and continuous-time linear systems in the complex ... - Semantic Scholar

3 downloads 0 Views 469KB Size Report
The stability of discrete-time linear systems has been studied in many ways [1]-[8], one of ..... [2] J. D. Markel and A. H. Gray, Linear Prediction of Speech. New ... of excitation, namely, voiced, unvoiced, and mixed (a combination of voiced and ...
1768

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. 37, NO. 11, NOVEMBER 1989

around the respectable values of for ARNORM and for AORTHO. Realizing that a difference of ten orders of magnitude for AORTHO between RITE and IMSL appears large, we also evaluated a measure of angle difference between IMSL and RITE eigenvectors for corresponding eigenvalues. This measure is defined as

Fig. 3 represents values obtained from the same ten trial runs. The actual eigenvector directions, obtained with the two different techniques, are thus very close. As an indication of numerical stability, note in Figs. 1 and 2 that the RITE algorithm, at higher dimensions, actually recovers from outlying errors at previous dimensions. The test sequences were computed from two incoming sources in white noise, thereby generating matrices with two large and distinct eigenvalues, while the others were small and clustered (especially for matrices of high dimension). If the algorithm is sensitive to the multiple eigenvalue situation, it is certainly being tested here. From these results, we see that error propagation does not seem to cause a marked degradation of the proposed algorithm for orders up to 50. IV. CONCLUSIONS The RITE algorithm finds recursively in order the eigendecomposition for a Hermitian Toeplitz matrix. This algorithm is especially well adapted for parallel implementation as it solves at each order a number of independent, structurally identical problems. Convergence of the algorithm is guaranteed, and very accurate results are obtained with fewer than 15 iterations of a restricted Newton algorithm for finding eigenvalues for the order increased matrix. Eigenvectors associated with distinct eigenvalues can be found efficiently using a fast Toeplitz solver. In the multiple minimum eigenvalue case, we use eigeninformation computed at the previous rank to instantaneously identify all but one of the clustered eigenvectors. The accumulation of errors in the RITE algorithm does not lead to a quick deterioration of good algorithm performance with increasing order. The performance is good and is shown to be well behaved for orders up to at least 50.

REFERENCES [ I ] R. Schmidt, “Multiple emitter location and signal parameter estimation,” presented at the RADC Spectrum Est. Workshop, 1979. 121 G. Bienvenu and L. Kopp, “Adaptivity to background noise spatial coherence for high resolution bearing estimation,’’ presented at ICASSP, Denver, CO, 1980. [3] Y. H. Hu and S.-Y. Kung, “Toeplitz eigensystem solver,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 12641271, Oct. 1985. [4] D. M. Wilkes and H. Hayes, “An eigenvalue recursion for Toeplitz matrices,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 907-909, June 1987. [5] G. Cybenko and C.F. Van Loan. “Computing the minimum eigenvalue of a symmetric positive definite Toeplitz matrix,” SIAM J . Sci. Statist. Comput. vol. 7 , pp. 123-131, Jan. 1986. [6] A. A. Beex, “Fast recursiveiiterative Toeplitz eigenspace decomposition,” in Proc. European Signal Processing Con$ (EUSIPCO), The Hague, The Netherlands, Sept. 1986, pp. 1001-1004. 171 F. Gianella and C . Gueguen, “Extraction des vecteurs propres de matrices de Toeplitz,” GRETSI, Nice, France, June 1981. [81 G . H . Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins University Press, 1983. [9l S . Zohar, “Fortran subroutines for the solution of Toeplitz sets of linear equations,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-27, pp. 656-658, Dec. 1979. [IO] - “The solution of a Toeplitz set of linear equations,” ACM J . , vol. 21, pp. 272-276, Apr. 1974. 11 11 P. Delsarte and Y . Genin, “Spectral properties of finite Toeplitz matrices,’’ in Proc. 1983 Inr. Symp. Math. Theory of Networks and Syst. Beer Sheva, Israel, pp. 194-213.

