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Real time vector quantization of LSP parameters
Speech Communication 29 (1999) 39±47
www.elsevier.nl/locate/specom
Real time vector quantization of LSP parameters Balazs K ovesi a, Samir Saoudi b,*, Jean Marc Boucher b, G abor Horv ath c b
a CNET, D ept. CMC, Av. P. Marzin, Lannion Cedex, France ENST-Bretagne, Groupe Signal, D ept. Signal et Communications, B.P. 832, 29285 Brest Cedex, France c TUB, D ept. MMT, M} uegyetem rkp. 9, 1521 Budapest, Hungary
Received 24 June 1998; received in revised form 2 June 1999; accepted 9 June 1999
Abstract The distance measure is of great importance in both the design and coding stage of a vector quantizer. Due to its complexity, however, the spectral distance which best correlates with the perceptual quality is seldom used. On the other hand, various weighted squared Euclidean distance measures give close or even accurate estimation of the meaningful spectral distance. Since they are in general mathematically more tractable, these weighted squared Euclidean distance measures are more commonly used. Signi®cant dierences can be found in the performance of dierent distance measures suggested in previous literatures. In this paper, a complete study and comparison of weighted squared Euclidean distance measures is given. This paper also proposes a new weighted squared Euclidean distance measure for vector quantization of Line Spectrum Pairs (LSP) or Cosine of LSP (CLSP) parameters. It also presents an ecient adaptation apparatus for using the proposed distance measure in the case of split or multi-stage vector quantizers. Ó 1999 Elsevier Science B.V. All rights reserved. Resume la fois pour la conLe choix d'une mesure de distance est tres important pour la quanti®cation vectorielle (QV) a struction du dictionnaire et pour la recherche de la valeur quanti®e. La distance spectrale, qui est la plus signi®cative, est peu souvent utilise a cause de sa complexite calculatoire. La distance Euclidienne quadratique ponderee est mathematiquement plus attractive et par consequent plus souvent utilisee. Selon la distance utilisee, on peut trouver des dierences signi®catives en terme de performances. Dans cet article, une comparaison entre les dierentes distances Euclidienne quadratique ponderee est etudiee. Une mesure de distance est proposee dans le cas de la QV des paires de raies spectrales (LSP) ou Cosinus des LSP (CLSP). Cette mesure permet d'obtenir une tres bonne estimation de la distance spectrale. Une adaptation, dans le cas de la QV eclatee ou multi-etage, est proposee. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Speech coding; Line spectrum pairs; Split vector quantization; Multi-stage vector quantization; Low bit rate; Spectral distance measure; Power spectral density
*
Corresponding author. Tel.: +33-298-00-11-79; e-mail: [email protected]
0167-6393/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 3 9 3 ( 9 9 ) 0 0 0 2 6 - 6
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B. K ovesi et al. / Speech Communication 29 (1999) 39±47
Nomenclature LSP CLSP xi 1i de ; dq ; dep ; dqp ; ds ; wai ; wbi ; wci ; wdi ; wei ; wfi ; wgi ; whi Sx ; nxi n 1i CE Cw
Line Spectrum Pairs Cosine of Line Spectrum Pairs the ith LSP parameter the ith CLSP parameter the Euclidean distance the squared Euclidean distance the weighted squared Euclidean distance the weighted Euclidean distance the spectral distance of two LSP vectors dierent suggested weights of the ith component of an LSP or CLSP vector the power spectral density function using LSP parameters the spectral distance sensitivity of the ith LSP parameter the spectral distance sensitivity of the ith CLSP parameter codebook designed using the squared Euclidean distance without weights codebook designed using the weighted (xgi ) squared Euclidean distance
1. Introduction Line Spectrum Pairs (LSP) or Cosine of LSP (CLSP) provide an ecient representation of the synthesis ®lter used in the Linear Predictive Coding (LPC) of speech (Sugamura and Itakura, 1986; Saoudi, 1990). These parameters are often encoded by a vector quantizer Q completely speci®ed by a codebook C fc1 ; c2 ; . . . ; cN g and a chosen distance measure d ; . Note that the distance measure is of great importance in both the design and coding stages of the vector quantizer.
The aim of speech coding is to obtain a synthesized speech signal perceptually as close as possible to the original. As the human ear is very sensitive to distortions of the spectrum, the spectral distortion is often used for measuring the performance of encoded LSP parameters. This spectral distance has been found to allow an accurate evaluation of the subjective LSP encoding quality (K ovesi, 1997, 1995). In particular, an average spectral distortion of no greater than 1 dB has traditionally been considered as transparent quality (Kleijn and Paliwal, 1995, Chapter. 12). When designing a vector quantizer, an essential problem is to ®nd a representative codevector ci , that minimizes the averaged distortion measure for all source vectors contained within the ith cell Ri . In addition, we need to choose among the quantizer codebook C a codevector ci that is closest to the input source vector according to the distortion measure during the coding stage. Theses problems become relatively complicated when the spectral distance is used as the distortion measure, due to its high computational complexity. To circumvent these problems, the spectral distance is often replaced by a weighted squared distortion measure whereby the codevector ci can be determined as the arithmetic mean of Ri . Its computational time is considerably shorter than that of the spectral distance. Hence, this distortion measure is more suitable for real-time applications. In this paper, a new weighted squared distortion measure for vector quantization of LSP (or CLSP) parameters is suggested.
