JP2014132739A - Digital notch filter - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a digital notch filter that can have all such characteristics at the same time that (1) the center frequency of a stop band can be accurately specified; (2) transmission characteristics can be "1" at a zero frequency and a maximum frequency; (3) an oscillatory transient response can be prevented from being generated; and (4) a filter degree including a minimum degree can be specified.SOLUTION: A digital notch filter transfer function has a zero point of conjugation set almost at exp(±jωh") on a unit circle of a complex plane comprising a real axis and an imaginary axis and also has (n) poles set nearby one point on the real axis, where ωh" is a normalization frequency indicative of the center frequency of the stop band.

Description

  The present invention relates to a design method for a notch filter that removes only a specific frequency component from a signal having a plurality of frequency components. In particular, the present invention relates to a direct design method of an IIR type digital notch filter.

  A request for removing only a specific frequency component from a signal having a plurality of frequency components can be found in various applications. As one of the band rejection filters (also called band stop filters) for this purpose, notch filters have been used for some time. A design method for an analog notch filter as one of the analog band rejection filters (that is, a method for designing a transfer function of an analog notch filter) has already been established.

  On the other hand, there is a demand for removing a specific frequency component from a digital signal generated by sampling an analog signal. The design method of the digital notch filter for this type is roughly divided into two types (refer to the prior art document column described later). The first design method designs a transfer function of an analog notch filter (generally, a rational function using a complex number s of Laplace transform), converts the designed analog notch filter transfer function into discrete time, and transmits the digital notch filter. This is a method of converting into a function (generally a rational function using a complex number z of z conversion). The second design method is a method of designing a transfer function of a digital low-pass filter and performing frequency conversion on the transfer function to obtain a digital notch filter transfer function.

  There are various methods for obtaining a digital filter transfer function through discrete-time conversion of a designed analog filter transfer function (that is, a digital filter design method). Typical methods include a zero-order hold method, a first-order hold method, an impulse invariant method, a Tustin transform method (bilinear transform method), and a Tustin transform method having frequency prewarping. The zero-order hold method, the first-order hold method, and the impulse invariant method relate the step response and impulse response, which are typical time responses, to bring the time response of the digital filter closer to the response of the original analog filter. On the other hand, the frequency response of a digital filter designed with this kind of conversion method does not necessarily approach the time response of an analog filter. The frequency response of the digital notch filter is basically required to have a characteristic of “1” at the zero frequency and the maximum frequency, but this basic characteristic is not necessarily achieved.

  In contrast, the Tustin transform method converts an analog filter transfer function into a digital filter transfer function so as to maintain the frequency response before and after the conversion. However, in the digital filter after conversion, a shift of the center frequency of the stop band (also called stop band) occurs, and the intended frequency component cannot always be accurately removed. This improvement was attempted by the Tustin transform method with frequency prewarping. The analog notch filter is designed to have a conjugate complex pole on the complex plane (s plane) in order to obtain a sharp notch characteristic. Similarly, the digital notch filter obtained by the Tustin transform of the analog notch filter having a conjugate complex pole also has a conjugate complex pole on the complex plane (z plane), and the transient response of the filter output becomes oscillatory.

  The method of obtaining the digital notch filter transfer function by performing frequency conversion on the digital low-pass filter transfer function is an effective method from the viewpoint of frequency response, but has a drawback that the filter order is doubled with frequency conversion.

Tatsuo Higuchi and Masayuki Kawamata: “Digital signal processing for MATLAB”, Shogodo (first edition, 15th printing, March 2012)

  The present invention has been made under the above background, without conversion by discrete time conversion of a conventional analog notch filter (discrete time conversion), and further without frequency conversion of a conventional digital low-pass filter, A new aim is to directly design a digital notch filter. The object of the present invention is (1) an accurate designation of the center frequency of the stop band, which could not be achieved by the conventional design method, and (2) a characteristic in which the transfer characteristic at the zero frequency and the maximum frequency is “1”. (3) Direct generation method of digital notch filter transfer function that can achieve characteristics such as (3) Occurrence of oscillatory transient response, (4) Filter order including minimum order can be specified at the same time Is newly given.