Wide Sense Stability of Discrete- and ContinuousTime Linear Systems in the Complex Case M . BENIDIR

AND

B. PICINBONO

Abstract-Canonical reflection coefficients are used to determine the number of roots appearing on the unit circle and their multiplicities for a stable polynomial (with no zero outside the unit circle). A criterion so that a stable polynomial has no multiple zeros on the unit circle is then deduced and transposed to the continuous-time linear systems in the complex case.

I. INTRODUCTION The stability of discrete-time linear systems has been studied in many ways [1]-[8], one of the simplest of which is to compute from a polynomial P associated to the system, assumed of degree n , the so-called reflection coefficients k, and to check that I k, 1 < 1 for 1 5 j 5 n [l], [2]. The more recent method of [3], of which many readers are now aware, is a method for testing polynomials stable in the strict sense (all zeros are located within the UC). In many problems, such as spectral analysis, it is interesting to test polynomials having no zero outside the unit circle (UC), which means that some zeros can be located on this circle. A criterion for testing this stability in the wide sense is given in [4] by using the continuedfraction expansion for the real case, and in [5] by using the canonical rejection coeficients for the complex case. The criterion for testing strict sense stability, as opposed to wide sense stability, requires that all zeros be located within the UC. Tests for wide sense stability are more complicated, and essentially deal with the singularities appearing in the stability tests. The test proposed in [3] for the real case and extended to the complex case in [6], and those established in [4] and [5], are analyzed and compared in ~71. This correspondence deals with counting the zeros appearing on the UC and their multiplicities for polynomials stable in the wide sense. The properties established in the z domain are transposed to the s domain, using the bilinear transform. In particular, we obtain a criterion for testing polynomials having neither multiple zeros on the imaginary axis nor zeros with positive real parts.

11. WIDE SENSESTABILITY OF DISCRETE-TIME SYSTEMS To any polynomial P,(z) = ciz’

+ c:z’-’ + . . . + c;,

(2.1)

of degree j with complex coefficients c:, we associate its reciprocal polynomial given by P,(z) = 7’0

+ ?,z’ + . . . + c y

=

z’F,(l/z)

(2.2) where C denotes the complex conjugate of c. If P, satisfies P, ( z ) + a p , ( z ) = 0 where a is a constant, it is referred to as a self-reciprocal (s.r.) polynomial and we have 1 a I = 1 . Denoting by P’ the derivative of P , we can associate to any stable polynomial P of degree n the sequence of polynomials P,, P n - ’ , . . . , P , [5] constructed as follows.

Algorithm A Initial condition: P , = P ; f o r j = n , n - 1,

...

Manuscript received July 28, 1988; revised February 11, 1989. The authors are with the Laboratoire des Signaux et S y s t e m s , ESE, 91 192, Gif-sur-Yvette-Cedex, France. IEEE Log Number 8930542.

0096-3518/89/1100-1768$01.00 0 1989 IEEE

Authorized licensed use limited to: Bernard Picinbono. Downloaded on December 15, 2008 at 09:13 from IEEE Xplore. Restrictions apply.

1769

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. 37, NO. 11, NOVEMBER 1989

=R

P j - , = Pj'

if

1 ~ #~ 11

if R = 0.

(2.4a) (2.4b)

Algorithm A ends if 1 K~ I = 1 and R # 0. For a large class of polynomials P , Algorithm A allows us to associate to a given P a unique vector, referred to as a canonical rejection vector,

Proof: 1) In this case, Pi-, satisfies ZPj-l(Z)

=

Pl(Z)

+ KjFj(Z).

(2.9)

If w is a root(q) of P j , we have

P,(Z) =

(Z

-

h = j - q , Ah(W) # 0.