2. Main distortion measures of interest The spectral distance of two pth order LSP ~y is de®ned as follows: ~x and x vectors x ~x ; x ~y ds x v u 92 u Z 8 u1 p < Sx x; ~ xx = dx; 10 log u t p 0 : ~y ; Sx x; x
dB; 1
B. K ovesi et al. / Speech Communication 29 (1999) 39±47
where 1 ~ Sx x; x 2 Ap;~x e jx
2
is the power spectral density function for the vec~ fx1 ; x2 ; . . . ; xp g. tor x The most known and most used distance is the Euclidean one: s p X de x; y 3 jxi ÿ yi j2 i1
dq x; y
p X
2 jxi ÿ yi j :
4
i1
By using the Euclidean distance or the squared Euclidean distance mentioned above, it is assumed that all components of the vectors have perceptually the same importance. In practice, this is rarely true. For example, in the case of the PARCOR (or the Log Area Ratio) parameters, the importance of a component depends directly on the order of the coecients. The spectrum of the synthesized signal is much more sensitive to the variation of the ®rst coecient than to that of the last one. This is why it is important to introduce a weighting function into the distance expression to re¯ect this phenomenon more precisely. The general expression of the weighted Euclidean distance is the following: q T 5 dep x; y x ÿ y W x ÿ y ; where W is a symmetric and positive de®nite weighting matrix. Let us note that this distance includes the usual squared Euclidean in the special case where W I, the identity matrix. In the case where W is a diagonal matrix with diagonal values wi P 0, the weighted squared Euclidean distance is given by dqp x; y
p X
wi jxi ÿ yi j2 :
previous publications, there are many proposals for weights which are all determined in an empirical way, for example: · In (Svendsen, 1994; Hagen and Hedelin, 1990), the distance is simply given by the usual squared Euclidean distance. · The weights used in (Collura and Tremain, 1993; Paliwal and Atal, 1991; Erzin and Cetin, 1993) are the power spectral densities calculated at the input LSP locations and raised to the power r: wi Sn xi;in r ;
or the squared Euclidean distance given by
6
i 1; . . . ; p:
7
In the case of this distance, the LPC parameters in the formant regions are better quantized than those in the non-formant regions. The value of r was chosen to 0.15 in (Collura and Tremain, 1993), the obtained weights will be referred to as wai . In (Paliwal and Atal, 1991; Erzin and Cetin, 1993) r 0:12 was used and the last two weights were modi®ed to de-emphasize the upper frequency range by multiplying by 0.8 and 0.4, respectively. These weights will be referred to as wbi . · The weights proposed in (Laroia et al., 1991) are based on the following property of LSP parameters: when an LSP parameter is close to one of its neighbours, the speech spectrum has a peak near that frequency. These LSP parameters have high spectral sensitivity and should be given higher weights: wci
1 1 ; xi ÿ xiÿ1 xi1 ÿ xi
i 1; . . . ; p;
8
with x0 0 and xp1 p. This weighting function is called Inverse Harmonic Mean (IHM) and was also used in (Pan and Fischer, 1994). · Following the same principle as above, the weights in (Deketelare et al., 1991) are de®ned as "
i1
There is no simple relationship between the spectral distance and the weighted squared distance. In
41
wdi max
1 xi ÿ xiÿ1
2 ;
1 xi1 ÿ xi
2 # ;
9
42
B. K ovesi et al. / Speech Communication 29 (1999) 39±47
with wd1 and wdp
1 x2 ÿ x1
2
1 xp ÿ xpÿ1
2 :
· In (Bruhn, 1994; Soong and Juang, 1990; Kuo et al., 1992), the used weighting factor wei is the so-called spectral sensitivity with respect to xi , which is de®ned as 2 Z p o log Sx x; x ~ dx: wei 10 oxi 0
Fig. 1. Spectral distance sensitivity of LSP parameters.
· The weights wfi introduced in (Gurgen et al., 1990) are equal to the inverse variance of the ith LPC parameter. Although all of the weighted distances mentioned above were de®ned originally for LSP coecients, they can be applied for CLSP coecients as well. The following section provides a weighting function that gives a good approximation of the spectral distance.
3. A suggested weighted squared Euclidean distance In this section, we suggest a new weighted Euclidean distance which computes the spectral distance more easily and which gives the same perceptual evaluation. In addition, this distance will give a distortion measurement directly, unlike the others which need a computation of this spectral distortion. Here, we focus on studying the relationship between the spectral and Euclidean distances. The following simulations were made: a 10th order LSP (respectively CLSP) vector was taken and one of its components xi (respectively 1i ) was modi®ed by dierent values to x0i (respectively 10i ). Then, the spectral distance of the original and the modi®ed vector was computed. The ten vertical dotted lines in Fig. 1 (respectively Fig. 2) ~ represent the values of the original vector x (respectively ~ 1). The ten solid lines illustrate the results of the ten simulations. In each of them, we
Fig. 2. Spectral distance sensitivity of CLSP parameters.
modi®ed only one of the ten parameters of the vector, while the other nine parameters of the two vectors were unchanged. Let us note x the value of the modi®cation x x0i ÿ xi (respectively x 10i ÿ 1i ), we found that the spectral distance between the original vector and the modi®ed vector is almost in direct proportion with x in both LSP and CLSP cases. So, the spectral distance of these vectors can be estimated by ÿ dbs x1 ; . . . ; xi ; . . . ; xp ; x1 ; . . . ; x0i ; . . . ; xp nxi x0i ÿ xi ; 11
B. K ovesi et al. / Speech Communication 29 (1999) 39±47
ÿ
dbs 11 ; . . . ; 1i ; . . . ; 1p ; 11 ; . . . ; 10i ; . . . ; 1p n1i 10i ÿ 1i ;
12
where nxi (respectively n1i ) denotes the spectral distance sensitivity of the ith LSP (respectively CLSP) parameter. The absolute value of the spectral distance sensitivity of a parameter LSP (or CLSP) was found to be independent of the sign of the modi®cation. The limits of the modi®ed coecients x0i (respectively 10i ) were determined by the well-known stability criteria of LSP and CLSP vectors: 0 < x1 < < xp < p;
13
ÿ2 < 11 < < 1p < 2:
14
Spectral distance sensitivity can be determined by linear regression. As the dependency is quasi-linear and one point of the straight line is known (when x 0), nxi or n1i can be estimated after computing just one more point of the straight line. We applied this second technique which is less complex. To compute the spectral distance sensitivity nxi (respectively n1i ), the value of the ith coecients xi (respectively 1i ), is ®rst perturbed by xi;1 xiÿ1 ÿ xi =2 (respectively xi;1 1iÿ1 ÿ1i =2) then by xi;2 xi1 ÿ xi =2 (respectively xi;2 1i1 ÿ 1i =2) with x0 0 and xp1 p (respectively 10 ÿ2 and 1p1 2). In these two cases, the spectral distances ds;i;1 and ds;i;2 of the original and the perturbed vector were computed. The spectral distance sensitivity nxi was determined as follows: nxi
ds;i;1 =jxi;1 j ds;i;2 =jxi;2 j ; 2
43
sponding squared spectral distance sensitivities, both in the case of LSP and CLSP: dbs x1 ; . . . ; xp ; x01 ; . . . ; x0p s p X 2 16 n2xi x0i ÿ xi ; i1
dbs 11 ; . . . ; 1p ; 101 ; . . . ; 10p s p X 2 n21i 10i ÿ 1i :
17
i1
Using the notation of Section 2, the proposed weights are equal to the squared spectral distance sensitivities: xgi n2xi :
18
Fig. 3 shows the spectral distance of two vectors LSP that have two dierent parameters. In this example, the third parameter of the modi®ed vector was ®xed to x03 x3 0:1. The spectral distance caused by this modi®cation is represented by the star, its estimated value using (16) is marked by the circle. Then, we modi®ed another parameter continuously, while the remaining eight parameters were unchanged. The nine solid lines illustrate the obtained spectral distance in the case of the nine possible simulations while the nine broken lines give the estimated spectral distance using the
i 1; . . . ; p: 15
n1i is obtained in the same way. After computing all spectral distance sensitivix (respectively ties nxi i1;...;p ; of an LSP vector ~ n1i i1;...;p of a CLSP vector ~ 1 ) in this way, we wished to ®nd out how the spectral distance of this ~0 (revector and a completely dierent vector x spectively CLSP ~ 10 ) could be estimated using the sensitivities. We found that the spectral distance could be estimated by the weighted Euclidean distance where the weights are equal to the corre-
Fig. 3. Spectral distance of LSP parameters versus the second modi®ed component.
