JP2009063761A - Linear prediction coefficient computing method, device, program, and recording medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a linear prediction coefficient computing method which reduces a residual code amount in linear regression analysis. <P>SOLUTION: The linear prediction coefficient computing method includes; a step (S13) in which a linear prediction coefficient A<SB>0</SB>is initialized; a step (S15) in which a residual vector is computed from an input signal and a linear prediction coefficient A<SB>k-1</SB>at the point of the input; a step (S16) of calculating an auxiliary variable B<SB>k</SB>for which an objective function, which represents a residual code amount with the linear prediction coefficient as the parameter, is equal to an auxiliary function which is equal to or larger than the objective function with the linear prediction coefficient A<SB>k-1</SB>and an auxiliary variable as the parameters; a step (S17) in which a linear prediction coefficient A<SB>k</SB>, which makes the auxiliary function minimum, is computed with the computed auxiliary variable; and a step (S18) in which it is determined whether a repetition satisfies prescribed conditions. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a method for calculating a prediction coefficient of a linear regression model, and is suitable for use in, for example, a speech prediction encoding apparatus that analyzes and encodes a time-series signal such as a speech acoustic signal using a linear prediction analysis technique. The present invention relates to a linear prediction coefficient calculation method, apparatus, program, and recording medium.

  Conventionally, autocorrelation method and covariance method are well known as one of linear prediction analysis methods. Both are aimed at minimizing the energy of the prediction residual (the square integral of the residual amplitude). As an example in which this linear prediction analysis method is used for a speech acoustic signal, a speech predictive coding device 130 is shown in FIG. 10 and its operation will be described.

The speech predictive coding apparatus 130 receives a discrete time series digital signal x _i (i: integer, hereinafter referred to as a digital signal x _i or an input signal x _i ) input from the input terminal 132 for each predetermined number of samples N. A frame dividing unit 134 that divides the frame into frames, performs linear prediction analysis of the digital signal x _i for each frame, calculates p linear prediction coefficients α _j , (j = 0,..., P), and calculates a linear prediction residual ( or a linear prediction analysis unit 136 outputs the linear prediction error) signals d _i, the residual coding unit 138 for coding the linear prediction residual signal linear prediction unit 136 outputs, linear to linear prediction unit 136 outputs The coefficient encoding unit 140 encodes the prediction coefficient α _j .

The linear prediction analysis unit 136 is an all-pole type (autoregressive type forward prediction) and calculates linear prediction coefficients α _j that minimize the energy (sum of squares) of the linear prediction residual signal d _i. A linear prediction unit 136b that calculates a prediction value from the linear prediction coefficient α _j calculated by the coefficient calculation unit 136a and the linear prediction coefficient calculation unit 136a and the digital signal x _i, and linearly subtracts the prediction value from the digital signal x _i composed of the subtraction unit 136c for generating a prediction residual signal d _i.

The linear prediction analysis 136 obtains a linear prediction residual signal from the digital signal x _i by p linear prediction coefficients α _j as shown in Expression (1).

  When the linear prediction residual signal sequence is written in the vector format, the following equation (2) is obtained.

  Where D is an N-dimensional column vector of the linear prediction residual signal sequence (Equation (3)), A is a p-dimensional column vector of the linear prediction coefficient sequence (Equation (4)), and X is the N dimension of the digital signal sequence. Column vector (equation (5)). Each is indicated by transposition. Y is a matrix of N rows and p columns (Formula (6)).

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 residual signal d _i shown in Expression (7).

  In order to minimize J, the partial differential with respect to A can be set to 0 with respect to J (formula (8)). That is, p simultaneous equations are solved.

  The solution of the simultaneous equations is given as a solution of a normal equation (Expression (9)) represented by an N-row p-column matrix Y and a linear prediction coefficient array A shown in Expression (6).

  The linear prediction coefficient sequence A which is a least square solution obtained from the normal equation (equation (9)) can be obtained by equation (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.

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

The linear prediction residual signal d _i generated by the subtraction unit 136c is entropy-coded by, for example, Rice coding or Huffman coding in the residual coding unit 138, and is output as a linear prediction residual 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 residual code and the linear prediction coefficient code are transmitted to a decoder (not shown), and include calculation in equation (12) (including rounding to the same integer as equation (11). In particular, no change is made to the integer), and the digital signal x _i is reproduced.

Kazuo Nakata, revised speech, 1994, Corona, page 53, appendix 4.2 Takehiro Moriya, Speech Coding, 1998, IEICE, 12-13

  However, the conventional linear prediction coefficient calculation method is not an optimization method based on a criterion for directly minimizing the residual code amount, such as obtaining a prediction coefficient so as to minimize the sum of squares of the residual amplitude. . Accordingly, there is not a little room for code amount reduction in the prediction coefficient calculation method. 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, Golomb-Rice code (hereinafter, “Rice code”) whose code length is approximately proportional to amplitude is used, and the code amount g (d) assigned to the residual amplitude value d is

Can be expressed as However, L · ”represents an integerization operator that rounds down the numerical value • to an integer value. R is referred to as a rice parameter. A relational expression between the residual amplitude value d and the code amount g (d) when r = 4 is shown in FIG. If the residual amplitude is d _i (i = 1,..., N), the total code amount required for the residual amplitude code is

In the linear prediction residual signal d _i obtained from the relationship that minimizes the square residual, it is apparent that the total code amount L (A) cannot be minimized. Although it is very difficult to obtain a linear prediction coefficient vector A that directly minimizes L (A), instead of L (A), for example, the absolute value of residual amplitude d _i (i = 1,..., N) is used. Sum of values (L1 norm of residual vector D (L1 norm is also called Manhattan distance))

There is a possibility that the linear prediction coefficient α _j that minimizes the code amount can be smaller than the linear prediction residual signal d _i obtained from the conventional relationship that minimizes the square residual. Designing a function that approximates g (d) in this way is referred to as “modeling the code amount”, and the approximate function g _n (d) is referred to as a “code amount model”. Similarly, L ₀ (A) is referred to as an “all code amount model”. That is, when n = 0, the code amount model indicates an absolute value function, and the entire code amount model indicates the L1 norm of D.

