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Inverse scattering for vowel articulation with frequency-domain data

An inverse scattering problem is analyzed for vowel articulation in the human vocal tract. When a unit amplitude, monochromatic, sinusoidal volume velocity is sent from the glottis towards the lips, various types of scattering data are used to examine whether the cross sectional area of the vocal tract can uniquely be determined by each data set. Among the data sets considered are the absolute value of the pressure measured at a microphone placed at some distance from the lips, the pressure at the lips, and the transfer function from the glottis to the lips. In case of nonuniqueness, it is indicated what additional information may be used for the unique determination.


INTRODUCTION
The fundamental inverse problem for vowel articulation is concerned [1][2][3][4][5] with the determination of the geometry of the human vocal tract from some data. In this paper, we consider various types of scattering data in the frequency domain resulting from a unit amplitude, monochromatic, sinusoidal volume velocity sent from the glottis, and we analyze whether each data set uniquely determines the shape of the vocal tract, or else, what additional information may be used for the unique recovery.
Let us use x to denote the distance from the glottis and l for the length of the vocal tract. Hence, the lips are located at a distance l from the glottis. Typically, l varies between 14 cm and 20 cm, usually smaller for children than for adults and smaller for females than for males [1,2,5]. Even though the vocal tract is not a right cylinder, to a good approximation it can be treated as one [3,4].
We will let A(x) denote the cross sectional area as a function of the distance from the glottis, and we suppose that A(x) is positive on [0, l]. Assuming that the propagation is lossless and planar (these assumptions are known [3,4] to be reasonable), the acoustics in the vocal tract is governed [1][2][3][4][5] by the first-order linear system of partial differential equations A(x) p x (x, t) + µ v t (x, t) = 0, A(x) p t (x, t) + µ c 2 v x (x, t) = 0, (1.1) where t is the time variable, the subscripts x and t denote the respective partial derivatives, µ is the air density, c is the speed of sound, v(x, t) is the volume velocity of the air flow, and p(x, t) is the pressure at location x and at time t.
The volume velocity is equal to the product of the cross sectional area with the average velocity of the air molecules crossing that area. The pressure is the force per unit cross sectional area and is exerted by the moving air molecules. The air density at room temperature is µ = 1.2 × 10 −3 gm/cm 3 . The speed of sound varies slightly with temperature, and c = 3.43 × 10 4 cm/sec in air at room temperature. In our analysis of the inverse problem, we assume that the values of µ and c are already known. There is no loss of generality to start the time at t = 0.
By using v xt = v tx , we can eliminate v in (1.1) and obtain Webster's horn equation we find that Φ satisfies the plasma-wave equation x ∈ (0, l), t > 0, (1.3) where we have defined , (1.4) with the prime denoting the x-derivative. The quantity Q is called the relative concavity of the vocal tract or the potential. Separating the variables as Φ(x, t) := ψ(k, x) e ikct , (1. 5) we find that ψ(k, x) satisfies the Schrödinger equation ψ ′′ (k, x) + k 2 ψ(k, x) = Q(x) ψ(k, x), x ∈ (0, l). (1.6) The frequency ν is measured in Hertz and related to k as ν = kc 2π . Informally, we can refer to k as the frequency even though the proper term for k is the angular wavenumber.
In order to recover A, we will consider various types of data for k ∈ R + resulting from the glottal volume velocity v(0, t) given in (4.1). As our data sets, we consider the absolute value of the impedance at the lips, the absolute value of the pressure measured at a microphone placed at some distance from the lips, the absolute value of the pressure at the lips, the absolute value of the transfer function from the glottis to the lips, the absolute value of the impedance at the glottis, the absolute value of a Green's function for (1.3) measured at the lips, and the real or imaginary part of the reflectance at the glottis.
The inverse problem of recovery of A can be analyzed either as an inverse spectral problem or as an inverse scattering problem. In the inverse-spectral formulation, in addition to a boundary condition at the the glottis such as (2.1), a boundary condition is also imposed at the lips. The imposition of the boundary conditions at both ends of the vocal tract results in standing waves that are related to an infinite sequence of discrete frequencies. It was established by Borg [6] that Q can be recovered by using two such infinite sequences of discrete frequencies corresponding to two sets of boundary conditions. It then follows [3,4,[7][8][9][10][11] that A can be recovered from two infinite sequences of constants. For example, such sequences can be chosen as the zeros and poles [7,8] of the input impedance or the poles and residues [9] of the input impedance.
In the the inverse-scattering formulation, a boundary condition is imposed at only one end of the vocal tract-either at the glottis or at the lips. Then, the measurement of the acoustic data used in the recovery of A is performed at the same end or at the opposite end. If the boundary condition and the measurement occur at the same end of the tract, the corresponding inverse problem is usually known as a reflection problem. On the other hand, if the boundary condition and the measurement occur at different ends, then we have a transmission problem. The methods based on the inverse scattering formulation may be applied either in the time domain or in the frequency domain, where the data set is a function of t in the former case and of k in the latter. We refer the reader to [3,4,[12][13][14][15] for some approaches as time-domain reflection problems and to [16] for an approach as a time-domain transmission problem. Our approach in this paper is a frequency-domain approach, where the analysis in Sections 6-10 may be viewed as that for a transmission problem and the analysis in Sections 11 and 12 may be viewed as that for a reflection problem.
Our paper is organized as follows. In Section 2 we review some preliminary material related to the Schrödinger equation and introduce the selfadjoint boundary condition involving cot α given in (2.1), the Jost solution f, the Jost function F α , and the scattering coefficients T, L, and R. In Section 3 we briefly review the recovery of Q, cot α, F α , f, T, L, and R from the data {|F α (k)| : k ∈ R + }. In Section 4 we obtain some explicit expressions for the pressure and the volume velocity in the vocal tract in terms of A, f, and F α , and we also show that cot α appearing in (2.1) is directly related to the physical parameters A(0) and A ′ (0). In Section 5 we introduce the relative area [η(x)] 2 and express it in terms of the Jost solution, cot α, and the scattering coefficients. In Sections 6-12 we analyze the recovery of Q, η, and A from various data sets. The data set used in Section 6 is the absolute value of the output impedance at the lips. In Section 7 it includes the absolute value of the pressure measured at a microphone placed at some distance from the lips. In Section 8 it is the absolute value of the pressure measured at the lips. In Section 9 the data set includes the absolute value of the transfer function from the glottis to the lips. In Section 10 it is the absolute value of an analog of the Green's function introduced in [17] for (1.3), in Section 11 the absolute value of the input impedance at the glottis, and in Section 12 the real or imaginary part of the reflectance. Finally, in Section 13 we present some examples to illustrate the theoretical results presented in the earlier sections.