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B. K ovesi et al. / Speech Communication 29 (1999) 39±47
suggested formula (16). This ®gure shows that (16) gives a good estimation of the spectral distance especially for reasonable values. Fig. 4 shows that a similar application is relevant for CLSP parameters. In Table 1, we give the mean values of the spectral distance sensitivities of 10th order LSP and CLSP parameters estimated on our learning sets. The variation of the spectral distance sensitivities of CLSP parameters is greater than that of the LSP parameters. In the case of CLSP, it is more important to use a weighted distance than the squared Euclidean distance. The weights can be de®ned as well as the squared mean spectral distance sensitivities. They will be referred to as xhi .
4. Experimental results To compare the dierent distance measure, ®rst, a codebook of 1024 codewords (denoted CE ) was designed by the k-means algorithm using a training set of 150 000 vectors. In the design stage of the codebook CE , the squared Euclidean distance was used without weights. We quantized then 35 000 testing vectors separated from the training set. All the individual weights were determined according to the codewords. Table 2 (respectively Table 3) gives the spectral distortion obtained versus the used distance in the case of the vectors LSP (respectively CLSP). Both the distance weighted by (xei ), (xci ) and our suggested distance (with xgi ) obtain a near equivalent performance to that of the meaningful spectral distance. However, while the suggested distance (xgi ) gives the same values as the distance weighted by (xei ) and (xci ), the former is easier to compute. Moreover, our Table 2 Spectral distortion of coding LSP vectors versus the distance measure
Fig. 4. Spectral distance of CLSP parameters versus the second modi®ed component.
Table 1 Mean spectral distance sensitivities of 10th order LSP and CLSP parameters
Distance
Spectral distortion in dB
Squared Euclidean (SE) Weighted SE : xai Weighted SE : xbi Weighted SE : xci Weighted SE : xdi Weighted SE : xei Weighted SE : xfi Weighted SE : xgi Weighted SE : xhi Spectral
2.84 2.80 2.83 2.77 2.96 2.74 2.92 2.74 2.82 2.72
Table 3 Spectral distortion of coding CLSP vectors versus the distance measure
No.
Sensibility LSP
Sensibility CLSP
Distance
Spectral distortion in dB
1 2 3 4 5 6 7 8 9 10
28.80 23.63 15.99 14.31 17.73 15.64 16.43 16.12 16.53 17.27
82.74 43.06 15.92 9.37 9.55 7.97 8.78 9.83 13.77 22.09
Squared Euclidean (SE) Weighted SE : xai Weighted SE : xbi Weighted SE : xci Weighted SE : xdi Weighted SE : xei Weighted SE : xfi Weighted SE : xgi Weighted SE : xhi Spectral
3.06 3.01 3.05 2.90 3.08 2.85 3.05 2.85 2.96 2.81
B. K ovesi et al. / Speech Communication 29 (1999) 39±47 Table 4 Spectral distortion of coding CLSP vectors versus the distance measure and the codebook Codebook and distance
Spectral distortion in dB
CE CE Cw Cw
3.06 2.85 3.15 2.77
and SE and weighted SE : xgi and SE and weighted SE : xgi
suggested distance allows the direct calculation of the distortion measure which is not the case for the classic ones. The weighted squared Euclidean distances can also be used in the learning and the coding stage. The following simulation shows that it is more important to use a correct distance measure in the coding stage. For the design of a codebook of 1024 CLSP vectors (denoted Cw ), we used the weighted squared Euclidean distance with xgi . In Table 4, four dierent cases can be compared. 5. Real time applications In the cases of the xfi weights, the values are constant, they have to be determined once before the coding stage, generally by using a training set. In the other cases, to compute the distance of two vectors, the weights depend on one of the two vectors. In vector quantization, generally, the distance of an input vector and a codeword is computed. Hence, there are three main possibilities to determine the individual weights: · The calculation of the weights can be based on the input vector. This solution was used in each paper mentioned above (Bruhn, 1994; Collura and Tremain, 1993; Deketelare et al., 1991; Erzin and Cetin, 1993; Gurgen et al., 1990; Hagen and Hedelin, 1990; Kuo et al., 1992; Laroia et al., 1991; Paliwal and Atal, 1991; Pan and Fischer, 1994; Soong and Juang, 1990; Svendsen, 1994). Its disadvantage is that the complexity of the coding increases seriously, especially in the case of the xei and xgi weights. Often, these calculations cannot be made in real time applications. It is less complex to calculate the spectral distance directly than to determine these weights for each input vector.
