A Novel Two-Level Method for the Computation of the ... - IEEE Xplore

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IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 5, NO. 2, MARCH 1997

A Novel Two-Level Method for the Computation of the LSP Frequencies Using a Decimation-in-Degree Algorithm Chung-Hsien Wu, Member, IEEE, and Jau-Hung Chen

Abstract— A novel two-level method is proposed in this study for rapidly and accurately computing the line spectrum pair (LSP) frequencies. An efficient decimation-in-degree (DID) algorithm is also proposed in the first level, which can transform any symmetric or antisymmetric polynomial with real coefficients into the other polynomials with lower degrees and without any transcendental functions. The DID algorithm not only can avoid prior storage or large calculation of transcendental functions but can also be easily applied toward those fast root-finding methods. In the second level, if the transformed polynomial is of degree 4 or less, employing closed-form formulas is the fastest procedure of quite high accuracy. If it is of a higher degree, a modified Newton–Raphson method with cubic convergence is applied. Additionally, the process of the modified Newton–Raphson method can be accelerated by adopting a deflation scheme along with Descartes rule of signs and the interlacing property of LSP frequencies for selecting the better initial values. Besides this, Horner’s method is extended to efficiently calculate the values of a polynomial and its first and second derivatives. A few conventional numerical methods are also implemented to make a comparison with the two-level method. Experimental results indicate that the two-level method is the fastest one. Furthermore, this method is more advantageous under the requirement of a high level of accuracy. Index Terms— Speech processing, decimation-in-degree algorithm

line

spectrum

pair,

I. INTRODUCTION

T

HE ANALYSIS and design of discrete time systems has been particularly noteworthy in the determination of the zero locations or the formant information of its symmetric or antisymmetric real characteristic polynomials. The characteristic polynomials with zeros on the unit circle have recently received significant attention. A well-known approach is the representation of line spectrum pair (LSP) [1] frequencies, which has been proposed as an alternative linear predictive coding (LPC) spectral representation. The LSP frequencies not only encode speech spectral information more efficiently than other transmission parameters [2]–[6], but also provide

Manuscript received September 11, 1994; revised August 25, 1996. This work was supported by the National Science Council of the Republic of China under Grant NSC82-0115-E-006-461. The associate editor coordinating the reivew of this manuscript and approving it for publication was Prof. John H. L. Hansen. The authors are with the Institute of Information Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C. (e-mail: [email protected]) Publisher Item Identifier S 1063-6676(97)01895-6.

good performance both in speech recognition [7] and speaker recognition [8], [9]. A few methods have estimated the LSP frequencies submitted to LPC analysis-and-synthesis [2], [3], [12]. Since the zeros of the LSP polynomials are on the unit circle [2], [10], [11], the other method proposed by [14] for estimating the zeros on the unit circle is also applicable. The methods proposed by Schmidt and Rabiner [14], Soong and Juang [2], and Kang and Fransen [3] first of all transform the characteristic polynomials into the sum of cosine functions without effort in order to avoid complex computation in the later processing. Most of them then evaluate on a reasonably large grid of points repeatedly to search for the subintervals, which isolate each zero. To locate the zeros of standard finite impulse response (FIR) linear phase digital filters, Schmidt and Rabinler found the Newton–Raphson method is more efficient than the bisection method and the modified false position method [15], in which a standard fast Fourier transform (FFT) was used to estimate the initial values on a fine grid. On estimating the LSP frequencies, however, Soong and Juang have adopted a discrete cosine transform (DCT) to evaluate the cosine functions on a fine grid using the bisection method [2]. Furthermore, they consider it a better approach to solve them through closed-form formulas when a 4th-degree polynomial is further reduced from those cosine functions. Also, Kang and Fransen proposed two other new methods, i.e., autocorrelation function method and allpass ratio filter method, for estimating the LSP frequencies. However, all of the above methods require large evaluations of trigonometric functions. Therefore, Kabal and Ramachandran [12] presented a backward recursion formulation to determine the values of the cosine functions on a fine grid without prior storage or large calculation of trigonometric functions. They applied it in the bisection method followed by either linear interpolation or a faster one such as inverse parabolic interpolation. However, they found that the performance of the latter combination is occasionally two or three times worse than that of the former. Besides this, being subject to the special expression of the backward recursion formulation, application to any other faster numerical methods is relatively difficult. Most of the above methods must first investigate the adequate size of adjacent grid points so that only an isolated zero is contained. A tradeoff occurs, since the variability of speech signals do not allow a larger size of grid and the performance will be significantly degraded as the size of grid is too small.