  In general linear regression analysis including linear prediction analysis (linear prediction analysis corresponds to a special case of linear regression analysis), in particular, the absolute value of the elements of the residual vector (corresponding to the linear prediction residual signal in linear prediction analysis) The solution of the regression coefficient that minimizes the sum of values has been studied for a long time. A typical example is based on the simplex method (linear programming), and can be a solution of the linear prediction analysis based on the above-described residual amplitude absolute value sum minimization criterion. However, this method has a feature that the amount of calculation becomes enormous when the size of the explanatory variable matrix and the regression coefficient vector of the linear regression model becomes large, and is not necessarily applied to compression coding that requires immediacy. There was a problem that it was not suitable. On the other hand, as a method for efficiently obtaining a regression coefficient vector that minimizes the L1 norm in a linear regression model, Green has proposed an iterative reweighted least square method (hereinafter referred to as IRLS method). However, the IRLS method has a problem that, although convergence is guaranteed for an objective function that satisfies a specific condition including the L1 norm of the residual vector D, generally, convergence is not guaranteed.

  In the case of r = 1, the L1 norm of the residual vector D is an objective function that exactly matches the code amount, but in reality, r> 1 is often set, and g (d) is As shown in FIG. 11, the function is a stepped function. It should be noted in FIG. 11 that the same code amount is assigned to values within a certain range around d = 0 (d = −8,..., +7 when r = 4). Rather than focusing on making the absolute value of the residual amplitude as small as possible, it is better to give the residual amplitude a moderate “play” and reduce the extreme outliers by the degree of freedom. It shows that the amount can be made smaller. Considering that the residual amplitude is normally concentrated around 0, it should be more preferable that g (d) around d = 0 be a good approximation when modeling the code amount function.

The present invention has been made on the basis of the above problem awareness, and provides a linear prediction coefficient calculation method, apparatus, program, and storage medium capable of reducing the residual code amount in linear regression analysis as compared with the conventional case. It is the purpose. More specifically, for example the L1 norm of the residual vector D and it is designed based on equation code amount was well approximated (13) Model _{g n (d) (n =} 1,2,3) than It is an object to provide a linear prediction coefficient calculation method, apparatus, program, and storage medium that can efficiently calculate a linear prediction coefficient that minimizes the objective function L _n (A) by iterative calculation with guaranteed convergence. And

  In order to solve the above problems, a linear prediction coefficient calculation method according to the present invention includes a coefficient initialization process for initializing a linear prediction coefficient, and a residual calculation process for calculating a residual vector from an input signal and a linear prediction coefficient at that time. Then, using the residual vector calculated in the residual calculation process, the linear prediction coefficient at that time, the objective function representing the residual code amount with the linear prediction coefficient as a parameter, and the linear prediction coefficient and auxiliary variable as parameters An auxiliary variable calculation process that calculates an auxiliary variable that is equal to or substantially equal to the auxiliary function that is equal to or larger than the objective function, and an auxiliary variable calculated in the auxiliary variable calculation process calculates a linear prediction coefficient that makes the auxiliary function substantially minimum. Coefficient calculation process, auxiliary variable calculation process, and repetition of coefficient calculation process to determine whether or not a predetermined condition is satisfied, and if not, residual calculation process, auxiliary variable calculation Characterized in that and a determination step of causing the degree and the coefficient calculating step.

  Another linear prediction coefficient calculation method according to the present invention is characterized in that the auxiliary function is based on a quadratic curve in contact with the objective function.

  In another linear prediction coefficient calculation method according to the present invention, when the objective function has a non-differentiable point, the linear prediction coefficient calculation method has a contact point that smoothly contacts the points before and after the non-differentiable point of the objective function. A function in which the contact section of the objective function is replaced by a function that monotonously increases or decreases in one contact section is defined as an objective function.

  The linear prediction coefficient calculation apparatus according to the present invention includes a coefficient initialization unit that initializes a linear prediction coefficient, a residual calculation unit that calculates a residual vector from an input signal and a linear prediction coefficient at that time, and a residual calculation Using the residual vector calculated by the means, the linear prediction coefficient at that time is equal to the objective function representing the residual code amount using the linear prediction coefficient as a parameter and the linear prediction coefficient and auxiliary variable as parameters. Or an auxiliary variable calculating means for calculating an auxiliary variable that is substantially equal to the auxiliary function to be increased, and a coefficient calculating means for calculating a linear prediction coefficient that makes the auxiliary function substantially minimum by the auxiliary variable calculated by the auxiliary variable calculating means. And determining whether or not the repetition of the auxiliary variable calculation by the auxiliary variable calculation means and the linear prediction coefficient calculation by the coefficient calculation means satisfies a predetermined condition, and if not, Difference calculating means, characterized by comprising a determination means for causing each calculated by the auxiliary variable calculation means and the coefficient calculating means.

  In another linear prediction coefficient calculation apparatus according to the present invention, the auxiliary function is based on a quadratic curve in contact with the objective function.

  Another linear prediction coefficient calculation apparatus according to the present invention has a contact point that smoothly contacts at points before and after the non-differentiable point of the objective function when the objective function has a non-differentiable point. A function in which the contact section of the objective function is replaced by a function that monotonously increases or decreases in one contact section is defined as an objective function.

