Sound Propagation in the Concatenated Tube Model

EXERCISES

4.4 A Discrete-Time Model Based on Tube Concatenation

4.4.1 Sound Propagation in the Concatenated Tube Model

Consider the concatenated tube model in Figure 4.14 in which thekth tube has cross-sectional areaA_k and length l_k. Because each tube is lossless and because we assume planar wave propagation, the pressure and volume velocity relations in each tube satisfy the wave equation, Equation (4.7), each quantity having a forward- and backward-traveling wave component as given in Equation (4.8). The volume velocity and presure solution for thekth tube is therefore written as

u_k(x, t ) = u⁺_k

t − x c

− u⁻_k

t + x c

p_k(x, t ) = ρc A_k

u⁺_k

t − x

c

+ u⁻_k

t + x c

(4.27) where the kth tube has length l_k, the spatial variable x falling in the range 0 ≤ x ≤ l_k, as illustrated in Figure 4.14. To give a specific solution requires boundary conditions that are provided by the pressure and volume velocity at two adjacent tubes. In particular, we use the physical principle that pressure and volume velocity must be continuous both in time and in space everywhere in the system [28]. Therefore, at thekth/(k+1)st junction we have

u_k(l_k, t ) = u_k+1(₀, t ) p_k(l_k, t ) = p_k+1(0, t ).

Applying the two boundary conditions with Equation (4.27), we then have u⁺_k

t − l_k

c

− u⁻_k

t + l_k c

= u⁺_k+1(t )− u⁻_k+1(t ) A_k+1

A_k

u⁺_k

t − l_k c

+u⁻_k

t + l_k

c

= u⁺_k+1(t )+ u⁻_k+1(t ). (4.28) Now defineτ_k = ^l_c^k, which is the time of propagation down the length of the tube. Then the first equation in the Equation (4.28) pair can be written as

u⁺_k(t −τ_k) = u⁻_k(t +τ_k)+u⁺_k+1(t ) −u⁻_k+1(t )

and substituting this expression foru⁺_k(t−τ_k)into the second equation of our pair, we obtain, after some algebra,

u⁺_k+1(t ) =

2A_k+1 A_k+1 +A_k

u⁺_k(t − τ_k)+

A_k+1 − A_k A_k+1 + A_k

u⁻_k+1(t ). (4.29) Then subtracting the top from the bottom component of the above modified equation pair, we obtain, after some rearranging,

u⁻_k(t + τ_k) = −

A_k+1 − A_k A_k+1 + A_k

u⁺_k(t −τ_k)+

2A_k

A_k+1 +A_k

u⁻_k+1(t ). _(4.30) Equations (4.29) and (4.30) illustrate the general rule that at a discontinuity alongxin the area functionA(x, t )there occurpropagation and reflection of the traveling wave, and so in each uniform tube, part of a traveling wave propagates to the next tube and part is reflected back (Figure 4.15). This is analogous to propagation and reflection due to a change in the impedance along an electrical transmission line.

From Equation (4.29), we can therefore interpretu⁺_k+1(t ), which is the forward-traveling wave in the(k+¹)st tube atx=0, as having two components:

1. A portion of the forward-traveling wave from the previous tube,u⁺_k(t−τ_k), propagates across the boundary.

2. A portion of the backward-traveling wave within the(k+1)_{st tube,}u⁻_k+1(t ), gets reflected back.

Observe thatu⁺_k(t−τ_k)occurs at the(k+¹)st junction and equals the forward waveu⁺_k(t ), which appeared at thekth junctionτ_k seconds earlier. This interpretation is applied generally to the forward/backward waves at each tube junction. Likewise, from Equation (4.30), we can interpretu⁻_k(t+τ_k), which is the backward-traveling wave in thekth tube atx=l_k, as having two components:

1. A portion of the forward-traveling wave in thek_{th tube,}u⁺_k(t−τ_k), gets reflected back.

2. A portion of the backward-traveling wave within the(k+1)st tube,u⁻_k+1(t ), propagates across the boundary.

Figure 4.15 further illustrates the forward- and backward-traveling wave components at the tube boundaries.