1063–6676/97$10.00  1997 IEEE

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In order to deal with the above problems, this study examine the techniques concerned with the following requirements: 1) avoiding the storage or the large computation of trigonometric functions; 2) avoiding the use of complex computation; 3) achieving good estimation of initial value to replace grid scheme; and 4) obtaining reliable performance even under the requirement of high accuracy. Additionally, a decimation-in-degree (DID) algorithm is proposed here to transform any symmetric or antisymmetric polynomials, i.e., an intrinsic property of LSP polynomials, to the other polynomials with lower degrees and without any transcendental functions. This algorithm satisfies the first two requirements. As for the rest of the requirements, better initial values are selected by a deflation scheme along with Descartes’ rule of signs and the interlacing property of LSP frequencies. Next, these values are refined by a modified Newton–Raphson method, in which Horner’s method is extended to efficiently calculate the values of a polynomial and its first and second derivatives. Additionally, closed-form formulas are adopted in order to achieve high efficiency and high accuracy. The rest of this paper is organized as follows. The LSP frequencies are reviewed in Section II. In Section III, a two-level method is proposed including the DID algorithm, followed by the closed-form formulas and the modified Newton–Raphson method to rapidly and accurately estimate the LSP frequencies. The performance evaluation for the two-level method is provided in Section IV. Concluding remarks are finally made in Section V. II. THE LINE SPECTRUM PAIR (LSP) FREQUENCIES

following reduced polynomials can be obtained, i.e., even (4) odd. and are with real Furthermore, since both coefficients, the zeros of them all appear in complex conjugate pairs. Associating this and the first property listed above, and can be expressed as follows: (5) (6) (7) (8) where even (9) odd The (since ) parameters, and , are referred to as LSP frequencies. The ordering property allows the LSP frequencies to be ordered as follows: even (10) odd.

A brief review of the LSP frequencies and some of the influential properties is provided in this section. For a given order and the LPC coefficients , the minimumphase LPC polynomial is expressed by either [12] or [18] as follows:

in and those By dropping the trivial factors in , respectively, two symmetric polynomials [10] of degree are obtained, i.e., or , which can be generally expressed by

(1) or [1] or [2] as follows: (11)

(2) and an antisymmetric polyA symmetric polynomial nomial are equated as follows: (3) and are referred to as LSP polynomials and possess the following interesting and significant properties [6]: • the zeros of LSP polynomials are on the unit circle; • the zeros of and are interlaced with each other; • the minimum phase property of can be easily preserved if the first two properties remain intact after quantization. Sugamura and Farvardin [13] referred to the second property as ordering property. It is evident that the LSP polynomials contain trivial zeros at 1 or at 1. These trivial zeros can be easily removed by synthetic division. Therefore, the

where are real coefficients with 1. Most of the methods described in the introduction rewrite (11) into a series in cosines:

(12) where is a zero of on the unit circle. The LSP frequencies are obtained by estimating the values of such that . III. THE TWO-LEVEL METHOD The process in our two-level method for estimating the LSP frequencies is described as follows.

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A. Level One in (12) is usually It is apparent that the evaluation of computationally intensive. Therefore, we propose a DID algorithm that yields a polynomial without transcendental functions to alleviate this problem. 1) The Derivation of the DID Algorithm: First exto be a new polynomial pressed in (11) is divided by

(13) Next, consider the following mapping (14) A sequence of polynomial functions are implicitly constructed as follows:

Fig. 1.

Plots of

F

Fig. 2.

Plots of

F

P (x)

= 0 and

F

P 6 (x) = 0 and

F

Q (x)

= 0:

(15) Since (16) we obtain the following recursive relation of functions: (17) is a variant of the In fact, polynomial [16], and first few polynomials are

th degree Chebyshev . The other

(18)

It can be proven by induction that degree . So we write

P 5 (x) = 0.

Substituting (15) and (19) into (13), the symmetric polynomial can be transformed to an th degree polynomial in defined as follows:

is a polynomial of

(24) are real constants. After some algebraic where manipulations, the coefficients of are determined by

(19) (25)

are real constants for . Note that the where recursive relation of functions yields the following recursive relation of coefficients: (20) given where, given , we define and for . Carrying out this recurrence, the leading coefficients and the constant terms are found to be explicitly given, respectively, by (21) and odd even.