W)'&(Z),

(2.10)

As 1 w I = 1, definition (2.2) implies

Comment I : The examination of the lowest and highest degree terms in (2.3) shows that polynomial R is of degree j - 1 iff I K~ 1 # 1. In addition, as Pj is of degree j , we have pj( 0) # 0 and the computation of K~ is possible. The computational complexity of Algorithm A and other algorithms for computing vector K, is analyzed in [8]. For the rest of this correspondence,- we need the two following properties established in [5] where P is denoted by P * . Property 1 (51: If polynomial Pi-, is deduced from Pj via Algorithm A, we have CXjPj(Z) = Zpj-l(Z) -

(2.6)

KjFj-,(Z)

with aj = 1 - 1 K~ l2 in case (2.4a) and cyj = j in case (2.4b). Property 2 (Generalized Stability Criterion) [5]: Polynomial P has no zero outside the UC iff Algorithm A allows us to compute from P vector K, with n components satisfying I K~ I 5 1 for 1 5 j

n. In this section, we give precise details on the generalized criterion above; in particular, the property below concerns both the zeros appearing on the UC and their multiplicities. Property 3: Let P be a polynomial with no zero outside the UC and K , , . . . , K , the sequence of the canonical rejection coeffcients computed from P via Algorithm A. Then, either 1 K~ I < 1 for 1 5 j 5 n and then P has no zero on the UC O r 1 Kj 1 = 1 for the increasing indexes j , < j 2 < . . . < j , , and then 1) P has j , zeros on the UC (including multiplicities), 2) the maximal multiplicity of the zeros on the UC is M , 3) there are j , different zeros with this multiplicity. Example 1: The canonical reflection vector computed via Algorithm A from polynomial 5

p j ( z ) = ( 1 - wz)'Ah(z) =

(-w)'(z

- w)'Ah(z).

(2.11)

Thus, relation (2.9) becomes

ZPj-I(Z)

=

( z - w)'S(z)

(2.12)

-K)'Ah.

(2.13)

where S = A,,

+

Kj(

As (Ah(X)I = IA),(x)I for 1x1 = 1, the assumption S ( w ) = 0 gives 1 K~ I = 1 by virtue of (2.13). This contradicts hypothesis I K~ I # 1; we then have S ( w ) # 0 and, according to (2.12), w is a root(q) of P j - , . Conversely, if w is a zero of P i - , , we get from (2.9) (2.14) However, as the considered case yields then implies

J P j ( W ) l=

1 K~ 1

JPj(W)l =

0.

# 1, relation (2.14)

(2.15)