45
· We propose here to determine the weights according to the codewords. As in non-adaptive coding, the codebook is ®xed during the coding stage, the calculations can be done before the quantization and the weights can be stored in memory. This solution increases slightly the memory needed but it is more suitable for real time applications. · To use the constant weights like xfi and xhi , the mean value of the weights estimated on a learning set can be taken. Of course, this solution is less accurate, but the increase of both computational time and memory needed is negligible. To be able to implement a vector quantizer for a real time application, often a sub-optimal solution has to be used like the MSVQ (Juang and Gray, 1982) or the SVQ (Paksoy et al., 1992). In this section, these solutions will be studied from the point of view of the used distance. It will be assumed that the individual weights can not be calculated according to the input vector because of the time constraint of a real time application. They have to be determined according to the codewords of the codebooks and stored in memory before the coding. The problem is that all individual weights (xai ; xbi ; xci ; xdi ; xei ; xgi ) are based on special properties of LSP (or CLSP) vectors and knowledge of the whole vector is needed to calculate them. 5.1. Multi-stage vector quantization In the case of the MSVQ, the codewords of the second and further stages represent the quantization error of the previous stages. These vectors do not have the same properties as the LSP vectors. The individual weights cannot be determined according to them. As the codeword chosen in the ®rst stage is near to the codeword corrected by the further stages, we propose to use the weights of the codeword chosen at the ®rst stage. The approximation is better if the quantization on the ®rst stage is more precise. That is why it is important to assign as many bits as possible to the ®rst stage.
46
B. K ovesi et al. / Speech Communication 29 (1999) 39±47
5.2. Split vector quantization In the case of SVQ, the codebooks contain only a part of the codewords that not allow the computation of individual weights. Our proposition is based on the following properties of LSP and CLSP parameters: · The spectral in¯uence of an LSP (or CLSP) parameters is localized in its region. · The closeness of two LSP parameters marks a spectrally important region. This is the reason we completed the sub-vectors in the codebooks by spectrally neutral ®ctitious subvectors and we determined the weights on this complete vector. For example, in the case of 10th order LSP vectors, if a codeword contains the last four coecients x7 ; . . . ; x10 , we de®ned the six ®ctitious coecients as xi xmin i x7 ÿ xmin =7; i 1; . . . ; 6, with xmin 0. The uniform distribution of ®ctitious LSP parameters assures the spectral neutrality of this part of the vector. In the case of CLSP parameters, we generated the ®ctitious CLSP parameters in the LSP domain to assure spectral neutrality, as the mean LSP vectors are more uniformly distributed. For example, if a codeword contains the ®rst six coecients 11 ; . . . ; 16 , we calculated the remaining four coecients as 1i ÿ2 cos arccos ÿ 16 =2 xmax ÿ arccos ÿ16 =2 ; 19 i ÿ 6 5 i 7; . . . ; 10, with xmax p. Of course, constant mean weights like xfi and xhi can be used in both cases without diculties. But the solutions proposed above give smaller spectral distortion. We constructed 17-bit SVQ's for 10th order CLSP parameters. One codebook contained 1024 codewords of the ®rst six coecients, the other contained 128 codewords of the last four coecients. During the design stage, the mean constant weights xhi were used. Then, in the coding stage, we ®rst used the same constant weights. The obtained spectral distance was 2.04 dB. In a second simulation, the squared
spectral distance sensitivities xgi were computed using the completed codewords. The obtained spectral distance was 1.95 dB. 6. Conclusions In this paper, we studied and compared the weighted squared Euclidean distances that can substitute the meaningful spectral distance. A weighted Euclidean distance was suggested which was found to give a good approximation of the spectral distance. We also showed how individual weights can be determined according to codewords of the codebook in the case of SVQ or MSVQ. By storing the obtained weights, computation is far less complex than that of the spectral distance, which is essential for a real time application. Acknowledgements The authors thank A. Gourves Hayward for her kind assistance with the English of this paper. The authors also thank the anonymous reviewers for their pertinent suggestions which improved the quality of this paper.
References Bruhn, S., 1994. Ecient interblock noiseless coding of speech LPC parameters. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing '94, Adelaide, Australia, April, pp. I.501±I.504. Collura, J.S., Tremain, T.E., 1993. Vector quantizer design for the coding of LSF parameters. In: Proceedings IEEE International Conference on Acoustics, Speech, and Signal Processing '93, Minneapolis, Minnesota, USA, April, pp. II.29±II.32. Deketelare, S., Leich, H., Boite, R., 1991. Application de la quanti®cation vectorielle au codage bas debit d'un signal de parole. In: Proceedings of Treizieme Colloque sur le Traitement du Signal et des Images'91, Juan-les-Pins, France, September, pp. 761±764. Erzin, E., Cetin, A.E., 1993. Interframe dierential vector coding of line spectrum frequencies. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing '93, Minneapolis, Minnesota, USA, April, pp. II.25±II.28.