(22)

Furthermore, we obtain for

and

(23)

(26) even are available, the corresponding zeros If the zeros of of can be easily obtained by the relation . Since we have , it becomes (27) Hence, the LSP frequencies are given by (28) is observed to be a real From (24), (26), and (27), polynomial in and all its zeros are real numbers. Therefore, evaluating in (24), as compared with in (12), is relatively simpler. 2) The DID Algorithm: The DID algorithm is established on the basis of (4) and (20)–(26). The main procedures involve

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determining the coefficients of from those of . For the sake of easy programming, ’s defined in (19) are implemented by the variables, ’s, which are yielded by dropping all ’s of zero values in (20) and by rearranging and reversing the indices. It is also noted that the notation denotes the floor function of . The precise DID algorithm then proceeds according to the following. Step 1) Given a symmetric or an antisymmetric polynomial, possible trivial zeros are removed at 1 or -1 [referring to (4)] by synthetic division to obtain a symmetric polynomial similar to expressed in (11). Step 2) Initialize ’s [referring to (21) and (22)]: Set for a)

set

. ’s [referring to (20) and (26)]: For

Step 3) Determine a) b)

Fig. 3. Illustration of the LSP frequencies for

set for

m = 12.

with manner. Given set set set

12 and the following LPC coefficients

(29) (30) (31)

Step 4) Stop. If the value of remains constant, however, a direct result of can be obtained by employing this algorithm previously. For purposes of reference, three cases of that are of frequent interest for finding the LSP frequencies are listed as follows: • If , we have

(35)

and can be obtained by (1) the coefficients of and (3). In Step 1), the upper part of (4) is selected to apply and are synthetic division. The coefficients of obtained, respectively, as follows:

(36) (32) • If

, we have After Steps 2) and 3), we have

(37) (33) • If

, we have

(34) 3) Example 1: For convenience of demonstrating the DID algorithm, only an illustrative example is provided for even order . The case of odd can be processed in a similar

The same results can also be obtained by directly applying (34) and are displayed after Step 1). The curves of in Fig. 1 for later reference. 4) Computational Complexity Analysis: The computational complexity is discussed as follows: addi• Step 1) requires no multiplication and at most tions, since this step invokes not more than two times of synthetic division, in which only the divisors at 1 or 1 are involved. • 2) and 4) clearly involve no multiplication and addition.

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• In Step 3), only the first iterations of the outer loop are under our consideration and they yield an amount of

Consequently, the three real roots become (47)

(38) or

Now if

4, and given the quartic equation (48)

(39) iterations of inner loop as is even or odd, respectively. Since (29), (30) and (31) require one multiplication and three additions for each inner loop, Step 3) requires

we implement Ferrari’s solution, whose performance is found to be slightly better than that of Descartes–Euler solution [19], as follows. First, solve one root, say in (47), for the following resolvent cubic equation corresponding to (48): (49)

even (40)

Next, let

odd

(50)

multiplications and

Then the four real roots of (48) are even (41)

(51)

odd (52)

additions. • As a result, this algorithm requires

(53) even (42)

(54)

odd and the inequalities of

multiplications and

(55)

even (43) odd additions at most, i.e., around O tions and O real additions.

real multiplica-

are obtained. The modified Newton–Raphson method that converges cubically is displayed as the following general terms: (56)

B. Level Two 1) Closed-Form Formulas and Modified Newton–Raphson Method: After applying the DID algorithm, two polynomials in are obtained, i.e., of degree and of degree , which are transformed from and , respectively. For convenience they are represented by a general polynomial of degree , where or . In the second level, the zeros of are estimated by either the closed-form formulas [19] as , such as the trigonometric solution and Ferrari’s solution, or a modified Newton-Raphson method [17] as . The procedure for is not discussed here since it is a trivial problem. Consider the cubic equation (i.e., 3) that follows: (44) Since only real roots are of concern here [see (27)], the trigonometric solution is adopted instead of using Cardan’s formula [19] and is simplified to be efficient as follows: Let (45) and let (46)

where , for , is an th-degree polyno1) mial; is constructed by deflating with the 2) largest zero of and, hence, the values of zeros are found to be decreasing; 3) and are the first and second derivatives of , respectively. Due to the interlacing property of (10), the symmetric polynomials never have multiple roots. However, in the special case of caused by close roots even though it hardly happens, we assume the polynomials have a root of multiplicity at . Our convergence criterion is defined to terminate the iterative procedure when the inequality (57) is satisfied for a specified value . An efficient method of calculating the three function values of and is described as follows. According to remainder theorem, we obtain (58)