共役の零点をもち、かつ次式

が示す実軸上の１点ａの近傍にｎ個の極をもつことを特徴とする。なお、零点は零とも呼ばれる。To achieve the above object, the invention of claim 1 is an IIR type digital notch filter having a transfer function described as a rational function of z in z-transform, wherein the imaginary unit is j and the time unit is second s. The phase unit is expressed in rad, the sampling period is Ts (s), and the stopband is

It has a conjugate zero, and

It has a feature that it has n poles in the vicinity of one point a on the real axis. The zero point is also called zero.

  A second aspect of the present invention is the IIR digital notch filter according to the first aspect, wherein the positive integer n is 1 or 2.

零点をもち、かつ正確に実軸上の１点ａにｎ個の極（すなわちｎ重極）をもつ場合（最も代表的な場合でもある）を例にとり、請求項１の発明の効果を説明する。本発明によるこの場合のデジタルノッチフィルタの伝達関数は、次式で与えられることになる。

上式におけるｂはフィルタゲインであり、簡単には、次式のように選定すればよい。

Hereinafter, the effect of the invention of claim 1 will be described clearly with reference to the drawings and mathematical expressions. The transfer function of the digital notch filter according to the invention of claim 1 is on the unit circle of the complex plane (z plane).

The effect of the invention of claim 1 will be described by taking, as an example, a case where there is a zero and there are exactly n poles (that is, n multipoles) at one point a on the real axis (which is the most representative case). To do. The transfer function of the digital notch filter in this case according to the present invention is given by:

In the above equation, b is a filter gain, and can be selected simply as the following equation.

次式で与えられる。

（５）式は、請求項１の発明によれば、第▲１▼の効果として、指定周波数ωｈにおいて正確なノッチ特性（アナログ角周波数ωｈに対応した成分の完全除去特性）が得られるという効果が得られることを意味している。The effect (1) of the invention of claim 1 will be described. Specify for the transfer function in equation (3)

It is given by

(5) According to the invention of claim 1, the effect of the first aspect is that an accurate notch characteristic (a complete removal characteristic of a component corresponding to the analog angular frequency ωh) can be obtained at the specified frequency ωh. Is obtained.

（６）式に、請求項１の発明に基づく（２）式を適用すれば、ゼロ周波数と最高周波数とにおける対称性を意味する次の関係を得る。

ここで、（４）式のようなゲイン調整を行なえば、すなわち、（４）式、（６）式、（７）式より、次の関係を得る。

（８）式は、請求項１の発明によれば、第▲２▼の効果として、▲２▼ゼロ周波数と最大周波数での伝達特性がともに「１」となる特性が達成されるという効果が得られることを意味している。The effect (2) of the invention of claim 1 will be described. In order to obtain frequency characteristics at the zero frequency and the maximum frequency, z = 1 and z = −1 may be applied to the transfer function of the expression (3) regarding z. Each is given by:

If the equation (2) based on the invention of claim 1 is applied to the equation (6), the following relationship indicating the symmetry between the zero frequency and the highest frequency is obtained.

Here, if gain adjustment is performed as in equation (4), that is, the following relationship is obtained from equations (4), (6), and (7).

According to the invention of claim 1, the expression (8) has the effect that the characteristic that the transfer characteristic at the zero frequency and the maximum frequency is both “1” is achieved as the effect (2). It means that it is obtained.

  The effect (3) of the invention of claim 1 will be described. As clearly shown by the transfer function of equation (3), according to the invention of claim 1, all the poles of the digital notch filter are on the real axis. The presence of a pole on the real axis means that the digital notch filter does not have a vibration mode. As a result, according to the first aspect of the invention, as the effect (3), there is obtained an effect that the transient response of the digital notch filter cannot be vibrated.

  The effect (4) of the invention of claim 1 will be described. As clearly shown by the transfer function of equation (3), in the digital notch filter according to the invention of claim 1, the value of the positive integer n can be arbitrarily selected. The positive integer n basically determines the order of the filter. Therefore, according to the first aspect of the invention, as the effect (4), there is obtained an effect that a digital notch filter in which the filter order can be arbitrarily designated can be designed.