B. K ovesi et al. / Speech Communication 29 (1999) 39±47 Gurgen, F.S., Sagayama, S., Furui, S., 1990. Line spectrum pair frequency based distance measures for speech recognition. In: Proceedings of International Conference on Spoken Language Processing '90, Kobe, Japan, November, pp. 512±524. Hagen, R., Hedelin, P., 1990. Low bit-rate spectral coding in CELP, a new LSP-method. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing '90, Albuquerque, New Mexico, USA, April, pp. 189±192. Juang, B.H., Gray, A.H., 1982. Multiple stage vector quantization for speech coding. In: Proceedings of IEEE International Conference of Acoustics, Speech, and Signal Processing '82, Paris, France, May, pp. 597±600. Kleijn, W.B., Paliwal, K.K., 1995. Speech Coding and Synthesis. Elsevier, Amsterdam. K ovesi, B., 1997. Quanti®cation vectorielle des paires de raies spectrales pour la compression de la parole a debit reduit. Application a la telephonie numerique sur canal acoustique sous-marin. These de Doctorat. Universite de Rennes I, January. K ovesi, B., Saoudi, S., Boucher, J.M., Reguly, Z., 1995. A fast robust stochastic algorithm for vector quantizer design for nonstationary channels. In: Proceedings of IEEE International Conference of Acoustics, Speech, and Signal Processing '95, Detroit, Michigan, USA, May, pp. 269±273. Kuo, C.C., Jean, F.R., Wang, H.C., 1992. Low bit-rate quantization of LSP parameters using two-dimensional dierential coding. In: Proceedings of IEEE International Conference of Acoustics, Speech, and Signal Processing '92, San Francisco, California, USA, March, pp. I.97±I.100. Laroia, R., Phamdo, N., Farvardin, N., 1991. Robust and ecient quantization of speech LSP parameters using
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structured vector quantizers. In: Proceedings of IEEE International Conference of Acoustics, Speech, and Signal Processing '91, Toronto, Canada, May, pp. 641±644. Paksoy, E., Chan, W.Y., Gersho, A., 1992. Vector quantization of speech LSP parameters with generalized product codes. In: Proceedings of International Conference on Spoken Language Processing '92, Ban, Alberta, Canada, October, pp. 33±36. Paliwal, K.K., Atal, B.S., 1991. Ecient vector quantization for LPC parameters at 24 bits/frame. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing '91, Toronto, Canada, May, pp. 661±664. Pan, J., Fischer, T.R., 1994. Vector quantization ± lattice quantization of speech LPC coecients. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing '94, Adelaide, Australia, April, pp. I.513±I.516. Saoudi, S., 1990. Codage de la parole par les paires de raies spectrales. Universite de Rennes I, December, These de Doctorat. Soong, F.K., Juang, B.H., 1990. Optimal quantization of LSP parameters using delayed decisions. In Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing '90, Albuquerque, New Mexico, USA, April, pp. 185±188. Sugamura, N., Itakura, F., 1986. Speech analysis and synthesis methods developed at ECL in NTT ± From LPC to LSP ±. Speech Communication 5 (2), 199±215. Svendsen, T., 1994. Segmental quantization of speech spectral information. In: Proceedings of IEEE International Conference of Acoustics, Speech, and Signal Processing '94, Adelaide, Australia, April, pp. I.517±I.520.
www.elsevier.nl/locate/specom
Real time vector quantization of LSP parameters Balazs K ovesi a, Samir Saoudi b,*, Jean Marc Boucher b, G abor Horv ath c b
a CNET, D ept. CMC, Av. P. Marzin, Lannion Cedex, France ENST-Bretagne, Groupe Signal, D ept. Signal et Communications, B.P. 832, 29285 Brest Cedex, France c TUB, D ept. MMT, M} uegyetem rkp. 9, 1521 Budapest, Hungary
Received 24 June 1998; received in revised form 2 June 1999; accepted 9 June 1999
Abstract The distance measure is of great importance in both the design and coding stage of a vector quantizer. Due to its complexity, however, the spectral distance which best correlates with the perceptual quality is seldom used. On the other hand, various weighted squared Euclidean distance measures give close or even accurate estimation of the meaningful spectral distance. Since they are in general mathematically more tractable, these weighted squared Euclidean distance measures are more commonly used. Signi®cant dierences can be found in the performance of dierent distance measures suggested in previous literatures. In this paper, a complete study and comparison of weighted squared Euclidean distance measures is given. This paper also proposes a new weighted squared Euclidean distance measure for vector quantization of Line Spectrum Pairs (LSP) or Cosine of LSP (CLSP) parameters. It also presents an ecient adaptation apparatus for using the proposed distance measure in the case of split or multi-stage vector quantizers. Ó 1999 Elsevier Science B.V. All rights reserved. Resume la fois pour la conLe choix d'une mesure de distance est tres important pour la quanti®cation vectorielle (QV) a struction du dictionnaire et pour la recherche de la valeur quanti®e. La distance spectrale, qui est la plus signi®cative, est peu souvent utilise a cause de sa complexite calculatoire. La distance Euclidienne quadratique ponderee est mathematiquement plus attractive et par consequent plus souvent utilisee. Selon la distance utilisee, on peut trouver des dierences signi®catives en terme de performances. Dans cet article, une comparaison entre les dierentes distances Euclidienne quadratique ponderee est etudiee. Une mesure de distance est proposee dans le cas de la QV des paires de raies spectrales (LSP) ou Cosinus des LSP (CLSP). Cette mesure permet d'obtenir une tres bonne estimation de la distance spectrale. Une adaptation, dans le cas de la QV eclatee ou multi-etage, est proposee. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Speech coding; Line spectrum pairs; Split vector quantization; Multi-stage vector quantization; Low bit rate; Spectral distance measure; Power spectral density
*
Corresponding author. Tel.: +33-298-00-11-79; e-mail: [email protected]
0167-6393/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 3 9 3 ( 9 9 ) 0 0 0 2 6 - 6
40
B. K ovesi et al. / Speech Communication 29 (1999) 39±47
Nomenclature LSP CLSP xi 1i de ; dq ; dep ; dqp ; ds ; wai ; wbi ; wci ; wdi ; wei ; wfi ; wgi ; whi Sx ; nxi n 1i CE Cw
Line Spectrum Pairs Cosine of Line Spectrum Pairs the ith LSP parameter the ith CLSP parameter the Euclidean distance the squared Euclidean distance the weighted squared Euclidean distance the weighted Euclidean distance the spectral distance of two LSP vectors dierent suggested weights of the ith component of an LSP or CLSP vector the power spectral density function using LSP parameters the spectral distance sensitivity of the ith LSP parameter the spectral distance sensitivity of the ith CLSP parameter codebook designed using the squared Euclidean distance without weights codebook designed using the weighted (xgi ) squared Euclidean distance
1. Introduction Line Spectrum Pairs (LSP) or Cosine of LSP (CLSP) provide an ecient representation of the synthesis ®lter used in the Linear Predictive Coding (LPC) of speech (Sugamura and Itakura, 1986; Saoudi, 1990). These parameters are often encoded by a vector quantizer Q completely speci®ed by a codebook C fc1 ; c2 ; . . . ; cN g and a chosen distance measure d ; . Note that the distance measure is of great importance in both the design and coding stages of the vector quantizer.
