JP4746532B2 - Linear prediction coefficient calculation method, apparatus thereof, program thereof, and storage medium thereof - Google Patents

Description

  The present invention relates to a linear prediction coefficient calculation method suitable for use in, for example, a speech prediction encoding apparatus that uses a linear prediction analysis technique for analysis and encoding of a speech sound signal and the like, an apparatus thereof, a program thereof, and a The present invention relates to a storage medium for storing a program.

Conventionally, autocorrelation method and covariance method are well known as one of linear prediction analysis methods. Both aim to minimize the energy of prediction errors. As an example of using this linear predictive analysis method for a speech acoustic signal, a speech predictive coding device 130 is shown in FIG. 14 and its operation will be described.
Speech predictive coding apparatus 130 divides discrete time-series digital signal x _i (i: integer, hereinafter referred to as digital signal x _i ) input from input terminal 132 into frames for each predetermined number of samples N. The frame division unit 134 performs linear prediction analysis of the digital signal x _i for each frame and calculates p linear prediction coefficients α _j , (j = 0,..., P) to obtain the linear prediction error signal d _i . The linear prediction analysis unit 136 to output, the error encoding unit 138 to encode the linear prediction error signal d _i output from the linear prediction analysis unit 136, and the linear prediction coefficient α _j output from the linear prediction analysis unit 136 are encoded. And a coefficient encoding unit 140.

線形予測誤差信号系列ｄ_ｉを行列で書くと式（２）のようになる。
Ｄ＝Ｘ−ＹＡ 式（２） The linear prediction analysis unit 136 is an all-pole type (autoregressive type forward prediction), linear prediction coefficient calculation unit 136a that calculates p linear prediction coefficients α _j that minimize the linear prediction error signal d _i , and linear a linear prediction coefficients alpha _j calculated by the prediction coefficient calculation unit 136a, a digital signal x and the linear prediction unit 136b that calculates a prediction value from the _i, digital signal by subtracting the predicted values from x _i the linear prediction error signal d _i It is comprised with the subtraction part 136c which produces | generates.
The linear prediction analysis unit 136 obtains a linear prediction error signal d _i for the digital signal x _i by using p linear prediction coefficients α _j as shown in Expression (1).

When the linear prediction error signal sequence d _i is written in a matrix, it becomes as shown in Equation (2).
D = X-YA Formula (2)

Here, D is a vector of N rows and 1 column of the linear prediction error signal sequence (Equation (3)), A is a vector of N rows and 1 column of the linear prediction coefficient sequence (Equation (4)), and X is N of the digital signal sequence. It is a vector of row 1 column (formula (5)). Each is shown as a transposed matrix. Y is a matrix of N rows and P columns (formula (6)).
D ^T = (d ₀ ,..., D _N−1 ) Formula (3)
A ^T = (α ₁ ,..., Α _p ) Equation (4)
X ^T = (x ₀ ,..., X _N−1 ) Expression (5)

線形予測係数算出部１３６ａは、式（７）に示す線形予測誤差信号ｄ_ｉの二乗和Ｊを最小にする線形予測係数α_ｊを計算する。

線形予測誤差信号ｄ_ｉを最小にするためには、Ｊについてα_ｊ（ｊ＝１，２，‥‥，ｐ）に関する偏微分を０とおいて解けばよい（式（８））。つまりｐ個の連立方程式を解く。

そのｐ個の連立方程式の解は、式（６）に示すＮ行Ｐ列の行列Ｙの転置行列Ｙ^Ｔを使った線形予測係数列Ａの未知ベクトルを推定する正規方程式（式（９））で与えられる。

The linear prediction coefficient calculation unit 136a calculates a linear prediction coefficient α _j that minimizes the sum of squares J of the linear prediction error signal d _i shown in Expression (7).

In order to minimize the linear prediction error signal d _i , the partial differentiation with respect to α _j (j = 1, 2,..., P) with respect to J may be set to 0 (formula (8)). That is, p simultaneous equations are solved.

Solution of the p number of simultaneous equations, normal equations for estimating the unknown vector of linear prediction coefficient string A using a transposed matrix Y ^T of the matrix Y of N rows P columns shown in Equation (6) (Equation (9)) Given in.

Y ^T X = Y ^T YA Formula (9)
A linear prediction coefficient sequence A (α _j : j = 1,..., P), which is a least square solution obtained from the normal equation (equation (9)), can be obtained by equation (10).
A = (Y ^T Y) ⁻¹ Y ^T X Formula (10)
Linear prediction unit 136b calculates the predictive value ^ x _i In this way, the linear prediction coefficient alpha _j obtained by the convolution shown in equation (11) in a digital signal x _i calculation.

減算部１３６ｃは、ディジタル信号ｘ_ｉから整数に四捨五入された予測値＾ｘ_ｉを引き算して線形予測誤差信号ｄ_ｉを生成する。

Subtraction unit 136c generates a linear prediction error signal d _i by subtracting the prediction value ^ x _i from digital signal x _i is rounded to an integer.

The linear prediction error signal d _i generated by the subtraction unit 136c is entropy-coded by, for example, Rice coding or Huffman coding in the error coding unit 138, and is output as a linear prediction error code.
The linear prediction coefficient α _j calculated by the linear prediction coefficient calculation unit 136a is also encoded by the coefficient encoding unit 140 and output.
The encoded linear prediction error code and the linear prediction coefficient code are transmitted to a decoder (not shown), and include the calculation shown in Expression (12) in the decoder (including rounding to the same integer as Expression (11). The digital signal x _i is regenerated by changing the integer number).