また、本願発明のデジタルノッチフィルタの伝達関数は、正整数ｎを２に選定する場合には、（２）〜（４）式より次の（１０）式となる。

（９）式、（１０）式より明白なように、正整数ｎを１または２に選定する場合には、対応のデジタルノッチフィルタ伝達関数のｚに関する次数は２次となる。デジタルノッチフィルタを設計可能な最小次数は２次である。これより、請求項２の発明によれば、最小次数によるデジタルノッチフィルタが設計できるようになるという効果が得られる。Next, the effect of the invention of claim 2 will be described. The invention of claim 2 selects the positive integer n as 1 or 2 in the invention of claim 1. When the positive integer n is selected as 1, the transfer function of the digital notch filter of the present invention is the following equation (9) from equations (2) to (4).

The transfer function of the digital notch filter of the present invention is expressed by the following equation (10) from equations (2) to (4) when the positive integer n is selected to be 2.

As is clear from the equations (9) and (10), when the positive integer n is selected to be 1 or 2, the order of the corresponding digital notch filter transfer function with respect to z is the second order. The minimum order in which the digital notch filter can be designed is second order. Thus, according to the second aspect of the invention, it is possible to obtain an effect that a digital notch filter having a minimum order can be designed.

  The figure which shows the rough position of the pole of a filter transfer function in one Example, and a zero point   The figure which shows the frequency response of the filter transfer function in 1 Example.   The figure which shows implementation | achievement of the filter transfer function in one Example   The figure which shows the rough position of the pole of a filter transfer function in one Example, and a zero point   The figure which shows the pole position of the filter transfer function in one Example   The figure which shows the frequency response of the filter transfer function in 1 Example.   The figure which shows implementation | achievement of the filter transfer function in one Example   The figure which shows the rough position of the pole of a filter transfer function in one Example, and a zero point   The figure which shows implementation | achievement of the filter transfer function in one Example   Example of serial connection of two digital notch filters   Example of parallel arrangement of two digital notch filters

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

となる。図１は、（９）式の伝達関数の零点と極の位置の例を図示したものである。The simplest embodiment of the present invention is shown. The simplest embodiment of the present invention is to select a positive integer n as 1. It has already been shown that a typical digital notch filter in this case is given by equation (9). The position a of the single pole on the real axis is exactly the same as the real part of the conjugate zero. sand

It becomes. FIG. 1 shows an example of the zero and pole positions of the transfer function of equation (9).

  The example of the frequency characteristic in a present Example is shown. FIG. 2 shows frequency characteristics when the analog angular frequency ωh of the component to be removed is selected as ωh = 800π (rad / s) and the sampling period is selected as Ts = 0.0001 (s). Due to the characteristics of the digital filter, the frequency display range is limited to the Nyquist frequency range ω = π / Ts = 10000 (rad / s). This frequency range is π if converted to the normalized frequency range. From the figure, the effects (1) and (2) of the present invention of claim 1 are confirmed.

  FIG. 3 shows an implementation example of the digital notch filter when the positive integer n is selected as 1. The zero part 1 (chain line block) in the figure realizes the designated conjugate zero. Moreover, the single pole designated by the pole part 2 (chain line block) is realized. In this embodiment, since the pole is single, there is only a single feedback loop having a coefficient a as the feedback loop as the IIR filter. From this example, the effect (4) of the invention of claim 1 is confirmed. 2 and 3, the effect of the invention of claim 2 is also confirmed.

中辺に基づく正確な関係を示しており，破線は同式右辺の近似直線に基づく関係を示している。二重極は，一般には、共役零点の実数部より原点側にある近似直線よりもさらに原点側に存在する，すなわち二重極の位置ａに関しては次の関係が成立している。

同図には参考までに、正規化周波数の余弦値すなわち正整数ｎを１とした場合の極位置を破線で示した。正整数の値により、一般に極位置は大きく変化することが、本図より確認される。The following is a simple example of the present invention. The next simplest embodiment of the present invention is to select the positive integer n to be 2. It has already been shown that a typical digital notch filter in this case is given by equation (10). FIG. 4 shows an example of the zero and pole positions of the transfer function of equation (10). The position of the double pole on the real axis is closer to the origin than the real part of the conjugate zero. Figure 5 shows the regular

The exact relationship based on the middle side is shown, and the broken line shows the relationship based on the approximate straight line on the right side of the equation. In general, the dipole exists on the origin side further than the approximate straight line on the origin side from the real part of the conjugate zero, that is, the following relation holds for the position a of the dipole.