The aim of speech coding is to obtain a synthesized speech signal perceptually as close as possible to the original. As the human ear is very sensitive to distortions of the spectrum, the spectral distortion is often used for measuring the performance of encoded LSP parameters. This spectral distance has been found to allow an accurate evaluation of the subjective LSP encoding quality (K ovesi, 1997, 1995). In particular, an average spectral distortion of no greater than 1 dB has traditionally been considered as transparent quality (Kleijn and Paliwal, 1995, Chapter. 12). When designing a vector quantizer, an essential problem is to ®nd a representative codevector ci , that minimizes the averaged distortion measure for all source vectors contained within the ith cell Ri . In addition, we need to choose among the quantizer codebook C a codevector ci that is closest to the input source vector according to the distortion measure during the coding stage. Theses problems become relatively complicated when the spectral distance is used as the distortion measure, due to its high computational complexity. To circumvent these problems, the spectral distance is often replaced by a weighted squared distortion measure whereby the codevector ci can be determined as the arithmetic mean of Ri . Its computational time is considerably shorter than that of the spectral distance. Hence, this distortion measure is more suitable for real-time applications. In this paper, a new weighted squared distortion measure for vector quantization of LSP (or CLSP) parameters is suggested.
2. Main distortion measures of interest The spectral distance of two pth order LSP ~y is de®ned as follows: ~x and x vectors x ~x ; x ~y ds x v u 92 u Z 8 u1 p < Sx x; ~ xx = dx; 10 log u t p 0 : ~y ; Sx x; x
dB; 1
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where 1 ~ Sx x; x 2 Ap;~x e jx
2
is the power spectral density function for the vec~ fx1 ; x2 ; . . . ; xp g. tor x The most known and most used distance is the Euclidean one: s p X de x; y 3 jxi ÿ yi j2 i1
dq x; y
p X
2 jxi ÿ yi j :
4
i1
By using the Euclidean distance or the squared Euclidean distance mentioned above, it is assumed that all components of the vectors have perceptually the same importance. In practice, this is rarely true. For example, in the case of the PARCOR (or the Log Area Ratio) parameters, the importance of a component depends directly on the order of the coecients. The spectrum of the synthesized signal is much more sensitive to the variation of the ®rst coecient than to that of the last one. This is why it is important to introduce a weighting function into the distance expression to re¯ect this phenomenon more precisely. The general expression of the weighted Euclidean distance is the following: q T 5 dep x; y x ÿ y W x ÿ y ; where W is a symmetric and positive de®nite weighting matrix. Let us note that this distance includes the usual squared Euclidean in the special case where W I, the identity matrix. In the case where W is a diagonal matrix with diagonal values wi P 0, the weighted squared Euclidean distance is given by dqp x; y
p X
wi jxi ÿ yi j2 :
previous publications, there are many proposals for weights which are all determined in an empirical way, for example: · In (Svendsen, 1994; Hagen and Hedelin, 1990), the distance is simply given by the usual squared Euclidean distance. · The weights used in (Collura and Tremain, 1993; Paliwal and Atal, 1991; Erzin and Cetin, 1993) are the power spectral densities calculated at the input LSP locations and raised to the power r: wi Sn xi;in r ;
or the squared Euclidean distance given by
6
i 1; . . . ; p:
7
In the case of this distance, the LPC parameters in the formant regions are better quantized than those in the non-formant regions. The value of r was chosen to 0.15 in (Collura and Tremain, 1993), the obtained weights will be referred to as wai . In (Paliwal and Atal, 1991; Erzin and Cetin, 1993) r 0:12 was used and the last two weights were modi®ed to de-emphasize the upper frequency range by multiplying by 0.8 and 0.4, respectively. These weights will be referred to as wbi . · The weights proposed in (Laroia et al., 1991) are based on the following property of LSP parameters: when an LSP parameter is close to one of its neighbours, the speech spectrum has a peak near that frequency. These LSP parameters have high spectral sensitivity and should be given higher weights: wci
1 1 ; xi ÿ xiÿ1 xi1 ÿ xi
i 1; . . . ; p;
8
with x0 0 and xp1 p. This weighting function is called Inverse Harmonic Mean (IHM) and was also used in (Pan and Fischer, 1994). · Following the same principle as above, the weights in (Deketelare et al., 1991) are de®ned as "
i1
There is no simple relationship between the spectral distance and the weighted squared distance. In
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wdi max
1 xi ÿ xiÿ1
2 ;
1 xi1 ÿ xi
2 # ;
9
42
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with wd1 and wdp
1 x2 ÿ x1
2
1 xp ÿ xpÿ1
2 :
· In (Bruhn, 1994; Soong and Juang, 1990; Kuo et al., 1992), the used weighting factor wei is the so-called spectral sensitivity with respect to xi , which is de®ned as 2 Z p o log Sx x; x ~ dx: wei 10 oxi 0
Fig. 1. Spectral distance sensitivity of LSP parameters.
· The weights wfi introduced in (Gurgen et al., 1990) are equal to the inverse variance of the ith LPC parameter. Although all of the weighted distances mentioned above were de®ned originally for LSP coecients, they can be applied for CLSP coecients as well. The following section provides a weighting function that gives a good approximation of the spectral distance.
