JPH02105713A - Method for designing digital filter - Google Patents

Abstract

PURPOSE:To make degrees of transmission functions of a digital and analog filters equal to each other, and approximate the digital filter to the analog filter with high accuracy by finding all factors of a 2nd transmission function by the method of least square based on a specific equation. CONSTITUTION:A 1st amplitude characteristics of an analog filter to be approximated are found from a 1st transmission function of the filter. Then the degree of the 2nd transmission function of a digital filter to be designed is made equal to that of the 1st transmission function and the 2nd amplitude characteristics are found from the 2nd transmission function. Then an equation which meets prescribed conditions is found and all factors of the 2nd transmission function are found by the method of least squares based on the equation. Therefore, the amplitude characteristics of the digital filter can be approximated to those of the analog filter with high accuracy by making the degrees of the transmission functions of the filters equal to each other.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a method for designing a digital filter, particularly a method for designing a digital filter having peaking characteristics.

(Summary of the invention) The present invention provides a method for designing a digital filter.
The first transfer function H of the analog filter to be approximated
(s), and the second transfer function H(z) of the digital filter to be designed is set to the same order as the first transfer function H(s). a step of determining a second amplitude characteristic of the function H(z);
The amplitude at the center frequency in the amplitude characteristic of
Find an equation that introduces the amplitude and conditions at each frequency so that the second amplitude characteristic has the condition that the amplitude at the center frequency of the amplitude characteristic is maximum, and use the least squares method with this equation as the condition. The step of calculating all the coefficients of the second transfer function H(z) makes it possible to approximate the digital filter to the analog filter with high accuracy by making the orders of the transfer functions of the analog filter and the digital filter the same. This greatly reduces the time required for design and calculation.

[Conventional technology]

Conventionally, the Laplace transform type multi-pole multi-zero transfer function H(s)
A digital filter design method is known in which a transfer function H(z) is determined by S-Z transformation from 0 to 1, and a digital filter is constructed based on this (Japanese Patent Publication No. 63-1836).
(See Publication No. 7).

[Problem to be solved by the invention]

However, when designing a digital filter by performing S-Z conversion on the transfer function of an analog filter as described above, although the order of the transfer function of the digital filter becomes the same, the frequency characteristics of the analog filter differ in the high frequency range. There is a problem in that the deviation increases the error and the accuracy decreases.

Additionally, in order to solve this problem, the order of the digital filter's transfer function H(z) has been designed to be higher than the order of the analog filter's transfer function H(s), but in this case, The problem with this method is that it becomes extremely time consuming due to the increase in the number of filters. Furthermore, since the coefficient data also increases, there is a problem in that a large number of ROMs must be provided.

Therefore, an object of the present invention is to provide a method for designing a digital filter that can approximate the digital filter to the analog filter with high accuracy by making the orders of the transfer functions of the digital filter and the analog filter equal.

[Means to solve the problem]

The digital filter design method according to the present invention is based on the first transfer function H(s
) and determining the order of the second transfer function H(z) of the digital filter to be designed.
A second transfer function H with the same order as the transfer function H(s) of
(z), the step of determining the second amplitude characteristic of (z), the amplitude at the center frequency in the first amplitude characteristic, the amplitude at the first frequency and the second frequency where the predetermined gain changes, and the first amplitude characteristic. Find the amplitude at each frequency and an equation that introduces the above conditions so that the second amplitude characteristic has the condition that the amplitude at the center frequency is maximum. The configuration consists of a step of determining all the coefficients of the transfer function H(z).

[Effect]

First, the first amplitude characteristic is determined from the first transfer function H(s) of the analog filter to be approximated.

Then, the order of the second transfer function H(z) of the digital filter to be designed is made equal to the order of the first transfer function H(s), and the order of the second transfer function H(z) is calculated from the second transfer function H(z). Find the amplitude characteristics.

Then, the second amplitude characteristic of the digital filter becomes equal to the first amplitude characteristic.
The amplitude at the center frequency in the amplitude characteristic of Find the equation in which the amplitude and conditions are introduced. All coefficients of the second transfer function H(z) are determined by the least square method using this equation as a condition.

Thereby, the orders of the transfer functions of the digital filter and the analog filter can be made equal, and the amplitude characteristics of the digital filter can be approximated to the amplitude characteristics of the analog filter with high accuracy.

