JPH10308650A - Filter design method and digital filter - Google Patents

Abstract

PROBLEM TO BE SOLVED: To realize an IIR filter which flattens the group delay of a pass area with a small number of taps. SOLUTION: In a CPU 1, a transmission function Hi(z) which makes group delay most flat and has only a pole is obtained. The transfer function Hs(z) or Hp(z) corresponding to the inhibition area or the pass area of the IIR filter is obtained by making an equation so that the inhibition area or the pass area of the IIR filter becomes a Chebyshev characteristic and solving it. The tap coefficient of the IIR filter is obtained based on the transfer function shown by an expression H(z)=Hi(z)Hs(z)Hp(z).

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a filter design method and a digital filter, and more particularly, to a filter design method capable of flattening a group delay in a pass band in an IIR type digital filter with a small number of taps. And digital filters.

[0002]

2. Description of the Related Art A digital filter employs an FIR (Finite Impulse Response) type and an IIR based on its impulse response.
It is classified into IR (Infinite Impulse Response) type. FIR filter (FIR type digital filter)
Has an impulse response of finite length, and an IIR filter (IIR type digital filter) has an impulse response of infinite length.

[0003]

By the way, according to the FIR filter, it is possible to make the group delay constant over the whole band. For example, when the pass band is narrow, If the stopband is wide, it is necessary to set a zero point over the wide stopband.
The number of taps of the filter becomes very large.

On the other hand, with an IIR filter, a narrow pass band can be realized with a small number of taps, but it is difficult to make the group delay constant over the entire band.

Here, even if the group delay is not constant over the entire band, it is generally sufficient if the group delay is flat in the pass band.

The present invention has been made in view of such a situation, and is intended to realize an IIR digital filter for flattening a group delay in a passband with a small number of taps. is there.

[0007]

According to a first aspect of the present invention, there is provided a filter design method comprising: obtaining a first transfer function having only a pole which maximizes a group delay to a maximum value; The corresponding second or third transfer function is determined by solving an equation so that the stopband or passband of the IIR filter has Chebyshev characteristics, and based on the first to third transfer functions, IIR It is characterized in that tap coefficients of a filter are obtained.

[0008] The digital filter according to claim 2 is
A tap coefficient is obtained by the filter design method according to claim 1.

In the filter design method according to the first aspect, a first transfer function having only a pole and maximizing the group delay is determined, and a second transfer function corresponding to a stop band or a pass band of the IIR filter is obtained. The third transfer function is given by II
A tap coefficient of the IIR filter is obtained based on the first to third transfer functions by obtaining and solving an equation so that the stop band or the pass band of the R filter has Chebyshev characteristics.

[0010] In the digital filter according to the second aspect, the tap coefficient is obtained by the filter design method according to the first aspect.

[0011]

FIG. 1 shows an example of the configuration of an embodiment of a filter design apparatus to which the present invention is applied. This filter design apparatus is configured based on, for example, a computer, and flattens the group delay in the passband. II
The tap coefficients of an R-type digital filter are obtained.

That is, a CPU (Central Processing Uniform)
t) 1 is also the auxiliary storage device 6 under the control of the operating system stored (recorded) in the auxiliary storage device 6.
By executing the application program stored in the .RTM., A predetermined process such as a filter design process for obtaining a tap coefficient of the IIR filter is performed. ROM (Read Only Me)
mory) 2 is, for example, an IPL (Initial Program Loadin).
g) programs and so on. RAM (Random
The Access Memory 3 stores programs executed by the CPU 1 and data necessary for the operation of the CPU 1. The input unit 4 includes, for example, a keyboard and a mouse, and is operated when inputting predetermined data and commands. The output unit 5 includes a display, a printer, and the like, and displays and prints predetermined information. The auxiliary storage device 6
For example, it is configured by a hard disk or the like, and stores an operating system, an application program for causing the CPU 1 to execute filter design processing, and the like. Further, the auxiliary storage device 6 is configured to store a processing result of the CPU 1 and other necessary data.

Next, referring to the flowchart of FIG.
The filter design processing performed in the filter design device of FIG. 1 will be described.

