1
Digital Speech Processing
Digital Speech Processing—
Lectures 5
Lectures 5-6
1
Sound Propagation in
the Vocal Tract
Basics
•
can use basic physics to formulate
air flow equations
for vocal tract
•
need to make
simplifying assumptions
about vocal tract shape
and energy losses to solve air flow equations
•
some complicating factors:
–
time variation
of the vocal tract shape (we will look mainly at fixed
shapes)
–
losses
in flow at vocal tract walls (we will first assume no loss, then a
simple model of loss)
softness of vocal tract walls
(leads to sound absorption issues)
2
–
–
radiation of sound
at lips (need to model how radiation occurs)
–
nasal coupling
(complicates the tube models as it leads to multi-tube
solutions)
–
excitation of sound
in the vocal tract (need to worry about vocal
source coupling to vocal tract as well as source-system interactions)
Bottom Line
: simplify as much as possible and see
what we can learn about the mechanics of sound
propagation in the human vocal tract
Sound in the Vocal Tract
• Issues in creating a detailed physical model
3
•
Issues in creating a detailed physical model
– time varying acoustic system
– losses due to heat conduction and friction in the
walls.
– radiation of sound at the lips and nostrils
– softness of the walls
– nasal coupling
– excitation of sound in the vocal tract
Schematic Vocal Tract
PDEs must
be solved in
this region
4
• simplified vocal tract area =>
non-uniform tube
with time varying
cross section
•
plane wave propagation
along the axis of the tube (this assumption
valid for frequencies below about 4000 Hz)
•
no losses at walls
Sound Wave Propagation
•
using the laws of conservation of mass, momentum and energy, it can be
shown that sound wave propagation in a lossless tube satisfies the
equations:
2
(
/
)
1
(
)
p
u A
x
t
u
pA
A
x
c
t
t
ρ
ρ
∂
∂
−
=
∂
∂
∂
∂
∂
−
=
+
∂
∂
∂
5
where
(
,
)
sound pressure in the tube at position
and time
(
,
)
volume velocity flow at position
and
p
p x t
x
t
u
u x t
x
•
=
=
=
=
time
the density of air in the tube
the velocity of sound
(
,
)
the 'area function' of the tube,
i.e., the cross-sectional area normal to the axis of the tube,
as a function of the distance alon
t
c
A
A x t
ρ
=
=
=
=
g the tube and as a function of time
Solutions to Wave Equation
•
no closed form solutions
exist for the
propagation equations
– need
boundary conditions
, namely
u(0,t)
(the
volume velocity flow at the glottis), and
p(l,t),
(the
sound pressure at the lips) to solve the equations
numerically (by a process of iteration)
– need
complete specification of A(x,t),
the vocal
tract area function; for simplification purposes we will
assume that there is no time variability in
A(x,t)
=> the
term related to the partial time derivative of
A
becomes 0
– even with these simplifying assumptions, numerical
solutions are very hard to compute
Consider simple cases and extrapolate results
to more complicated cases

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