KR100903110B1 - The Quantizer and method of LSF coefficient in wide-band speech coder using Trellis Coded Quantization algorithm - Google Patents

Abstract

The present invention relates to an LSF coefficient quantization apparatus and method for a wideband speech coder using a trellis code quantization algorithm, and to quantize an LSF coefficient vector to calculate a candidate vector for quantization, and to predict the candidate vector of the LSF coefficient vector. A predictive structure quantizer for trellis coded quantization (TCQ) with reference to an LSF vector to calculate a predicted quantized final vector of the LSF coefficient vector, and a quantized candidate vector to be quantized by quantizing the LSF coefficient vector, A non-prediction structure quantizer for calculating an unpredicted quantization final vector of the LSF coefficient vector by trellis code quantization (TCQ), and the difference between the LSF coefficient vector among the predicted quantization final vector and the unpredicted quantization final vector is small; Safe quantization, consisting of a switching unit that determines the final quantization vector of the LSF coefficient vector To provide a group which is capable to improve and SD bits allocated Performance for reducing the effective transfer phenomenon LSF quantization error.

Quantizer, BCTCQ Algorithm, Predictive Vector, Unpredictable Vector

Description

The Quantizer and method of LSF coefficient in wide-band speech coder using Trellis Coded Quantization algorithm

1 is a diagram illustrating an S-MSVQ structure as an LPC coefficient quantizer of a conventional AMR wideband speech coder proposed by 3GPP.

FIG. 2 is a diagram illustrating an LSF coefficient quantizer using VQ-BCTCQ (Vector Quantization-Block Constrained Trellis Coded Quantization) in a form in which a non-memory vector quantizer and a memory vector quantizer are coupled in parallel according to an embodiment of the present invention. .

FIG. 3 is a diagram illustrating a trellis path to be considered in a single viterbi encoding process according to an initial state when using a BCTCQ algorithm in a 4-state trellis structure according to an embodiment of the present invention.

4 is a diagram illustrating a viterbi encoding process according to an embodiment of the present invention.

5 is a flowchart illustrating a quantization process of an LSF coefficient quantizer according to an exemplary embodiment of the present invention.

The present invention relates to an LSF coefficient quantization apparatus and method for a wideband speech encoder using a trellis code quantization algorithm, and more particularly, to an LSF coefficient quantization apparatus and method using a predictive vector quantizer and a BC trellis code quantizer. .

In the speech encoder, LPC coefficients are used to express short-term frequency characteristics of speech. The LPC coefficient value is obtained by the concept of dividing an input speech signal into frames and minimizing energy of prediction error for each frame.

In the wideband speech coder, a 16th order all-pole filter is used as an LPC filter, and many bits are allocated for quantization of the linear prediction coefficients used. LPC coefficients have a large dynamic range, and the filter characteristics are very sensitive to coefficient quantization error.

That is, since the position of the pole of the filter is sensitively moved by the quantization error, there is a problem that the stability of the LPC filter is not guaranteed due to the quantization error.

Due to these problems, LPC coefficients are transformed into other coefficients that are easy to check the stability of the filter and have good quantization characteristics, and quantization is mainly preferred by converting them into reflection coefficient or line spectral frequency (LSF). . In particular, the LSF coefficient values are closely related to the frequency characteristics of speech, so most recently developed standard speech coders use LSF coefficients.

The quantization method of LSF coefficients has been introduced a number of modified methods based on scalar quantization and vector quantization.

The LSF quantization technique uses interframe correlation (short-term correlation) for efficiency of quantization, that is, the LSF of the current frame is predicted from the LSF information of the past frame without directly quantizing the LSF of the current frame, and then the prediction is performed. A technique for quantizing the error of.

As the prediction method, the AR (Auto Regressive) method and the moving average (MA) method are used. The former has excellent prediction performance while the effect of coefficient transfer error is continuously transmitted to the receiving side. In comparison, the predictive performance is inferior, but the influence of propagation error is limited.

Therefore, the MA method is used in an environment where a lot of transmission errors occur, such as wireless communication.

Since scalar quantizers use a lot of bits compared to their performance, vector quantization is frequently used for more efficient quantization. The vector quantization technique shows much better quantization performance than the scalar quantization technique, but has a very complicated problem.