4.4 A Discrete-Time Model Based on Tube Concatenation 139

The above interpretations allow us to see how the forward- and backward-traveling waves propagate across a junction. As a consequence, we can think of^A_A^k+1−Ak

k+1+Akas areflection coefficient at the kth junction, i.e., the boundary between the kth and(k +1)st tube, that we denote byr_k:

r_k = A_k+1 −A_k

A_k+1 +A_k ^(4.31)

which is the amount of u⁻_k+

1(t ) _[oru⁺_k(t −τ_k)] that is reflected at thekth junction. It is straightforward to show that because A_k > 0, it follows that −1 ≤ r_k ≤ 1. From our definition Equation (4.31), we can write Equations (4.29) and (4.30) as

u⁺_k+1(t ) = (1+ r_k)u⁺_k(t −τ_k)+r_ku⁻_k+1(t )

u⁻_k(t +τ_k) = −r_ku⁺_k(t −τ_k)+(1− r_k)u⁻_k+1(t ) (4.32) which can be envisioned by way of a signal flow graph illustrated in Figure 4.16a. We will see shortly that these equationsat the junctionslead to the volume velocity relations for multiple concatenated tubes. Before doing so, however, we consider the boundary conditions at the lips and at the glottis.

k + 1 k – 1

k u_k⁺(t)

+ +

+ –

u_k⁺(t –τk) u_k+1⁺ (t) u_k+1⁺ (t –τk+1) u_k+1^– (t + τk+1) u_k+1^– (t)

u_k^–(t + τk) u_k^–(t)

l_k x 0

u_k(t) = u_k(t – 0/c)

– –

u_k(t) = u_k(t + 0/c) u_k(0, t) = u_k(t) – u_k(t)

+ +

– +

u_k(t –τk) = u_k(t – l_k/c)

– –

u_k(t + τk) = u_k(t + l_k/c) u_k(l_k, t) = u_k(t –τk) – u_k(t +τk) Figure 4.15 Sound waves in the concatenated tube model consist of forward- and backward-going traveling waves that arise from reflection and transmission at a tube junction.

u_k(t)

u_N(t) u_N(t –τN)

u_N(t + τN)

u_N(l_N, t) (1 + r_L)

u_k(t –τk) u_k+1(t)

u_k+1(t –τk+1)

u_k+1(t + τk+1) u_k+1(t)

u_k(t +τk)

+

+

u_g(t) +

1 + r_g

2 r_g

u₁(t)

u–₁(t)

u₁(t + τ1) u₁(t –τ1)

+

u_N(t) u_N(l_N, t)

u_g(t) u₁(t)

u₁(t)

u_N(t –τN)

u_N(t + τN) ^{– –}

+ +

+

– –

u_k^–(t) –

τk

τk

τk+1

τk+1

τN

τN

–r_L

kth Tube (k + 1)st Tube

(a)

(b)

(c) (1 – r_k)

τ1

τ1

Tube N

Tube 1

+

–

+

– +

–

(1 + r_k)

–r_k +r_k

Figure 4.16 Signal flow graphs of (a) two concatenated tubes; (b) lip boundary condition; (c) glottal boundary condition. An open circle denotes addition.

SOURCE: L.R. Rabiner and R.W. Schafer,Digital Processing of Speech Signals[28]. ©1978, Pearson Education, Inc. Used by permission.

Suppose there are a total ofN tubes. Then the boundary conditions at the lips relate pressure p_N(l_N, t ) and the volume velocity u_N(l_N, t ) at the output of the Nth tube [28].

From continuity p_N(l_N, t ) = p_L(t ), which is the pressure at the lips output. Likewise, u_N(l_N, t ) = u_L(t ), which is the volume velocity at the output of the lips. Suppose now that the radiation impedance is real, corresponding to a very large frequencyand thus, from Equation (4.24), a purely resistive load, i.e., Z_r() = R_r. Then, from Figure 4.11, we can write the pressure/volume velocity relation at the lips as

p_N(l_N, t ) = Z_ru_N(l_N, t ), (4.33) whereZ_r is not a function of. From our earlier expressions for pressure and volume velocity within a tube, we then have

ρc

A_N^[u⁺_N(t − τ_N)+u⁻_k(t +τ_N)] = Z_r[u⁺_N(t −τ_N)−u⁻_N(t +τ_N)].

4.4 A Discrete-Time Model Based on Tube Concatenation 141

This expression can be written as

ρc

A_N +Z_r

u⁻_N(t + τ_N) = −

ρc

A_N −Z_r

u⁺_N(t −τ_N) and thus

u⁻_N(t +τ_N) = − ρc

A_N − Z_r ρc

A_N + Z_r

u⁺_N(t − τ_N)

= −r_Lu⁺_N(t −τ_N) (4.34)

where the reflection coefficient at the lips is defined as r_L =

ρc A_N − Z_r

ρc A_N + Z_r.