PRELIMINARIES
In this section we review the scattering data related to the potential Q appearing in the Schrödinger equation on the half line R + with the selfadjoint boundary condition [18][19][20][21] sin α · ϕ ′ (k, 0) + cos α · ϕ(k, 0) = 0, where α is a number in the interval (0, π) identifying the boundary condition at x = 0. We can relate the half-line Schrödinger equation to (1.6) by assuming that Q(x) ≡ 0 for x > l.
Note that the mapping α → cot α is one-to-one and from (0, π) onto R.
Let f denote the Jost solution [18][19][20][21][22][23] to the half-line Schrödinger equation. It is uniquely determined by the asymptotic conditions Since Q vanishes when x > l, we have The Jost function F α associated with the half-line Schrödinger equation with the boundary condition (2.1) is defined as [18][19][20][21] Let us emphasize that the subscript in F α identifies the boundary condition at x = 0 and it does not indicate any partial derivative. It is known [18][19][20][21] that where the asterisk denotes complex conjugation.
We assume that Q is real valued and integrable on (0, l) and that there are no bound states for the half-line Schrödinger equation with the boundary condition (2.1). The absence of bound states is equivalent [18][19][20][21] to assuming that F α (k) has no zeros on I + , where I + := i(0, +∞) is the positive imaginary axis in the complex plane. It is known [19][20][21] that either F α (0) = 0 or F α (k) has a simple zero at k = 0; the former is known as the generic case and the latter as the exceptional case. The exceptional case corresponds to the threshold where the number of bound states can be changed by one under a small perturbation of the potential.
By using the extension Q(x) ≡ 0 when x < 0, we can relate f (k, 0) and f ′ (k, 0) to the scattering coefficients in the full-line Schrödinger equation. We have [19,20,22,24] f (k, 0) = 1 + L(k) where T and L denote the transmission coefficient and the left reflection coefficient, respectively, associated with Q. The right reflection coefficient R is given by It is known [19,20,22,24] that The absence of bound states for the full-line Schrödinger equation is equivalent [19,20,22,24] for T (k) not to have any poles on I + , and this is also equivalent [25] for F π/2 (k) not to have any zeros on I + .