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(a)

(b)

(c)

(d)

(e)

(f)

m

Fig. 4. Histograms of the number of iterations for each corresponding to Table VIII (count versus number of iterations). (a) 12. (c) 14. (d) 16. (e) 18. (f) 20.

m=

m=

m=

m=

The simplified results of taking the first and second derivatives , respectively, are of (58) with respect to at (59) Let

after the other in the listed order by the following extended Horner’s method. Step 1) Set . Step 2) For a) b) c)

be represented as (60)

Using the two equations in (59) and extending Horner’s method (or synthetic-division algorithm) [16]–[17], the three and , which are function values of and , respectively, are determined one represented as

m = 10. (b) m =

set set set

; ; . ;

Step 3) Set a) b) c)

set set set

Step 4) Stop.

; ; .

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EXAMPLE

RESULTS

OF THE

OF THE

TABLE II CLOSED-FORM FORMULAS

TABLE I MODIFIED NEWTON–RAPHSON METHOD

FOR

F4P (x)

Instead of searching for the zeros on a fine grid, the following criteria are used to obtain a better initial value : C1) The inequalities in (10) are equivalent to (61) and , for , where are zeros of and , respectively. An illustration of the interlace property in (61) is displayed in Fig. 1. Therefore, the zeros of must first be found and then they can be used as the initial values . for the corresponding zeros of C2) From (27) or (61) we found 2 2. Hence, let 2 and then find zeros backward since has a nearest zero at 2. C3) Assume is an approximate zero of which satisfies (57). Using (59) and applying the original Newton–Raphson method, the initial value for the largest zero of is given by (62) C4) Descartes’ rule of signs [19] is used to obtain the number, say , of positive real zeros of . Now if given in (62) is positive when 1, then we reassign 0. If is determined for each polynomial , the corresponding largest zero is iteratively obtained by (56). Notice that the use of deflation may increasingly lead toward an worse

FOR

F6P (x)

AND

F5P (x)

accuracy of each subsequent zero, i.e., the zeros of are not equivalent to that of the undeflated polynomial . It is thus necessary to refine the zeros of by using (56) with respect to the undeflated polynomial more and again. After obtaining the refined zero, is then deflated to by this refined zero. If is of a degree greater than 4, the above procedures are then repeated again for ; otherwise, the closed-form formulas are used to determine all the zeros of , which are also fed to (56) for obtaining the refined zeros. Finally, the corresponding LSP frequencies are given by . 2) Example 2: Example 1 taken in Section III-A is continued. The is assumed here to be 0.001. Now consider the polynomial represented in (37). The results of using the modified Newton–Raphson method are summarized in Table I. We begin with 2.0 by the criterion C2). After successively applying (56) two times, the largest zero is obtained at 1.9778, which corresponds to the LSP frequency at 0.1490. Next, is yielded by deflating at 1.9778. By criterion C3), the initial value for is calculated as follows: (63) and along with the Fig. 2 depicts the curves of initial guess . After three iterations, a zero is obtained at 1.6289. However, this zero must be refined since is a deflated polynomial. After this, the second zero, 1.6289, and the LSP frequency, 0.6190, are obtained. Similarly, is obtained by deflating at . Next, the closed-form formulas are adopted to approximately solve the four zeros of , followed by using the modified Newton–Raphson method to refine them. Those results are displayed in Table II. Similar processing is applied to and those results are listed in Table III and Table IV. Furand for Example 1 and thermore, the zeros of their corresponding LSP frequencies found here are plotted in Fig. 3, where the amplitude indicates the LSP frequency.

WU AND CHEN: COMPUTATION OF THE LSP FREQUENCIES

EXAMPLE

IV. EXPERIMENTS

AND

OF THE

113

TABLE III MODIFIED NEWTON–RAPHSON METHOD

FOR

Q

Q

F6 (x) AND F5 (x)

RESULTS RESULTS

OF THE

TABLE IV CLOSED-FORM FORMULAS

FOR

Q

F4 (x)