  The linear prediction coefficient calculation program of the present invention includes a coefficient initialization process for initializing a linear prediction coefficient, a residual calculation process for calculating a residual vector from an input signal and a linear prediction coefficient at that time, and a residual calculation. Using the residual vector calculated in the process, whether or not the current linear prediction coefficient is equal to the objective function that represents the residual code amount with the linear prediction coefficient as a parameter and the linear prediction coefficient and auxiliary variables as parameters Or an auxiliary variable calculating process for calculating an auxiliary variable that is substantially equal to an increasing auxiliary function, and a coefficient calculating process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function by the auxiliary variable calculated in the auxiliary variable calculating process. Determining whether the repetition of the auxiliary variable calculation process and the coefficient calculation process satisfies a predetermined condition, and if not, the residual calculation process, the auxiliary variable calculation process, and the coefficient Characterized in that it contains a command to execute a determination step of causing the exit process by the computer.

  The recording medium of the present invention is calculated by a coefficient initialization process for initializing the linear prediction coefficient, a residual calculation process for calculating a residual vector from the input signal and the linear prediction coefficient at that time, and a residual calculation process. Using the generated residual vector, an objective function that represents the residual code amount with the linear prediction coefficient as a parameter and a linear prediction coefficient at that time, and equal to or larger than the objective function with the linear prediction coefficient and an auxiliary variable as parameters Auxiliary variable calculation process for calculating an auxiliary variable that is substantially equal to the auxiliary function, a coefficient calculation process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function with the auxiliary variable calculated in the auxiliary variable calculation process, and an auxiliary variable It is determined whether the repetition of the calculation process and the coefficient calculation process satisfies a predetermined condition, and if not, the residual calculation process, the auxiliary variable calculation process, and the coefficient calculation process are performed. Records the linear prediction coefficient calculation program, characterized in that it includes a command to execute a constant process by a computer.

  According to the present invention, using a residual vector, an objective function representing a residual code amount using a linear prediction coefficient as a parameter, and a linear prediction coefficient at that time, and an objective function using a linear prediction coefficient and an auxiliary variable as parameters. An auxiliary variable calculation process for calculating an auxiliary variable that is substantially equal to an auxiliary function that increases, or a coefficient calculation process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function by the auxiliary variable calculated in the auxiliary variable calculation process. Determining whether or not the repetition of the auxiliary variable calculation process and the coefficient calculation process satisfies a predetermined condition, and if not, a determination process for performing a residual calculation process, an auxiliary variable calculation process, and a coefficient calculation process. Therefore, by appropriately designing the objective function and auxiliary function, the residual code amount in linear regression analysis can be reduced compared to conventional ones such as those based on the square residual. It is possible to fence.

  Embodiments of the present invention will be described below with reference to the drawings. FIG. 1 is a block diagram showing the configuration of the embodiment of the present invention, and FIG. 2 is a flowchart for explaining the operation thereof.

  The linear prediction coefficient calculation apparatus shown in FIG. 1 includes a residual calculation unit 1, a coefficient initialization unit 2, an auxiliary variable calculation unit 3, a coefficient calculation unit 4, and an iteration number determination unit 5. Each configuration shown in FIG. 1 includes, for example, a computer and its peripheral devices and a program executed by the computer. Each unit is controlled by a central processing unit (not shown).

  The residual calculation unit 1 receives the N-dimensional column vector X (see Equation (5)) of the digital signal sequence, and also receives the p-dimensional column vector A (linear prediction coefficient) (Equation (4) from the coefficient initialization unit 2. (Reference) is received, and an N-dimensional column vector D (residual vector) (equation (3)) of the linear prediction residual signal sequence is calculated and output by equation (2). On the other hand, the coefficient initialization unit 2 initializes the linear prediction coefficient A to a predetermined value and outputs it.

  The auxiliary variable calculation unit 3 uses the auxiliary function such that the objective function becomes equal to the auxiliary function based on the value of each element of the residual vector D calculated by the residual calculation unit 1 and the linear prediction coefficient A at that time. Calculate variables analytically. The objective function is a function representing the residual code amount that is the total code amount necessary for the residual amplitude code with the linear prediction coefficient A as a parameter, and the auxiliary function is an object with the linear prediction coefficient A and the auxiliary variable B as parameters. A function that is equal to or greater than a function. That is, the relationship of objective function ≦ auxiliary function is established between the objective function and the auxiliary function. However, since an error is actually included, the auxiliary variable is calculated so that the objective function and the auxiliary function are substantially equal.

  The objective function and auxiliary function will be described in detail later.

  The coefficient calculation unit 4 analytically calculates a linear prediction coefficient A that substantially minimizes the auxiliary function based on the auxiliary variable B calculated by the auxiliary variable calculation unit 3. The iteration number determination unit 5 determines whether the number of iterations of the calculation of the auxiliary variable B by the auxiliary variable calculation unit 3 and the calculation of the linear prediction coefficient A by the coefficient calculation unit 4 exceeds a predetermined number. If the condition is satisfied (exceeded), the coefficient A is output. If the determination condition is not satisfied, the residual calculation unit 1, the auxiliary variable calculation unit 3, and the coefficient calculation unit 4 are output. Each calculation is performed again.

Next, the operation of the linear prediction coefficient calculation apparatus shown in FIG. 1 will be described with reference to FIG.
The linear prediction coefficient A and the auxiliary variable B are calculated a plurality of times, and the linear prediction coefficient and the auxiliary variable calculated for the kth time are represented by the linear prediction coefficient A _k and the auxiliary variable B _k , respectively.

