JP2007104528A - Filter design method and filter design system, filter design program and filter - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a filter design method, filter design system, and filter design program for designing a filter having a desired amplitude characteristic and group delay characteristic, which can perform a filter design in consideration of the amplitude characteristic and group delay characteristic. <P>SOLUTION: When designing a filter having the amplitude characteristic and group delay characteristic corresponding to a desired filter factor, this filter design method updates the filter factor so that the amplitude characteristic approximates a previously set range and updates the filter factor so that the group delay characteristic approximates the previously set range. The update of the filter factor is repeated. Thus, the filter factor is optimized so as to be the previously set amplitude characteristic and/or previously set group delay characteristic. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a filter design method, a filter design system, a filter design program, and a filter, and more particularly to a filter design method, a filter design system, a filter design program, and a filter for designing a filter having desired amplitude characteristics and group delay characteristics.

  In recent years, many design methods for reducing the order of a filter by providing an approximate linear phase characteristic only in the passband have been proposed (Non-Patent Documents 1 and 2).

  However, these methods are approximation algorithms for amplitude and phase characteristics, and no direct consideration has been given to the group delay that actually represents the signal delay time.

Also, the amplitude and group delay characteristics are specified at the specified points on the frequency as the compensation characteristics from the actually measured characteristics, and the inverse FFT transform is performed to obtain the filter coefficients of the FIR filter, thereby compensating for the amplitude and group delay distortion. There has been proposed a distortion compensation method using a filter for performing (see Patent Document 1).
Seiji Fukae, Naoyuki Aikawa, Masamitsu Sato, "Complex Chebyshev approximation using the sequential projection method," IEICE theory (A), Vol.J80, A, No.7, pp.1192-1196, 1997 LJKaram and JHMcClellan, "Complex Chebyshev Approximation for FIR Filter Design", IEEE Trans. Circuits and Systems, Vol. 42, NO. 3, pp 207-216, March 1995. JP-A-9-307401

  However, the conventional filter design method does not consider the group delay characteristics, and if the flatness of the group delay characteristics cannot be obtained as a result of the design, problems such as waveform distortion occur during waveform transmission. There was a point.

  The present invention has been made in view of the above points, and an object of the present invention is to provide a filter design method, a filter design system, a filter design program, and a filter that can perform filter design in consideration of amplitude characteristics and group delay characteristics.

  The present invention is a filter design method for designing a filter having desired amplitude characteristics and group delay characteristics, and updates filter coefficients so that the amplitude characteristics approximate a preset range, and group delay characteristics. The filter coefficient is updated so as to approximate the preset range, and the filter coefficient is optimized so that the preset amplitude characteristic and / or the preset group delay characteristic are obtained by repeating the update of the filter coefficient. It is characterized by becoming.

  At this time, the filter coefficient, when the amplitude characteristic is normalized angular frequency ω,

で表されており、
Ｍ(ω)とτd(ω)とはそれぞれ正規化角周波数ωにおける振幅と群遅延としたとき、所望の周波数特性が

It is represented by
M (ω) and τd (ω) are the amplitude and group delay at the normalized angular frequency ω, respectively.

で与えられており、
λ(ω)を振幅特性の最大許容誤差としたときに、

Is given in
When λ (ω) is the maximum allowable error of amplitude characteristics,

を満足させるように更新されるとともに、群遅延特性が振幅特性を自然対数ｌn｛Ｈ(ω)｝をωで微分することで、

And the group delay characteristic differentiates the amplitude characteristic from the natural logarithm ln {H (ω)} by ω,

ただし、−dθ(ω)／d(ω)＝τ(ω)
で表されており、
μ(ω)を群遅延の最大許容誤差としたときに、

However, −dθ (ω) / d (ω) = τ (ω)
It is represented by
When μ (ω) is the maximum allowable group delay error,

を満足するように更新されることを特徴とする。

It is characterized by being updated to satisfy

  Further, the present invention is a filter in which a filter coefficient is determined by the above filter design method.

  According to the present invention, the filter coefficient is updated so that the amplitude characteristic approximates a preset range, the filter coefficient is updated so that the group delay characteristic approximates a preset range, and the filter coefficient is updated. To obtain a filter coefficient that takes into account both the amplitude characteristic and the group delay characteristic by optimizing the filter coefficient so as to obtain a preset amplitude characteristic and / or a preset group delay characteristic. Generation of waveform distortion and the like can be reduced.

