Keywords: Vocal Tract; Lattice model; Reflection coefficients; Linear Prediction; Levinson algorithm.

Transcription

1 Volume 3, Issue 6, June 213 ISSN: X International Journal of Advanced Research in Comuter Science and Software Engineering Research Paer Available online at: Lattice Filter Model of Human Vocal Tract Neha Garg 1, Rakesh Garg 2 ECE Deartment, KITM, Kurukeshtra, India Abstract- This aer describes the generalized lattice model of human vocal tract for seech analysis, relating it to the all ole tye linear rediction. A more natural synthesized seech can be achieved using Auto Regressive (AR) lattice model. Seech analysis is done using linear rediction. The structure of AR lattice model is numerically stable, which is main requirement in seech analysis and synthesis alications. Lattice model uses reflection coefficients; relation of these reflection coefficients with LPC redicted coefficients is given by Levinson Durbin algorithm. Simulation results shows that roots of the olynomial in the denominator of transfer function of lattice model is always guarantee to be inside the unit circle and the stability of the system is achieved. Keywords: Vocal Tract; Lattice model; Reflection coefficients; Linear Prediction; Levinson algorithm. I. INTRODUCTION The vocal tract is the cavity in human beings where sound is roduced at the sound source and filtered. Human seech is roduced in the vocal tract which can be aroximated as a variable diameter tube [1]. The seech mechanism can be modelled as a time-varying filter which acts as the vocal tract excited by an oscillator as the vocal folds, with different oututs as shown in figure 1. where G is the gain, x(n) the inut signal, a(n) the amlified signal and s(n) the outut signal; n is a digitized time variable. When voiced sound is roduced, the filter is excited by an imulse sequence. When unvoiced sound is roduced, the filter is excited by random white noise [2]. Figure 1- Block diagram of the source-filter model of seech The linear redictive coding (LPC) model is based on a mathematical aroximation of the vocal tract. Vocal tract and glottal excitation can be reresented by a time-varying digital filter [3] whose steady state system function is of the form as: H (Z) = S(Z) = G (1) U (Z) 1+ a k =1 k Z k Generally the all-ole model is referred for most alications because it is comutationally more efficient. Moreover, it is easy to solve an all-ole model [4]. The basic roblem of linear rediction here is to determine a set of redictor coefficients directly from seech signal in such a manner as to obtain a good estimate of sectral roerties of seech signal..in section 2 we have described the basic rincile of linear rediction and LPC analysis stes with brief introduction. LPC redicted coefficients comutation is described using Levinson Durbin algorithm in section 3. Lattice imlementation is introduced in section 4. At last simulation results is shown in section 5 and at the end conclusion in section 6 and references. II. LINEAR PREDICTION BASIC PRINCIPLE Linear redictive coding is defined as a digital method for encoding an analog signal in which a articular value is redicted by a linear function of the ast values of the signal [5]. The seech samles s(n) can be given by using simle difference equation (2). Where a k is the redicted LPC coefficients and G is the gain and u(n) is unknown inut signal. 213, IJARCSSE All Rights Reserved Page 676

2 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, s(n) = a k s(n k) +G u(n) (2) The signal s (n) can be redicted only aroximately from a linearly weighted summation of ast samles as in equation (3). s (n) = - a k s(n k) (3) Linear Prediction (LP) residual is the rediction error e(n) obtained as the difference between the redicted seech samle s (n) and the current samle s(n). This is shown in equation (4). e(n) = s(n) s (n) (4) e(n) = s(n) + a k s(n k) (5) In the frequency domain, the equation (5) can be reresented as, E (Z) = S (Z) + a k S(Z) Z k (6) By this we can obtained: A (Z) = E(Z) = 1+ a S(Z) k Z k (7) Now by comaring the equation (1) and equation (7) by taking gain (G) equals to one, we can obtained LP residual by filtering the seech signal with A (z) as indicated in figure 2. Figure 2: Comuting the LP residual by inverse filtering As A (z) is the recirocal of H(z), LP residual is obtained by the inverse filtering of seech [14]. A. LPC Analysis Stes Figure 3 shows a block diagram of the main comutational stes required for the LPC front-end rocessor, Figure 3: LPC Coefficients Extraction Process It is combines activities like, re-emhasis, act as a low-order digital filter, increases the ower in the uer frequency bands of the seech signal [8]. The FIR filter is similar to a high ass filter.frame blocking ste blocks the reemhasized seech signal into frames. In order to minimize the signal discontinuities at the beginning and end of each frame, windowing ste is alied. Autocorrelation analysis identifies the zeroth autocorrelation which is the energy of each frame which is very imortant arameter for seech detection systems [12]. After auto correlation difference equations are solved using Levinson Durbin algorithm and LPC redicted coefficients a k and reflection coefficients k i are obtained. As residual rediction error is reresented by equation (5) then the method of least squares is alied to obtained coefficients a k [7]. 213, IJARCSSE All Rights Reserved Page 677