以上説明した音声予測符号化装置１３０と復号器の動作を模式的に図１５に示す。ディジタル信号ｘ_ｉは、音声アナログ信号をディジタル信号ｘ_ｉに変換する際のサンプリング周期の時間間隔を空けた離散的時系列情報として音声予測符号化装置１３０に入力される。線形予測係数算出部１３６ａは、過去のフレーム内のＮ個（作図の都合により図１５ではｐ個のみ表記）のディジタル信号ｘ_ｉから、次のフレームのディジタル信号ｘ_ｉを予測するｐ個の線形予測係数α_ｊを計算する。線形予測部１３６ｂは、そのｐ個の線形予測係数α_ｊと順次入力されるディジタル信号ｘ_ｉとを畳み込み演算して予測値＾ｘ_ｉを生成する。減算部１３６ｃは、ディジタル信号ｘ_ｉから予測値＾ｘ_ｉを引き算して線形予測誤差信号ｄ_ｉを出力する。

The operations of the speech predictive coding apparatus 130 and the decoder described above are schematically shown in FIG. Digital signal x _i is input to the audio prediction encoding apparatus 130 as a discrete time series information at time intervals of a sampling period when converting an audio analog signal into a digital signal x _i. Linear prediction coefficient calculation unit 136a from the digital signal x _i of N in a past frame (only p pieces in FIG. 15 for convenience of drawing notation), p number of linear predicting a digital signal x _i of the next frame A prediction coefficient α _j is calculated. Linear prediction unit 136b generates the p number of the predicted value and the digital signal x _i convolution operation to sequentially input the LPC coefficients alpha _j ^ x _i. Subtraction unit 136c outputs the linear prediction error signal _{d i} by subtracting the prediction value ^ _{x i} from the digital signal _{x i.}

The linear prediction error signal d _i and the linear prediction coefficient α _j are encoded and transmitted to the decoder. In the decoder, the digital signal output x _si is reproduced from the p linear prediction coefficients α _j and the linear prediction error signal d _i sequentially input. This operation is repeated after p linear prediction coefficients α _j are calculated for each frame.
Kazuo Nakata, revised voice, 1994, Corona, page 53, appendix 4.2 Takehiro Moriya, Speech Coding, 1998, IEICE, 12-13

However, in the conventional method, the code amount of the entropy-coded linear prediction error signal d _i is not minimized. As is well known, entropy coding is such that the code length is proportional to the logarithm of occurrence frequency. That is, this is a method of reducing the overall code amount by assigning a code with a small number of bits to an input with a high occurrence frequency. In normal speech compression, a Rice code whose code length is proportional to the amplitude is used, and in order to minimize the amount of code, it is necessary to minimize the sum of the absolute values of the amplitudes, and to minimize the square error Even if the linear prediction error signal d _i obtained from the above is entropy-encoded, the code amount is not minimized.
The present invention has been made in view of such a problem. The linear prediction error signal d is approximated by minimizing the absolute value of the amplitude of the linear prediction error signal d _i instead of minimizing the square error. It is an object of the present invention to provide a linear prediction coefficient calculation method capable of reducing the code amount when entropy encoding _i , an apparatus thereof, a program thereof, and a storage medium thereof.

An apparatus for calculating a linear prediction coefficient according to the present invention receives an output signal of a frame division unit that divides a discrete time-series digital signal into frames for each predetermined number of samples, and performs all-pole linear prediction from the time-series signal for each frame. A coefficient is generated, and includes a provisional prediction error generation unit and a prediction coefficient calculation unit.
The provisional prediction error generation unit obtains a linear prediction coefficient for each frame, generates a provisional prediction coefficient, obtains a prediction error signal based on the provisional prediction coefficient, and outputs it as a provisional prediction error signal.
The prediction coefficient calculation unit receives the discrete time-series digital signal for each frame and the provisional prediction error signal as input, and calculates the linear prediction coefficient of each order of a differentiable continuous function that approximates the absolute value of the amplitude of the provisional prediction error signal. A linear prediction coefficient in which the partial differentiation is all 0 is generated.

  According to the present invention, when obtaining a linear prediction coefficient, a continuous function that obtains a prediction error obtained by a conventional criterion for minimizing a square error as a provisional prediction error signal and that can differentiate the absolute value of the amplitude of the provisional prediction error signal. , And the linear derivative of each successive linear prediction coefficient of the continuous function generates a linear prediction coefficient that is all zero, so that the code amount is reduced from the code amount of the prediction error based on the conventional square error. I can do it.

  Embodiments of the present invention will be described below with reference to the drawings. The same reference numerals are given to the same components in a plurality of drawings, and the description will not be repeated.

FIG. 1 shows a functional configuration example of the first embodiment of the linear prediction coefficient calculation apparatus 10 according to the present invention. The linear prediction coefficient calculation device 10 of the present invention corresponds to the linear prediction coefficient calculation unit 136a constituting the linear prediction analysis unit 136 in the speech prediction encoding device 130 shown in FIG. 14 as an example of the prior art.
The linear prediction coefficient calculation device 10 includes a frame division unit 134 and a linear prediction coefficient calculation unit 50. The linear prediction coefficient calculation unit 50 includes a provisional prediction error generation unit 20 and a prediction coefficient calculation unit 30.
The digital signal x _i , (i: integer) input from the input terminal 132 is input to the provisional prediction error generation unit 20. The provisional prediction error generation unit 20 is the linear prediction analysis unit 136 itself illustrated in FIG. The linear prediction coefficient calculation unit 136a in FIG. 14 is the provisional prediction coefficient calculation unit 22. The linear prediction unit 136b is the provisional linear prediction unit 24. The subtraction unit 136 c is the provisional subtraction unit 26. That is, in the first embodiment, the linear prediction error signal d _i for each frame is obtained by a method of temporarily minimizing the square error.
Linear prediction error signal d _i obtained by the provisional prediction error generation unit 20 is identical to the linear prediction error signal d _i calculated in the prior art.

The frequency distribution of the amplitude of each sample (d ₀ , d ₂ ,..., D _N−1 ) of the provisional prediction error signal d _i tentatively obtained can be approximated by an exponential distribution. The frequency distribution of the amplitude of the provisional prediction error signal d _i is schematically shown in FIG. 2A (hereinafter, the amplitude value is also expressed by the same reference sign d _i ). The horizontal axis in FIG. 2 (a), the amplitude of the provisional prediction error signal d _i, the vertical axis represents its frequency. Frequency distribution of the amplitude of the provisional prediction error signal d _i is the frequency F is higher amplitude is small, the frequency F is decreased the amplitude is large. That is, the logarithm of the frequency can be approximated by an exponential function such as e- ^ax , which is proportional to the amplitude.
The horizontal axis represents time this and schematically represented in a time series on the vertical axis ± amplitude interim prediction error signal d _i, expressed as in FIG. 2 (b). That is, there are many provisional prediction error signals with small amplitude and few provisional prediction error signals with large amplitude.