〔System configuration〕
FIG. 1 shows a system configuration diagram of an embodiment of the present invention.

  The filter design system 100 according to this embodiment includes an input device 111, a processing device 112, a storage device 113, a memory 114, and a display device 115.

  The input device 111 includes a keyboard and a mouse. For example, the maximum amplitude range and the maximum group delay range are set by operating the input device 111.

  The processing device 112 includes a CPU and the like, and executes processing for filter design based on a filter design program installed in the storage device 113. The storage device 113 is composed of a hard disk drive or the like, and a filter design program is installed in advance.

  The memory 114 is composed of a RAM or the like, and is used as a working storage area for the processing device 112. The display device 115 is composed of a CRT, an LCD, etc., and is controlled by the processing device 112 to display the input window of the maximum amplitude range and the maximum group delay range, the designed filter amplitude characteristics, group delay characteristics, and the like.

[Design program]
FIG. 2 shows a process flowchart of the design program.

  The design program of this embodiment realizes the design of a filter having arbitrary amplitude and group delay characteristics by alternately repeating the amplitude / phase approximation algorithm and the group delay approximation algorithm.

  First, in step S1-1, the input device 111 is operated by the user, and the processing device 112 has a frequency characteristic D (ω), a maximum amplitude range λ (ω), and a maximum group delay range μ (ω). Is set. In step S1-2, the processing device 112 obtains a filter coefficient that satisfies the set frequency characteristic D (ω) and the maximum amplitude range λ (ω).

  Next, in step S1-3, the processing device 112 changes the filter coefficient so as to satisfy the set frequency characteristic D (ω) and the maximum group delay range μ (ω).

  When the processing unit 112 finds a filter coefficient that satisfies both the maximum amplitude range λ (ω) and the maximum group delay range μ (ω) in step S1-4, the processing unit 112 should design the filter coefficient obtained in step S1-5. The filter coefficient of the FIR filter is stored in the storage device 113 or the like.

  By realizing the FIR filter using the filter coefficient stored in the storage device 113, an FIR filter having a desired frequency characteristic satisfying a desired amplitude characteristic and group delay characteristic can be obtained.

[Amplitude phase approximation algorithm]
First, the amplitude phase approximation algorithm will be described.

  In general, complex coefficients

を持つ、フィルタ長ＮのＦＩＲフィルタの周波数特性は、以下の式（１）で表すことができる。

The frequency characteristic of an FIR filter with a filter length N having the following equation (1) can be expressed by the following equation (1).

式（１）においてωは正規化された角周波数である。

In equation (1), ω is a normalized angular frequency.

  Now, the desired frequency characteristics

で与える。式（２）においてＭ(ω)とτd(ω)とはそれぞれ正規化角周波数ωにおける振幅と群遅延とである。

Give in. In equation (2), M (ω) and τd (ω) are the amplitude and group delay at the normalized angular frequency ω, respectively.

  At this time, the filter design problem that approximates the amplitude and phase characteristics is

を満足するフィルタ係数ｈｉを求めることに帰着する。ここで、λ(ω)は所望の周波数特性からの振幅の最大許容誤差を示している。

This results in finding a filter coefficient hi that satisfies Here, λ (ω) represents the maximum allowable error in amplitude from the desired frequency characteristic.

  By the way, when the sequential projection method extended to the complex Chebyshev approximation is used in equation (3), the nth iteration error function of the sequential projection method which is an iterative approximation method is given by the following equation (4).

式（４）において、ｔは複素近似問題を実近似問題に変換するための回転子であり、ｔのとり方によって、

In Equation (4), t is a rotator for converting a complex approximation problem into a real approximation problem.

が保障される。また、(n)は繰り返し回数を示している。

Is guaranteed. (N) indicates the number of repetitions.

  At this time, the update formula of the (n + 1) th filter coefficient hi in the sequential projection method can be expressed as follows.

(For complex coefficients)
First, the case of complex coefficients will be described.

  The coefficient update formula for each of the real part and imaginary part of the complex coefficient hi is

となる。ここで、ωMは所望特性からの誤差が最大となる周波数点であり、次式を満足する。

It becomes. Here, ωM is a frequency point at which the error from the desired characteristic is maximized, and satisfies the following equation.

