JPH1131944A - Stability determination method of iir type periodic temporal change filter - Google Patents

Abstract

PROBLEM TO BE SOLVED: To determine the stability of an IIR(infinite impulse response) type periodic temporal change filter whose output equation is represented by a difference equation. SOLUTION: A filter coefficient Bm' constituting a difference equation of an equivalent temporal change filter is found first by using filter coefficients (a) and (b) constituting the difference equation of the IIR type periodic temporal change filter. Then a matrix A including the filter coefficient Bm' as an element is found and the absolute value of the characteristic value of the matrix A is obtained. Lastly, it is checked whether or not the absolute value of the characteristics value is smaller than 1 to determine the stability of the IIR type periodic temporal change filter.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an IIR (Infinite Impedance) used for a communication system, audio processing and image processing.
ulse Response) type periodic time-varying filter,
The present invention relates to a method for determining the stability of an IIR periodic time-varying filter.

[0002]

2. Description of the Related Art In recent years, studies on periodic time-varying filters have been advanced. In particular, in the field of communication and the field of voice / image processing, frequency scrambling, time division multiplexing / frequency division multiplexing converters, modems, Furthermore, QM
It is being realized in the form of an F (Quadrature Mirror filter) filter. This periodic time-varying filter has an FIR
(Finite Impulse Response) type periodic time-varying filter and IIR type periodic time-varying filter. Generally, the latter IIR-type periodic time-varying filter is realized with a lower order than that of the former FIR-type periodic time-varying filter.
Therefore, the IIR-type periodic time-varying filter can be processed at high speed, and is suitable for a device or a system that requires real-time performance.

As a typical example of such a method of designing an IIR-type periodic time-varying filter, an IIR-type time-invariant filter which can be regarded as equivalent with respect to the operation at each time is prepared by the number of the periods. A time-invariant filter is used to construct an IIR periodic time-varying filter,
There is a method of designing based on the specification in the frequency domain of the R-type periodic time-varying filter. The determination of stability of an IIR periodic time-varying filter designed by this method is equivalent to determining whether all of its constituent IIR time-invariant filters are stable or unstable. Therefore, in order to know the stability of the IIR type time-varying filter, it is necessary to check the stability of all IIR type time-invariant filters. When the number of IIR time-invariant filters is small, much time is not spent, but when the number is large, extremely large time is required.

[0004]

Against the background of the conventional design method and stability determination method for such an IIR-type periodic time-varying filter, the present inventors have realized high-speed processing, low power consumption, and a simple configuration. A new IIR-type periodic time-varying filter to be realized and a method for designing the same are proposed in Japanese Patent Application No. Hei 8-058567. It is an object of the present application to provide a method for determining the stability of an IIR periodic time-varying filter proposed in the above-mentioned application.

[0005]

In order to achieve the above-mentioned object, according to the present invention, an input / output change P times (where P is an arbitrary positive integer) is defined as one cycle, and this one cycle is repeated. And the output equation is based on the input, the output, the filter coefficient for the input, and the stability of the IIR periodic time-varying filter represented by the difference equation using the filter coefficient for the output, from P inputs Means is adopted in which a determination is made based on a stability determination based on an output equation of an equivalent time-invariant filter, which is generated based on the configured input vector and the output vector configured with P outputs. By adopting such means,
The stability determination of the IIR-type periodic time-varying filter can be replaced with the stability determination of the time-invariant filter. This makes it possible to easily determine the stability of the IIR periodic time-varying filter.

[0006]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A specific example of a method for determining the stability of an IIR-type periodic time-varying filter according to the present invention will be described. FIG. 1 is a flowchart of a method for determining the stability of an IIR-type periodic time-varying filter in a specific example, FIG. 2 is a diagram for explaining the stability determination method, and FIG. It is a figure showing the composition of the IIR type time invariant filter equivalent to a variable filter. Prior to the description of the method of determining the stability of the IIR-type periodic time-varying filter in the specific example, the IIR-type periodic time-varying filter itself will be described.