Therefore, for the purpose of minimizing the code amount, approximation is performed to reduce the code length of the provisional prediction error signal having a small amplitude and to increase the code length of the provisional prediction error signal having a large amplitude. The relationship is schematically shown in FIG. In FIG. 2C, the horizontal axis indicates 0 as the center, one is + amplitude, the other is -amplitude, and the vertical axis indicates the code length. The code length does not become 0 even when the amplitude of the provisional prediction error signal is 0 and has a predetermined value.
If the absolute value of the amplitude increases, the code length increases in proportion to it. The code amount of the entire frame is obtained from the code length approximated in this way, and the linear prediction coefficient is obtained so as to minimize the code amount. When the prediction error is obtained from the linear prediction coefficient thus obtained, the code amount of the entire frame can be minimized.

そこでこの発明では、暫定予測誤差信号の振幅の絶対値を近似する微分可能な連続関数を用いる。この実施例１では、その連続関数の一つとして式（１５）に示す関数を使うことを考える。
Ｅ（ｄ_ｉ）＝２（δ＋ｄ_ｉ ^ｑ）^{（１/ｑ）} 式（１５）
ここでδは定数である。特にｑ＝２の場合が計算上簡便である（式（１６））。 For example, when entropy coding is performed using a Rice code, the code length per sample of the provisional prediction error signal can be approximated by twice the amplitude d _i of the provisional prediction error signal. This is because the code length E (d _i ) of the Rice code has the relationship shown in the equation (13) with respect to the amplitude d _i .
E (d _i ) = 2 | d _i +0.25 | +0.5 Formula (13)
Therefore, the code amount J of the entire frame can be calculated by Expression (14). However, the absolute value of the absolute value is always constant and cannot be differentiated at the origin, so it cannot be used as a minimization standard based on differentiation.

Therefore, in the present invention, a differentiable continuous function that approximates the absolute value of the amplitude of the provisional prediction error signal is used. In the first embodiment, it is considered to use the function shown in Expression (15) as one of the continuous functions.
E (d _i ) = 2 (δ + d _i ^q ) ^{(1 / q)} Equation (15)
Here, δ is a constant. In particular, the case of q = 2 is simple in calculation (formula (16)).

Ｅ（ｄ_ｉ）を振幅ｄ_ｉで微分する（式（１７））。

式（１６）を線形予測係数α_ｊで微分する（式１８）。

式（１６）からｄ_ｉがδより十分大きいときは、２ｄ_ｉで近似出来る。ｄ_ｉがδより小さい時は、式（１７）からｄ_ｉの微分が一次関数２ｄ_ｉで近似できるからｄ_ｉは２次関数で近似できる。
式（１６）のδをδ＝０.５とした符号長の特性を図３に示す。横軸は暫定予測誤差信号ｄ_ｉの振幅、縦軸は符号長Ｅ（ｄ_ｉ）である。振幅＝０の時の符号長は１．４１４、振幅＝±１の符号長は２．４４、振幅＝±２の符号長は４．２４、振幅±７以上の振幅は、２×ｄ_ｉの一次関数である。つまり、暫定予測誤差信号ｄ_ｉの振幅の絶対値に比例した符号長を得ることが出来る。

E (d _i ) is differentiated by amplitude d _i (formula (17)).

Equation (16) is differentiated by the linear prediction coefficient α _j (Equation 18).

When d _i is sufficiently larger than δ from equation (16), it can be approximated by 2d _i . when d _i is less than δ is, _{d i} since differentiation can be approximated by a linear function 2d _i of _{d i} from equation (17) can be approximated by a quadratic function.
FIG. 3 shows the characteristics of the code length when δ in Expression (16) is δ = 0.5. The horizontal axis represents the amplitude of the provisional prediction error signal d _i , and the vertical axis represents the code length E (d _i ). The code length when amplitude = 0 is 1.414, the code length when amplitude = ± 1 is 2.44, the code length when amplitude = ± 2 is 4.24, and the amplitude greater than ± 7 is 2 × d _i It is a linear function. That is, a code length proportional to the absolute value of the amplitude of the provisional prediction error signal d _i can be obtained.

絶対値近似誤差を最小にするためには、Ｊについてα_ｊ（ｊ＝１，２，‥‥，ｐ）に関する偏微分を０とおいて解けばよい（式（２０））。

暫定予測誤差信号ｄ_ｉは、式（１）に示したようにα_ｊの関数である。 The code amount J of the entire frame due to the approximate absolute value approximation error can be calculated by Expression (19).

In order to minimize the absolute value approximation error, the partial differentiation with respect to α _j (j = 1, 2,..., P) with respect to J may be set to 0 (formula (20)).

The provisional prediction error signal d _i is a function of α _j as shown in Equation (1).

That is, p simultaneous equations may be solved for α _j from J = 1 to j = p where dJ / dα _j = 0. This concept is the same as that in the case of minimizing the square error described in the prior art, and p linear prediction coefficients can be obtained by solving the normal equation shown in Equation (9).
Since the linear prediction coefficient obtained in this way minimizes the amplitude of the absolute value approximation error, the code amount when the entropy coding of the prediction error signal obtained based on the linear prediction coefficient is minimized. I can do it.
The linear prediction coefficient calculation apparatus according to the present invention calculates the provisional prediction error signal d _i obtained with the idea of minimizing the square error as shown in the first embodiment as the sum of absolute values of the amplitudes of the provisional prediction error signal d _i. A linear prediction coefficient that approximates and minimizes the sum of absolute values of amplitudes is obtained. A linear prediction coefficient that minimizes the absolute value average value obtained by dividing the sum of the absolute values of the amplitudes by the number of samples N may be obtained. That is, 1 / N for obtaining the average value is a constant, and even if 1 / N is multiplied, the relationship in which the code length is proportional to the absolute value of the amplitude does not change.

The denominator of the coefficient 2d _i / √ (δ + d ² _i ) of the equation (19) obtained by partial differentiation of the code amount J of the entire frame due to the absolute value approximation error of the amplitude of the provisional prediction error signal by the linear prediction coefficient α _j is α _j Therefore, the normal equation of Equation (9) cannot be solved without using time-consuming numerical analysis.
Therefore, the normal equation is multiplied by the weight coefficient matrix W (equation (21)) of only the diagonal component that neglects the samples (d ₁ , d ₂ ,..., D _i ) whose amplitude of the provisional prediction error signal d _i is large. (Formula (20)). The weight is a value that determines how much each sample affects the estimation of the linear prediction coefficient.