3 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, Where E Denote the total squared error by: 2 E = n e n = n ( s n + a k s n k ) 2 (8) E is minimized by setting: E =, 1 k (9) a k The set of equations in unknowns can be solved for the redictor coefficients which minimize E.The minimum total squared error, denoted by E is obtained by: E = s 2 n n + a k n ( s n s n k ) (1) Now auto correlation analysis is alied, here we assume that the error in E is minimized over the infinite duration. a k R (i-k) = - R (i), 1 i (11) E = R () + a k R(k) (12) Where R (i) is the autocorrelation function of the signal s (n): R (i) = n= s n s n+i (13) Note that R (i) is an even function of i :- R (-i) = R (i) (14) III. COMPUTATION OF PREDICTED COEFFICIENTS The redicted coefficients can be comuted by solving a set of equations with unknowns. Note that the X auto correlation matrix is symmetric and the elements along any diagonal are identical (i.e., a Toelitz matrix). This simlification causes the redictor coefficients to be comuted efficiently using the Levinson-Durbin recursion as [1: E () = R () (15) For 1 i and 1 j i-1 i 1 (i 1) c i =[R (i) j=1 R i j a j ] (16) c k i = i E (i 1) (17) a i i = k i (18) i a j =a (i 1) (i 1) j - k i a (j 1) (19) E (i) = (1- k i 2 ). E (i 1) (2) Equations (16) to (2) are solved recursively for i = 1, 2,..., and finally the redictor coefficients a j i and reflection coefficients k i are calculated. IV. LATTICE IMPLEMENTATION OF LPC FILTERS Any LPC analysis filter can be imlemented as a lattice filter. The relationshi between a k and k i is given by the Levinson-Durbin recursion. The first order rediction coefficient a11 is the same as the reflection coefficient k1. The mth order linear rediction coefficients are obtained from the (m 1)th order rediction coefficients and the reflection 213, IJARCSSE All Rights Reserved Page 678

4 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, coefficients k m [11]. Thus, M linear rediction coefficients are equivalent to M reflection coefficients. Quantization of reflection coefficients is easier because of the well-defined range of values that they take on. Note that the absolute value of reflection coefficients is never greater than one [9].This is the reason why reflection coefficients instead of linear rediction coefficients are often used to reresent a vocal tract filter. Figure 4 shows the lattice imlementation of LPC analysis filter. Figure 4: Lattice Imlementation of LPC Analysis Filter From each frame of seech samles, M reflection coefficients are comuted. Because imortant information about the vocal tract model is extracted in the form of reflection coefficients, the outut of the LPC analysis filter using reflection coefficients will have less redundancy than the original seech. Thus, less number of bits is required to quantize this socalled residual error. This quantized residual error along with the quantized reflection coefficients are transmitted or stored. To lay back, a lattice imlementation of the LPC synthesis filter is required. In this case, the inut is the residual error and the outut is the reconstructed seech. V. SIMULATION SET UP AND RESULTS For analysis of seech signal we have used MATLAB software. Digital seech rocessing tool box is used. Here we have taken a seech signal of duration 5 seconds. We have erformed four stes. First is the calculation of residue signal, second is the conversion of redicted LPC coefficients in to reflection coefficients,third ste is transfer function reresentation of vocal tract filter and fourth is the ole-zero lot. Technical Details: TABLE 1 TECHNICAL DETAILS OF SPEECH SIGNAL Start time: seconds NO. of samles: 485 End time:.5165 seconds Samling eriod:.125 sec Total Duration:.5165 Samling frequency: 8 seconds Hz Prediction order: 1 Number of frames: 17 Figure 5 shows the seech signal waveform and.figure 6 shows the outut waveforms LPC analysis of 1st frame duration [ 1 : 24 and table 2 shows the reflection coefficients for 17 frames. At last average is taken and transfer function is obtained. Figure 7 shows the ole zero lot of transfer function. Figure 5: Seech Signal Waveform 213, IJARCSSE All Rights Reserved Page 679