First, in step S1, a transfer function Hi having only poles for maximizing the group delay is provided.
Initial values of the order ni of (z) and the magnitude τ of the group delay are set. That is, in step S1, a message prompting the user to input ni and τ is displayed on the output unit 5. In response to this, when the user operates the input unit 4 to input ni and τ, Is set as an initial value.

Here, the transfer function Hi (z) is a function represented by the following equation and having a coefficient only in the denominator.

[0016]

(Equation 1) (1) where a (k) represents a coefficient of the transfer function Hi (z), and n represents an order of the transfer function Hi (z).
Therefore, here, n = ni.

Thereafter, the process proceeds to step S2, in which the CPU 1 calculates a coefficient a (k) of the transfer function Hi (z) in accordance with, for example, a recurrence formula represented by the formula (2).

[0018]

(Equation 2) ... (2)

That is, the CPU 1 determines, for example, that a (0) = 1
By using the values set in step S1 as n (ni) and τ, formula (2) is calculated,
Find the coefficient a (k).

Here, the above-described method of calculating the transfer function Hi (z) having only a pole and maximizing the group delay is called the silane (Thiran) method.

The coefficient a (k) is, for example,
It is also possible to obtain the transfer function Hi (z) by performing equiripple approximation so that the group delay becomes Chebyshev characteristics (equiripple characteristics).

After obtaining the coefficient a (k), step S3
The CPU 1 determines whether the group delay of the pass band of the filter represented by the transfer function Hi (z) having the coefficient a (k) is flat. If it is determined in step S3 that the group delay of the passband of the filter represented by the transfer function Hi (z) is not flat, the process proceeds to step S4, where the CPU 1 changes ni or τ, and changes the value of ni or τ.
Return to 2. That is, when the group delay in the pass band of the filter represented by the transfer function Hi (z) does not become flat, the order ni of the transfer function Hi (z) is increased or the magnitude τ of the group delay is reduced. Is done.

In step S3, the transfer function H
If it is determined that the group delay of the pass band of the filter represented by i (z) is flat, the process proceeds to step S5, and the number of taps ns of the transfer function corresponding to the stop band or the pass band of the IIR filter to be obtained. Alternatively, np is set.
That is, in step S5, a message prompting the user to input ns and np is displayed on the output unit 5. In response to this, when the user operates the input unit 4 to input two values, Is set as ns or np. Note that ns and np need to be integers that satisfy the following equation.

Ns + np <ni, 1 <ns, np (3)

Thereafter, in step S6 and subsequent steps, transfer functions respectively corresponding to the stopband or passband of the IIR filter to be obtained are obtained.

That is, the transfer function corresponding to the stop band (stop band) or pass band (pass band) of the transfer function of the IIR filter to be obtained is Hs (z) or Hs (z), respectively.
When expressed as p (z), the transfer function H (z) of the IIR filter to be obtained is given by, for example, the following equation.

H (z) = Hp (z) Hs (z) Hi (z) (4) where the transfer function Hi (z) is the denominator of the transfer function H (z), and the transfer function Hp (z) ) And Hs (z) are transfer functions H
Each corresponds to the molecule of (z).

Here, the transfer functions Hp (z) and Hs
Regarding (z), in order to make the group delay flat, the tap coefficients are set to be symmetrical. In this case, the amplitude characteristics obtained from the transfer functions Hp (z) and Hs (z) can be both represented by cosine polynomials.

When the tap number np or ns of the transfer function Hp (z) or Hs (z) is input by operating the input unit 4, the transfer function Hp (z) or Hs (z)
A coefficient of a cosine polynomial (hereinafter, appropriately referred to as a cosine polynomial) representing the amplitude characteristic obtained from each of them is obtained as follows, for example.

That is, in step S6, the CPU 1
First finds the initial values of the coefficients of the cosine polynomial for the transfer function Hp (z), and further calculates the transfer function Hs (z)
Find the initial value of the cosine polynomial coefficient for.

Here, these initial values can be obtained, for example, as follows.