（実係数の場合）
次に実係数の場合について説明する。

(In the case of real coefficient)
Next, the case of a real coefficient will be described.

In the case of a real coefficient, since the imaginary part of the filter coefficient is all 0, when the real coefficient is represented by hi,
hi = ai. At this time, the (n + 1) th coefficient update formula is

となる。だたし、

It becomes. However,

は式（４）において

Is in equation (4)

とおいたものである。なお、詳細な導出過程については「深江聖司、相川直幸、佐藤正光、"逐次射影法を用いた複素チェビシェフ近似、"信学論（Ａ）、vol.J80,A、No.7,pp.1192-1196、1997」に記載がある。

It is what I put. As for the detailed derivation process, see “Seiji Fukae, Naoyuki Aikawa, Masamitsu Sato,“ Complex Chebyshev Approximation Using Sequential Projection Method ”,“ Science Theory (A), vol. J80, A, No. 7, pp. 1192 -1196, 1997 ".

  The conventionally proposed design method is an approximation algorithm of amplitude and phase characteristics, and group delay is not considered.

  Therefore, the group delay characteristic of the filter obtained using this algorithm may cause a large error from a desired value. Next, consider the coefficient update for approximating the group delay characteristic.

[Group delay approximation algorithm]
Next, the group delay approximation algorithm will be described.

  The group delay τ (ω) of the designed filter can be obtained as follows by differentiating the natural logarithm ln {H (ω)} of the equation (1) by ω.

ただし、−dθ(ω)／d(ω)＝τ(ω)である。

However, −dθ (ω) / d (ω) = τ (ω).

  At this time, the design problem of the filter approximating the group delay characteristic is

を満足するフィルタ係数ｈiを求めることに帰着される。

This results in finding a filter coefficient hi that satisfies.

  Here, the non-negative function μ (ω) is a maximum allowable error from a desired group delay characteristic. However, since Equation (9) is a rational function, it is difficult to solve it using the sequential projection method as it is.

  Therefore, for the denominator H (ω), the amplitude before the coefficient update is approximately

と位相

And phase

を用いる。このとき、群遅延における繰り返し誤差関数は、以下のように与えられる。

Is used. At this time, the repetition error function in the group delay is given as follows.

逐次射影法によるフィルタ係数の更新は、以下のような最小化問題として表すことができる。

The update of the filter coefficient by the sequential projection method can be expressed as the following minimization problem.

ここで、ωMは群遅延誤差が最大となる周波数点であり、次式を満足する。

Here, ωM is a frequency point at which the group delay error is maximum, and satisfies the following equation.

この結果、逐次射影法におけるｎ＋１回目のフィルタ係数ｈiの更新式は、以下のように求めることができる。

As a result, the update formula of the (n + 1) th filter coefficient hi in the sequential projection method can be obtained as follows.

(For complex coefficients)
The coefficient update formulas for the real part and the imaginary part of the complex coefficient hi are as follows.

（実係数の場合）
実係数の場合、フィルタ係数の虚数部はすべて０となるので、実係数をｈiで表すと、
ｈi＝ａiとなる。このとき、ｎ＋１回目の係数更新式は以下のようになる。

(In the case of real coefficient)
In the case of a real coefficient, since the imaginary part of the filter coefficient is all 0, when the real coefficient is represented by hi,
hi = ai. At this time, the (n + 1) th coefficient update formula is as follows.

ただし、

However,

は式（１１）において、

Is in equation (11):

とおいたものとなる。

It will be what you put.

[Design example]
A design example using the above algorithm will be described.

とした場合の低域通過実係数ＦＩＲフィルタの設計例を説明する。

A design example of the low-pass real coefficient FIR filter in the case of the above will be described.

  3 is a diagram showing the maximum allowable error of the design example, FIG. 4 is a diagram showing the amplitude characteristic, and FIG. 5 is a diagram showing the group delay characteristic.

  In FIGS. 3, 4, and 5, design examples ex.1 and ex.2 are

として、本実施例のフィルタ設計方法によりフィルタを設計したものであり、設計例ref.1は

The filter is designed by the filter design method of this embodiment, and the design example ref.1 is

として、特許文献１に記載のフィルタ設計方法により設計を行なった特性を示している。

As shown, the characteristics designed by the filter design method described in Patent Document 1 are shown.