The calculation process of the linear prediction coefficient A is first started from a process of initializing the repetition count k of the calculation process to “0” by a central processing unit (not shown) that controls the respective units 1 to 5 in FIG. Step S11). Next, it is determined whether k is equal to “0” (step S12). When k is equal to “0” (“Y” in step S12), the coefficient predicting unit 2 initializes the linear prediction coefficient A ₀ to a predetermined value (step S13). On the other hand, when k is not equal to “0” (“N” in step S12) and when the linear prediction coefficient A ₀ is initialized to a predetermined value in step S13, k is incremented by 1 (step S14).

Next, a residual vector D is calculated from the input signal X and the linear prediction coefficient A _k− 1 by the residual calculation unit 1 (step S15). Next, the auxiliary variable calculation unit 3 calculates an auxiliary variable B _k that makes the objective function and the auxiliary function substantially equal with the linear prediction coefficient A _k−1 (step S16).

Next, the coefficient calculation unit 4 calculates a linear prediction coefficient A _k that substantially minimizes the auxiliary function with the auxiliary variable B _k calculated in step S16 (step S17). Here, the linear prediction coefficient A _k is the next calculation result of the linear prediction coefficient A _k .

Next, it is determined whether the number of repetitions k has exceeded the number of repetitions setting value H (step S18). If k does not exceed H (“N” in step S18), the process returns to step S12, and the subsequent processing is executed again as described above. On the other hand, if k exceeds H (“Y” in step S18), the linear prediction coefficient A _k−1 is output (step S19), and the process ends.

  Next, with reference to FIG. 3, the objective function and auxiliary function used in the present embodiment will be first described qualitatively.

If the objective function to be minimized with respect to the parameter Θ = (Θ ₁ ,... Θ _J ) is Φ (Θ),

When satisfying, Φ ⁺ (Θ, m) is defined as an auxiliary function of the objective function Φ (Θ), and m = (m ₁ ,..., _{M L} ) is defined as an auxiliary variable. The right side of the above equation means Θ with m as a variable and the lower limit of Φ ⁺ (Θ, m) for m, and the symbol “+” indicates that Φ ⁺ is a closure of Φ.

From the above definition, the following inequality (F1) is established between the objective function and the auxiliary function. Therefore, as shown in FIG. 3, the objective function Φ (Θ ⁽ⁱ⁾ ) is usually smaller than the auxiliary function Φ ⁺ (Θ ⁽ⁱ⁾ , m ⁽ⁱ⁾ ). FIG. 3 shows that the objective function Φ (Θ) and the auxiliary function Φ ⁺ (Θ, m) decrease in the direction opposite to the direction of the arrow. The variable i represents the number of parameter updates. For example, i + 1 represents parameter update one time after i.

When the auxiliary variable m is updated so as to minimize the auxiliary function Φ ⁺ (Θ ⁽ⁱ⁾ , m ⁽ⁱ⁾ ), the auxiliary function Φ ⁺ (Θ ⁽ⁱ⁾ , m ^{(i + 1)} ) after the parameter update and An equal sign holds between the objective function Φ (Θ ⁽ⁱ⁾ ). Here, when the parameter Θ is updated so as to minimize the auxiliary function Φ ⁺ (Θ ⁽ⁱ⁾ , m ^{(i + 1)} ), the objective function Φ (Θ ⁽ⁱ⁾ ) also changes as the parameter Θ changes. Will do. However, the inequality (F1) is established between the updated auxiliary function Φ ⁺ (Θ ^{(i + 1)} , m ^{(i + 1)} ) and the objective function Φ (Θ ^{(i + 1)} ). The function Φ (Θ ^{(i + 1)} ) is smaller than the auxiliary function Φ ⁺ (Θ ^{(i + 1)} , m ^{(i + 1)} ), that is, it automatically drops as the parameter is updated. .

  If the above update is repeated, the objective function Φ (Θ) can be indirectly reduced. The optimization algorithm of this embodiment is one of iterative optimization methods. Another example of the iterative optimization method is an EM algorithm (Expectation Maximization algorithm).

  The requirements for designing an auxiliary function that satisfies the relationship between the objective function and the auxiliary function are as follows. (1) Design auxiliary functions so that objective function ≤ auxiliary function. (2) Ensure that the auxiliary variable when the equal sign holds is obtained analytically. (3) Ensure that Θ that minimizes the auxiliary function is obtained analytically.

  Next, a specific configuration when the iterative optimization method using the auxiliary function is applied to the present embodiment will be described. An input signal X and a linear prediction coefficient A are respectively N-dimensional and P-dimensional vectors, and Y is a matrix of N rows and P columns.

  think of. Here, D is a probability vector (residual (error) vector) representing noise. Based on this, the problem of obtaining the maximum likelihood estimate of the linear prediction coefficient A will be considered. If the elements of the residual vector D are independent Gaussian noise, this problem is the L2 norm (the sum of squares of the elements of the residual vector D (L2 norm is also called Euclidean distance)).

  Is a problem with respect to the linear prediction coefficient A, and if N ≧ p

  Is the optimal solution for the linear prediction coefficient A.

On the other hand, the first problem is to find a linear prediction coefficient A that minimizes the sum of absolute values of each element of the residual vector D. That is, the code amount model g ₀ (d) is set as an absolute value function | d |

  And want to obtain a linear prediction coefficient A that minimizes this.

There is a linear prediction model as an example of a special case of the linear regression model expressed by Equation (16). The linear prediction model is a model in which a sample x _i of a digital signal at time i is expressed by a linear combination of p samples x _i−1 , x _i−2 ,..., X _ip from time i−1.

It is also called autoregressive model. Based on the above model, the problem of estimating p linear prediction coefficients (elements) α _j of a linear prediction model that best represents _N consecutive sample sequences x ₁ , x ₂ ,. This is called predictive analysis. Corresponding to equation (16), the linear prediction model is

  It is equivalent to placing.