That is, zero points of the number corresponding to the number of taps ns are arranged, for example, at equal intervals on the frequency of the stop band of the IIR filter to be obtained, and the transfer function Hs (z) for obtaining such zero points is obtained. Is determined as its initial value. Next, if the coefficients of the cosine polynomial are known, the tap coefficients of the transfer function can be obtained.
From the initial value of the coefficient for the transfer function Hs (z), a tap coefficient is obtained, and the transfer function Hs (z) and the transfer function Hi (z) obtained in step S2 are substituted into the equation (4). The transfer function Hs is selected so that the resulting amplitude characteristic in the passband of the transfer function H (z) becomes a desired characteristic.
An equation is made for the coefficients of the cosine polynomial for (z). Note that the equation relating to the coefficient of the cosine polynomial for the transfer function Hs (z) is obtained by taking, for example, at equal intervals points corresponding to the number of taps np on the frequency of the passband and measuring the amplitude value at that point. Te

By solving the above equation, the initial value of the coefficient of the cosine polynomial for the transfer function Hs (z) is obtained.

As described above, the initial value of the coefficient of the cosine polynomial for the transfer function Hp (z) is obtained, and the initial value of the coefficient of the cosine polynomial for Hs (z) is calculated using the initial value. After the determination, the process proceeds to step S7, where the CPU 1 determines the tap coefficient from the initial value of the coefficient for the transfer function Hp (z) determined in step S6, and determines the transfer function Hp (z) and step S2. Substituting the transfer function Hi (z) obtained by
An equation is established for the coefficients of the cosine polynomial for the transfer function Hs (z) such that the resulting amplitude characteristic of the transfer function H (z) in the stopband is a Chebyshev characteristic (equiripple characteristic).

That is, the transfer functions Hp (z) and Hi
(Z) is known and the transfer function Hs (z) is unknown,
In the stop band of the transfer function H (z) represented by the formula Hp (z) Hs (z) Hi (z), the local maximum value and the local minimum value of the amplitude characteristic are obtained.
An equation is set so that the deviation amount (magnitude) from a desired value becomes a predetermined value.

Then, by solving the equation, the coefficient of the cosine polynomial for the transfer function Hs (z) is obtained.

Thereafter, the process proceeds to step S8, where the CPU 1
Is the transfer function Hs whose coefficient has been determined in step S7.
(Z) and the transfer function Hi obtained in step S2.
(Z) is substituted into equation (4), and the transfer function Hp is set so that the amplitude characteristic in the pass band of the transfer function H (z) obtained as a result is a Chebyshev characteristic (equal ripple characteristic).
An equation is made for the coefficients of the cosine polynomial for (z).

That is, the transfer functions Hs (z) and Hi
(Z) is known, and the transfer function Hp (z) is unknown,
In the stop band of the transfer function H (z) represented by the formula Hp (z) Hs (z) Hi (z), the local maximum value and the local minimum value of the amplitude characteristic are obtained.
An equation is set so that the deviation amount (magnitude) from a desired value becomes a predetermined value.

Then, by solving the equation, the coefficient of the cosine polynomial for the transfer function Hp (z) is obtained.

Thereafter, the process proceeds to step S9, where the CPU 1 determines whether the transfer function Hp (z) or Hs
An amount of change between each coefficient of the cosine polynomial for (z) and each coefficient of the cosine polynomial for the transfer function Hp (z) or Hs (z) obtained last time is obtained, and the amount of change is equal to or less than a predetermined threshold. Is determined. If it is determined in step S9 that the amount of change is not equal to or smaller than the predetermined threshold, that is, if it cannot be said that the coefficients have converged, the process returns to step S7, and the same processing is repeated. In this case, in step S7, an equation is established using the coefficient obtained in step S8 instead of the initial value of the coefficient of the cosine polynomial for the transfer function Hp (z) obtained in step S6.

If it is determined in step S9 that the amount of change is equal to or smaller than the predetermined threshold, that is, if the coefficients have converged, the process proceeds to step S10, where the CPU 1 determines the converged coefficients and the values obtained in step S2. From the obtained transfer function Hi (z), it is determined whether or not the amplitude characteristic of the transfer function H (z) obtained according to the equation (4) is a desired characteristic. In step S10,
If it is determined that the amplitude characteristic of the transfer function H (z) is not the desired characteristic, the process proceeds to step S11, where the CPU
At 1, it is determined whether there is room to change np and ns to other values.