The unstructured VQ quantizer designed with the "Linde, Buzo and Gray" algorithm quantizes the entire vector at once, increasing the complexity by the exponential multiplier of the product of the vector size and the encoding rate.

To solve this problem, structured vector quantization techniques such as tree-structured VQ, split VQ, multistage VQ, or shape-gain VQ have been introduced.

Split VQ, which divides an entire vector into several subvectors and quantizes it, and multistage VQ, which is quantized through multiple stages, are used as LPC coefficient quantizer structures of some standardized speech coders.

These schemes cause some performance degradation compared to full-search VQ due to the constraints of the structure, but they are widely used as quantization methods of VQ because of the advantage of using the concept of product code to greatly reduce the computation and memory.

1 is a diagram illustrating an S-MSVQ structure as an LPC coefficient quantizer of a conventional AMR wideband speech coder proposed by 3GPP.

FIG. 1 is a block diagram of a linear prediction coefficient quantizer used in a wideband speech coder having an S-MSVQ (Split-Multi Stage Vector Quantization) structure according to the 3GPP standard, which is a quantization to which split VQ and multistage VQ concepts are simultaneously applied. Qi.

The S-MSVQ structure is designed to reduce the memory and code table search time required for quantization of the LSF coefficients allocated with 46 bits, and to reduce the memory and code table search time required for the quantization of the entire vector. It has much less memory and code table search computations than quantization for, but still requires a large amount of computation due to the large amount of memory and complexity of the code table search.

Trellis coded quantization is a kind of vector quantization, which consists of a vector codebook required for coding, a scalar codebook corresponding to each element of a vector, and a trellis structure with a convolutional coder. The number of states in the entire trellis structure is N = 2 ^v , and there are two incoming or outgoing branches to each trellis state.

The trellis path for optimal encoding is found using the Viterbi algorithm, and the trellis coded quantization technique is much smaller in complexity than unstructured VQ. The traditional trellis coded quantization technique has to transmit the initial state of the trellis path determined as additional information after searching for the viterbi algorithm.

This additional transmission information does not have a significant effect on a large source vector, but is a significant obstacle in terms of rate-distiortion for a small source. However, the introduction of 'tail-biting TCQ' solved the disadvantage of TCQ side information transmission.

Block-constrained TCQ has 2 ^k , 0 ≦ ^k ≦ ^v initial states among N-trellis states, and has a trellis structure constrained to have 2 ^vk last states for each initial state. The encoding process proceeds as follows.

Step1) The path metrics for the ^2k initial states are initialized to '0', and the path metrics for the remaining states are initialized to '∞'.

Step 2) One viterbi algorithm search proceeds to the n-v stage, and the trellis path to the particular state of the last stage is determined according to the initial state of the trellis path determined by the viterbi algorithm search to the n-v stage.

Step3) The trellis path information determined in step2 is selected from among the nv bits representing the k-bit initial state information and the state transition information in each trellis state up to the nv stage, and finally the 2 ^vk states of the last stage in the nv stage. It consists of the vk bits to represent the state.

This encoding method consists of one search of the viterbi algorithm, regardless of the value of k.

An object of the present invention is to provide a method for efficiently quantizing the LSF coefficients of a wideband speech encoder in VQ-BCTCQ, which is a structure in which a vector quantizer and a BCTCQ are serially coupled.

Another technical task of the present invention is to construct a predictive VQ-BCTCQ quantization system including a predictive structure to improve memory requirements and calculation amount and to provide improved SD performance.

In order to solve the above technical problem, the LSF coefficient quantization apparatus for wideband speech coder using the trellis code quantization algorithm proposed in the present invention calculates a candidate vector quantized by quantizing an LSF coefficient vector, and converts the candidate vector into the LSF coefficient. A predictive structure quantizer configured to trellis code quantize (TCQ) the predicted quantization final vector of the LSF coefficient vector by referring to the predicted LSF vector of the vector, and quantize the LSF coefficient vector to calculate a quantization candidate vector; A non-predictive structure quantizer configured to trellis code quantize the candidate vector to yield an unpredicted quantized final vector of the LSF coefficient vector, and the predicted quantized final vector and the predicted quantized final vector to the LSF coefficient vector. A switching unit for determining a small difference as the final quantization vector of the LSF coefficient vector It characterized in that it comprises.

The predictive structure quantization unit and the non-prediction structure quantization unit may have a structure connected in parallel.