3. A suggested weighted squared Euclidean distance In this section, we suggest a new weighted Euclidean distance which computes the spectral distance more easily and which gives the same perceptual evaluation. In addition, this distance will give a distortion measurement directly, unlike the others which need a computation of this spectral distortion. Here, we focus on studying the relationship between the spectral and Euclidean distances. The following simulations were made: a 10th order LSP (respectively CLSP) vector was taken and one of its components xi (respectively 1i ) was modi®ed by dierent values to x0i (respectively 10i ). Then, the spectral distance of the original and the modi®ed vector was computed. The ten vertical dotted lines in Fig. 1 (respectively Fig. 2) ~ represent the values of the original vector x (respectively ~ 1). The ten solid lines illustrate the results of the ten simulations. In each of them, we
Fig. 2. Spectral distance sensitivity of CLSP parameters.
modi®ed only one of the ten parameters of the vector, while the other nine parameters of the two vectors were unchanged. Let us note x the value of the modi®cation x x0i ÿ xi (respectively x 10i ÿ 1i ), we found that the spectral distance between the original vector and the modi®ed vector is almost in direct proportion with x in both LSP and CLSP cases. So, the spectral distance of these vectors can be estimated by ÿ dbs x1 ; . . . ; xi ; . . . ; xp ; x1 ; . . . ; x0i ; . . . ; xp nxi x0i ÿ xi ; 11
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ÿ
dbs 11 ; . . . ; 1i ; . . . ; 1p ; 11 ; . . . ; 10i ; . . . ; 1p n1i 10i ÿ 1i ;
12
where nxi (respectively n1i ) denotes the spectral distance sensitivity of the ith LSP (respectively CLSP) parameter. The absolute value of the spectral distance sensitivity of a parameter LSP (or CLSP) was found to be independent of the sign of the modi®cation. The limits of the modi®ed coecients x0i (respectively 10i ) were determined by the well-known stability criteria of LSP and CLSP vectors: 0 < x1 < < xp < p;
13
ÿ2 < 11 < < 1p < 2:
14
Spectral distance sensitivity can be determined by linear regression. As the dependency is quasi-linear and one point of the straight line is known (when x 0), nxi or n1i can be estimated after computing just one more point of the straight line. We applied this second technique which is less complex. To compute the spectral distance sensitivity nxi (respectively n1i ), the value of the ith coecients xi (respectively 1i ), is ®rst perturbed by xi;1 xiÿ1 ÿ xi =2 (respectively xi;1 1iÿ1 ÿ1i =2) then by xi;2 xi1 ÿ xi =2 (respectively xi;2 1i1 ÿ 1i =2) with x0 0 and xp1 p (respectively 10 ÿ2 and 1p1 2). In these two cases, the spectral distances ds;i;1 and ds;i;2 of the original and the perturbed vector were computed. The spectral distance sensitivity nxi was determined as follows: nxi
ds;i;1 =jxi;1 j ds;i;2 =jxi;2 j ; 2
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sponding squared spectral distance sensitivities, both in the case of LSP and CLSP: dbs x1 ; . . . ; xp ; x01 ; . . . ; x0p s p X 2 16 n2xi x0i ÿ xi ; i1
dbs 11 ; . . . ; 1p ; 101 ; . . . ; 10p s p X 2 n21i 10i ÿ 1i :
17
i1
Using the notation of Section 2, the proposed weights are equal to the squared spectral distance sensitivities: xgi n2xi :
18
Fig. 3 shows the spectral distance of two vectors LSP that have two dierent parameters. In this example, the third parameter of the modi®ed vector was ®xed to x03 x3 0:1. The spectral distance caused by this modi®cation is represented by the star, its estimated value using (16) is marked by the circle. Then, we modi®ed another parameter continuously, while the remaining eight parameters were unchanged. The nine solid lines illustrate the obtained spectral distance in the case of the nine possible simulations while the nine broken lines give the estimated spectral distance using the
i 1; . . . ; p: 15
n1i is obtained in the same way. After computing all spectral distance sensitivix (respectively ties nxi i1;...;p ; of an LSP vector ~ n1i i1;...;p of a CLSP vector ~ 1 ) in this way, we wished to ®nd out how the spectral distance of this ~0 (revector and a completely dierent vector x spectively CLSP ~ 10 ) could be estimated using the sensitivities. We found that the spectral distance could be estimated by the weighted Euclidean distance where the weights are equal to the corre-
Fig. 3. Spectral distance of LSP parameters versus the second modi®ed component.
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suggested formula (16). This ®gure shows that (16) gives a good estimation of the spectral distance especially for reasonable values. Fig. 4 shows that a similar application is relevant for CLSP parameters. In Table 1, we give the mean values of the spectral distance sensitivities of 10th order LSP and CLSP parameters estimated on our learning sets. The variation of the spectral distance sensitivities of CLSP parameters is greater than that of the LSP parameters. In the case of CLSP, it is more important to use a weighted distance than the squared Euclidean distance. The weights can be de®ned as well as the squared mean spectral distance sensitivities. They will be referred to as xhi .
4. Experimental results To compare the dierent distance measure, ®rst, a codebook of 1024 codewords (denoted CE ) was designed by the k-means algorithm using a training set of 150 000 vectors. In the design stage of the codebook CE , the squared Euclidean distance was used without weights. We quantized then 35 000 testing vectors separated from the training set. All the individual weights were determined according to the codewords. Table 2 (respectively Table 3) gives the spectral distortion obtained versus the used distance in the case of the vectors LSP (respectively CLSP). Both the distance weighted by (xei ), (xci ) and our suggested distance (with xgi ) obtain a near equivalent performance to that of the meaningful spectral distance. However, while the suggested distance (xgi ) gives the same values as the distance weighted by (xei ) and (xci ), the former is easier to compute. Moreover, our Table 2 Spectral distortion of coding LSP vectors versus the distance measure
Fig. 4. Spectral distance of CLSP parameters versus the second modi®ed component.
Table 1 Mean spectral distance sensitivities of 10th order LSP and CLSP parameters
Distance
Spectral distortion in dB
Squared Euclidean (SE) Weighted SE : xai Weighted SE : xbi Weighted SE : xci Weighted SE : xdi Weighted SE : xei Weighted SE : xfi Weighted SE : xgi Weighted SE : xhi Spectral
2.84 2.80 2.83 2.77 2.96 2.74 2.92 2.74 2.82 2.72
Table 3 Spectral distortion of coding CLSP vectors versus the distance measure
No.
