N2 = …
Second Arrival, Open Cylicone
Closed Cylinder Comparison
Open Cylinder Comparison
Closed Cylicone Comparison
Open Cylicone Comparison
Summary• We observed the behaviour of the
following waveguide filter elements, from measured impulse responses:– Open end reflection (cylinder and cone)– Propagation losses (cylinder and cone)– Junction reflection and transmission
(cylicone)
• We confirmed that the impulse response measurements matched the responses of the waveguide models.
Conclusions
• We observed and verified theoretical waveguide filter elements using our measurements.
• The measurement system yields very good data at relatively low cost.
• The validation of the measurement system implies it can be extended to any tube structure.
Acknowlegements
• We would like to thank Theresa Leonard and the Banff Centre for Performing Arts.
• Natural Sciences and Engineering Research Council (NSERC).