〔Example〕

An embodiment of the present invention will be described below.

In this embodiment, the present invention is applied to a second-order IIR digital filter that has peaking characteristics. Note that this peaking characteristic exhibits a maximum amplitude at the center frequency f0, and an amplitude with a predetermined gain reduction at frequencies f, ... ft before and after the center frequency f0.

(Step 1) Amplitude characteristics based on transfer function of analog filter The transfer function of a second-order analog filter having peaking characteristics is expressed by equation (1).

however to=1 0″-1 The square of the amplitude characteristic of Eq. It is shown by the formula.

The above formula Differentiate it by and set it to 0, find the maximum value f.

fl"-1.L-io"=0 "i6) Equation (6) is a quadratic equation regarding the frequency f, and by solving this equation (6), the following equation is obtained.

If we take 10 in the above equation, The formula is obtained.

If we square the expression, we get The formula is obtained.

From the formula Subtract f and t (9) Get the formula.

Equation (2) takes the maximum value at f=f. Its maximum value (
4) Show the formula.

Hage. ) l ”=(1+k)”=(10”)”・
...(4) Here, consider frequencies f, , f, as shown in equation (5).

From equation (5), The formula is obtained.

the expression (2) Substitute into equation.

(lO) The formula is frequency f.

is the amplitude at .

Similarly, frequency f! (11) Equation % Equation % (Step 2) Amplitude characteristics based on transfer function of digital filter The transfer function of the second-order ■■R digital filter having the above-mentioned peaking characteristic is expressed by the following equation.

This equation uses a quadratic equation of 2 because the transfer function H(s) of the analog filter expressed by equation (1) is expressed by a quadratic equation of S. In order to express the amplitude characteristics with real numbers, create the following equation from (12) 2.

In equation (13), f is a standardized frequency, and is an angular frequency that also includes 2π.

Z=e −J f First, co, cl, which approximates equation (13) to equation (2),
In order to obtain cz, do, and dI+di, set d0=1 (the same applies if co=i). At this time, step l
The center frequency f0, each amplitude of the frequencies f1 and f2, and the amplitude of the center frequency f0 obtained in the above are set to satisfy the condition that the amplitude is the maximum.

Since the amplitude of the center frequency f0 and equation (13) are equal,
From equations (4) and (13), equation (14) can be changed.

Next, the amplitude of the frequency f expressed by equation (10) and (1
3) Since the equations are equal, equation (15) can be changed from equation (10) and (13).

Furthermore, the amplitude of frequency f2 expressed by equation (11) and (1
3) Since 2 is equal, equation (16) can be changed from equations (11) and (13).

Moreover, since the equation (2) takes the maximum value at the center frequency f0, when (13) 2 is differentiated and the center frequency f0 is substituted, it becomes 0, and the equation (17) is obtained.

(-c, 5info-2czsin2fo) (1+d
, coSfo+d2cos2fo)+(d r s
1nfo+2dzs 1n2fo) (co+c +
cosfo+czcos2fo)=0--- (17)
By substituting equation (14) into equation (17), equation (18) is obtained.

c Is 1nfo+2czs 1n2fO-(1+k
)” (ds i info+2dzs 1n2fo)
=0--- (18) (14) , (15) ,
Equations (16) and (18) are C6+ CI + C2+
Simultaneous linear equations regarding d 1 + d2 (variables are 5
There are four equations). When this is represented in a matrix, equation (19) is obtained.

AX-a...(19) Here, R is the sum of the squares of the error, 靝□ is the weighting coefficient, and H''(
fi) is a value obtained by substituting f into equation (2).

The digital signal is folded back at f=π. (7) The second frequency f may be larger than π depending on the center frequency f0 and the value of Q. In this case, since equation (15) cannot be established, the second row of equation (20) is excluded and treated as a matrix of 3 rows and 5 columns.

(Step 3) Determination of coefficients by least squares method Approximate equation (13) to equation (2). First, consider the following equation by dividing the frequency into n parts.

Here, if equation (23) is used, it becomes the square of the error when the amplitude characteristics at each frequency are expressed in decibels.

This is done here to simplify the calculation. L and dz give initial values. α is a small value that prevents calculation from becoming impossible when H(ft) or 1+d, cosft+dtcos2ft becomes small.