RECOVERY OF Q FROM |F α |
In the absence of bound states, the fundamental inverse scattering problem for the half-line Schrödinger equation with the selfadjoint boundary condition (2.1) consists of determining Q and cot α from various types of scattering data. In this section we review the solution to this inverse problem when the data set is {|F α (k)| : k ∈ R + }.
Theorem 3.1 Assume that Q is real valued, measurable, and integrable for x ∈ (0, l).
The same data set also uniquely determines the corresponding Jost solution f (k, x) and the scattering coefficients T (k), R(k), and L(k).
Below we outline some steps involved in the solution to the inverse problem stated in Theorem 3.1. As seen from (2.4), |F α (k)| is an even function of k ∈ R, and hence the data sets {|F α (k)| : k ∈ R + } and {|F α (k)| : k ∈ R} are equivalent. By using the data {|F α (k)| : k ∈ R} as input in the Gel'fand-Levitan method [18][19][20][21], we form the kernel and then solve the Gel'fand-Levitan integral equation The solution to (3.1) is known [19][20][21] to exist and to be unique. Once h α (x, y) is obtained, we recover the potential as where x − indicates that the limit from the left must be used in the evaluation. We also recover the boundary condition as Alternatively, we can proceed [21] as follows. Let where we use C + for the upper half complex plane and C + for C + ∪ R. Then From the data {|F α (k)| : k ∈ R} we first construct the function Λ α (k) via the Schwarz integral formula as where the quantity i0 + indicates that the values for real k should be obtained as limits Then, we have where the limit in (3.8) can be evaluated in any manner in C + . Having both f (k, 0) and f ′ (k, 0) in hand, we can construct all the quantities that are relevant in the scattering theory for the Schrödinger equation. For example, the scattering coefficients for the fullline Schrödinger equation can be obtained as Having obtained such quantities, we can construct the potential by using any one of the various methods available [19,20,22,24]. For example, we can use the Faddeev-Marchenko method [19,20,22,24] and get where K(x, y) is obtained by solving the (left) Faddeev-Marchenko integral equation with the kernelR The Jost solution f (k, x) can directly be obtained from K(x, y) as Let us remark that that, in order to obtain instead of using (3.4) we can equivalently construct the real and imaginary parts of Λ α (k) via (3.3) and where CPV indicates that the integral must be evaluated as a Cauchy principal value.
Consequently, cot α can be recovered from the equivalent form of (3.8) given by