  The feature of this embodiment is a method of minimizing various total code amount models by an efficient iterative estimation algorithm using an auxiliary function (hereinafter referred to as an auxiliary function method). Therefore, first, the principle of the auxiliary function method (the principle of the auxiliary function method itself is an existing one) is shown below in accordance with the present embodiment, and then, first, the minimization of Expression (19) is performed. A method of performing the above by the auxiliary function method will be described.

To the objective function L _n (A) to the linear prediction coefficients A and parameter, when _{L n (A) ≦ L n} + (A, B) ^{_{holds, L n + (A, B}} ) that the L _n ( A) is called an auxiliary function, and B is called an auxiliary variable.

  From the above definition, the following theorem holds.

Theorem 1: The objective function L _n (A) obtains B that minimizes the auxiliary function L _n ⁺ (A, B) with respect to A and L _n (A) = L _n ⁺ (A, B). It can be decreased monotonically by repeating the steps.

Proof: _Let B be L such that L _n (A ₀ ) = L _n ⁺ (A ₀ , B), that is, B where the objective function and auxiliary function are equal,

And That is, A that minimizes the auxiliary function L _n ⁺ (A, B) is represented by A.

Change from A = A ₀ , B = B ₀ to A = ^ A, B = ^ B to show that each update of A and B indicated by Theorem 1 decreases the objective function L _n (A) Confirm that L _n (A) is not increased.

Obviously L _n (A ₀ ) = L _n ⁺ (A ₀ , ^ B), L _n ⁺ (A ₀ , ^ B) ≧ L _n ⁺ (^ A, ^ B), and further by definition, Since L _n ⁺ (^ A, ^ B) ≧ L _n (^ A), L _n (A ₀ ) ≧ L _n (^ A) is finally obtained (end of proof).

  The key to applying the auxiliary function method to an optimization problem is whether or not an auxiliary function that satisfies the following points can be designed.

Requirement 1: B that satisfies L _n (A) = L _n ⁺ (A, B) is analytically obtained.
Requirement 2: A that minimizes the auxiliary function L _n ⁺ (A, B) is analytically determined.

  If this point can be satisfied positively, the effectiveness of the application to the optimization problem is great. In designing an auxiliary function that satisfies the above properties in the minimization problem of the entire code quantity model, it never becomes smaller than the code quantity model (objective function) (from the definition of the auxiliary function) and touches the code quantity model. (From Requirement 1) The basic policy of auxiliary function design is to find a quadratic function (from Requirement 2) that minimizes the quadratic coefficient.

  Hereinafter, specific examples of objective function and auxiliary function design will be described.

[First embodiment]
In this embodiment, the objective function L (A) is a function g ₀ (d) composed of an absolute value function | d |, and the auxiliary function L ₀ ⁺ (A, B) is an objective function and (b, g ₀ (b)). A quadratic function (parabola) that touches at.

A code amount model represented by an absolute value function as shown in Expression (22) is expressed as g ₀ (d)

The quadratic coefficient is the smallest among the quadratic functions that contact at the point (b, g ₀ (b)) without being smaller than g ₀ (d).

And ∀d holds that g ₀ (d) ≦ h ₀ (d, b). FIG. 4 shows h ₀ (d, b) when g ₀ (d) and b = 0.1, 3, 5. When the linear prediction residual signal d _i is made to correspond to (X) _i − (YA) _i ,

An inequality such as The matrix M and the vector W are collectively expressed as an auxiliary variable B, and the right side of the inequality is set as L ₀ ⁺ (A, B). The matrix M represents the coefficient of the quadratic term of the auxiliary function, and the vector W represents the center position (the same applies hereinafter). However,

  It is. From equation (24)

Therefore, L ₀ ⁺ (A, B) satisfies the definition of the auxiliary function.

Using the above auxiliary functions, the two steps of Theorem 1 are specifically shown. First, when B is fixed, A that minimizes L ₀ ⁺ (A, B) is obtained.

And solving

Get. Since Y ^T MY is a positive definite symmetric matrix, (Y ^T MY) ⁻¹ is preferably obtained by Cholesky decomposition. Next, the auxiliary variable B that satisfies L ₀ (A) = L ₀ ⁺ (A, B) is

It is.

In summary, L ₀ (A) can be monotonously decreased by appropriately setting the initial value of A and repeatedly calculating Equation (29) and Equation (30). Since this optimization problem is a convex programming problem, that is, a problem that minimizes a convex function on a convex set, the convergence solution should match the global optimal solution.

[Second Embodiment]
In this embodiment, the objective function L ₁ (A) is divided into three intervals (for example, intervals a 1, a 2, a 3) by dividing the absolute value function | d | into the non-differentiable point (z = 0), the absolute value function | d | is used as it is for the section a1 and the section a3, and the section a2 touches the dividing points of the sections a1 and a2 and the dividing points of the sections a2 and a3. A function g ₁ (d) with a quadratic curve (parabola: where | Z | <ε) is assumed. The auxiliary function L ₁ ⁺ (A, B) is a quadratic function (parabola) in contact with the objective function at (b, g ₁ (b)).

That is, in this embodiment, a function g ₁ (d) in which the absolute value function is replaced with a quadratic curve in the vicinity of the linear prediction residual signal d = 0 as shown in Expression (31) is considered as the code amount model.

  However, c> 0. That is, the total code amount model is

It is equivalent to placing. Here, a method for obtaining A that minimizes this will be described.

The quadratic coefficient is the smallest among the quadratic functions which are never smaller than g ₁ (d) and touch at the point (b, g ₁ (b)).

And g ₁ (d) ≦ h ₁ (d, b) holds at ｂd. When the linear prediction residual signal d _i is made to correspond to (X) _i − (YA) _i ,

An inequality such as The matrix M and the vector W are collectively expressed as an auxiliary variable B, and the right side of the upper inequality is set as L ₁ ⁺ (A, B). However,

  It is. From equation (34)

Therefore, L ₁ ⁺ (A, B) satisfies the definition of the auxiliary function of L ₁ (A).

