JPH0261807B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention relates to a method of designing a filter configured by cascade connection.

The Betzel function in a filter is designed so that its delay characteristics are maximally flat within the passband of the filter.

As a result, in the case of waveform transmission, the phase shift experienced by each frequency component of the transmitted signal waveform becomes proportional to the frequency, and waveform distortion is considerable for signal inputs that have fast rising parts such as square waves. It is especially effective when the waveform is a problem.

By the way, when high-order Betzel characteristics are realized by cascading filters having first-order and second-order transfer functions, each factorized first-order and second-order transfer function is
Parameters that determine the next-order transfer function include the cutoff frequency parameter f _ci of each stage and its selectivity Qi. However, since the parameter fci of each stage is different and is not the same as the cutoff frequency _fc of the overall characteristic, it has the following practical disadvantages.

(a) In the case of a variable cut-off frequency filter that has both maximum amplitude flatness (Butterworth characteristic) and phase linearity (delay flatness characteristic),
Since the cutoff frequency fci of each stage differs between when the amplitude is maximum flat and when the phase is straight, the _fci of each stage is different each time.
The CR product must be switched to determine .
This not only makes the switching mechanism complicated, but also considerably increases the number of parts used, leading to an increase in the cost of the device.

(b) In order to avoid the above-mentioned inconvenience, it is conceivable to adjust only the selectivity Qi to the Betzel characteristic Qi without changing the f _ci of each stage, but this would result in the phase straight line, that is, the delay characteristic shown by the broken line c in Figure 3. becomes quite different from the ideal Betssel delay characteristic shown by the solid line a in the figure, causing problems such as overshoot in waveform transmission.

This invention was made to eliminate the above-mentioned drawbacks, and the cut-off frequency fci of each stage of filters constituting a high-order filter is set to the overall cut-off frequency.
By determining the selectivity Qi of each stage to bring the phase linear characteristic closer to the ideal Betsu cell characteristic of the same order on the premise of matching fc, the CR product can be changed to switch between the phase linear characteristic and the maximum amplitude flatness. It is an object of the present invention to provide a method for designing a variable filter that can be easily performed using only the selectivity Qi without having to do so, thereby reducing the number of parts and reducing costs.

By the way, first, the idea of this invention will be explained.

A high-order filter based on the cascade connection method has primary and secondary filters as basic elements, and is constructed by cascading these filters. However, conventional high-order Betzel function synthesis methods start from a high-order transfer function that satisfies maximum delay flatness and factorize it into first-order or second-order transfer functions. Therefore, the cutoff frequency fci of each basic element becomes different.

Therefore, in this invention, in order to avoid the above-mentioned inconvenience, the group delay of the higher-order filter is determined by starting from the group delay of each basic element, on the premise that the cutoff frequency fci of each basic element is made to match the cutoff frequency fc of the overall characteristic from the beginning. Then, the obtained high-order group delay function is expanded into a Pine Chlorin series so that the delay characteristic is maximally flat at direct current, and the number of differentials with respect to frequency is set to zero by the same number as the free selectivity parameter to create a nonlinear By obtaining simultaneous equations and solving them, the selectivity Qi of each basic element is determined.

That is, the n-th order low-pass transfer function is expressed as a first-order or second-order transfer function as follows.

(a) When n is an even number T(s)= _o/2 〓 ⁱ⁼¹ H/s ² +ais+bi …(1) (b) When n is an odd number T(s)=H/s+bo _{(o-1) /2} 〓 ⁱ⁼¹ 1/s ² +ais+bi (2) where bi ai is the cutoff frequency of the i-th stage, a parameter that determines the selectivity, and H is a constant that determines the overall gain.

Here, in this invention, as described above, the cutoff frequency fci of each stage is made to match the cutoff frequency fc of the overall characteristic, so all bi's become 1.

As a result, the group delay τ(ω) can be expressed as follows from equations (1) and (2).

(a) When n is an even number τ(ω)= _o/2 〓 ⁱ⁼¹ ai(1+Ω)/(1−Ω) ² +ai ² Ω …(3) (b) When n is a parsimonious number τ(w )=1/1+Ω+ _(o-1)/2 〓 ⁱ⁼¹ ai(1+Ω)/(1-Ω) ² +a ² Ω …(4) However, Ω=ω ² .

Next, rearranging equations (3) and (4) results in the following.