PRESSURE AND VOLUME VELOCITY
It is known [19,20,22] that f (k, ·) and f (−k, ·) are linearly independent for each k ∈ C + \ {0}. Hence, the general solution to (1.6) can be written as a linear combination of f (k, ·) and f (−k, ·). From (1.2), (1.5), and (1.6), we see that the pressure has the form where a(k) and b(k) are coefficients to be determined. When x ≥ l, the pressure p(x, t) should be a wave traveling outward from the lips and should not contain the part proportional to e ik(ct+x) traveling into the mouth. Thus, with the help of (2.2) we see that we Our aim is to determine the value of a(k) in terms of the pressure P (k, l + r) measured by a microphone placed at a radial distance r from the lips. The relationship between the pressure measured at the microphone and the volume velocity at the lips is explicitly known and is given by (cf. (3.1) of [5]) where we recall that µ is the air density and c is the sound speed.
From (4.1), (4.2), and the first line of (1.1), for the x-derivative of the pressure at the glottis we get Note that from (4.4) through differentiation we obtain where we have used A comparison of (4.7) with (2.3) shows that, by choosing we can write (4.7) as Comparing (4.6) and (4.10) we get and hence we can write (4.4) in the equivalent form Using (4.2) and (4.11) in the first line of (1.1), we get In particular, from (4.12) we obtain v t (l, t − r/c) = ck A(l) e ikc(t−r/c) Finally, comparing (4.5) and (4.14) we get 15) or equivalently, with the help of (4.2), we have From (4.16) we can conclude that k appears as ik in P (k, l + r) and hence P (−k, l + r) = P (k, l + r) * , k ∈ R. (4.17) We emphasize that P (k, x) given in (4.11) is valid only when x ∈ [0, l], and hence P (k, l+r) is not obtained from (4.11) by replacing x by l + r there. Finally, we remark that, with the help of (4.1) and (4.12), we obtain

AREA AND RELATIVE AREA
Let us view (1.4) as the zero-energy Schrödinger equation, and consider the initialvalue problem y ′′ = Q(y) y, x ∈ (0, l), where cot α is the quantity in (4.9). Let y 1 (x) and y 2 (x) be any two linearly independent solutions to (5.1) on the interval [0, l]. Then, the unique solution to (5.1) can be written as We will refer to [η(x)] 2 as the relative area of the vocal tract. Let us remark that we can write (5.3) in the equivalent form on the interval (0, l), is zero when x < 0, but is a real-valued, measurable, integrable function with a finite first moment when x > l. Let τ (k), ℓ(k), and ρ(k) be the corresponding transmission coefficient, the left reflection coefficient, and the right reflection coefficient, respectively. Generically, we have τ (0) = 0 or equivalently [g l (0, x); g r (0, x)] = 0. In the exceptional case, we have τ (0) = 0 or equivalently [g l (0, x); g r (0, x)] = 0.
In the exceptional case, we can choose In this case, we have with y 1 (0) = 0 because [19,20,22,24] we have −1 < ℓ(0) < 1. Hence, from (5.2) we get It is possible to construct η(x) from {|F α (k)| : k ∈ R + } as follows. With the help of (3.4)-(3.12), we can construct the corresponding right reflection coefficient R(k), the transmission coefficient T (k), and the Jost solution f (k, x). Hence, in the generic case every term appearing on the right hand side of (5.6) can be constructed from {|F α (k)| : k ∈ R + }, and we get In the exceptional case, from (5.7) we get . Note that we can write the Jost solution f (k, x) as a linear combination of g l (k, x) and g l (−k, x), where g l (k, x) is the quantity appearing in (5.5). With the help of (2.2) we obtain f (k, x) = e ikl 2ik 0 g l (k, x) g l (−k, x) 1 g l (k, l) g l (−k, l) ik g ′ l (k, l) g ′ l (−k, l) .

(5.9)
We can also express F α (k) with the help of g l (k, x). To do so, we can obtain f (k, 0) and f ′ (k, 0) from (5.9) and use (2.3) to get F α (k). Alternatively, by using [cf. (2.5)] we can write F α (k) with the help of the transmission and left reflection coefficients associated with g l (k, x).

RECOVERY FROM THE IMPEDANCE AT THE LIPS
The impedance at the lips is defined as Z(k, l) := p(l, t) v(l, t) .
(6.1) Using (4.2), (4.11), and (4.18) in (6.1), we get Thus, we can only hope to get A(l) and A ′ (l) from Z(k, l). We can refer to Z(k, l) as the output impedance because the volume velocity is from the glottis to the lips. In Section 11 we will analyze the impedance at the glottis, which we can identify as the input impedance.
Note that from (6.2) we get By using (6.2) at two distinct real k values, say k 1 and k 2 , we can recover A(l) and A ′ (l) by solving a linear algebraic system and get On the other hand, if we only know |Z(k, l)| without knowing its phase, then from (6.3) we get A(l) and |A ′ (l)| as As seen from (6.2), Z(k, l) by itself contains no other information related to Q, η, or A.