For reference, the cosine value of the normalized frequency, that is, the pole position when the positive integer n is 1, is shown by a broken line in FIG. It can be confirmed from this figure that the pole position generally varies greatly depending on the value of a positive integer.

  The example of the frequency characteristic in a present Example is shown. FIG. 6 shows frequency characteristics when the analog angular frequency ωh of the component to be removed is selected as ωh = 800π (rad / s) and the sampling period is selected as Ts = 0.0001 (s). Due to the characteristics of the digital filter, the frequency display range is limited to the Nyquist frequency range ω = π / Ts = 10000 (rad / s). This frequency range is π if converted to the normalized frequency range. From the figure, the effects (1) and (2) of the present invention of claim 1 are confirmed.

  FIG. 7 shows an implementation example of the digital notch filter when the positive integer n is selected as 2. The zero part 1 (chain line block) in the figure realizes the designated conjugate zero. Further, the double pole specified by the pole part 2 (chain line block) is realized. Since this embodiment is an example of a double pole, there are two feedback loops as a feedback loop as an IIR filter. From this example, the effect (4) of the invention of claim 1 is confirmed. 2 and 3, the effect of the invention of claim 2 is also confirmed.

  An n-th embodiment of the present invention is shown. It has already been shown that a typical digital notch filter in this case is given by equation (3). FIG. 8 shows an example of the positions of the zeros and poles of the transfer function of equation (3). The position of the n-pole on the real axis specified by equation (2) moves closer to the origin as the order n increases.

  FIG. 9 shows an implementation example of an nth-order digital notch filter. The zero part 1 (chain line block) in the figure realizes the designated conjugate zero. In addition, the n-pole specified by the pole 2 (chain line block) is realized. Since the present embodiment is an example of n-pole, n feedback loops exist as feedback loops as IIR filters.

  In the embodiment described with reference to FIGS. 1 to 9, the two conjugate zeros of the transfer function are accurately arranged at points on the unit circle corresponding to the normalized frequency. According to the present invention, the arrangement positions of the two conjugate zeros of the transfer function do not necessarily have to be exact points corresponding to the normalized frequency. The arrangement position of the conjugate zero point may be in the vicinity of this exact point.

  In the embodiment described with reference to FIGS. 4 to 9, all the poles of the transfer function are accurately arranged at one point a on the real axis specified by the equation (2). According to the present invention, the poles of the transfer function do not necessarily have to be exactly poles on the real axis. The n poles may be disposed in the vicinity of one point a on the real axis, and the n poles may have a slight difference from each other.

  The realization of the filter shown in FIGS. 3, 7 and 9 is only an example of realization of a digital notch filter according to the present invention. It should be pointed out that there are various other methods for realizing the digital notch filter according to the present invention.

  FIG. 10 shows an application example of the digital notch filter according to the present invention. FIG. 2A shows a state in which two digital notch filters according to the present invention are connected in series. The two digital notch filters do not necessarily have to be the same. FIG. 2B shows an example in which two digital notch filters according to the present invention are arranged in parallel and used as a two-input two-output filter.

  The example of the motor drive control apparatus using the digital notch filter of this invention is shown. FIG. 11 shows an embodiment in which a drive control device having a digital notch filter according to the present invention is applied to a synchronous motor as a typical AC motor. Although the main point is in the drive control apparatus provided with the digital notch filter of the present invention, the entire motor drive control system including the drive control apparatus will be described in order to clarify the position of the drive control apparatus in the entire motor drive control system. 13 is a synchronous motor, 14 is a power converter, 3 is a current detector, 4a and 4b are 3 phase 2 phase converters, 2 phase 3 phase converters, 5a and 5b are both vector rotators, 6 is a current controller, 7 is a command converter, 8 is a speed controller, 9 is a digital notch filter (2-input 2-output filter illustrated in FIG. 10B), and 10 is a phase speed estimator. , 11 indicates a coefficient unit, and 12 indicates a cosine sine signal generator. In FIG. 11, various devices other than the electric motor 13 constitute a drive control device. In this figure, a 2 × 1 vector signal is represented by one thick signal line to ensure simplicity.