A. Experimental Environment Speech data of 40 speakers (29 male and 11 female) were used to assess the performance of the two-level method. At least 225 utterances consisting of isolated and continuous syllables which cover all of the phonemes of Mandarin speech were recorded for each speaker. The speech signals were digitized by a 12-b A/D converter at a 10-kHz sampling rate. A 25.6-ms Hamming window was performed for computing an th-order LPC analysis using the autocorrelation method and the window was shifted every 12.8 ms. Next, the first autocorrelation coefficients were used to derive the thorder LSP frequencies. Finally, an average amount of 6695 speech frames were obtained for each speaker. Besides this, our programs are implemented on a personal computer (486 DX-33) using C language. B. Performance of the Two-Level Method We first investigate the maximum values of the fine grid for and with respect to three values of the LSP order 8, 10, and 12, namely, 4, n = 5, and 6, respectively. Table V indicates that the larger value of implies the smaller value of . Furthermore, the zeros of are more separated than those of . However, how a large value of can be chosen if the other value of is given remains unknown. On the other hand, the large variability of speech signals causes us to select a smaller size of grid, which will require much more computation. Note that the reasonable value of is investigated for the methods employed here except for the two-level method and the Newton–Raphson method. Next, the computation time of the computer is compared for the two-level method and a few conventional methods. As for the methods mentioned in the introduction, only the method of Kabal and Ramachandran is implemented in our experiments, since it is the most efficient method. Besides this, the performance of the DID algorithm is evaluated when combined with

TABLE V

MAXIMUM VALUES

OF

 FOR F P (x) AND F Q (x), RESPECTIVELY

the other methods [16], e.g., the Newton–Raphson method, the secant method, and the bisection method. Two requirements and , are given to of accuracy , where estimate the LSP frequencies for the 40 speakers’ speech data. The average computation time for each speaker is summarized in Table VI and Table VII corresponding to 8 and 12 (i.e., 4 and ) 6, respectively. These tables indicate that the two-level, which adopts the closed-form formulas as 4 and employs the modified Newton–Raphson method as 5, is the fastest one among them. This result also confirms that the DID algorithm can save much more computation time against the transformation time spent by itself. Besides this, the bisection method and the method of Kabal and Ramachandran inevitably take more computation time under high accuracy since they converge linearly. Finally, the stability and the convergence of the two-level method are investigated numerically for even with 10. Because the computation time of the closed-form formulas is given, the modified Newton–Raphson is constant as method is under consideration in the following experiments. The maximum and minimum numbers of iterations, including

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TABLE VII AVERAGE COMPUTATION TIME (S) OF THE LSP FREQUENCIES FOR EACH SPEAKER GIVEN m = 12

TABLE VIII MINIMUM NUMBERS OF ITERATIONS REQUIRED FOR THE MODIFIED NEWTON–RAPHSON METHOD TO ESTIMATE EACH ZERO OF Fn (x) UNDER  = 1006 MAXIMUM

(a)

(b) Fig. 5. Maximum errors between the true roots and the approximate roots before and after refinement as a function of the LSP order under (a)  1003 and (b)  = 1006

=

TABLE VI AVERAGE COMPUTATION TIME (S) OF THE LSP FREQUENCIES FOR EACH SPEAKER GIVEN m = 8

AND

estimated around four iterations as 10 12 or five iterations as 14 20. Therefore, the two-level method is guaranteed to converge rapidly and perform stably. On the other hand, in order to check if the modified Newton–Raphson method will fail to converge or converge to a root that is not closest to the initial guess using higher order iterative deflated equation, the maximum errors between the true roots and the approximate roots before and after refinement were investigated, respectively. Fig. 5 shows the maximum errors before and after refinement as a function of the LSP order under the tolerances of and . It is not surprising that the errors of roots caused by iterative deflation are monotone increasing with respect to the LSP order. However, both the approximate roots before and after refinement will converge to the true roots closest to the initial guess and the maximum errors are always smaller than the defined tolerance for the LSP order under 20. Generally, the LSP orders which are most frequently of interest for finding the LSP frequencies in speech processing are under 16. Accordingly, the convergence problem in the proposed method is insignificant for the LSP order under 20. V. CONCLUSIONS

those iterations for refinement, required for this method to estimate each zero of are displayed in Table VIII. It can be seen that at most eight iterations were required as 20 while the minimum number of iterations is one. Besides this, the maximum number of iterations increases slowly with respect to . Additionally, Fig. 4 depicts the histograms of the number of iterations for each corresponding to Table VIII. Most of the zeros are observed to be estimated at one iteration, which occurred at the process of refinement after using the closed-form formulas, while the others are