In addition, the prediction structure quantization unit is a prediction unit for calculating a predicted LSF vector of the LSF coefficient vector using a calculated predictive quantization vector, the quantization target candidate by quantizing the difference vector between the LSF coefficient vector and the predicted LSF vector A first vector quantizer for calculating a vector and a first trellis quantizer for calculating a predictive quantized final vector of the LSF coefficient vector by trellis-signing quantizing a difference vector between the difference vector and the candidate vector. It is done.

The first vector quantization unit and the first trellis code quantization unit may have a structure connected in series.

The first trellis code quantization unit may be a blocked constrained (BC) trellis code quantizer.

The non-prediction structure quantization unit quantizes the LSF coefficient vector to obtain a candidate vector to be quantized, and trellis-quantizes the difference vector between the LSF coefficient vector and the candidate vector to perform the LSF coefficient vector. And a second trellis-encoded quantization unit for calculating an unpredicted quantization final vector of the vector.

The second vector quantization unit and the second trellis code quantization unit may have a structure connected in series.

The second trellis code quantization unit is a blocked constrained (BC) trellis code quantizer.

The switching unit may determine a difference between the candidate vectors quantized in the prediction structure quantization unit and the non-prediction structure quantization unit and the LSF coefficient vector by using an Euclidean distance algorithm, and the difference The small candidate vector is determined as the final quantization vector.

In order to solve the above technical problem, the LSF coefficient quantization method for a wideband speech coder using the trellis code quantization algorithm proposed by the present invention includes (a) quantizing an LSF coefficient vector to calculate a candidate vector for quantization, and selecting the candidate vector. Calculating a predicted quantized final vector of the LSF coefficient vector by trellis code quantization (TCQ) with reference to the predicted LSF vector of the LSF coefficient vector, (b) quantizing the LSF coefficient vector to obtain a candidate vector to be quantized Calculating a trellis code quantization (TCQ) of the candidate vector to calculate an unpredicted quantized final vector of the LSF coefficient vector; and (c) the LSF coefficient vector among the predicted quantized final vector and the unpredicted quantized final vector. Determining that the difference between and is small as the final quantization vector of the LSF coefficient vector. .

In the step (a) and the step (b), the predictive quantization final vector and the unpredictable quantization final vector are simultaneously calculated.

In addition, the step (a) is a step of calculating a predicted LSF vector of the LSF coefficient vector using a calculated predictive quantization vector, the quantization target candidate by quantizing the difference vector of the LSF coefficient vector and the predicted LSF vector Calculating a vector and calculating a predictive quantization final vector of the LSF coefficient vector by trellis code quantizing the difference vector between the difference vector and the candidate vector.

In the step (b), the LSF coefficient vector is quantized to produce a quantization candidate vector, and the difference vector between the LSF coefficient vector and the candidate vector is trellis coded to unpredicted quantization of the LSF coefficient vector. Calculating a final vector.

In the step (c), the difference between the candidate vectors quantized in the step (a) and the step (b) and the LSF coefficient vector is determined using an Euclidean distance algorithm. The candidate vector having the small difference is determined as the final quantization vector.

Hereinafter, with reference to the accompanying drawings will be described in detail a preferred embodiment of the present invention.

FIG. 2 is a diagram illustrating an LSF coefficient quantizer using VQ-BCTCQ (Vector Quantization-Block Constrained Trellis Coded Quantization) in a form in which a non-memory vector quantizer and a memory vector quantizer are coupled in parallel according to an embodiment of the present invention. . Referring to FIG. 2, the quantizer includes a prediction structure quantizer 200, an unpredicted structure quantizer 210, and a switching unit 220.

)을 벡터 양자화하여 양자화 대상 후보 벡터(

)를 산출하는 제1 벡터 양자화부(202)와 그 후보 벡터를 BCTCQ로 양자화시켜 상기 LSF 계수 벡터의 예측 양자화 최종 벡터를 산출하는 제1 BCTCQ부(203)로 구성되어 있다.The prediction structure quantization unit 200 calculates the prediction error value of the prediction unit 201 that calculates the predicted LSF vector of the LSF coefficient vector.

) To vector quantize the candidate vector (

The first vector quantization unit 202 for calculating the C quantization unit 202 and the candidate vector quantize the candidate vector with BCTCQ to calculate the predictive quantization final vector of the LSF coefficient vector.