Thus, w is also zero of Pj and, due to the if part, the multiplicity of w is the same for Pi-, and P I . 2) In this caselPj-, = Pj and P j - , satisfies (2.6). If P j - , ( w ) = 0, we also have P j - , ( w ) = 0 and (2.6) yields P j ( w ) = 0. Furthermore, if x is a common zero of a polynomial Q and its derivative Q', then x is root(q) of Q' iff it is a root(q + 1 ) of Q. Combining these results yields the proof. Lemma 2: If Algorithm A allows us to compute, from polynomial P , vector K, with n components satisfying 1 K~ 1 # 1, 1 I j 5 n , then P has no zero on the UC. Proof: According to hypothesis 1 K~ 1 # 1, 1 5 j 5 n, the P ( ~ =) ( z - l f ( z I Q = z 5 + z4 - 2z3 - 2z2 1 construction of K, does not introduce the step (2.4b). Assume P ( w ) = 0 for 1 w I = 1. Due to Lemma 1, w is also a zero of P , - , , (2.7) . . . , P I . But P l ( w ) = a ( w - K ~ and, ) according to hypothesis is 1 K , 1 # 1, w cannot be a zero of P I . This contradicts the assumption, and thus P has no zero on the UC. K5 = [ K I , K 2 , K 3 , K4, K5]' = [ - I , -1/5, 1, 1/3, -11'. (2.8) Proof ofproperty 3: If 1 K] 1 < 1 for 1 s j s n, Property 3 According to Property 2, P has no zero outside the UC. In addition, is a direct consequence of Property 2 and Lemma 2. We then asas ( j , , j 2 ,j 3 ) = (1, 3, S), Property 3 thus implies that P has j 3 = sume that M L 1. 5 zeros on the UC, the maximal multiplicity is M = 3, and there Case I: According to Property 1, polynomial P , is generated are j , = 1 zeros with this multiplicity. from _K, via (2.6). It is easy to see that recursion Q j ( z ) = z Q j - , ( z ) Comment 2: If polynomial P of degree n is computed from a - K] Ql- ,(z ) allows us to generate from K, a polynomial Q, having sequence of reflection coefficients k l , k2, * , k,, it is worth notthe same zeros as from P,. It is shown [5, Proposition 3.51 that ing that Property 3 cannot be applied with these coefficients. In this Q,, denoted P , in [5], has j, zeros on the UC and this gives 1. case, we begin with polynomial P and apply Algorithm A to find Cases 2 and 3: Polynomial P can be constructed from Po = the new and unique sequence of the canonical reflection coefficients Po = 1 and the sequence K , , . . . , K , via relation (2.6), and we for the property to hold. have P = PjlP,-j, where Pjl is generated from K , , . , K~~ and has Example 2: The polynomial (2.7) can be constructed from many only distinct zeros w i , 1 s i 5 j , , located on the UC [5]. Accordsequences of reflection coefficients; for example, this polynomial ing to Lemma 1, w i , 1 s i s j , are root( 1 ) of each polynomial can be generated from ( 1, 1, - 1, - 1, - 1 ). It is clear that this P j l , Pjl+ P,, + 2 , . . . , Ph- I and root(2) of Pj2. Indeed, Pj2-I is sequence corresponds to j , = 1, j , = 5, and M = 5. The maximal deduced from Pj2 via relation (2.4b), and thus wi is root( 1 ) of multiplicity of the zeros on the UC is 3 and Property 3 does not P j 2 - , and root(2) of Pi,; in addition, the construction of the sehold with this sequence. quence P j 2 - 3 , ' ' , Pj, from Pj2- introduces steps of the The proof of Property 3 is given below as a direct consequence kind (2.4a) only, and thus wi is root( 1 ) of polynomials of the following two lemmas. Pj2-l, Pj2-2, . . . , Pll. Iterating this procedure and according to the fact that the computation of P passes M times by a coefficient Lemma I: Let P j - , , j L 2 be deduced from Pj via Algorithm A, and let root ( q ) denote a root with multiplicity q 2 1. Then K~ such that I K~ I = 1, we deduce that w , , 1 5 i 5 j , are root ( M ) 1) in case (2.4a), w E UC is root ( q ) of Pj iff w is root ( q ) of of P = P, and M is the maximal multiplicity of the zeros of P appearing on the UC. Pj- 1 2) in case (2.4b), w E UC is root ( q 1 ) of Pj iff w is root (4) From Property 2 and Property 3, we can easily deduce the folof Pilowing criterion.

+

+ +

-

,

7

,.

+

Authorized licensed use limited to: Bernard Picinbono. Downloaded on December 15, 2008 at 09:13 from IEEE Xplore. Restrictions apply.

1770

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING, VOL. 37, NO. I I , NOVEMBER 1989

Stability Criterion in the z Domain: Polynomial P of degree n has U distinct zeros on the UC and n - U zeros inside this circle iff Algorithm A allows us to associate to P a vector K,, with components satisfying 1 K , 1 = 1 and I K~ I < 1 for; # U . To establish an extension of Property 3, we give below the multiplicities of the zeros of P located on the UC in terms of the indexes of the K ~ ’ Ssatisfying 1 K , I = 1. For this, let p,, 1 5 J 5 M denote the number of roots of P on the UC having j as multiplicity, and consider polynomials PI appearing in the construction of K,,. To obtain the numbers CL,, we evaluate the numbers, including multiplicities, of zeros appearing on the UC for polynomials P,, j = j , , j,, . . . ,J,. According to Property 3, it is clear that PI, has j, distinct zeros on the UC which are the zeros of P having the maximal multiplicity M . Thus, pM = J , . Polynomial PJZ has;, zeros on the UC, among which pM have a multiplicity 2 and p M - , a multiplicity 1 , Thus, p M - , 2pM = j,. Repeating this procedure for the other polynomials, we get the following equations:

Algorithm B Initial condition: Q, = Q; f o r j = n , n - 1, K,

(s QJ-i

..