重みを考慮した線形予測係数列Ａは式（２２）で求められる。
Ａ＝（Ｙ^ＴＷＹ）^−１Ｙ^ＴＷＸ 式（２２）
式（２２）の形は、従来からの重み付き線形予測の式と同じである。
この重み係数行列Ｗは、図１の予測係数算出部３０の中に破線で示す重み係数行列生成部３２で生成される。重み係数行列Ｗは、式（２１）に示す様に対角成分のみの対角行列である。各対角要素は、振幅大のサンプルの影響度を小さくする目的から、例えば、暫定予測誤差信号ｄ_ｉの振幅の逆数（１/ｄ_ｉ）又は、振幅の平方根の逆数（１/√ｄ_ｉ）が使用できる。
重み係数行列生成部３２は、予測係数算出部３０内の図示しないレジスタに記憶された暫定予測誤差信号ｄ_ｉを、読み出して暫定予測誤差信号ｄ_ｉの逆数若しくは、平方根の逆数を、重み係数行列Ｗの対角要素に割り付ける動作を行う。

The linear prediction coefficient sequence A in consideration of the weight is obtained by Expression (22).
^{A = (Y T WY) -1} Y T WX formula (22)
The form of equation (22) is the same as the conventional weighted linear prediction equation.
This weighting coefficient matrix W is generated by a weighting coefficient matrix generation unit 32 indicated by a broken line in the prediction coefficient calculation unit 30 of FIG. The weighting coefficient matrix W is a diagonal matrix having only diagonal components as shown in Expression (21). For the purpose of reducing the influence of a sample having a large amplitude, each diagonal element is, for example, the reciprocal of the amplitude (1 / d _i ) of the provisional prediction error signal d _i or the reciprocal of the square root of the amplitude (1 / √d _i ) Can be used.
Weighting coefficient matrix generating unit 32, a provisional prediction error signal d _i stored in the register (not shown) in the prediction coefficient calculation unit 30 reads out the inverse of the provisional prediction error signal d _i with or the reciprocal of the square root, the weighting factor matrix The operation of assigning to the diagonal elements of W is performed.

However, even if a weighting coefficient matrix W that neglects a sample with a large amplitude of the provisional prediction error signal d _i is used, an appropriate linear prediction coefficient may not be obtained.
Therefore, the prediction coefficient calculation unit 30 included in the linear prediction coefficient calculation unit 50 illustrated in FIG. 4 as the second embodiment includes a repetition unit 34 that repeatedly performs an operation for obtaining a linear prediction coefficient for each frame. The prediction coefficient calculation unit 30 repeatedly performs an operation for obtaining a linear prediction coefficient until the end determination unit 36 in the repetition unit 34 outputs an end signal for ending the repetition operation. In addition, since the prediction coefficient calculation unit 30 repeatedly calculates linear prediction coefficients, a buffer unit 28 that stores at least one frame of digital signal output from the frame division unit 134 is provided in the provisional prediction error generation unit 20. did. The buffer unit 28 sequentially outputs the digital signals of the frames specified by the repetition unit 34 to the provisional prediction coefficient calculation unit 22, the provisional linear prediction unit 24, and the provisional subtraction unit 26.

The main processing flow of the second embodiment is shown in FIG. The digital signal divided into frames for each predetermined number of samples by the frame dividing unit 134 is stored in the buffer unit 28 in units of frames (step S10 in FIG. 5). The buffer unit 28 stores and outputs at least one frame of digital signal.
When a new frame L is obtained (step S12), the count value K is initialized to 0 (step S14), and the count value K is incremented by 1 (step S16). Normally, the digital signal x _Li of the latest frame L in the buffer unit 28 is sequentially read, and the provisional prediction coefficient calculation unit 22, the provisional linear prediction unit 24, and the provisional subtraction unit 26 constituting the provisional prediction error generation unit 20. (Step S18).
Preliminary prediction error generation unit 20 performs the calculation shown in the equations (8) to (10), to generate a provisional prediction error signal _{d i} (step S20). The prediction coefficient calculation unit 30 to which the provisional prediction error signal d _i is input performs the calculation shown in the above equations (19) to (22), and calculates a linear prediction coefficient that minimizes the code amount of the prediction error signal. (Step S22).

  Until the end determination unit 36 determines that the processing from step S16 to step S22 described above is complete, the provisional prediction error generation unit 20 repeats the operation of recalculating the provisional prediction error based on the prediction coefficient obtained in step 22. (Step S24), an appropriate linear prediction coefficient can be obtained. The end determination unit 36 may determine whether or not the end count unit 36a in the end determination unit 36 determines whether or not the count value K is equal to the predetermined number N. Alternatively, the absolute value sum of the prediction coefficients obtained in step S22 is calculated in step S26, and whether the difference between the previous and current prediction coefficient sums is smaller than the predetermined value Th stored in the register 38 or no longer changes. You may carry out by the result judged by the convergence determination part 36b. Alternatively, the code amount of the prediction error signal in the frame is calculated in step S28 from the linear prediction coefficient obtained in step S22 and the digital signal, and the previous and current prediction error signals are calculated by the code amount determination unit 36c in the end determination unit 36. And the code amount may be compared to determine when the code amount is minimized.

  As described above, it is possible to obtain an appropriate linear prediction coefficient by providing the repetition unit 34 in the prediction coefficient calculation unit 30 and repeatedly obtaining the prediction coefficient. However, in the present invention, the repeating section 34 is not an essential component. As described in the first embodiment, the prediction coefficient that minimizes the code amount of the prediction error signal may be calculated only once using laborious numerical analysis, and the linear prediction coefficient may be used. In that case, a provisional prediction coefficient is generated for each frame, a prediction error signal based on the provisional prediction coefficient is obtained and a provisional prediction error signal is output. There are only two processes, shown in step S22, in which the linear differential coefficients of all the linear predictive coefficients of the differentiable continuous functions that approximate the absolute value of the error amplitude are all zero. .

In Example 2, the idea of multiplying the weighting coefficient matrix W (equation (22)) of only the diagonal component was introduced for the purpose of easily solving the normal equation (equation (9)) for obtaining the linear prediction coefficient. For the purpose of neglecting a sample having a large amplitude of the provisional prediction error signal, each element of the weighting coefficient has shown a relatively simple example such as the reciprocal of the amplitude of the provisional prediction error signal.
Another example in which each element of the weighting factor is calculated by the weighting factor calculating unit 31 will be described as a third embodiment. The weight coefficient calculation unit 31 is indicated by a broken line in the prediction coefficient calculation unit 30 of FIG.
The third embodiment is different from the second embodiment only in that the weight coefficient calculating unit 31 is provided. Other configurations are the same. The weighting coefficient calculation unit 31 is provided with a register for storing a constant (not shown), and is characterized in that a weighting coefficient more complicated than that in the second embodiment can be calculated.