RECOVERY FROM PRESSURE AT A MICROPHONE
Let us place a microphone at a radial distance r from the lips and measure at that microphone the absolute value of the pressure resulting from the glottal volume velocity given in (4.1). With the help of (4.2) it follows that this is equivalent to having |P (k, l +r)| at hand for k ∈ R + . From (4.17) we see that |P (k, l + r)| is an even function of k ∈ R, and from (4.16) we get In this section we show that the data set {|P (k, l + r)| : k ∈ R + } by itself does not uniquely determine any of Q(x), η(x), or A(x), and how additional data may be used for the unique determination.
PROOF: It is known (cf. (3.9) of [21]) that for any fixed α ∈ (0, π) we have Thus, from (7.1) we obtain or equivalently From (4.17) and (7.4) we see that our data set uniquely determines |F α (k)| for k ∈ R, and hence, as indicated in Theorem 3.1, Q(x) is uniquely determined for x ∈ (0, l). Next, from Theorem 5.1 it follows that η(x) is also uniquely determined for x ∈ [0, l]. Finally, from (7.3) we see that A(0) is also determined by our data set, and thus we recover A(x) for x ∈ [0, l] uniquely by using (5.4).
Note that we assume that A(l) and |A ′ (l)| do not change with k, and hence they are constants. As indicated in Section 6 they can be obtained via (6.5) and (6.6) by measuring the absolute value of the impedance at the lips at two different frequencies. With the help of (7.4) and Theorem 7.2 we have the following conclusions.

RECOVERY FROM THE PRESSURE AT THE LIPS
Let us consider the recovery of A(x) for x ∈ [0, l] from the absolute value of the pressure at the lips resulting from the glottal volume velocity in (4.1). From (4.2) we see that our data set is equivalent to {|P (k, l)| : k ∈ R + }. With the help of (2.4), (2.5), (2.7), and (4.11), we notice that |P (k, l)| is an even function of k ∈ R, and hence we have our data actually available for k ∈ R. In this section we show that this data set uniquely recovers each of Q(x), η(x), and A(x) for x ∈ (0, l), and we outline an explicit procedure to determine these quantities. and hence As seen from (2.4) and (8.3), by measuring the absolute value of the pressure at the lips for k ∈ R + , we get |F α (k)| for k ∈ R. Then, by proceeding as in Section 3, we can recover Q(x) for x ∈ (0, l) and the constant cot α appearing in (4.9). Next, by proceeding as in Section 5, we determine η(x) for x ∈ (0, l). Note that A(0) A(l) is uniquely determined from our data via (8.2). Furthermore, as seen from (5.3), we have η(l) = A(l)/A(0).

Thus, we obtain
and hence we get the area function uniquely via (5.4).

RECOVERY FROM THE TRANSFER FUNCTION
The transfer function T(k, l) from the glottis to the lips is defined as and as we see from (4.1), (4.18), and (9.1), we have Hence, with the help of (2.4), we get 3) Using (7.2) in (9.2) we obtain and as a result we can recover A(0) as Thus, from (9.2) and (9.4), with the help of (2.4), we have Comparing (9.3) with (7.1) we see that and hence we have the following conclusion.
Corollary 9.1 For each fixed r > 0, the information contained in the data set {|T(k, l)| : k ∈ R + } is equivalent to that in {|P (k, l + r)| : k ∈ R + }.
In other words, measuring the absolute value of the pressure at a microphone placed at some distance from the lips is equivalent to measuring the absolute value of the transfer function from the glottis to the lips. Consequently, we have the following analogs of the results of Section 7.  PROOF: From (2.4) and (10.1) it follows that |G(k, l; t)| is independent of t and is an even function of k on R, and hence our data can be extended from k ∈ R + to k ∈ R. Using (7.2) in (10.1) we get and hence |k| |F α (k)| = |G(k, l; t)| lim k→+∞ |G(k, l; t)| , k ∈ R.
Thus, we get |F α (k)| for k ∈ R whenever we have |G(k, l; t)| for k ∈ R + . Then, as in Section 3 we construct Q(x) for x ∈ (0, l) and cot α. Next, as in Section 5, we construct η(x) for x ∈ [0, l]. Finally, with the help of (5.4) and (10.2) we obtain Thus, the proof is complete.