)를 벡터 양자화하여 양자화 대상 후보 벡터(

)를 산출하는 제2 벡터 양자화부(211)와 상기 후보 벡터를 BCTCQ로 양자화하여 상기 LSF 계수 벡터의 비예측 양자화시켜 최종 벡터를 산출하는 제2 BCTCQ부(212)로 구성되어 있다.Unpredicted structure quantization unit 210 is the LSF coefficient vector (

) To vector quantize the candidate vector (

A second vector quantization unit 211 that calculates the ") and a second BCTCQ unit 212 that quantizes the candidate vector by BCTCQ to unpredictably quantize the LSF coefficient vector to produce a final vector.

The switching unit 220 selects the smallest difference between the LSF coefficient vector among the predicted quantization final vector and the unpredicted quantization final vector as the final quantization vector of the LSF coefficient vector.

The present invention uses a vector quantizer and a Block Constructed Trellis Coded Quantization in a safety-net structure that selects only one according to the performance of spectral distortion (SD) after quantizing at the same time in two ways by using a prediction structure and a non-prediction structure in parallel. ) Is a structure that combines a dual-stage quantizer that combines in series.

Predictive and Unpredicted Structure The internal structure of the dual stage quantizer constituting the quantizer is described. The first stage quantizer performs vector quantization.

Vector quantization quantized entire vectors to make the most of the correlation between vector elements.

By quantizing the entire vector, it is necessary to consider the size of the code table. In the present invention, by assigning less than 1/7 of the allotted bits to the vector quantizer, the burden of code table memory and search time required by the vector quantizer is reduced.

The BCTCQ algorithm, which is the second stage quantizer, uses an N (= 2 ^v ) -state trellis structure based on the 1 / 2-rate Kimsam coder and feedback-free encoder structure.

As mentioned above, the BCTCQ algorithm restricts the initial state of the selectable trellis path, and the state of the last stage is also limited by the initial state of the trellis path.

FIG. 3 is a diagram illustrating a trellis path to be considered in a single viterbi encoding process according to an initial state when using a BCTCQ algorithm in a 4-state trellis structure according to an embodiment of the present invention.

3 illustrates a trellis path considered when the BCTCQ algorithm of k = 1 is applied to the entire 4-state trellis structure.

If '00' is determined as the state in the L-log ₂ 4 stage, the initial state of the survivor path (bold dotted line) is '00', so the trellis paths that can be selected in the remaining stages have states of '00' and '01'. It is indicated by the bold solid-dotted line.

4 is a diagram illustrating a viterbi encoding process according to an embodiment of the present invention.

The Viterbi encoding process in the j-th stage of FIG. 4 is as follows.

을 이용해서 양자화 distortion을 다음과 같이 구하여 distance meric

에 저장한다.Subcodebook assigned to two branches connected to the p state of the jth stage

Find the quantization distortion using

Store in

는 j번째 stage의 p state와 j-1번째 stage의 i' state사이의 branch에 할당된 부 코드북을 나타낸다.here

Represents a sub-codebook assigned to a branch between the p state of the j th stage and the i 'state of the j-1 stage.

The selection of one of the two trellis paths connected to the p state of the jth stage and the cumulative distortion update process are performed as follows.

5 is a flowchart illustrating a quantization process of an LSF coefficient quantizer according to an exemplary embodiment of the present invention.

FIG. 5 illustrates the operation of the quantizer shown in FIG. 2 and is performed as follows.

)는 예측 구조와 비 예측 구조로 입력되어 각각 양자화된다.In the speech encoder, LPC is converted to LSF, and the converted LSF (

) Is input to the prediction structure and the non-prediction structure and quantized, respectively.

In each structure, first, the average value of the LSF is removed from the LSF value as shown in Equation (3) (S500).

)와 원래 벡터(

)와의 차이 값을 수학식(4)와 같이 구하여(S511) 도 2에서의 제 1벡터 양자화부를 통해 양자화를 한다(S512, S513).In the predictive structure quantization unit, a vector predicted using a fourth-order MA predictor (

) And the original vector (

) And the difference value is calculated as in Equation (4) (S511) and quantized through the first vector quantizer in FIG. 2 (S512 and S513).