Comparing the conventional equation (8) for obtaining the linear prediction coefficient that minimizes the prediction error with the equation (20) for obtaining the linear prediction coefficient that minimizes the absolute value of the amplitude of the provisional prediction error signal according to the present invention, 1 The only difference is that / √ (δ + d ² _i ) is multiplied. Therefore, by considering this coefficient 1 / √ (δ + d ² _i ) as a weighting coefficient, the absolute value of the amplitude is minimized as in the case of the linear prediction coefficient obtained by the least square method that minimizes the conventional prediction error. It is considered that a linear prediction coefficient is obtained.

生成された重み係数行列Ｗを正規方程式に掛けて、最小二乗解を式（２２）で求めることで、不適当な重み係数になることがなく常に好ましい線形予測係数を求めることが出来る。 Therefore, the weighting factor calculation unit 31 obtains each weighting factor by the calculation shown in Expression (23), and the weighting factor matrix generation unit 32 assigns each calculated weighting factor to a diagonal element.

By multiplying the generated weight coefficient matrix W by a normal equation and obtaining the least square solution by the equation (22), it is possible to always obtain a preferable linear prediction coefficient without becoming an inappropriate weight coefficient.

When the prediction error signal using Rice codes to entropy coding, the code length is given by Equation (13) mentioned above, as shown in FIG. 6, for example the amplitude d _i provisional prediction error signal . The code length is 3 for amplitude 1, code length 2 for amplitude-1, code length 5 for amplitude 2, and code length 4 for amplitude-2. -The code length of the amplitude is one less than the code length of the + amplitude.
As the amplitude of the provisional prediction error signal, for example, a number such as 10,000 (1 × 10,000) is assumed. Therefore, when all the 10,000 amplitudes are subjected to the rice encoding, the code amount becomes an extremely large value of 20,000 bits. turn into. Therefore, when the amplitude is large, a method in which the lower bits are fixed-length encoded and a predetermined bit or more is entropy encoded (variable length encoding) can be considered. FIG. 7 shows a functional configuration example of the fourth embodiment in which the fixed length coding and the variable length coding are combined.

ただし、｛ ｝は切捨てを表わす。
この可変長符号化開始ビットβを求める方法は、ＩＳＯ規格（ＩＳＯ/ＩＥＣ14496-５ 2001/AMD 10 Reference Software
）に例示されている。 The fourth embodiment is different from the third embodiment in that a variable length coding start bit setting unit 70 is provided in the prediction coefficient calculation unit 30. Other configurations are the same as those of the second embodiment.
The variable length coding start bit setting unit 70 includes an absolute value averaging unit 72 and a variable length coding start bit calculation unit 74. The absolute value average unit 72 calculates the absolute value average value τ of the provisional prediction error sequence from the provisional prediction error signal d _i (formula (24)). The absolute value average value τ is input to the variable-length code bit calculation unit 74, and β, which is a variable-length encoding start bit, is calculated based on, for example, Expression (25).

However, {} represents truncation.
The method for obtaining the variable length coding start bit β is defined by the ISO standard (ISO / IEC14496-5 2001 / AMD 10 Reference Software).
).

符号長Ｅ（ｄ_ｉ）の近似式である式（２７）で符号長を計算した例を図８に示す。図８の横軸と縦軸の交点は、２^β未満の振幅を意味し、交点を中心に振幅の絶対値が２^β以上の範囲を±で表わす。縦軸は符号長である。図８は、ライス符号の例を示した図６と対比することで容易に理解出来る。２^β未満の振幅は、β＋１ビットであり縦軸上の○で表わされる。２^βの振幅はβ＋３ビットとなる。図６からも明らかなように、ライス符号では、−側の振幅の符号長が１ビット少ないので、２^β未満の振幅に対して符号長は左右対称ではない。振幅が−２^βの時の符号長は、β＋２ビットとなる。振幅がその倍（−２^β×２）の時の符号長はβ＋４ビットとなる。 Using the variable length coding start bit β thus obtained, the code length of the provisional prediction error signal obtained by, for example, rice-encoding bits of β bits or more can be calculated by Expression (26).
E (d _i ) = 2 | (d _i / 2 ^β ) +0.25 | + β + 0.5 Equation (26)
The code length E (d _i ) can be approximated by equation (27).

FIG. 8 shows an example in which the code length is calculated by Expression (27), which is an approximate expression of the code length E (d _i ). Intersection of the horizontal and vertical axes of FIG. 8 denotes an amplitude of less than 2 ^beta, the absolute value of the amplitude at the center of the intersection represents the range of more than 2 ^beta at ±. The vertical axis is the code length. FIG. 8 can be easily understood by comparing with FIG. 6 showing an example of the Rice code. The amplitude of less than 2 ^beta is represented by ○ on and the vertical axis a beta + 1 bits. The amplitude of 2 ^beta is the beta + 3 bits. As it is apparent from FIG. 6, the Rice code, - since the code length of the amplitude of the side one bit less, code length relative to less than 2 ^beta amplitude is not symmetrical. Code length when the amplitude -2 ^beta becomes beta + 2 bits. Amplitude code length at the time of its multiples (-2 ^β × 2) becomes beta + 4 bits.

式（２７）を線形予測係数α_ｊで微分する（式（２９））。

絶対値近似誤差を最小にするためには、Ｊについてα_ｊ（ｊ＝１，２，‥‥，ｐ）に関する偏微分を０とおいて解けばよい（式（３０））。

ここでも分母の２^β√（δ＋（ｄ_ｉ/２^β＋０．２５）^２）（式（３１））は、定数とみなし、ｄｄ_ｉ/ｄα_ｊは各ｉとα_ｊに対してｘ_ｉ−ｊとなるので、分子第一項のｄ_ｉに対しては、実施例１と同じになる。 E (d _i ) approximated in this way is differentiated by the amplitude d _i (formula (28)).

Equation (27) is differentiated by the linear prediction coefficient α _j (equation (29)).

In order to minimize the absolute value approximation error, the partial differential with respect to α _j (j = 1, 2,..., P) with respect to J may be set to 0 (formula (30)).