RECOVERY FROM THE IMPEDANCE AT THE GLOTTIS
The impedance at the glottis is defined as Using (4.1), (4.2), and (4.11) in (11.1) we get From (2.4), (2.5), (2.7), and (11.2) we see that |Z(k, 0)| is an even function of k ∈ R, and hence |Z(k, 0)| is known for k ∈ R if it is known for k ∈ R + . In this section we show that the information contained in {|Z(k, 0)| : k ∈ R + } enables us to uniquely construct Q, η, and A.
Recall that we assume that the half-line Schrödinger equation with the boundary condition (2.1) does not have any bound states. In that case, it is known [21] that kf (k, 0)/F α (k) is analytic in C + , continuous in C + , nonzero in C + \ {0}, and it is either nonzero at k = 0 or has a simple zero there, and As a result, we can recover kf (k, 0)/F α (k) for k ∈ C + from its amplitude known for k ∈ R via k f (k, 0) Having constructed kf (k, 0)/F α (k) for k ∈ R, we can use (3.2) and obtain Λ α (k) for k ∈ R.

RECOVERY FROM THE REFLECTANCE AT THE GLOTTIS
The reflectance at the glottis is defined [3,4] as the ratio of the right-moving (from the glottis towards the lips) pressure wave to the left-moving pressure wave. From (4.11) that reflectance is seen to be equal to L(−k) because the Jost solution to the full-line Schrödinger equation has the extension where we recall that L(k) is the left reflection coefficient appearing in (2.5). In this section we show that either of the real and imaginary parts of the reflectance at the glottis known for k ∈ R + enables us to construct Q(x) uniquely for x ∈ (0, l). On the other hand, it cannot uniquely determine either η(x) or A(x).
We can then obtain Q uniquely, for example, via [22,24] the Faddeev-Marchenko method, as where B r (x, y) is obtained by solving the (right) Faddeev-Marchenko integral equation It is already known [19,20,22,24] that T (k) and R(k) can be constructed from L(k) with the help of (2.6) and and hence, as we see from (5.8), η(x) is not uniquely determined and we have the corresponding one-parameter family for η(x) with cot α being the parameter. From (5.1) we see that we have the corresponding two-parameter family for A(x), where the parameters can be chosen, for example, as A(0) and A ′ (0), or as A(l) and A ′ (l).
Finally, let us remark that, as indicated in (6.4), we can obtain A(l) and A ′ (l) if we know the impedance at the lips at two distinct real k values.

EXAMPLES
In this section we illustrate the theoretical results presented in the previous sections with some examples.
Let us use l = 17.5 cm, c = 3.43 × 10 4 cm/sec, µ = 1.2 × 10 −3 gm/cm 3 , A(0) = 5 cm 2 , and A ′ (0) = −0.52 cm, and When Q is viewed as a potential of the full-line Schrödinger equation with support on R + , the corresponding scattering coefficients τ (k), ρ(k), ℓ(k) and the left Jost solution g l (k, x) introduced in Section 5 are rational functions of k, and it can be verified that , ℓ(k) = 2 (k + i)(k + 2i) , All the quantities related to (1.1), (1.    Let us remark that in Fig. 13.7 the absolute value of the pressure is measured at a distance of 20 cm from the lips. In Fig. 13.9 the real and imaginary parts of the reflectance are displayed, which can be told apart from Re[L(0)] = −1 and Im[L(0)] = 0.
It is known from Section 6 that the graph in Fig. 13 We know from Section 12 that either graph in Fig. 13.9 uniquely constructs the graph in Fig. 13.1, but not that in Fig. 13.2.