FIG. 4 is a diagram showing the configuration of an IIR-type periodic time-varying filter, and FIG. 5 is a diagram for explaining characteristics of the IIR-type periodic time-varying filter. IIR shown in FIG.
Type periodic time-varying filter is disclosed in Japanese Patent Application No. Hei.
This is an IIR-type periodic time-varying filter (the order is 2) proposed in Japanese Patent Publication No. 67-67. Its input / output characteristics are equivalent to those of a conventional filter including a plurality of time-invariant filter circuits, a plurality of thinning circuits, and a plurality of encoding circuits, but have the same expression (1) in FIG. (Where M represents the order of the filter).

This IIR-type periodic time-varying filter has filter coefficients a (k) and b (k) (where k represents time) with P times as one cycle. More specifically,
The filter coefficients a (k) and b (k) are represented by equations (2) and (3) in FIG.

Next, an outline of a method for determining the stability of the IIR-type periodic time-varying filter of the specific example will be described with reference to the flowchart of FIG. The main point of this stability determination method is that a difference equation representing an IIR time-invariant filter is obtained from a difference equation representing an IIR-type periodic time-varying filter, and the obtained II
The purpose of the present invention is to determine the stability of the R-type time-invariant filter and determine the stability of the IIR-type periodic time-variable filter based on the result of the determination. However, in the process of obtaining the final difference equation (Equation (10) in FIG. 2) of the IIR time-invariant filter, an intermediate difference equation (Equation (3) in FIG. 2) is obtained.

Step S10: A filter coefficient Bm for the output vector Y (km) in the intermediate difference equation of the IIR time-invariant filter is calculated. Step S20: In the final difference equation of the IIR time-invariant filter, a filter coefficient Bm ′ for the output vector Y (km) is calculated. Step S30: A matrix A including the filter coefficient Bm 'as an element is calculated. Step S40: The eigenvalue of the matrix A is calculated based on the theorem of stability determination of the time-invariant filter. Step S50: It is examined whether or not the absolute values of all eigenvalues are smaller than 1. Step S60: If the absolute values of all eigenvalues are smaller than 1, it is determined that the IIR periodic time-varying filter is stable. Step S70: If one of the absolute values of the eigenvalues is greater than 1, it is determined that the IIR periodic time-varying filter is unstable.

Next, a method for determining the stability of the IIR-type periodic time-varying filter will be described with reference to the mathematical formula shown in FIG. First, as shown in equation (1) of FIG.
By summing the output y (k) of the difference equation of the IIR-type periodic time-varying filter shown in the equation (1) into P units,
An input vector Y (k) for each time k is generated. Similarly, as shown in equation (2) of FIG. 2, the outputs u (k) of the difference equation of the IIR-type periodic time-varying filter are grouped in P units, so that the output vector U (k ).

Next, the output vector Y (k) shown in the equation (1) in FIG. 2 is transformed into the equation (3) in FIG. 2 (corresponding to the above-mentioned intermediate difference equation). Where Am,
Bm is a matrix having a size of P × P, and is represented by the matrices shown in Expressions (4) and (5) in FIG. However, the filter coefficients a (k) and b (k) in the equation (1) in FIG. 5, which are the elements of the matrices Am and Bm, are expressed by the equations (6) and (7) in FIG. Further, the integer W is calculated by the equation (8) in FIG.
It is a positive integer that satisfies (9) and is the minimum.

Further, equation (3) in FIG. 2 is replaced with equation (1) in FIG.
0) (corresponding to the final difference equation described above). However, Am ′ and Bm ′ are calculated by the equation (1) in FIG.
1) and expressed by equation (12). A time-invariant filter for realizing this difference equation is a time-invariant filter as shown in FIG. 3 (however, the order is 2). The stability determination of the time-invariant filter obtained in this way is determined using the general theorem of stability determination of the time-invariant filter. Specifically, it is examined whether or not all the absolute values of the eigenvalues of the matrix A having the size of PW × PW represented by the equation (13) in FIG. 2 are smaller than 1.
Here, I is a unit matrix having a size of P × P.

The difference equation shown in equation (13) of FIG.
Is the same input u (k) as the difference equation shown in equation (3).
Give the same output y (k). Therefore, both difference equations have an equivalent relationship. Further, the difference equation shown in equation (3) of FIG. 2 and the difference equation shown in FIG. 5 (1) also have an equivalent relationship. Therefore, performing the stability determination regarding the difference equation of Expression (13) in FIG. 2 is equivalent to performing the stability determination regarding the difference equation of Expression (1) in FIG. Therefore, if the difference equation of equation (13) in FIG. 2 is stable, it is determined that the difference equation of equation (1) in FIG. 5 is also stable, and conversely, the difference equation of equation (13) in FIG. If it is unstable, it is also determined that the difference equation of equation (1) in FIG. 5 is unstable.

