Transformation of Spoken to Written Forms of (Natural) Languages via Spectral and Pseudo-spectral Methods

MATERIALS AND METHODS

Wave Equation for Alphabetization and Syllabication of a Natural Language
Since speech is generated by the vibration of the human vocal chord, movement of tongue and its touching various parts of the mouth and involvement of lips and nose and the generated sound propagates in the air in the form of a wave with velocity c, it is logical to start with the wave equation of the form:

(1)

in which u is the displacement, t is time and x is spatial coordinate. The angular frequency, ω = 2πf, f being the frequency, while the wave number, k = 2π/λ, λ being the wavelength. The wave velocity is given by

(2)

The initial condition is given by

(3)

In addition, the boundary conditions, which depend on the form of a written language, must be specified.

On applying the separation of variables,

(4)

Eq. 1 yields the following:

(5)

in which C is a constant. In order to obtain the desired solution in the form of e^{i(kx - ωt)} and e^{-i(kx - ωt)} (or equivalently sine and cosine), C must be negative, which is taken to be equal to -1, without any loss of generality. This operation yields the following two ordinary differential equations (ODE) in x and t, respectively:

(6)

(7)

It may be noted that the separated wave equation in x, given by Eq. 6, is consistent with the underlying hypothesis of the present research that a spoken natural language (speech, word form) can be mathematically represented in the syllabified and alphabetized written forms of the same. This justifies the concept of the spatial wave function, Ψ(x), wherein x is given in terms of phoneme/alphabet units. The wave number k represents the corresponding eigenvalue.

Syllabic Form of Writing
In the syllabic form of writing, a symbol (representing a syllable) is abruptly terminated before a new syllable starts. This is idealized by one dimensional well (in analogy to quantum mechanics). The boundary conditions are specified at the start (x = 0) and end (x = L) of the syllable, where L is given in terms of phoneme units. These are given by

(8)

The solution to the ODE (6) subjected to the boundary condition (8) can then be easily obtained as follows:

(9)

(10)

which is the solution for a standing wave. The solution to the ODE (7) can also be easily obtained as follows:

(11)

The complete solution is given by

(12)

in which

(13)

After applying the initial conditions, the constant coefficients can be obtained as follows:

(14)

D_n can easily be determined by expanding g(x) in the form of Fourier sine series in x. The syllabic form of the written language needs to capture the initial condition (14) of the spoken language. Here the computed wave function is localized within the box (well), while in a speech, the wave function is extended all the way to "infinity".

Phonemic Analysis of Speech Sounds and Alphabetic Form of Writing
Morpheme is the smallest part of a word that has meaning of its own. Morphemes may be words, prefixes, suffixes or endings that show inflection (Barnhart and Barnhart, 1991). In the word carelessness, the morphemes are care, -less and -ness. A morpheme, consisting of alphabets, "does not necessarily consist of phonemes, but all morphemes are statable in terms of phonemes" (H.A. Gleason, Jr., quoted by Barnhart and Barnhart, 1991). In the present work, morpheme/syllable is assumed to be comprised of a number of phonemes (approximated by phonetic alphabets). A phoneme acts as a low-pass filter in the frequency domain. It follows that the corresponding alphabet acts as a long-wave filter or low wave number (k) filter/sampler in the wave number domain.

The characteristics of an ideal low-pass filter, in the frequency and time domains is discussed in standard texts e.g., Gajic (2003). An ideal low-pass filter transfer function is given by Gajic (2003).

(15)

The phase of the ideal filter is assumed to change linearly in frequency, which corresponds to the time shift of the filter input signals by t_d, known as the time delay. The impulse response of the ideal low-pass filter is obtained as follows. Using the time domain Fourier equivalent of a rectangular frequency domain pulse of unit height given by the Fourier transform pair

(16)

the ideal low-pass filter transfer function is given by Gajic (2003)

(17)

in which the time shift property of the Fourier transform has been utilized and the Sinc function is defined as:

(18)

The characteristics of the same ideal low-pass filter, in the wave number and space domains can be described in a similar manner. An ideal low-pass filter space transfer function can be written as follows:

(19)

The phase of the ideal filter is again assumed to change linearly in wave number, which corresponds to the space shift of the filter input signals by x_d, to be termed here as the space shift (delay). The spatial impulse response of the ideal low-pass filter is then obtained as follows. Using the space domain Fourier equivalent of a rectangular wave number domain pulse of height, g₀, given by the Fourier transform pair

(20)

the ideal low-pass filter spatial transfer function can now be written follows:

(21)

in which the spatial shift property of the Fourier transform has been utilized and the Sinc function is defined as before. Generalizing the above, g(x) can be expressed in standard sinc series (Stenger, 1993; Lund and Bowers, 1992) as follows:

(22a)

(22b)

in which

and g₀ = g(0) corresponding to j = 0.

Sampling with an Ideal/Physical Sampler and the Discrete Time Fourier Transform (DTFT)/Discrete Fourier Transform (DFT)
Shannon's Sampling Theorem (Gajic, 2003) states that a continuous-time band-limited signal, such as the Sinc function, sinc(t), with bandwidth frequency f_max can be uniquely reconstructed from its sampled values at sinc (jT_s), j = 0, ±1, ±2, ±3,..., if the sampling frequency f_s = 1/ T_s satisfies

(23)

The frequency 2f_max is called the Nyquist frequency and the frequency interval [-f_max, f_max] is called the Nyquist interval. This theorem can be easily extended to the space and wave number domains.