In step S11, np and ns are set as follows:
If it is determined that there is room to change to another value, that is,
Np, ns that have not been used yet satisfying equation (3)
If exists, the process proceeds to step S12, where the CPU 1
Change np and ns to such values, and return to step S6. In step S11, np and ns are set as follows.
If it is determined that there is no room to change to another value, that is,
Np, ns that have not been used yet satisfying equation (3)
Does not exist, the process proceeds to step S13, and the CPU 1
Causes the output unit 5 to display a message prompting the user to change ni and τ to another value. In response to this, the user operates the input unit 4 to input ni and τ. And returns to step S2.

On the other hand, if it is determined in step S10 that the amplitude characteristic of the transfer function H (z) is the desired characteristic, the process proceeds to step S14, where the CPU 1 determines the transfer function from the converged cosine polynomial coefficient. The functions Hs (z) and Hp (z) are obtained, and these and
From the transfer function Hi (z) obtained in step 2, the transfer function H (z) is obtained according to equation (4). And CPU
1 outputs and stores the coefficient of the transfer function H (z) as the tap coefficient of the IIR filter to be obtained, for example, to the auxiliary storage device 6, and ends the processing.

FIG. 3 shows a configuration example of an embodiment of an IIR filter to which the present invention is applied.

The calculation unit 11 is, for example, a DSP (Digital
Signal Processor) and performs product-sum operation and other digital signal processing. The coefficient storage unit 12 stores tap coefficients obtained according to the flowchart of FIG.

In the IIR filter configured as described above, when a digital signal is input to the operation unit 11, the operation unit 11 performs a product-sum operation using the tap coefficients stored in the coefficient storage unit 12. By doing so, the input digital signal is filtered and output.

Next, according to the filter design processing described with reference to FIG. 2, the number of taps of the IIR filter obtained thereby can be made very small as compared with a case where the FIR filter is used. For example, how much the frequency f becomes 0 to f1, f1 to f2, or f
In the range of 2 to fs / 2, the gain is 1
((Cos (π (f−f1) / (f2−f1)) + 1)
/ 2) An example in which a digital filter having an amplitude characteristic of ^1/2 or 0 is designed (gain is 0)
Means, for example, -40 dB or less here).

Note that fs represents a sampling frequency,
Here, for example, the frequency is 24.576 Hz. Also, f
1 and f2 are represented by the following equations.

F1 = fd (1−r) / 2 f2 = fd (1 + r) / 2 where fd and r are, for example,
0.64 Hz or 0.3.

When a digital filter having the above-described amplitude characteristic is realized by an FIR filter, the stop band (f
2 to fs / 2) is set to −40 dB, and the ripple in the pass band (0 to f1) is reduced to 0.50 dB.
According to the simulation by the inventor of the present invention, the number of taps is required to be 270 for B. Here, FIG. 4 shows the amplitude characteristics of the FIR filter.

On the other hand, according to the filter design processing described with reference to FIG. 2, first, the frequency characteristics of the transfer function Hi (z) are as shown in FIG. Here, the magnitude of the group delay (the magnitude of the group delay in the passband for the frequency f = 0) τ is 67, and the number of taps ni is 10.

Here, in FIG. 5, the frequency on the horizontal axis is represented by a normalized frequency (a value obtained by dividing the frequency by the sampling frequency fs). The group delay (Grou
p Delay), the sampling frequency fs
The values normalized (divided) by are shown.

As can be seen from FIG. 5, the group delay is flat when the frequency f is in the range of 0 to fd / 2 (= 0.32 Hz) (0.013028 in the normalized frequency).
Its value is about 2.76 (= 67 / 24.76). Since the program used by the present inventor for the simulation has the accuracy of double precision, the number of taps ni is set to 19, which is approximately twice as large as 10, and the group delay magnitude τ is set to 67 times. If the value is doubled to 134, the accuracy becomes worse. So, in the simulation,
The digital filters are used in a longitudinal connection.