Again, the denominator 2 ^β √ (δ + (d _i / 2 ^β +0.25) ² ) (Equation (31)) is regarded as a constant, and dd _i / dα _j is x _i− for each i and α _j . _j , the first term d _i of the molecule is the same as in Example 1.

更に定数０.２５に対するｄｄ_ｉ/ｄα_ｊの成分ｕは各α_ｊ毎に式（３２）で表わされる。

ｕ_ｊを音声信号と仮定すると、交流波形なのでフレーム内の平均値は０と近似出来る。したがって、式（３０）は、実施例１に対して重み係数行列Ｗが異なるだけである。
重み係数行列Ｗは、重み係数算出部３１でフレームの各サンプル毎に式（３３）で重みを計算し、その重みを重み係数行列生成部３２で対角要素に割り付けることで得られる。

この重みは、可変長符号化開始ビットβ以上の暫定予測誤差信号ｄ_ｉに適用される。可変長符号化開始ビットβ未満の符号化ビット数は、振幅の大小とは無関係なので、重みを小さくして軽視すればよい。

Further, the component u of dd _i / dα _{j with} respect to the constant 0.25 is expressed by equation (32) for each α _j .

Assuming that u _j is an audio signal, the average value in the frame can be approximated to 0 because it is an AC waveform. Therefore, the expression (30) is different from the first embodiment only in the weight coefficient matrix W.
The weighting coefficient matrix W is obtained by calculating weights for each sample of the frame by the weighting coefficient calculation unit 31 using Equation (33) and assigning the weights to diagonal elements by the weighting coefficient matrix generation unit 32.

This weight is applied to the provisional prediction error signal d _i that is greater than or equal to the variable length coding start bit β. Since the number of encoded bits less than the variable-length encoding start bit β is irrelevant to the magnitude of the amplitude, it may be neglected by reducing the weight.

However provisional prediction error signal close tentative amplitude to 2 ^beta when the prediction coefficients are updated, since it is conceivable if the amplitude is greater than 2 ^beta, may use the weighting factor γ as shown in Fig. The horizontal axis in FIG. 9 is the amplitude of the provisional prediction error signal, and the vertical axis is the weight γ. The provisional prediction error signal equal to or greater than the variable length coding start bit β is multiplied by the weighting coefficient calculated by Expression (33). That is, the provisional prediction error signal having a large amplitude is multiplied by a small weighting factor so as to be neglected. A provisional prediction error sample less than the variable length coding start bit β is multiplied by a weight of 1. Alternatively, the provisional prediction error less than the variable length coding start bit β may be multiplied by a weighting factor smaller than the weighting factor for the code length 2 ^β as indicated by a broken line in FIG.
Further, instead of calculating the weight for each sample as shown in Expression (33), a discrete weight coefficient for determining a predetermined weight for each code length range as shown in FIG. 10 may be used. . By determining the weight for each predetermined range of the code length, the weight coefficient calculating unit 31 can save time and effort for calculating the weight.
The main processing flow of the fourth embodiment is shown in FIG. In the fourth embodiment, the absolute value of the temporary prediction error signal between step S20 for generating the temporary prediction error signal and step S22 for calculating the prediction coefficient in FIG. 5 showing the main processing flow of the second embodiment. The only difference is that step S80 for calculating the average value τ and step S82 for calculating the variable length coding start bit β from the absolute value average value τ are provided. The process of step S80 is to calculate the above equation (24), and step S82 is to calculate the above equation (25). Therefore, explanation is omitted.

〔simulation result〕
For example, in the MPEG standard, it is decided to use a rice code. Therefore, the simulation is performed on the assumption that the Rice code is used. When the Rice code is used, the code amount of the total code length for each prediction error signal in the frame is proportional to the absolute value of the amplitude of each prediction error signal. Minimizing the sum of the code amounts in the frame can be approximated by minimizing the sum of the absolute values of the amplitudes of the prediction error signals.
In conventional linear prediction, a prediction coefficient is obtained on the basis of minimizing the square error, and a prediction error signal is obtained from the prediction coefficient. Therefore, the sum of the code amounts of the prediction error signal is not necessarily minimized.

Therefore, an example of an extreme input sequence is shown to confirm the effect of the present invention. For example, the number of samples in a frame is 1024 as an input sequence, and the first sample x ₀ = 10, the second sample x ₁ = −1000, and the subsequent samples have an amplitude of 10 or −10 as shown in FIG. Consider a series of alternating lines.
A specific example is shown using d _i = x _i −αx _i−1 . Since the normal first-order linear prediction coefficient α is a first-order correlation coefficient / energy (sum of squares), α = (− 10000−10000 + (− 100) × 1021) / (1000 × 1000 + 10 × 10 × 1023) = − 122100/1102300 = −0.1108
When the prediction error is obtained using the prediction coefficient α, that is, when the above-described equation (1) is calculated and converted into an integer, d ₀ = 10, d ₁ = −1000 + 10 × 0.1108 = −999, d ₂ = 10− ( -1000) × (-0.1108) = - 101, d 3 = -9, d 4 = 9, ‥‥, d 1023 = -9, the square sum that is integer by rounding the _{d i} becomes 1,091,003.

The prediction error is obtained using the approximation that minimizes the sum of the absolute values of the amplitudes of the present invention. When the linear prediction coefficient α is obtained by setting the weighting coefficient W to 0.1, for example, α = (− 10000 × W−10000 × W + (− 100) × 1021) / (1000 × 1000 × W + 10 × 10 × 1023) = − 0.51458 It becomes. The prediction error is obtained with α = −0.5. d ₀ = 10, d ₁ = −1000 + 10 × 0.5 = −995, d ₂ = 10 − (− 1000) × (−0.5) = 490, d ₃ = −5, d ₄ = 5,..., d ₁₀₂₃ = -5, square sum of the _{d i} becomes 1,255,750.
Comparing the sum of squares of prediction errors obtained by the method of minimizing the sum of square errors with the sum of squares of prediction errors obtained by the method of minimizing the sum of absolute values, it is 1255750 to 1091003. The method of minimizing the sum of the absolute values of the invention is larger.
However, when the sum of the absolute values of the prediction errors d _i is compared to a rounded value, the method that minimizes the sum of the square errors is 10299, and the method that minimizes the sum of the absolute values is 6600. Thus, the code amount obtained by the method of minimizing the sum of the absolute values of the provisional prediction errors (prediction errors) of the present invention is smaller. Therefore, the amount of code encoded by Rice is also smaller in the present invention.