5 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, seech signal [1 : Aftr re-emhasis Aftr windowing Auto Correlation Residue signal Figure 6: LPC Analysis waveform [1: 24 TABLE 2 LATTICE MODEL REFLECTION COEFFICIENTS Frame [1: Frame Frame Frame [72 : Frame [12 24 [24:48 [48:72 96 Frame[96:12 : Frame[144:168 Frame[168:192 Frame[192:216 Frame[216:24 Frame[24:264 Frame[264: Frame[288:312 Frame[312:336 Frame[336:36 Frame[36:384 Frame[384:48 Average[1: , IJARCSSE All Rights Reserved Page 68

6 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, Figure 7: Pole zero lot VI. CONCLUSION In this study, we have used the AR lattice model to model vocal tract for glottal source excited seech analysis. Using LPC analysis we calculated the residue signal which is the comressed seech signal and can be reresented by fewer bits for transmission uroses. Levinson Durbin algorithm comutes the redicted coefficients and reflection coefficients efficiently. Using the AR lattice model, it can be observed from ole zero lot that all oles lies inside the unit circle, which is the essential condition for the system to be stable. Lattice filters are used rather than direct-form filters since the lattice filter coefficients have magnitude less than one and, conveniently, are available directly as a result of the Levinson- Durbin algorithm. REFERENCES [1]. G. Fant, Acoustic Theory of Seech Production. The Hague: Mouton & Co.,196; Walter de Gruyter, 197. [2]. B. S. Atal and M. R. Schroeder, Linear rediction analysis of seech based on a ole-zero reresentation, J. Acoust. Soc. Am., vol. 64, no. 5, , [3]. E.M. Hofstetter, An Introduction to the Mathematics of Linear Predictive Filtering as Alied to Seech Analysis and Synthesis, Technical Note , MIT Lincoln Labs, July [4]. J.D. Markel and A.H. Gray Jr., A Linear Prediction Vocoder Simulation Based Uon the Autocorrelation Method, IEEE Transactions on Acoustics, Seech, and Signal Processing, vol. ASSP-22, no. 2, , Aril [5]. J. Makhoul. Linear rediction: A tutorial review, Proc. IEEE, vol 63, no. 4, , Ar [6]. J.D. Markel and A.H. Gray Jr., Linear Prediction of Seech, Sringer-Verlag, Berlin Heidelberg, New York, USA, [7]. L.R. Rabiner, B.S. Atal and M.R. Sambur, LPC Prediction Error Analysis of Its Variation with the Position of the Analysis Frame, IEEE Transactions on Acoustics, Seech, and Signal Processing, vol. ASSP-25, no. 5, , October [8]. P. Thevenaz, and H. Hugli. Usefulness of the LPC-residue in text-indeendent seaker verification, Seech Commun., vol. 17, , Aug [9]. Kari Roth And Ismo Kauinen Frequency wared burg s method for AR-modeling 23 IEEE Worksho on Alications of Signal Processing to Audio and Acoustics. [1.Ziad A. Al Bawab, An Analysis-by- Synthesis Aroach to Vocal Tract Modeling for Robust Seech Recognition, Ph.D, December 17, 29. [11]. loan Tabus Interleaved Quantization-Otimization and Predictor Structure Selection for Lossless Comression of Audio Comanded Signals Proceedings of the 4th International Symosium on Communications, Control and Signal Processing, ISCCSP , IJARCSSE All Rights Reserved Page 681

7 Garg et al., International Journal of Advanced Research in Comuter Science and Software Engineering 3(6), June - 213, [12]. Thiang and Suryo Wijoyo Seech Recognition Using Linear Predictive Coding and Artificial Neural Network for Controlling Movement of Mobile Robot 211 International Conference on Information and Electronics Engineering IPCSIT vol.6 (211) IACSIT Press, Singaore. [13]. Chian-Hong Wong 211 Wared Linear Predictive Seech Coding IEEE International Conference on Signal and Image Processing Alications (ICSIPA211). [14]. Christoh R. Norrenbrock1 On the Use of Vocal-Tract Aroximations for Instrumental Quality Assessment ITG-Fachbericht 236: Srachkommunikation Setember 28, , IJARCSSE All Rights Reserved Page 682