Next, using the transfer function corresponding to the characteristic of FIG. 5, the number of taps ns or np is set to 11 or 8 respectively.
When the transfer functions Hs (z) and Hp (z) were obtained, the frequency characteristics were as shown in FIG.
As shown in the figure, the gain in the stop band is -40 dB or less. The ripple in the pass band is 0.48 dB.
It became.

The transfer functions Hi (z) and Hs as described above
The transfer function H (z) obtained from (z) and Hp (z) according to the equation (4) has 17 or 18 zero points or 18 poles, respectively, which are shown in Table 1 or Table 2, respectively. .

[0056]

[Table 1]

[Table 2]

FIG. 7 shows these zero points and poles on the z-plane. In FIG. 7, the zero point is indicated by a circle and the pole is indicated by a cross.

FIG. 8 shows an impulse response corresponding to the transfer function H (z). Regarding the impulse response, the symmetry is slightly lost at the first time (where the time is close to 0).
The impulse response of a digital filter (IIR filter) represented by a transfer function obtained by cascading a digital filter represented by (z), that is, a transfer function obtained by squaring the transfer function H (z), was obtained. It began to show. The impulse response is almost symmetrical as shown in FIG. 3, because the group delay is almost flat in the pass band.

As described above, in the FIR filter, 2
According to the filter design processing described with reference to FIG. 2, a 70-tap required digital filter having the same amplitude characteristics and a flat passband group delay, a 17th-order numerator and a denominator Is represented by an 18th-order transfer function.
This can be realized by a filter, that is, an IIR filter having a very small number of taps.

The stop band and pass band of the IIR filter,
Furthermore, the magnitude of the group delay in the passband is not limited to the above. However, if the pass area is wide,
The present invention is particularly useful when the passband is narrow, since the number of poles increases and consequently the number of taps in the IIR filter increases.

[0061]

According to the filter design method of the present invention, the first transfer function having only a pole and maximizing the group delay is obtained, and corresponds to the stop band or the pass band of the IIR filter. The second or third transfer function is
It is determined by solving an equation so that the stopband or passband of the IIR filter has Chebyshev characteristics. Then, a tap coefficient of the IIR filter is obtained based on the first to third transfer functions. Also,
According to the digital filter of the second aspect, the tap coefficient obtained by the filter design method of the first aspect is used. Therefore, an IIR digital filter for flattening the group delay in the passband can be realized with a small number of taps.

[Brief description of the drawings]

FIG. 1 is a block diagram illustrating a configuration example of an embodiment of a filter design apparatus to which the present invention has been applied.

FIG. 2 is a flowchart illustrating a filter design process performed by the filter design device of FIG. 1;

FIG. 3 is a block diagram illustrating a configuration example of an embodiment of an IIR filter to which the present invention has been applied.

FIG. 4 is a diagram illustrating a frequency characteristic of a 270-tap FIR filter.

FIG. 5 is a diagram illustrating a frequency characteristic of a filter represented by a transfer function Hi (z).

FIG. 6 is a diagram illustrating frequency characteristics of a filter represented by transfer functions Hs (z) and Hp (z).

FIG. 7 is a diagram showing z in which the zero point and the pole of the transfer function H (z) are arranged.
It is a figure showing a plane.

FIG. 8 is a diagram illustrating an impulse response corresponding to a transfer function H (z).

FIG. 9 is a diagram showing an impulse response corresponding to a square of the transfer function H (z).

[Explanation of symbols]

1 CPU, 2 ROM, 3 RAM, 4 input unit, 5 output unit, 6 auxiliary storage device, 11 arithmetic unit, 12 coefficient storage unit

Claims (2)

[Claims] An IIR (Infini-band) having a flat group delay in a passband.
te Impulse Response) A filter design method for determining tap coefficients of a filter, wherein a first transfer function having only poles for maximizing a group delay is determined, and corresponds to a stopband or a passband of the IIR filter, respectively. A second or third transfer function is obtained by solving an equation so that a stop band or a pass band of the IIR filter has Chebyshev characteristics, and based on the first to third transfer functions, A filter design method characterized in that a tap coefficient of an IIR filter is obtained. 2. An IIR (Infini-band) having a flat group delay in a passband.
A digital filter, wherein tap coefficients are obtained by the filter design method according to claim 1.