Next, FIG. 13 shows the result of comparison between the conventional method for minimizing the sum of squared errors and the code amount according to the present invention for minimizing the sum of absolute values while changing the prediction order. The horizontal axis in FIG. 13 indicates the predicted order, and the vertical axis indicates the compression rate in%. The compression rate is a value in which the denominator is a code amount in the case of a normal PCM code.
Simulation conditions: 51 files of 30 seconds each with predicted orders of 8th, 16th, 32nd, 64th and sampling rates of 48kHz, 16bit, 48kHz, 24bit, 96kHz, 24bit, 192kHz, 24bit The code amount of the minute was compared.
FIG. 13 shows a comparison of average values of code amounts of 51 files. In the predicted order 8, the compression rate of the present invention is 49.29% compared to the compression rate of the conventional method 49.32%, and the code amount is reduced by about 0.03%. In the prediction order 16th, the compression rate of the conventional method is 47.64% as compared with the compression rate of the conventional method≈47.68%, and the code amount is reduced by about 0.04%. In the predicted order 32, the compression rate of the present invention is 47.45% compared to the compression rate of the conventional method≈47.5%, and the code amount is reduced by about 0.05%. In the predicted order 64, the compression rate of the present invention is 47.41% compared to the compression rate of the conventional method≈47.5%, and the code amount is reduced by about 0.09%.

Thus, it can be seen that the code amount can be reduced at any order. In addition, although the impression that the reduction rate of the code amount with respect to the conventional method is small is received, it is generally difficult to reduce the code amount by 1% in a speech predictive encoding device or the like. Therefore, the reduction of the code amount of 0.03 to 0.09% according to the present invention is a great effect in practical use. Further, as shown in the first specific example, the effect is very large when a small number of samples have a very large value.
In addition to the above embodiments, each apparatus and method according to the present invention is not limited to the above-described embodiment, and can be appropriately changed without departing from the spirit of the present invention. Further, the processes described in the above apparatus and method are not only executed in time series according to the order of description, but also may be executed in parallel or individually as required by the processing capability of the apparatus that executes the process. Good.

In addition, when the processing functions in the above devices are realized by a computer, the processing contents of the functions that the linear prediction coefficient calculation device should have are described by a program. Then, by executing this program on a computer, the processing function in the linear prediction coefficient calculation device is realized on the computer.
The program describing the processing contents can be stored in a computer-readable storage medium. The computer-readable storage medium may be any medium such as a magnetic storage device, an optical disk, a magneto-optical storage medium, and a semiconductor memory. Specifically, for example, as a magnetic storage device, a hard disk device, a flexible disk, a magnetic tape, etc., and as an optical disk, a DVD (Digital Versatile Disc), a DVD-RAM (Random Access Memory), a CD-ROM (Compact Disc Read Only) Memory), CD-R (Recordable) / RW (ReWritable), etc., magneto-optical storage media, MO (Magneto Optical disc), etc., semiconductor memory, EEP-ROM (Electronically Erasable and Programmable-Read Only Memory), etc. Can be used.

The program is distributed by selling, transferring, or lending a portable storage medium such as a DVD or CD-ROM storing the program, for example. Furthermore, the program may be distributed by storing the program in a storage device of the server computer and transferring the program from the server computer to another computer via a network.
For example, a computer that executes such a program first stores a program stored in a portable storage medium or a program transferred from a server computer in its own storage device. When executing the process, this computer reads a program stored in its own storage medium and executes a process according to the read program. As another execution form of the program, the computer may read the program directly from the portable storage medium and execute processing according to the program, and the program is transferred from the server computer to the computer. Each time, the processing according to the received program may be executed sequentially. Also, the program is not transferred from the server computer to the computer, and the above-described processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition. It is good. Note that the program in this embodiment includes information that is used for processing by an electronic computer and that conforms to the program (data that is not a direct command to a computer but has a property that defines the processing of the computer).

  In this embodiment, each apparatus is configured by executing a predetermined program on a computer. However, at least a part of these processing contents may be realized by hardware.

The figure which shows the function structural example of Example 1 of the linear prediction coefficient calculation apparatus 10 of this invention. FIG. 2A is a diagram showing a prediction error signal d _i , FIG. 2A is a diagram showing a relationship between amplitude and frequency, FIG. 2B is a diagram showing a prediction error signal d _i generated in time series, and FIG. ) Is a diagram showing the relationship between amplitude and code length. The figure which shows the change of the code length with respect to the amplitude of a temporary prediction error signal. The figure which shows the function structural example of Example 2 of the linear prediction coefficient calculation apparatus 10 of this invention. FIG. 6 is an operation flowchart of the second embodiment. The figure which shows the example of a rice code | symbol. The figure which shows the function structural example of Example 4 of the linear prediction coefficient calculation apparatus 10 of this invention. The figure which shows an example of the code length calculated by Formula (27). FIG. 10 is a diagram illustrating an example of weighting factors used in the fourth embodiment. The figure which shows the example of a discrete weighting coefficient. FIG. 10 is an operation flowchart of the fourth embodiment. The figure which shows the comparison of the code amount of this invention and the conventional method. The figure which shows an example of a discrete time series digital signal. The figure which shows the structure of the conventional audio | voice prediction encoding apparatus 130. FIG. The figure which shows typically operation | movement of the conventional audio | voice prediction encoding apparatus 130 and a decoder.