A two-level method was proposed in this study for rapidly and accurately estimating the LSP frequencies. In the first level, a novel DID algorithm was successfully demonstrated for the transformation of symmetric or antisymmetric polynomials. It only requires about O multiplications and O additions. Besides this, it obviates prior storage or the large calculation of transcendental functions and complex computation and, of importance, provides more selections of rapid numerical methods. Next, the reduced polynomials were fed into the closed-form formulas and the modified

WU AND CHEN: COMPUTATION OF THE LSP FREQUENCIES

Newton–Raphson method in the second level. Experimental results indicated that the two-level method is appreciated more because it is not involved with the decision on the size of fine grid. Besides this, our criteria for obtaining better initial values could not only save the computation time but could also be conductive to numerical robustness. Furthermore, the twolevel method was demonstrated to be the most suitable one for estimating the LSP frequencies with a high level of accuracy.

REFERENCES [1] F. Itakura, “Line spectrum representation of linear predictive coefficients of speech signals,” J. Acoust. Soc. Amer., vol. 57, 1975. [2] F. K. Soong and B. H. Juang, “Line spectrum pair (LSP) and speech data compression,” in Proc. ICASSP-84, pp. 1.10.1–1.10.4. [3] G. S. Kang and L. J. Fransen, “Low bit rate speech encoders based on line spectrum frequencies (LSF’s),” Naval Res. Lab., Washington, DC, Rep. 8857, Nov. 1984. , “Application of Line-Spectrum Pairs to low-bit rate speech [4] encoders,” in Proc. ICASSP-85, pp. 244–247. [5] J. R. Crosmer and T. P. Barnwell, III, “A low bit rate segment vocoder base on line spectrum pairs,” in Proc. ICASSP-85, pp. 7.2.1–7.2.4. [6] F. K. Soong and B. H. Juang, “Optimal quantization of LSP parameters,” IEEE Trans. Speech Audio Processing, vol. 1, pp. 15-24, Jan. 1993. [7] K. K. Paliwal, “A study of line spectrum pair frequencies for speech recognition,” in Proc. ICASSP-88, pp. 485–488. [8] C. S. Liu, M. T. Lin, W. J. Wang, and H. C. Wang, “Study of line spectrum pair frequencies for speaker recognition,” in Proc. ICASSP-90, pp. 277–280. [9] Z. X. Yuan, C. Z. Yu, and Y. Fang, “Text independent speaker identification using fuzzy mathematical algorithm,” in Proc. ICASSP, 1993, pp. II.403–II.406, [10] Y. Bistritz, “Zero location with respect to the unit circle of discrete-time linear system polynomials,” Proc. IEEE, vol. 72, pp. 1131–1142, Sept. 1984. [11] P. Stoica and A. Nehorai, “The poles of symmetric linear Prediction models lie on the unit circle,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1344–1346, Oct. 1986. [12] P. Kabal and R. P. Ramachandran, “The computation of line spectral frequencies using Chebyshev polynomials,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1419–1426, Dec. 1986.

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[13] N. Sugamura and N. Farvardin, “Quantizer design in LSP speech analysis-synthesis,” IEEE Select. Areas Commun., vol. 6, pp. 432–440, 1988. [14] C. E. Schmidt and L. R. Rabiner, “A study of techniques for finding the zeros of linear phase FIR digital filters,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 96–98, Feb. 1977. [15] R. W. Hamming, Introduction to Applied Numerical Analysis. New York: McGraw-Hill, 1971, ch. 2, pp. 36–51. [16] C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis. Reading, MA: Addison-Wesley, 1994, pp. 33–72 and p. 702. [17] P. Henrici, Elements of Numerical Analysis. New York: Wiley, 1964, pp. 51–52 and pp. 84–85. [18] L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals. Englewood Cliffs, NJ: Prentice-Hall, 1978, p. 399. [19] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1968, p. 17 and pp. 23–24.

Chung-Hsein Wu (M’88) received the B.S. degree in electronic engineering from National Chiao Tung University, Hsinchu, Taiwan, in 1981, and the M.S. and Ph.D. degrees in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 1987 and 1991, respective. He is currently Associate Professor with the Institute of Information Engineering at National Cheng Kung University, Tainan, Taiwan. His research interests include speech recognition, text-to-speech, natural language, and neural networks.

Jau-Hung Chen received the B.S. degree in mathemtics from National Taiwan Normal University, Taipei, in 1991, and the M.S. degree in information engineering from National Cheng Kung University, Tainan, Taiwan, in 1993. He is currently a doctoral student in the Institute of Information Engineering. His research interests include digital signal processing, speaker recognition, and text-to-speech.