As described above, in the method of determining the stability of the IIR-type periodic time-varying filter according to the specific example, first, a difference equation representing the IIR-type periodic time-varying filter (Equation (1) in FIG. 5)
Is a difference equation representing a time-invariant filter (formula (1) in FIG. 2).
0)), and then calculates a filter coefficient Bm ′ necessary for calculating a filter coefficient Bm forming the latter difference equation by using the filter coefficient b (k) forming the former difference equation. Then, a filter coefficient Bm is calculated using the calculated filter coefficient Bm ′, and
A matrix A is calculated using the filter coefficient Bm ', and finally, an eigenvalue is calculated from the calculated matrix A on the basis of the stability determination theorem of the time-invariant filter, and all of the absolute values of the eigenvalues are 1. We examine whether or not. Therefore, the determination of the stability of the IIR-type periodic time-varying filter can be easily performed without determining the stability of the IIR-type periodic time-variable filter itself, which is a complicated method of determination. This can be realized by determining the stability of the time-invariant filter.

Note that such a stability determination method contributes to the design of a stable IIR-type periodic time-varying filter, and in particular, a computer having a finite word length and a DSP (Digital Signal Processor).
This can contribute to the study of the presence or absence of oscillation in the filter.

[Brief description of the drawings]

FIG. 1 is a flowchart of a method for determining the stability of an IIR periodic time-varying filter according to a specific example.

FIG. 2 is an explanatory diagram of a method of determining the stability of a specific example IIR-type periodic time-varying filter.

FIG. 3 is a diagram showing a configuration of a time-invariant filter.

FIG. 4 is a diagram illustrating a configuration of an IIR-type periodic time-varying filter.

FIG. 5 is an explanatory diagram of characteristics of an IIR-type periodic time-varying filter.

[Explanation of symbols]

 S10 Calculate matrix Bm S20 Calculate matrix Bm 'S30 Calculate matrix A S40 Calculate eigenvalue of matrix A S50 Examining absolute value of eigenvalue S60 Judgment of stability S70 Judgment of instability

Claims (4)

[Claims] 1. An input / output change of P times (where P is an arbitrary positive integer) is defined as one cycle, and the one cycle is repeated, and its output equation depends on an input, an output, and a time for the input. IIR expressed by a difference equation using a filter coefficient to be applied and a time-dependent filter coefficient for the output
(Infinite Impulse Response) A stability determination method for determining the stability of a periodic time-varying filter, wherein a difference equation representing an output equation of an equivalent time-invariant filter is generated from the difference equation of the IIR periodic time-varying filter. And a determining step of determining the stability of the IIR periodic time-varying filter by determining the stability of the time-invariant filter, wherein the generating step determines P inputs in the one cycle. An input vector creating step of creating an input vector by using an output vector creating step of creating an output vector by using the P outputs in the one cycle; and a time for the input vector, the output vector, and the input vector. Independent filter coefficients and time independent filter coefficients for the output vector There are represented, stability determination method of IIR periodic time-variant filter; and a difference equation generating step of generating a difference equation of the time-invariant filter. 2. The method according to claim 1, wherein the determining step includes a filter coefficient calculating step of calculating a filter coefficient for an output vector in the difference equation of the time-invariant filter, and a matrix generating step of generating a matrix including the filter coefficient. 2. The method according to claim 1, wherein the stability of the IIR-type periodic time-varying filter is determined. 3. The method according to claim 1, wherein the filter coefficient calculating step uses a time-dependent filter coefficient for the output.
3. The method of claim 2, further comprising the step of calculating the filter coefficients for the output vector.
A method for determining the stability of an IR type periodic time-varying filter. 4. The method according to claim 1, wherein the determining step includes a step of obtaining an absolute value of the eigenvalue of the matrix, and a step of determining whether all of the absolute values of the eigenvalue are smaller than 1. 2. The method for judging the stability of an IIR periodic time-varying filter according to 2.