Using the above auxiliary functions, the two steps of Theorem 1 are specifically shown. First, when B is fixed, A that minimizes L ₁ ⁺ (A, B) is obtained.

  And solving

Get. Since Y ^T MY is a positive definite symmetric matrix, (Y ^T MY) ⁻¹ is preferably obtained by Cholesky decomposition. Next, the auxiliary variable B that satisfies L ₁ (A) = L ₁ ⁺ (A, B) is

  It is.

To summarize the above, if the initial value of the linear prediction coefficient A is appropriately set and the equations (38) and (39) are repeatedly calculated, the total code amount model L ₁ (A ) Can be monotonously decreased.

[Third embodiment]
In this embodiment, the objective function L ₂ (A) is a function g ₂ (d) representing a hyperbola asymptotic to the absolute value function | d |. The auxiliary function L ₂ ⁺ (A, B) is a quadratic function (parabola) in contact with the objective function at (b, g ₂ (b)).

That is, a function g ₂ (d) as shown in Expression (40) is considered as a code amount model. A graph of the absolute value function g ₀ (d) and the hyperbolic function g ₂ (d) (c = 1) is shown together in FIG.

  That is, the total code amount model is

  It is equivalent to placing. Here, a method for obtaining A that minimizes this will be described.

It is never smaller than g ₂ (d), and the quadratic coefficient is the smallest among the quadratic functions that touch at the point (b, g ₂ (b)).

And g ₂ (d) ≦ h ₂ (d, b) holds at ∀d. FIG. 6 shows g ₂ (d) and h ₂ (d, b) when b = 0, 3, and 5. When the linear prediction residual signal d _i is made to correspond to (X) _i − (YA) _i ,

An inequality such as The matrix M and the vector W are collectively expressed as an auxiliary variable B, and the right side of the upper inequality is set as L ₂ ⁺ (A, B). However,

  It is. From equation (43)

Therefore, L ₂ ⁺ (A, B) satisfies the definition of the auxiliary function of L ₂ (A).

Using the above auxiliary functions, the two steps of Theorem 1 are specifically shown. First, when the auxiliary variable B is fixed, A that minimizes L ₂ ⁺ (A, B) is obtained.

  And solving

Get. Since Y ^T MY is a positive definite symmetric matrix, (Y ^T MY) ⁻¹ is preferably obtained by Cholesky decomposition. Next, the auxiliary variable B that satisfies L ₂ (A) = L ₂ ⁺ (A, B) is

  It is.

To summarize the above, if the initial value of the linear prediction coefficient A is appropriately set and the equations (47) and (48) are repeatedly calculated, the total code amount model L ₂ (A ) Can be monotonously decreased.

  In order to increase the efficiency of the iterative algorithm based on the auxiliary function method described above, it is desirable that the initial value of A is a value obtained by Equation (18).

[Fourth embodiment]
In this embodiment, the objective function L ₃ (A) is divided into a plurality of sections focusing on the non-differentiable points composed of a plurality of line segments, and the section including the non-differentiable points is defined as the preceding and following sections. A function g ₃ (d) representing a quadratic curve (parabola) that touches at the boundary of. The auxiliary function L ₃ ⁺ (A, B) is a quadratic function (parabola) that contacts the objective function at (b, g ₃ (b)).

That is, a piece function g ₃ (d) in which 0th-order, first-order, and second-order functions are smoothly connected piecewise as shown in Expression (49) is considered as the code amount model.

However, c is a constant that can be freely determined within a range satisfying 0 <c ≦ β, and when c → 0, g ₃ (d) gradually approaches g ₀ (d) = | d |. FIG. 7 shows an example of a line segment focused on a point that cannot be differentiated, and FIG. 8 shows an outline of the function g ₃ (d). From the above code quantity model, the total code quantity model is

Given in. It is never smaller than g ₃ (d), and the quadratic coefficient is the smallest among the quadratic functions that touch at the point (b, g ₃ (b)).

And g ₃ (d) ≦ h ₃ (d, b) holds for ∀d (proof omitted). From this inequality and Equation (49), Equation (50), Equation (51),

An inequality such as As in the above embodiment, both the matrix M and the vector W depend on N auxiliary variables b _i (i = 1,..., N), and are collectively referred to as auxiliary variables B. , The right side of the upper inequality is set as L ₃ ⁺ (A, B). However,

  It is. From equation (52)

Therefore, L ₃ ⁺ (A, B) satisfies the definition of the auxiliary function of L ₃ (A).

Using the above auxiliary functions, the two steps of Theorem 1 are specifically shown. First, when the auxiliary variable B is fixed, A that minimizes L ₃ ⁺ (A, B) is obtained.

  And solving

Get. Since Y ^T MY is a positive definite symmetric matrix, (Y ^T MY) ⁻¹ is preferably obtained by Cholesky decomposition. Next, an auxiliary variable B satisfying L ₃ (A) = L ₃ ⁺ (A, B) is a residual (X) _i − (YA) obtained by A obtained in the previous stage as shown in Expression (57). _It is obtained by substituting _i into b _i and calculating the matrix M and the vector W.

From the above, if the initial value of A is set and the equations (56) and (57) are repeatedly calculated, the total code amount model L ₃ (A) given by the equation (50) is monotonously decreased. be able to. In the embodiment, a quadratic curve has been described as an example of a curve that touches before and after the discontinuous point of the objective function, but a curve other than the quadratic curve is used as long as it is a function that monotonously increases or decreases monotonously between two contact points. However, the present invention can be implemented. That is, when the objective function has a non-differentiable point, the objective function has a contact point that touches smoothly at the points before and after the non-differentiable point of the objective function, and the objective function increases or decreases monotonically in these two contact sections. A function obtained by replacing the contact section of the function can be set as an objective function.