  The three-phase stator current sampled and detected at the sampling frequency Ts in the current detector 3 (the three-phase current is a digital signal at the time of detection) is fixed in the fixed αβ coordinate system by the three-phase two-phase converter 4a. After being converted into the above two-phase current, the vector rotator 5a is converted into a two-phase current of a quasi-synchronous coordinate system aimed at phase synchronization with a zero phase difference to the rotor phase. Using the digital notch filter 9 from the converted current, a high-frequency component contained in the converted two-phase current is removed to extract a driving current which is a low-frequency component, and this is sent to the current controller 6. The current controller 6 generates a driving two-phase voltage command value on the quasi-synchronous coordinate system so that the driving two-phase current on the quasi-synchronous coordinate system follows the current command value of each phase. Here, the high-frequency voltage command value received from the phase velocity estimator 10 is superposed on the driving two-phase voltage command value, and the superposed and synthesized two-phase voltage command value is sent to the vector rotator 5b. In 5b, the voltage command value for superposition and synthesis on the quasi-synchronous coordinate system is converted into a two-phase voltage command value in the fixed αβ coordinate system and sent to the two-phase three-phase converter 4b. In 4b, the two-phase voltage command value is converted into a three-phase voltage command value and output as a command value to the power converter 14. The power converter 14 generates electric power according to the command value, applies it to the synchronous motor 13 and drives it.

  The phase speed estimator 10 receives the stator current output from the vector rotator 5a, and outputs a rotor phase estimation value, a rotor electric speed estimation value, and a high frequency voltage command value of a constant high frequency ωh. ing. The rotor phase estimation value is converted into a cosine / sine signal by the cosine sine signal generator 12, and then passed to the vector rotators 5a and 5b which determine the quasi-synchronous coordinate system.

As is well known to those skilled in the art, the two-phase current command value on the quasi-synchronous coordinate system is obtained by converting the torque command value through the command converter 7. In the speed controller 8, the rotor speed estimated value (electric speed estimated value), which is one of the output signals from the phase speed estimator 10, is sent through the coefficient unit 11 as the inverse of the pole pair number Np, which is a constant value. It is sent after being multiplied and converted to a machine speed estimate. In the present example of FIG. 5, an example in which the speed control system is configured is shown, and thus the torque command value is obtained as the output of the speed controller 8. As is well known to those skilled in the art, the speed controller 8 is unnecessary when the control purpose is torque control and the speed control system is not configured. In this case, the torque command value is directly applied from the outside.

  The high frequency voltage command value output from the phase velocity estimator 10 is finally input to the electric motor 13 as a high frequency voltage by the power converter 2 as described above. As a result, the stator current that is the response of the high-frequency voltage contains a component of the frequency ωh of the high-frequency voltage command value, that is, a high-frequency component. In the drive control system of FIG. 11, it is required to avoid that the high frequency component is controlled by the current controller 6. The digital notch filter 9 plays a role in avoiding control of the high-frequency component of the stator current. Since the high frequency ωh is already known, if the digital notch filter 9 is configured in the form of two inputs and two outputs as shown in FIG.

  The present invention can be widely used for digital signal processing intended to remove specific frequency components. A typical one is a drive control apparatus for an AC motor that performs rotor phase speed estimation via high-frequency signal application.

1 Zero part 2 Polar part 9 Digital notch filter

Claims (2)

an IIR digital notch filter having a transfer function described as a rational function of z in the z-transform,
The imaginary unit is j, the time unit is seconds s, the phase unit is rad, the sampling period is Ts (s), the target center analog angular frequency of the stopband is ωh (rad / s), and n is positive. An integer,

It has a conjugate zero, and

An IIR type digital notch filter having n poles in the vicinity of one point a on the real axis indicated by. 2. The IIR digital notch filter according to claim 1, wherein the positive integer n is 1 or 2.

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