Using the round equation of equation (21), equation (22) becomes R=Σ−! ”
Cco+c+cosf=+czcos2ft-H”
(f=)X (1+dtcosm+dzcos2ft
))”= ΣCellCo+ettCI+e3tCz+
ea+dI+esrdz-g=>""125) Here, the equation for solving this using the least squares method is expressed by the following equation.

E3X=b...(27) Element B of B is expressed by the following formula.

Bjk””Σe J! e ht...(28)
Element S of vector b is expressed by the following equation.

b, = Σ e J! g = ... (29
) [-1 Vector X is the same as equation (21), and the old one is 5 rows
It is a square matrix of columns.

In order to solve this least squares method with the condition (19)2, it is sufficient to introduce Lagrange's undetermined coefficient k and solve the following equation.

This allows X (Co, C++ CZ, d+, dz,)
, k(k+, kZ, kff, kn), and therefore Co, C1, C2 in equation (13)
, dl+dz, (do=1) can be obtained.

From this value, “O+a+, az+t)o, t” in equation (12)
)1. The method for determining bz will be described.

From the comparison of the denominator of equation (3) and the denominator of equation (12), we find that D
(f)=do+d, cosf+d, cos2f, --
(31)=(bzz-”+b+z-'+bo)(bo+
b+z+bzz”) ...(33) This,, b,
, b2 are determined as follows. First, set O to equation (32) and find the zero point. There are four zero points. If these four zero points are two outside the unit circle and two inside, then (33
) can be obtained by finding the polynomial using the two inner zeros.However, if the zero point of the equation 32) is set to 0, it is not a double plate on the unit circle. When they exist, bo, b+, and bz cannot be determined.

The four roots of equation (32) set to 0 are determined as follows.

This is a quadratic equation for 20-, and when this is solved, equation (36) can be changed.

Equation (36) becomes a quadratic equation for Z and Z respectively. By solving this with decryption (32)
We can find the roots of the symmetry coefficients of the equation.

If two roots are inside the unit circle and two roots are outside the unit circle, then by creating a polynomial where the two roots on the inside are zeros, b
,,b,,b,can be found.

a 6+ a 1. a! Similarly to , b, , b, it can be found from C(++ CInCZ.

In this way, the analog filter transfer function H(s) shown in equation (1) and the digital filter transfer function H(
By making the orders of z) the same, the amplitude characteristics of the digital filter can be approximated to those of the analog filter with high accuracy. Furthermore, this can significantly reduce the time required to design a digital filter, and there is no need to provide a large number of ROMs as in the past.

Calculation Example The result of applying the above method to the design of a digital graphic equalizer is shown in the figure. The figure shows the characteristics of a digital graphic equalizer. in this case,
The results are shown when the gain B=±12 dB and Q=0.67. Note that the result for B=-12 dB was obtained by reversing the denominator and numerator of the transfer function.

〔Effect of the invention〕

According to the digital filter design method according to the present invention, it is possible to approximate the digital filter to the analog filter with high accuracy by making the orders of the transfer functions of the analog filter and the digital filter equal, and to create a highly accurate digital filter with desired characteristics. This has the effect of providing a filter.

Further, since there is no need to increase the order of the transfer function of the digital filter and design it as in the past, there is an effect that the time required for design can be significantly shortened. Further, since the coefficient data does not increase, there is an advantage that there is no need to prepare a large number of 120M as in the conventional case.

[Brief explanation of drawings]

The figure is a characteristic diagram of a digital filter according to an embodiment of the present invention. Gain d8 Explanation of main symbols in the drawings fo: Center frequency.

Claims (1)

[Claims] First transfer function H of an analog filter to be approximated
(s), and a second transfer function H(z) of the digital filter to be designed.
obtaining a second amplitude characteristic of the second transfer function H(z) by changing the order of the first transfer function H(s) to the same order as the first transfer function H(s); and the second amplitude characteristic has conditions such that the amplitude at the first frequency and the second frequency where the predetermined gain changes, and the amplitude at the center frequency of the first amplitude characteristic is maximum. The design of a digital filter consists of the steps of finding an equation introducing the amplitude at each frequency and the above conditions, and finding all the coefficients of the second transfer function H(z) by the least squares method using this equation as a condition. Method.