[Effects of the invention]
When the method according to the present invention is used, the maximum likelihood solution of the regression coefficient vector (the absolute value sum of the elements of the residual (error) vector) under the assumption that the elements of the error vector independently follow the Laplace distribution in the linear regression model. The regression coefficient vector to be minimized) or its approximate solution can be obtained efficiently. The application scene in which the present invention is particularly effective is in the problem of large-scale linear regression analysis (the size of the explanatory variable matrix and the regression coefficient vector is large) in which the amount of calculation becomes unrealistically enormous in the simplex method. is there. A specific example directly applicable to such an application situation is distortion-free (lossless) compression coding based on linear prediction analysis, as shown in the above-mentioned “problem to be solved by the invention”.

  Other possible problems include the introduction of sparsity as a measure of optimality, such as multi-channel blind source separation based on speech sparsity, sparse coding, and sparse non-negative matrix factorization (elements of some vector or matrix). When all but a few of them are nearly zero, the vector or matrix is said to be "sparse.") Sparse coding and sparse non-negative matrix factorization are methodologies that try to represent the observed signal well by linear combination of the basis by sparse coefficient vectors (only as few basis as possible), such as image recognition and music pitch recognition. It is applied to various fields. Since the sparsity of a vector can be measured by the L1 norm (the sparsity of a matrix can be measured by the sum of the L1 norms of each column vector of the matrix), the present invention is applied in the above application. Is expected to be effective.

[Points of the present invention]
The present invention is a method for calculating a prediction coefficient that reduces the code amount of a prediction error signal in linear prediction analysis. The objective function is obtained by using a function that closely resembles the function of the code amount of the Rice code with respect to the amplitude value. It is a method for efficiently obtaining a prediction coefficient that is designed and minimizes this. In this way,
1. The use of the auxiliary function method (the auxiliary function can be designed based on a quadratic function (quadratic curve) in contact with the objective function according to the basic policy of the auxiliary function design described above),
2. The objective function (a standard for optimization)
(A) A code amount function expressed as a sum of absolute values of elements of a prediction error vector, or
(B) Code amount function designed based on trapezoidal function (formula (46)), or
(C) When the objective function has a non-differentiable point, it has a contact point that touches smoothly at points before and after the non-differentiable point of the objective function, The function that replaces the contact section of the objective function is the objective function,
As a specific example,
(C-1) An approximation function of (a) designed based on a smooth function that is a good approximation of an absolute value function and has no discontinuities.
(C-2) There is an approximation function of (a) designed based on a smooth function that is a good approximation of a trapezoidal function and has no discontinuities,
3. In the case of (c-1) above, as a smooth function approximating the absolute value function, a function (formula (28)) in which only the discontinuity of the absolute value function is replaced with a quadratic curve, or a hyperbola Point that adopts function (formula (37))
4). In the case of (c-2) above, as a smooth function approximating a trapezoidal function, a piecewise function (equation (48)) in which 0th-order, first-order, and second-order functions are smoothly connected is adopted. ,
5). The auxiliary function is designed based on a quadratic curve that touches the above functions (a) to (c),
These are the main points of the present invention.

[Results of evaluation experiment]
In order to confirm the effect of the proposed method of the present invention, an evaluation experiment of compression performance was performed for acoustic signals. The purpose of this experiment is not only to demonstrate the principle of the proposed method, but also how much room is left to reduce the residual code amount in lossless compression coding based on LPC (Linear Predictive Coding). It is to determine whether there is. For the evaluation experiment, 10 stereo music files recorded at a sampling frequency of 44.1 kHz and 16 bits were used as experimental data (total of about 432 MB). The LPC analysis frame length was 2048 samples.

Conventional methods conventional method which performs LPC in which the L2 norm criterion by Levinson-Durbin algorithm, prediction coefficients obtained by the conventional method as an initial value, and the criteria for L ₂ (A) and L ₃ (A) LPC Proposed method 1 and proposed method are respectively performed by the iterative optimization algorithm.
Abbreviated as 2. The number of iterations for proposed method 1 and proposed method 2 is 10. In each method, compression based on inter-channel correlation is not performed. The predicted order of LPC (p in equation (21) was fixed).

Since L ₂ (A) is an objective function in accordance with the Rice code amount as compared with the L ₂ norm, the compression rate of the proposed method 1 is expected to be improved as compared with the conventional method. On the other hand, since L ₃ (A) is an objective function that is more in line with the Rice code amount than L ₂ (A), the compression rate of Proposed Method 2 is expected to be slightly improved compared to Proposed Method 1. Is done.

  FIG. 9 shows the compression rates when the predicted orders are 7, 15, and 31 for the conventional method, the proposed method 1, and the proposed method 2, respectively. The compression rate was calculated as follows. Compression rate (%) = compressed file size / original file size × 100. From FIG. 9, as expected, it was confirmed that the compression rate by the proposed method 2 was the highest, and then the compression rate by the proposed method 1 was high.

  The embodiment of the present invention is not limited to the above, but can be used not only for linear prediction calculation in speech predictive coding but also for general linear regression models, and for example, the units shown in FIG. Or, it can be further divided, or the determination content of the repetition number determination unit 5 can be changed to one that determines whether or not a predetermined condition is satisfied based on the number of repetitions of calculation and the calculation result. Etc. are possible. The linear prediction coefficient calculation apparatus of the present invention can be configured using a computer and a program executed by the computer. The program in that case is distributed via a computer-readable recording medium or a communication line. Is possible.