*

= (-1)’+’Q,(-l)/G,(1);

=

+ l ) U ( s ) = Q,(s) +

U

K,(

-l)’Q,(

(3.6)

-s)

(3.7a)

ifIK,( # 1

+

(3.7b) if U = 0. = jQ, ( 1 - s)Q,’ Algorithm B ends if I K, I = 1 and U # 0. For a large class of polynomials Q, Algorithm B allows us to compute vector K, from the given polynomial Q. As in the bilinear transformation

+

PM PM-l

+ 2Ph4

=.A = J2

...

+ 2p3 + . . . + ( M - l)pM = jM-, p, + 2p2 + 3p3 + + M ~ L ,= j,.

p2

*

(2.16)

# U.

The linear system above can be written in matrix form as

TN = J

the unit circle maps into the imaginary axis, and the inside or outside of this circle maps, respectively, into the left or right halfplane with respect to the imaginary axis, the results of Section I1 can easily be transposed to the z domain. We obtain, in particular, the following stability criterion for the continuous-time linear systems in complex and real cases. Stability Criterion in the s Domain: Polynomial Q of degree n has U distinct zeros on the imaginary axis and n - U zeros with negative real parts iff Algorithm B allows us to associate to Q a vector K, with components satisfying 1 K, 1 = 1 and 1 K, 1 < 1 for j

(2.17)

where

Example 3: Algorithm B allows us to compute vector ciated with polynomial Q(S) = s(s

N = ( P M , P M - ~ ,. . . , PI)^; J = ( j l , j,, . . . ,jM)T (2.18) and T denotes the lower triangular Toeplitz matrix having 1, 2, . . . , M as elements of the first column. Thus,

N

=

T-’J

(2.19)

where T ’is the lower triangular Toeplitz matrix with 1 , -2, 1, 0, * . . , 0 as elements of the first column. 111. WIDE SENSESTABILITY OF CONTINUOUS-TIME SYSTEMS

Let Q ( s ) be a polynomial of degree n with complex coefficients such that Q ( 1 ) # 0. Defining (3.1) we get

+ I)(S

-

i ) = s3

+ (1 - i)s2 - is,

K~

i2

asso-

=

I.

(3.9) We obtain K3 = [ K i , K2, K 3 I T =

[ - 1 / ( 1 -k i ) , -1, 01’.

(3.10)

As I K, I 5 1 f o r j = 1, 2, 3, polynomial Q has no zero with positive real part. In addition, the application of Property 3, transposed to the s domain, shows that Q has j, = 2 zeros on the imaginary axis, the maximal multiplicity is M = 1, and there are jl = 2 zeros with this multiplicity. These results can be directly deduced from the factorization Q(s) =

S(S

+ l ) ( s - i).

(3.11)

Comrnenr 3: Algorithm B can be inverted and give an algorithm which allows us to compute polynomial Q of degree n from a sequence of reflection coefficients k , , k2, . . . , k,. In this case, we have to take into account Comment 2 in the s-domain.

IV. CONCLUSION where 03 ( . ) denotes the bilinear polynomial transformation. As Q( 1 ) # 0, it is easy to see that polynomial P is also of degree n. Algorithm A, given in Section 11, allows us to compute from P the sequence of polynomials P j , j, = n , n - 1 , . . * for which we can associate the new sequence given by Q,(s) =

@[PI],

j = n, n - 1,

... .