Claims (20)

In a linear prediction coefficient calculation apparatus that receives an output signal of a frame division unit that divides a discrete time series digital signal into frames for each predetermined number of samples and generates an all-polar linear prediction coefficient from the time series signal for each frame. ,
A provisional prediction coefficient generating unit that obtains a linear prediction coefficient for each frame to generate a provisional prediction coefficient, obtains a prediction error signal based on the provisional prediction coefficient, and outputs the prediction error signal;
The partial differential of each linear predictive coefficient of a differentiable continuous function that takes the discrete time series digital signal for each frame and the provisional prediction error signal as inputs and approximates the absolute value of the amplitude of the provisional prediction error signal is A prediction coefficient calculation unit that generates linear prediction coefficients that are all 0,
A linear prediction coefficient calculation device comprising linear prediction coefficient calculation means. The linear prediction coefficient calculation apparatus according to claim 1,
The prediction coefficient calculation unit includes a weight coefficient matrix generation unit that generates a weight coefficient matrix,
The prediction coefficient calculation unit generates the linear prediction coefficient by multiplying a normal equation for obtaining each linear prediction coefficient by the weighting coefficient matrix of only a diagonal component. . In the linear prediction coefficient calculation apparatus according to claim 2,
The prediction coefficient calculation unit includes a weight coefficient calculation unit that calculates a weight coefficient from the provisional prediction error signal, and the weight coefficient matrix generation unit uses the weight coefficient calculated by the weight coefficient calculation unit as the weight coefficient matrix. A linear prediction coefficient calculation apparatus characterized by being assigned to a diagonal component. In the linear prediction coefficient calculation apparatus in any one of Claims 1 thru | or 3,
The prediction coefficient calculation unit includes a variable length code including an absolute value average unit that calculates an absolute value average value of the provisional prediction error signal and a variable length encoding start bit calculation unit that determines the number of variable length encoding start bits. A start bit setting unit,
The prediction coefficient calculation unit is configured to calculate each order of a differentiable continuous function that approximates the absolute value of the amplitude of the provisional prediction error signal that is equal to or greater than the number of variable length coding start bits set by the variable length coding start bit setting unit. A linear prediction coefficient calculation apparatus that generates a linear prediction coefficient in which partial differentiation of the linear prediction coefficient is all zero. In the linear prediction coefficient calculation apparatus according to claim 4,
The linear prediction coefficient calculation apparatus according to claim 1, wherein a weight applied to the provisional prediction error signal less than the number of variable length coding start bits is equal to or less than a weight applied to the variable length coding start bit. The linear prediction coefficient calculation apparatus according to any one of claims 1 to 5,
The provisional prediction error generation unit stores at least one frame of the discrete time-series digital signal output from the frame division unit, and sequentially stores the discrete time-series digital signals of designated frames in the plurality of frames. A buffer unit for output;
The prediction coefficient calculation unit further determines a repetition unit that repeats an operation for obtaining a linear prediction coefficient for each frame, and determines whether or not to end the repetition operation of the repetition unit, and outputs an end signal to the repetition unit when the operation ends. A linear prediction coefficient calculation apparatus comprising: an end determination unit that performs the determination. The linear prediction coefficient calculation apparatus according to claim 6,
The linear prediction coefficient calculation apparatus according to claim 1, wherein the end determination unit includes a number determination unit that outputs an end signal when the number of repetitions of the repetition unit reaches a predetermined number. The linear prediction coefficient calculation apparatus according to claim 6,
The end determination unit includes a convergence determination unit that outputs an end signal when the difference between the absolute value sum of the previous linear prediction coefficient obtained repeatedly and the absolute value sum of the current linear prediction coefficient is equal to or less than a predetermined value. An apparatus for calculating a linear prediction coefficient, wherein The linear prediction coefficient calculation apparatus according to claim 6,
The termination determination unit compares a code amount calculation unit that calculates a code amount in a frame each time the repetition unit repeats an operation for obtaining a linear prediction coefficient, and compares the calculated code amount with a code amount calculated so far. And a linear prediction coefficient calculation apparatus comprising a code amount determination unit that outputs an end signal when the code amount becomes minimum. In the linear prediction coefficient calculation method for generating an all-pole prediction coefficient from the time series signal for each frame, using the output signal of the frame division unit that divides the discrete time series digital signal into frames for each predetermined number of samples.
The provisional prediction count generation unit includes a process of obtaining a linear prediction coefficient for each frame, generating a provisional prediction coefficient, obtaining a prediction error signal based on the provisional prediction coefficient, and outputting the prediction error signal as a provisional prediction error signal,
The prediction coefficient calculation unit receives the discrete time series digital signal for each frame and the provisional prediction error signal, and each linear prediction coefficient of a differentiable continuous function that approximates the absolute value of the amplitude of the prediction error signal. A method for calculating a linear prediction coefficient, comprising a step of generating a linear prediction coefficient in which partial differentiation of all becomes zero. The linear prediction coefficient calculation method according to claim 10,
The method of generating a linear prediction coefficient includes a step of multiplying a normal equation for obtaining each linear prediction coefficient by a weight coefficient matrix of only diagonal components. The linear prediction coefficient calculation method according to claim 10,
The method of generating a linear prediction coefficient includes a weighting coefficient calculation process of calculating a weighting coefficient from the provisional prediction error signal. The linear prediction coefficient calculation method according to any one of claims 10 to 12,
The provisional prediction error generation unit stores the discrete time series digital signal over a plurality of frames output from the frame division unit, and the discrete time series digital signal of a specified frame in the plurality of frames. Have the process of outputting sequentially,
The process of generating the linear prediction coefficient includes a process of repeating the operation of obtaining the linear prediction coefficient for each frame until the end determination unit outputs an end signal. The linear prediction coefficient calculation method according to claim 13,
The method of calculating a linear prediction coefficient, wherein the process of outputting the end signal is a process of outputting an end signal when the number of repetitions of the repeating process reaches a predetermined number. The linear prediction coefficient calculation method according to claim 13,
The process of outputting the end signal includes a process of calculating the sum of absolute values of the previous and current linear prediction coefficients obtained repeatedly. When the difference between the sum of absolute values of the previous time and the current time is less than a predetermined value, the end signal is output. A linear prediction coefficient calculation method characterized by being a process of outputting. The linear prediction coefficient calculation method according to claim 13,
The process of outputting the end signal includes a process of calculating the code amount of the prediction error signal in the frame, and ends when the code amount based on the previous and current linear prediction coefficients obtained repeatedly is minimized. A method for calculating a linear prediction coefficient, which is a process of outputting a signal. The linear prediction coefficient calculation method according to any one of claims 13 to 16,
The step of generating the linear prediction coefficient includes a step of determining a variable length coding start bit number from the provisional prediction error signal,
A linear prediction coefficient calculation method comprising a step of calculating a code length from the variable length coding start bit determined in the step of determining the variable length coding start bit number and the provisional prediction error signal. The linear prediction coefficient calculation method according to claim 17,
The process of calculating the code length includes a process of applying a weight to a code having the number of variable length coding start bits or less, wherein the weight is less than or equal to a weight to be applied to the variable length coding start bit. Linear prediction coefficient calculation method.   An apparatus program for causing a computer to function as each apparatus according to claim 1.   A computer-readable storage medium storing the program according to claim 19.

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