It is a block diagram which shows the structural example of the linear prediction coefficient calculation apparatus of this invention. 2 is a flowchart for explaining the operation of the configuration of FIG. 1. It is a figure for demonstrating the objective function and auxiliary function which are utilized in the structure of FIG. It is a figure for demonstrating the auxiliary function method in 1st Example of this invention. It is a figure for demonstrating the auxiliary function method in the 3rd Example of this invention. It is a figure for demonstrating the auxiliary function method in the 3rd Example of this invention. It is a figure for demonstrating the auxiliary function method in the 4th Example of this invention. It is a figure for demonstrating the auxiliary function method in the 4th Example of this invention. It is a figure which shows the comparison of the code amount of this invention and the conventional method. It is a figure which shows the structure of the conventional audio | voice prediction encoding apparatus. It is a figure which shows the code amount g (d) allocated with respect to the amplitude value d with a Rice code.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Residual calculation part 2 Coefficient initialization part 3 Auxiliary variable calculation part 4 Coefficient calculation part 5 Iteration number determination part

Claims (8)

A coefficient initialization process to initialize the linear prediction coefficients;
A residual calculation process for calculating a residual vector from the input signal and the linear prediction coefficient at that time;
Using the residual vector calculated in the residual calculation process, the linear prediction coefficient at that time, the objective function representing the residual code amount with the linear prediction coefficient as a parameter, and the objective function with the linear prediction coefficient and auxiliary variable as parameters An auxiliary variable calculation process for calculating an auxiliary variable that is substantially equal to an auxiliary function that is equal to or greater than
A coefficient calculation process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function with the auxiliary variable calculated in the auxiliary variable calculation process,
It is determined whether or not the repetition of the auxiliary variable calculation process and the coefficient calculation process satisfies a predetermined condition, and if not, the residual calculation process, the auxiliary variable calculation process, and the determination process for performing the coefficient calculation process are included. A linear prediction coefficient calculation method characterized by The linear prediction coefficient calculation method according to claim 1, wherein the auxiliary function is based on a quadratic curve in contact with the objective function.   When the objective function has a non-differentiable point, the objective function has a contact point that touches smoothly at points before and after the non-differentiable point of the objective function, and the objective function is a function that monotonously increases or decreases monotonically in these two contact points. The linear prediction coefficient calculation method according to claim 1, wherein a function obtained by replacing the contact section is an objective function. Coefficient initialization means for initializing linear prediction coefficients;
A residual calculation means for calculating a residual vector from the input signal and the linear prediction coefficient at that time;
Using the residual vector calculated by the residual calculation means, the objective function representing the residual code amount using the linear prediction coefficient as a parameter, the linear prediction coefficient as a parameter, and the linear prediction coefficient and auxiliary variable as parameters An auxiliary variable calculating means for calculating an auxiliary variable that is substantially equal to an auxiliary function that is equal to or larger than the function;
Coefficient calculation means for calculating a linear prediction coefficient that makes the auxiliary function substantially minimum with the auxiliary variable calculated by the auxiliary variable calculation means;
It is determined whether the repetition of the calculation of the auxiliary variable by the auxiliary variable calculation unit and the calculation of the linear prediction coefficient by the coefficient calculation unit satisfies a predetermined condition, and if not, the residual calculation unit, the auxiliary variable calculation unit, A linear prediction coefficient calculation apparatus comprising: determination means for performing each calculation by a coefficient calculation means. The linear prediction coefficient calculation apparatus according to claim 4, wherein the auxiliary function is based on a quadratic curve in contact with the objective function.   When the objective function has a non-differentiable point, the objective function has a contact point that touches smoothly at points before and after the non-differentiable point of the objective function, and the objective function is a function that monotonously increases or decreases monotonically in these two contact points. 6. The linear prediction coefficient calculation apparatus according to claim 4, wherein a function obtained by replacing the contact section is an objective function. A coefficient initialization process to initialize the linear prediction coefficients;
A residual calculation process for calculating a residual vector from the input signal and the linear prediction coefficient at that time;
Using the residual vector calculated in the residual calculation process, the linear prediction coefficient at that time, the objective function representing the residual code amount with the linear prediction coefficient as a parameter, and the objective function with the linear prediction coefficient and auxiliary variable as parameters An auxiliary variable calculation process for calculating an auxiliary variable that is substantially equal to an auxiliary function that is equal to or greater than
A coefficient calculation process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function with the auxiliary variable calculated in the auxiliary variable calculation process,
It is determined whether the repetition of the auxiliary variable calculation process and the coefficient calculation process satisfies a predetermined condition. If not, the residual calculation process, the auxiliary variable calculation process, and the determination process for performing the coefficient calculation process are performed by a computer. A linear prediction coefficient calculation program characterized by including a command to be executed by. A coefficient initialization process to initialize the linear prediction coefficients;
A residual calculation process for calculating a residual vector from the input signal and the linear prediction coefficient at that time;
Using the residual vector calculated in the residual calculation process, the linear prediction coefficient at that time, the objective function representing the residual code amount with the linear prediction coefficient as a parameter, and the objective function with the linear prediction coefficient and auxiliary variable as parameters An auxiliary variable calculation process for calculating an auxiliary variable that is substantially equal to an auxiliary function that is equal to or greater than
A coefficient calculation process for calculating a linear prediction coefficient that substantially minimizes the auxiliary function with the auxiliary variable calculated in the auxiliary variable calculation process,
It is determined whether the repetition of the auxiliary variable calculation process and the coefficient calculation process satisfies a predetermined condition. If not, the residual calculation process, the auxiliary variable calculation process, and the determination process for performing the coefficient calculation process are performed by a computer. The computer-readable recording medium which recorded the linear prediction coefficient calculation program characterized by including the instruction | command for performing by this.

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