(3.3)

The wide sense stability criterion of discrete-time linear systems is discussed, and the number of zeros appearing on the unit circle and their multiplicities for a polynomial with no zero outside the unit circle is determined. A criterion so that a polynomial has neither a zero outside the unit circle nor a multiple zero on this circle is then deduced and transposed to the continuous-time linear systems in the complex case. REFERENCES

After simple calculations, we obtain

@ [ P , ] = (-l)lG,(

-s)

(3.4)

and

@[p,’l= {jQ,+ ( 1 - s)Q,’}/2.

(3.5)

Algorithm A can therefore be transposed to the s domain and gives the following.

[l]M. Marden, “The geometry of the zeros of a polynomial in a complex variable,” Amer. Math. Soc., New York, NY, 1949. [2] J. D. Markel and A . H . Gray, Linear Prediction of Speech. New York: Springer-Verlag, 1976, ch. 5 .

[3] Y. Bistritz, “Zero location with respect to the unit circle of discretetime linear system polynomials,” Proc. IEEE, vol. 72, pp. 1131-1142, Sept. 1984. [4] Y. S . Lai, “ A generalized test for discrete system stability,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 12421243, Dec. 1984.

Authorized licensed use limited to: Bernard Picinbono. Downloaded on December 15, 2008 at 09:13 from IEEE Xplore. Restrictions apply.

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. 31, NO. 11, NOVEMBER 1989

[5] M. Benidir and B. Picinbono, “Extensions of the stability criterion for ARMA filters,” IEEE Trans. Acousr., Speech, Signal Processing, vol. ASSP-35, pp. 425-431, Apr. 1987. [6] Y. Bistritz, “A circular stability test for general polynomials,” Sysr. Conrr. Left., vol. 7, pp. 89-91, 1986. [7] M. Benidir and B. Picinbono, “Comparison between some stability criteria of discrete-time filters,” IEEE Trans. Acousr., Speech, Signal Processing, vol. 36, pp. 993-1001, July 1988. [8] H. Krishna, B . Krishna, and S. D. Morgera, “Efficient procedure for stepup and stepdown computations,” in Proc. ICASSP’88, New York, NY, Apr. 1988, pp. 1651-1654.

Silent and Voiced/Unvoiced/Mixed Excitation (FourWay) Classification of Speech D. G. CHILDERS, M. HAHN,

AND

J. N. LARAR

Abstract-We present an algorithm for automatically classifying speech into four categories: silent and speech produced by three types of excitation, namely, voiced, unvoiced, and mixed (a combination of voiced and unvoiced). The algorithm uses two-channel (speech and electroglottogram) signal analysis and has been tested on data from six speakers (three male and three female), each speaking five sentences. An overall correct classification accuracy of approximately 98.2 percent was achieved when compared to skilled manual classification. This is superior to previously reported automatic classification schemes. It’ word boundary errors, including the beginning and ending of sentences, are excluded, then the algorithm’s performance improves to 99.5 percent.

INTRODUCTION In previous work [ l ] , we described a two-channel, two-way (V/ U-S) algorithm for automatically classifying speech. This algorithm used the speech and electroglottogram (EGG) signals. One of our objectives has been to demonstrate that two-channel-based algorithms can lead to computational and performance improvements over algorithms based on acoustic-signal-only analysis methods. We recognize that in many situations, the EGG signal is either unavailable or cannot be used. However, both the speech and EGG signals can be used in the laboratory to help benchmark the performance of numerous speech systems. We advocate this approach. Previous research has focused on three-way speech classification, i.e., either V/U/S or V/U/M [2]-[13]. The speech classification problem is important because its solution affects other speech analysis, synthesis, and recognition problems. For example, speech classification can help reduce the number of lexical candidates in speech (word) recognition, improve speech synthesis by selecting the proper excitation, and improve the performance of phoneme boundary detection in speech analysis. Consider the large vocabManuscript received March 8, 1988; revised February 13, 1989. This work was supported in part by NSF Grant ECE-8413583, NIH Grants NINCDS R01 NS17078 and NS 27022, the University of Florida Center of Excellence Program in Information Transfer and Processing, the Florida High Technology and Industry Council under Grant 4505208-12, and MindMachine Interaction Research Center. D. G. Childers and M. Hahn are with the Department of Electrical Engineering, University of Florida, Gainesville, FL 3261 1. J. N. Larar was with AT&T Bell Laboratories, Murray Hill, NJ. He is now with the Department of Medicine, University of Miami, Miami, FL 33101. IEEE Log Number 8930546.