과 원래 벡터(

)간의 차이 벡터(

)를 두 번째 양자화기인 도 2에서의 제 1 BCTCQ부의 입력으로 사용한다.A vector quantized (S514) as shown in Equation (5)

And the original vector (

Difference vector between

) Is used as an input of the first BCTCQ unit in FIG. 2, which is a second quantizer.

)과(S516) 예측 벡터(

), 그리고 LSF의 평균값을 더하여(S517) 최종적인 양자화 값

을 구하게 된다(S518).After quantization by the BCTCQ unit (S515), the quantization value through the first vector quantization unit (

) And (S516) prediction vector (

) And the final quantization value by adding the average value of the LSF (S517).

It is obtained (S518).

)를 예측 구조와 같은 형태의 양자화기로 양자화한다(S520).On the other hand, in the non-prediction structure quantization unit, the LSF vector (only the mean value is removed without performing prediction)

) Is quantized by a quantizer having the same shape as the prediction structure (S520).

The quantization error, which is the difference between the quantized candidate vectors and the original input vector in the prediction and non-prediction structures, is determined by using the Euclidean distance at the switching unit, and the candidate vector having the small quantization error is converted into the quantization vector. Select (S530).

The invention can also be embodied as computer readable code on a computer readable recording medium. Computer-readable recording media include all types of recording devices that store data that can be read by a computer system. Examples of computer-readable recording media include ROM, RAM, CD_ROM, magnetic tape, floppy disks, and optical data storage, and may also include those implemented in the form of carrier waves (e.g., transmission over the Internet). . The computer readable recording medium can also be distributed over network coupled computer systems so that the computer readable code is stored and executed in a distributed fashion.

As described above, optimal embodiments have been disclosed in the drawings and the specification. Although specific terms have been used herein, they are used only for the purpose of describing the present invention and are not intended to limit the scope of the invention as defined in the claims or the claims. Therefore, those skilled in the art will understand that various modifications and equivalent other embodiments are possible from this. Therefore, the true technical protection scope of the present invention will be defined by the technical spirit of the appended claims.

As described above, the LSF coefficient quantizer of the existing speech coder is replaced with a new type of unstructured vector quantizer and BCTCQ in series to reduce the amount of memory and search time required, and to parallel the predicted and unpredicted structures. The new LSF quantizer can be used to improve the performance of allocated bits and SD by reducing the safety quantization and error propagation.

In order to evaluate the performance of the present invention, a wideband voice sample provided by NTT was used, and the voice sample was 13 minutes in total and composed of Korean male, female and English male and female voice.

Table 1 and Table 2 show spectral distortion (SD) performances and calculations between the split and multi-stage vector quantization (S-MSVQ) method used in the LPC quantizer of AMR-WB and the method proposed in the present invention, respectively. (46 bits / frame).

As shown in the table below, the average SD value was improved by 0.03dB compared to AMR-WB S-MSVQ, and the outer point between 3dB and 5dB decreased by 0.2% and the outer point above 5dB decreased by 0.001%. As shown in Table 2, the calculation yielded a decrease of approximately 57%.

AMR-WB S-MSVQ Predictive VQ-BCTCQ Average SD [dB] 0.7933 0.7618 3 dB to 5 dB [%] 0.4099 0.2184 5 dB or more [%] Z 0.0026 0.0013

AMR-WB S-MSVQ Predictive VQ-BCTCQ Remarks ADD 15624 4219 73% reduction MULT 8832 5411 39% reduction Comp 3570 2379 33% reduction Total 28026 12009 57% reduction

Claims (15)