Standard texts on digital signal processing consider two types of samplers: ideal sampler and the physical sampler (Gajic, 2003). It is shown that an ideal sampler gives rise to discrete time Fourier transform (DTFT), while a real or physical sampler produces discrete Fourier transform (DFT). Applying ideal sampling operation to syllable (sound) based written language, such as Hieroglyphic or Hieratic, these written symbols can be mathematically represented by DSFT (discrete space Fourier transform), the spatial counterpart of the DTFT. This is expressed by the spatial counterpart of Eq. (9.16) of Gajic (2003), which suggests that the wave number spectrum of the ideal sampled "signal" (here the symbols) is periodic.

As has been commented by Gajic (2003), the DTFT or in our case DSFT is the Fourier transform of a discrete time or space "signal", useful for representing a syllable based written language and obtained by sampling a continuous-time or -space "signal" by an ideal sampler. It has been shown (Gajic, 2003) that DTFT and its inverse as defined in Eq. 9.23 and 9.27 of that book can be derived using the discrete-time or -space Fourier series. By analogy to the continuous-time or -space Fourier series, the discrete-time or -space Fourier series are applicable to discrete-time or space periodic signals. A discrete-time or -space aperiodic signal can be considered in the limit as a discrete-time or -space periodic "signal" whose period tends to infinity and thus DTFT or DSFT tends to the corresponding DFT (discrete Fourier transform), discussed in standard treatises (Gajic, 2003; Press et al., 1992).

Application of the physical sampling to the phonemic signal analysis produces an alphabet-based written language. Alphabet serves as the physical or real sampler of narrow width, 'a'. The alphabet sampler inside the morphemic "well" model is shown in Fig. 1. Assuming the number of alphabets inside a morpheme to be N, the length of the syllable, L = Na, where a is the width of the phoneme/alphabet. The boundary conditions are given by Eq. (8) at x = ± L.

A rectangular pulse is defined as follows:

(24)

Since in the present alphabetic physical sampler inside the morphemic box (well) problem, the wave function is permitted neither to extend all the way to "infinity", nor to be localized in length scale of the order of "sampler width", a (<< L), the computed eigenfunctions cannot be orthonormalized with respect to either the Dirac or Kronecker δ functions. Consequently, these eigenfunctions need to be orthonormalized with respect to the reverse Cantor set based rational δ function, discussed in the Appendix. Therefore, the correct solution must be of the form of discrete Fourier transform, as given by Eq. (A10) and (A11).

(25a)

(25b)

where Ψ_j is the physically sampled spatial wave function. A_k, with k being the eigenvalue, need to be numerically evaluated by using the DFT (or FFT, DWT, etc.) technique and must satisfy the various boundary conditions of the problem. This scheme is currently being implemented and numerical results will be reported in future.

The fast Fourier transform (FFT) is an algorithm that computes in O (N log₂N) operations as compared to the DFT's O (N²). The discrete wavelet transform (DWT) is a more recently developed fast computational tool that linearly operates on a data vector, the length of which is an integral power of 2 and transforms it into a numerically different vector without altering the length (Daubechies, 1992). Like the FFT, the DWT is invertible and orthogonal, its inverse transform, when viewed as a large matrix, being the transpose of the transform. Both the FFT and DWT can, therefore, be viewed as rotations in function space, from the input space to a new domain. In the case of FFT, the rotated domain has basis functions in the form of standard sines and cosines, while in the wavelet domain the basis functions are somewhat novel, with names like "mother functions" and "wavelets". The details of the FFT and DWT algorithms are available in Press et al. (1992) and will not be repeated here.

CONCLUSIONS

The concept of wave function is employed to mathematically model the transformation of a natural language from its spoken to both written forms – syllable (symbol) based and alphabet based writings. The fundamental steps to solving this problem are concerned with the solution to the wave equation subjected to initial and boundary conditions and derivation of these solutions for syllables and phonemes via Fourier series and Sinc series expansion techniques for symbol and alphabet-based writings, respectively. These, in turn, through ideal and physical sampling, reduce to discrete time (space) Fourier series (DTFT or DSFT) for symbol based writing and discrete Fourier transform (DFT) for alphabet based writing.

In this work, the syllable is approximated to be ideally sampled version of morpheme (represented by a symbol in the case of Hieroglyphics), while the phonetic alphabet is the physically sampled version of the corresponding phoneme. In regards to the relationship between morphemes (symbols) and syllables, the above assumption is justified on the ground that in ancient Egyptians' writings, morphemes were represented as syllables and vice versa. The assumption pertaining to the relationship between phonetic alphabets and phonemes is justified on the ground of ancient Phoenicians' pioneering usage to the same effect.

A rigorous set theoretic basis for mathematical modeling of a generic alphabet based written language is provided through application of the reverse Cantor set based rational delta function, first introduced by Chaudhuri (2006). Alphabetization bridges the gap between the two situations that arise in phonetic/linguistic research, namely the syllable based written language with discrete eigenvalues and the spoken language with a continuous spectrum of eigenvalues.

Most important, this novel rational delta function, δ_n(x), provides a rigorous set theoretic basis for permitting the resulting computed wave function to be expressed in the form of the discrete Fourier transform (DFT) in the Fourier domain and recover the sampled wave function in the physical domain by employing the inverse discrete Fourier transform (IDFT). The relatively straightforward transition to faster techniques, such as the fast Fourier transform (FFT), Sinc and discrete wavelet transform (DWT), will be the subject of future research.

This research helps bridge the long-standing gap between phonetic/linguistic research and modern development in computational spectral and pseudo-spectral methods, such as DFT, FFT, Sinc and DWT. The ease of the present method and its relative accuracy will help make challenging language problems numerically solvable.

ACKNOWLEDGEMENT

The authors wish to thank Dr. J.A. Laursen and an anonymous reviewer for their helpful suggestions on an earlier version of the manuscript.