1771

ulary isolated word recognition problem. By using only four-way (V/U/M/S) classification and stress analysis, one can define an equivalence class of words having the same representation or “coding” [9], [IO], [14]. For example, the words speed, steep, scout, and stop all belong to the same equivalence class of U/S/ stressed-U/V/U. In an isolated word recognition system, the search to identify a test word among all possible candidates can be reduced by using such a simple coding technique. Following such a reduction of the lexical candidates, one may perform other, more detailed analyses to match the test word with one of the remaining words. Some of the problems with classifying speech as V/U using acoustic-signal-based algorithms are caused by the use of a large analysis frame, a low level of voicing, or even the strength of the first formant. Classification of U/S segments is even more difficult for such algorithms. Typically, researchers have adopted sophisticated approaches to overcome these problems, using additional features, a statistical approach, or an optimized set of parameters 121, ~ 3 1 ,[71, [121, t131.

A SPEECH-EGG-BASED ALGORITHMFOR V/U/M/S CLASSIFICATION A. Algorithm Overview The properties and some applications of the EGG to speech analysis appear in [l], [SI, [15]-[18]. The EGG offers advantages not readily available from a microphone, even a throat contact microphone. The EGG is not susceptible to environment noise, providing instead a direct measure of vocal fold contact [19], while the throat contact microphone provides au acoustic signal similar in form to that provided by other microphones. The EGG amplitude varies both within and across speakers. Baseline variations in the EGG may be removed by differentiating the EGG. For voiced segments, the EGG usually has only two zero crossings per fundamental (pitch) period of voicing. One exception is vocal fry. For unvoiced segments, the electroglottograph output is a very low-level high-frequency noise-like signal generated by the internal electronics of the device that is easily distinguished from the excitation for voiced speech. Thus, V/U-S classification is achieved using a combination of EGG amplitude and level-crossing rate [l], [8]. Mixed excitation detection is accomplished by noting that the EGG signal appears similar to that for voiced sounds, but the speech signal is small in amplitude and has a high levelcrossing rate (see Fig. 2 and other examples in [ 11). Silent intervals are detected by observing that the EGG waveform appears as it does for unvoiced speech and that the speech signal is below a predetermined energy threshold. The two-channel, four-way speech classification algorithm appears in Fig. 1. Note that the algorithm does not use endpoint classification, but this could be added if desired [4], [ l l ] .

B. Algorithm Details Fig. 2 depicts both illustrative results and some difficulties encountered in attempting to evaluate the level-crossing rate (LCR) and energy of the EGG signal. Some fluctuations in the EGG data may be removed by simple differentiation, a procedure which also yields a waveform with enhanced positive and negative peaks. These peaks occur approximately at the instants of glottal opening and closure, respectively [I], [8], [18]. The differentiation is implemented as a backward difference equation. The differentiated EGG is then normalized by dividing by its maximum positive value. The resulting waveform is denoted as the DEGG and is shown at the bottom of Fig. 2. The two-way, V-M and U-S classification uses the LCR and energy information from the DEGG as follows. The DEGG is segmented into 10 ms frames of 100 samples each (10 kHz sampling

0096-3518/89/1100-1771$01.00 0 1989 IEEE

Authorized licensed use limited to: Bernard Picinbono. Downloaded on December 15, 2008 at 09:13 from IEEE Xplore. Restrictions apply.