The characteristics of design examples ex.1 and ex.2 designed by the filter design method of the present embodiment are characteristics in which both the amplitude and the group delay are optimized as shown in FIGS. On the other hand, it can be seen that the characteristics of the design example ref.

  Thus, according to the present embodiment, it is possible to obtain a filter coefficient that optimizes both in consideration of both the amplitude and the group delay characteristic. As a result, the flatness of the group delay characteristic can be ensured, and therefore waveform distortion during waveform transmission can be reduced.

It is a system configuration figure of one example of the present invention. It is a process flowchart of a design program. It is a figure which shows the maximum permissible error of a design example. It is a figure which shows an amplitude characteristic. It is a figure which shows a group delay characteristic.

Explanation of symbols

100 Filter Design System 111 Input Device, 112 Processing Device, 113 Storage Device, 114 Memory 115 Display Device

Claims (7)

A filter design method for designing a filter having desired amplitude characteristics and group delay characteristics,
Updating the filter coefficient so that the amplitude characteristic approximates a preset range, and updating the filter coefficient so that the group delay characteristic approximates a preset range;
A filter design method comprising: optimizing the filter coefficient so as to obtain a preset amplitude characteristic and / or a preset group delay characteristic by repeating the update of the filter coefficient. The filter coefficient, when the amplitude characteristic is a normalized angular frequency ω,

It is represented by
M (ω) and τd (ω) are the amplitude and group delay at the normalized angular frequency ω, respectively.

Is given in
When λ (ω) is the maximum allowable error of amplitude characteristics,

Is updated to satisfy
The group delay characteristic is obtained by differentiating the amplitude characteristic from a natural logarithm ln {H (ω)} by ω,

However, −dθ (ω) / d (ω) = τ (ω)
It is represented by
When μ (ω) is the maximum allowable group delay error,

The filter design method according to claim 1, wherein the filter design method is updated to satisfy the following. A filter design system for designing a filter having desired amplitude characteristics and group delay characteristics,
Amplitude characteristic approximating means for updating the filter coefficient so that the amplitude characteristic approximates a preset range;
Group delay characteristic approximating means for updating the filter coefficient so that the group delay characteristic approximates a preset range;
Optimizing the filter coefficient so as to obtain a preset amplitude characteristic and / or a preset group delay characteristic by repeatedly updating the filter coefficient by the amplitude characteristic approximating means and / or the group delay characteristic approximating means And a filter design system characterized by comprising optimization means for converting into a filter. The amplitude characteristic approximating means, when the angular frequency normalized amplitude characteristic is ω,

It is represented by
M (ω) and τd (ω) are the amplitude and group delay at the normalized angular frequency ω, respectively.

Is given in
When λ (ω) is the maximum allowable error of amplitude characteristics,

Update the filter coefficients to satisfy
The group delay characteristic approximating means differentiates the amplitude characteristic of the group delay characteristic from a natural logarithm ln {H (ω)} by ω,

However, −dθ (ω) / d (ω) = τ (ω)
It is represented by
When μ (ω) is the maximum allowable group delay error,

The filter design system according to claim 3, wherein a filter coefficient satisfying the above is updated. On the computer,
A filter coefficient update procedure for updating the filter coefficient so that the amplitude characteristic approximates a preset range, and updating the filter coefficient so that the group delay characteristic approximates a preset range;
A computer that performs an optimization procedure for optimizing the filter coefficient so as to obtain a preset amplitude characteristic and / or a preset group delay characteristic by repeatedly updating the filter coefficient A readable filter design program. In the filter coefficient update procedure, when the amplitude characteristic is ω as the normalized angular frequency,

It is represented by
M (ω) and τd (ω) are the amplitude and group delay at the normalized angular frequency ω, respectively.

Is given in
When λ (ω) is the maximum allowable error of amplitude characteristics,

Update the filter coefficients to satisfy
The group delay characteristic is obtained by differentiating the amplitude characteristic from a natural logarithm ln {H (ω)} by ω,

However, −dθ (ω) / d (ω) = τ (ω)
It is represented by
When μ (ω) is the maximum allowable group delay error,

6. The filter design program according to claim 5, wherein a filter coefficient satisfying the above is updated. A filter whose filter coefficients are determined by the filter design method according to claim 1.

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