Sensibility LSP
Sensibility CLSP
Distance
Spectral distortion in dB
1 2 3 4 5 6 7 8 9 10
28.80 23.63 15.99 14.31 17.73 15.64 16.43 16.12 16.53 17.27
82.74 43.06 15.92 9.37 9.55 7.97 8.78 9.83 13.77 22.09
Squared Euclidean (SE) Weighted SE : xai Weighted SE : xbi Weighted SE : xci Weighted SE : xdi Weighted SE : xei Weighted SE : xfi Weighted SE : xgi Weighted SE : xhi Spectral
3.06 3.01 3.05 2.90 3.08 2.85 3.05 2.85 2.96 2.81
B. K ovesi et al. / Speech Communication 29 (1999) 39±47 Table 4 Spectral distortion of coding CLSP vectors versus the distance measure and the codebook Codebook and distance
Spectral distortion in dB
CE CE Cw Cw
3.06 2.85 3.15 2.77
and SE and weighted SE : xgi and SE and weighted SE : xgi
suggested distance allows the direct calculation of the distortion measure which is not the case for the classic ones. The weighted squared Euclidean distances can also be used in the learning and the coding stage. The following simulation shows that it is more important to use a correct distance measure in the coding stage. For the design of a codebook of 1024 CLSP vectors (denoted Cw ), we used the weighted squared Euclidean distance with xgi . In Table 4, four dierent cases can be compared. 5. Real time applications In the cases of the xfi weights, the values are constant, they have to be determined once before the coding stage, generally by using a training set. In the other cases, to compute the distance of two vectors, the weights depend on one of the two vectors. In vector quantization, generally, the distance of an input vector and a codeword is computed. Hence, there are three main possibilities to determine the individual weights: · The calculation of the weights can be based on the input vector. This solution was used in each paper mentioned above (Bruhn, 1994; Collura and Tremain, 1993; Deketelare et al., 1991; Erzin and Cetin, 1993; Gurgen et al., 1990; Hagen and Hedelin, 1990; Kuo et al., 1992; Laroia et al., 1991; Paliwal and Atal, 1991; Pan and Fischer, 1994; Soong and Juang, 1990; Svendsen, 1994). Its disadvantage is that the complexity of the coding increases seriously, especially in the case of the xei and xgi weights. Often, these calculations cannot be made in real time applications. It is less complex to calculate the spectral distance directly than to determine these weights for each input vector.
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· We propose here to determine the weights according to the codewords. As in non-adaptive coding, the codebook is ®xed during the coding stage, the calculations can be done before the quantization and the weights can be stored in memory. This solution increases slightly the memory needed but it is more suitable for real time applications. · To use the constant weights like xfi and xhi , the mean value of the weights estimated on a learning set can be taken. Of course, this solution is less accurate, but the increase of both computational time and memory needed is negligible. To be able to implement a vector quantizer for a real time application, often a sub-optimal solution has to be used like the MSVQ (Juang and Gray, 1982) or the SVQ (Paksoy et al., 1992). In this section, these solutions will be studied from the point of view of the used distance. It will be assumed that the individual weights can not be calculated according to the input vector because of the time constraint of a real time application. They have to be determined according to the codewords of the codebooks and stored in memory before the coding. The problem is that all individual weights (xai ; xbi ; xci ; xdi ; xei ; xgi ) are based on special properties of LSP (or CLSP) vectors and knowledge of the whole vector is needed to calculate them. 5.1. Multi-stage vector quantization In the case of the MSVQ, the codewords of the second and further stages represent the quantization error of the previous stages. These vectors do not have the same properties as the LSP vectors. The individual weights cannot be determined according to them. As the codeword chosen in the ®rst stage is near to the codeword corrected by the further stages, we propose to use the weights of the codeword chosen at the ®rst stage. The approximation is better if the quantization on the ®rst stage is more precise. That is why it is important to assign as many bits as possible to the ®rst stage.
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5.2. Split vector quantization In the case of SVQ, the codebooks contain only a part of the codewords that not allow the computation of individual weights. Our proposition is based on the following properties of LSP and CLSP parameters: · The spectral in¯uence of an LSP (or CLSP) parameters is localized in its region. · The closeness of two LSP parameters marks a spectrally important region. This is the reason we completed the sub-vectors in the codebooks by spectrally neutral ®ctitious subvectors and we determined the weights on this complete vector. For example, in the case of 10th order LSP vectors, if a codeword contains the last four coecients x7 ; . . . ; x10 , we de®ned the six ®ctitious coecients as xi xmin i x7 ÿ xmin =7; i 1; . . . ; 6, with xmin 0. The uniform distribution of ®ctitious LSP parameters assures the spectral neutrality of this part of the vector. In the case of CLSP parameters, we generated the ®ctitious CLSP parameters in the LSP domain to assure spectral neutrality, as the mean LSP vectors are more uniformly distributed. For example, if a codeword contains the ®rst six coecients 11 ; . . . ; 16 , we calculated the remaining four coecients as 1i ÿ2 cos arccos ÿ 16 =2 xmax ÿ arccos ÿ16 =2 ; 19 i ÿ 6 5 i 7; . . . ; 10, with xmax p. Of course, constant mean weights like xfi and xhi can be used in both cases without diculties. But the solutions proposed above give smaller spectral distortion. We constructed 17-bit SVQ's for 10th order CLSP parameters. One codebook contained 1024 codewords of the ®rst six coecients, the other contained 128 codewords of the last four coecients. During the design stage, the mean constant weights xhi were used. Then, in the coding stage, we ®rst used the same constant weights. The obtained spectral distance was 2.04 dB. In a second simulation, the squared
spectral distance sensitivities xgi were computed using the completed codewords. The obtained spectral distance was 1.95 dB. 6. Conclusions In this paper, we studied and compared the weighted squared Euclidean distances that can substitute the meaningful spectral distance. A weighted Euclidean distance was suggested which was found to give a good approximation of the spectral distance. We also showed how individual weights can be determined according to codewords of the codebook in the case of SVQ or MSVQ. By storing the obtained weights, computation is far less complex than that of the spectral distance, which is essential for a real time application. Acknowledgements The authors thank A. Gourves Hayward for her kind assistance with the English of this paper. The authors also thank the anonymous reviewers for their pertinent suggestions which improved the quality of this paper.
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