Equation (4.34) states that the forward-going wave is reflected back at the lips boundary with reflection coefficientr_L. There is no backward-going wave contribution from free space to the right. The volume velocity at the lips output can then be expressed as

u_N(l_N, t ) = u⁺_N(t −τ_N)− u⁻_k(t + τ_N)

= (₁ +r_L)u⁺_N(t −τ_N)

which is shown schematically in Figure 4.16b. A complex radiation impedanceZ_r()can be invoked by using a frequency-domain version of the previous derivation or by using a differential equation, the inductance resulting in a first-order ordinary differential equation.

We can also model the radiation from the lips with an additional tube of cross-section A_N+1 and infinite length. The tube has no backward-propagating wave because of its infinite length, which is consistent with sound radiating out from the lips. In addition, if we selectA_N+1 such thatZ_r = _A^ρc_N+1, then the expression forr_Lbecomes

r_L =

ρc

AN − _A^ρc_N+1

ρc

A_N + _A^ρc_N+1

= A_N+1 −A_N

A_N+1 +A_N ^(4.35)

which is our original definition of reflection coefficient r_N for two adjacent tubes of cross- sectional areaA_{N+1 and}A_N. Observe that whenZ_r()=0 thenr_L=1, the pressure drop is zero, and the imaginary added tube becomes infinite in cross-section becauseA_N must be finite.

The boundary condition model for the glottis is shown in Figure 4.16c where the glottal impedance is assumed approximated as a series resistance and inductance. We can think of the glottal slit, anatomically, as sitting just below (schematically, to the left of) the first tube of the

concatenated tube model, and so we denote the pressure at the output of the glottis byp₁(0, t ), i.e., the pressure at the left edge of the first tube, and the volume velocity output of the glottis byu₁(0, t ), i.e., the volume velocity at the left edge of the first tube. Suppose that the glottal impedance is real (purely resistive), i.e.,Z_g =R_g, then using Kirchoff’s current law, we can write in the time domain [28]

u₁(₀, t ) = u_g(t ) − p₁(₀, t ) Z_g .

Then from our earlier expressions for volume velocity and pressure within a uniform tube [Equation (4.27)], we have

u⁺₁(t )−u⁻₁(t ) = u_g(t ) − ρc A₁

u⁺₁(t ) +u⁻₁(t ) Z_g

. Solving for the forward-going traveling wave

u⁺₁(t ) = (1+ r_g)

2 u_g(t ) +r_gu⁻₁(t ) (4.36) where

r_g = Z_g − ^ρc_A1 Z_g + ^ρc_A1.

IfZ_g()is complex, then, as with a complexZ_r(), a differential equation realization of the glottal boundary condition is required. A flow diagram of the flow relations in Equation (4.36) is illustrated in Figure 4.16c. As with the radiation impedance, we can also model the effect of glottal impedance with an additional tube of cross-sectionA₀ and infinite length. (We are looking down into the lungs.) If we makeA₀ such thatZ_g = ^ρc_A0, then our expression forr_g becomes

r_g =

ρc A0 − _A^ρc₁

ρc A1 + _A^ρc₀

= A₁ −A₀ A₁ +A₀.

Consider now the special case of two concatenated lossless tubes of equal length, with radiation and glottal boundary conditions, depicted in the flow diagram of Figure 4.18a. In Exercise 4.9, using this flow diagram, you are asked to show that the transfer function relating the volume velocity at the lips to the glottis is given by

V_a(s) = be^−s2τ

1+a₁e^−s2τ +a₂e^{−s4τ (4.37)} whereb=(1+r_g)(1+r_L)(1+r₁)/2,a₁=r₁r_g+r₁r_L, anda₂ =r_Lr_g. In this model, as well as more general models with an arbitrary number of lossless tubes, all loss in the system

4.4 A Discrete-Time Model Based on Tube Concatenation 143

occurs at the two boundary conditions. Nevertheless, it is possible to select loss at the glottis and lips to give observed formant bandwidths. Furthermore, with this configuration Fant [7]

and Flanagan [8] have shown that with appropriate choice of section lengths and cross-sectional areas, realistic formant frequencies can be obtained for vowels. As we noted earlier, however, the resulting model is not necessarily consistent with the underlying physics. In the following section, we describe a means of converting the analog concatenated tube model to discrete time.

In this discrete-time realization, as in the analog case, all loss, and hence control of formant bandwidths, is introduced at the two boundaries.