A quantization candidate vector is obtained by quantizing an LSF coefficient vector, and the candidate vector is trellis coded quantized (TCQ) with reference to a predicted LSF vector of the LSF coefficient vector to calculate a predicted quantization final vector of the LSF coefficient vector. A predictive structure quantization unit; A non-prediction structure quantizer configured to quantize the LSF coefficient vector to yield a candidate vector to be quantized, and to trellis code quantize the candidate vector to calculate an unpredicted quantized final vector of the LSF coefficient vector; And And a switching unit configured to determine, as a final quantization vector of the LSF coefficient vectors, that a difference between the predicted quantization final vector and the unpredicted quantization final vector is smaller than that of the LSF coefficient vector. The prediction structure quantization unit and the unpredicted structure LSF coefficient quantization apparatus for a wideband speech coder using a trellis code quantization algorithm, characterized in that the quantization unit has a structure connected in parallel. delete The method of claim 1, wherein the prediction structure quantization unit A predicting unit configured to calculate a predicted LSF vector of the LSF coefficient vector using a calculated predictive quantization vector; A first vector quantizer configured to quantize the difference vector between the LSF coefficient vector and the predicted LSF vector and calculate a quantization candidate vector; And A trellis quantization unit configured to trellis code quantize the difference vector between the difference vector and the candidate vector to calculate a predicted quantized final vector of the LSF coefficient vector; LSF Coefficient Quantization Apparatus for Wideband Speech Coder. The method of claim 3, wherein The first vector quantization unit and the first trellis code quantization unit has a structure connected in series, LSF coefficient quantization apparatus for a wideband speech coder using a trellis code quantization algorithm. The method of claim 3, wherein The first trellis code quantization unit is a blocked constrained (BC) trellis code quantizer, LSF coefficient quantization apparatus for a wideband speech coder using a trellis code quantization algorithm. The method of claim 1, wherein the non-predictive structure quantization unit A second vector quantizer configured to quantize the LSF coefficient vector to calculate a quantization candidate vector; And A trellis coded quantization unit configured to trellis code quantize the difference vector between the LSF coefficient vector and the candidate vector to produce an unpredicted quantized final vector of the LSF coefficient vector; LSF coefficient quantization apparatus for wideband speech coder using code quantization algorithm. The method of claim 6, The second vector quantization unit and the second trellis code quantization unit has a structure connected in series, LSF coefficient quantization apparatus for a wideband speech encoder using a trellis code quantization algorithm. The method of claim 6, The second trellis code quantization unit is a blocked constrained (BC) trellis code quantizer, LSF coefficient quantization apparatus for a wideband speech coder using a trellis code quantization algorithm. The method of claim 1, wherein the switching unit The difference between the candidate vectors quantized in the prediction structure quantization unit and the non-prediction structure quantization unit and the LSF coefficient vector is determined using an Euclidean distance algorithm, and the candidate vector having the small difference is determined. An LSF coefficient quantization apparatus for a wideband speech coder using a trellis code quantization algorithm characterized in that it is determined as a final quantization vector. (a) Quantizing an LSF coefficient vector to yield a quantization candidate vector, and performing a trellis code quantization (TCQ) on the candidate vector with reference to the predicted LSF vector of the LSF coefficient vector to obtain a final quantization of the LSF coefficient vector. Calculating a vector; (b) quantizing the LSF coefficient vector to obtain a candidate vector to be quantized, and trellis code quantizing the candidate vector to calculate an unpredicted quantized final vector of the LSF coefficient vector; And (c) determining that the difference between the LSF coefficient vector among the prediction quantization final vector and the unpredicted quantization final vector is small as the final quantization vector of the LSF coefficient vector; Steps (a) and (b) are LSF for wideband speech coder using a trellis code quantization algorithm, which simultaneously calculates the predicted quantization final vector and the unpredicted quantization final vector. Coefficient Quantization Method. delete The method of claim 10, wherein step (a) Calculating a predicted LSF vector of the LSF coefficient vector using a calculated predictive quantization vector; Quantizing the difference vector between the LSF coefficient vector and the predicted LSF vector to calculate a quantization candidate vector; And Trellis code quantizing the difference vector between the difference vector and the candidate vector to produce a predictive quantized final vector of the LSF coefficient vector; for a wideband speech coder using a trellis code quantization algorithm LSF coefficient quantization method. The method of claim 10, wherein step (b) Quantizing the LSF coefficient vector to produce a quantization candidate vector; And Comprising a trellis code quantization of the difference vector between the LSF coefficient vector and the candidate vector to calculate an unpredicted quantized final vector of the LSF coefficient vector; wideband speech using a trellis code quantization algorithm LSF coefficient quantization method for an encoder. The method of claim 10, wherein step (c) In step (a) and step (b), the difference between the quantized candidate vectors and the LSF coefficient vector is determined by using an Euclidean distance algorithm, and the candidate vector having the smallest difference is final. A LSF coefficient quantization method for a wideband speech coder using a trellis code quantization algorithm characterized in that it is determined by a quantization vector. A computer readable recording medium having recorded thereon a program capable of performing the method of claim 10.

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