τ(Ω)=A ₀ (1+A ₁ Ω+A ₂ Ω ² +A ₃ Ω ³ +A ₄ Ω ⁴ …+Am
Ω ^m )/1+B ₁ Ω+B ₂ Ω ² +B ₃ Ω ³ +B ₄ Ω ⁴ +…+BmΩ ^m …(
5) However, Ai Bi is determined only by ai.

Furthermore, when formula (5) is expanded to pine chlorin, the following formula is obtained.

τ(Ω)=A ₀ {1+(A ₁ −B ₁ )Ω + [(A ₂ −B ₂ )−B ₁ (A ₁ −B ₁ )}Ω ² …}…(6) Here, τ(Ω ) becomes maximally flat at the point ω=0 by setting the first m derivatives with respect to Ω to be zero by the same number of m free parameters ai, so A ₁ = B ₁ , A ₂ = B ₂ , A ₃ =B ₃ , (7) is obtained, and by solving the same equation, m pieces of ai are obtained, and thereby the selectivity Qi of each stage is obtained.

Next, an embodiment of the present invention based on this idea will be described with reference to the drawings.

In this case, an example is shown in which the present invention is applied to a fourth-order variable low-pass filter that can switch between the maximum amplitude characteristic and the phase straight line.

In Fig. 1, 1 is an input terminal to which an input waveform is applied, and to this input terminal 1, secondary basic elements 2 and 3 are connected in cascade, making it possible to switch the selectivity Qi in accordance with the maximum amplitude characteristic and the phase line. Furthermore, an output terminal 4 is connected to the basic element 3. Here, various circuit types can be considered as the secondary basic elements 2 and 3. For example, as shown in Fig. 2a, a circuit in which positive feedback is applied to the amplifier _A11 , or as shown in Fig. 2b, As shown, there is a state variable circuit that is constructed of two anti-phase integrators A ₂₁ and A ₂₂ and one differential amplifier A ₂₃ and can obtain high-pass, band-pass, and low-pass simultaneously. Since the circuits shown in FIG. 2a or b are well known, their explanation will be omitted here.

Here, assuming that each of the secondary basic elements 2 and 3 is configured using the positive feedback circuit shown in Fig. 2a, the input signal ei is applied to the resistance side R _F of each basic element 2 and ₃ , and the After guiding, it was further formed with R _F and C.
The output voltage is derived through a 6 dB/oct low-pass filter to t ₂ and passed through a positive phase amplifier A ₁₁ with a gain K.
e ₀ will be extracted. In this case, capacitor C
The cutoff frequency fci can be changed without changing the selectivity of the fundamental section by interlocking the and two resistors R _F , and the selectivity Qi can be changed by using the gain K, that is, the resistor _R2 .

Therefore, the overall transfer function T of the circuit shown in FIG.
(s) is given by the following equation.

T(s)= _o/2 〓 ⁱ⁼¹ 1/s ² +ais+1 …(8) Here, since the order n (n=4) is an even number, the group delay τ(Ω) in Fig. 1 is lower from equation (3). It becomes like the formula.

τ(Ω)=(1+Ω) [a ₁ /(1+Ω) ² +a ² Ω +a ₂ /(1−Ω) ² +a ² Ω …(9) To rearrange this, τ(Ω)=(a+a ₂ )×1+ (a ₁ a ₂ -1) Ω + (a ₁ a ₂ -1
)Ω ² Ω ³ /1+(a ₁ ² +a ₂ ² -4)Ω+[6-2(a ₁ ² +a
₂ ² ) + a ₁ ² a ₂ ² ] Ω ² + (a ₁ ² + a ₂ ² −4) Ω ² + Ω ⁴ …(10). Further, from the above consideration, in order to make the group delay τ (Ω) maximum flat at the point where ω=0, referring to equations (5), (6), and (7), the following simultaneous equations are obtained.

a ₁ a ₂ −1=a ₁ ² +a ₂ ² −4 a ₁ a ₂ −1=6−2(a ₁ ² +a ₂ ² )+a ₁ ² a ₂ ² } …(11) By solving this Therefore, a ₁ =1.9562952 and a ₂ =1.3382612 are obtained, and therefore the selectivity Qi of each basic element 2 and 3 in FIG. 1 is given as follows.

Q ₁ = 1/a ₁ = 0.5111703 Q ₂ = 1/a ₂ = 0.7472383 (12) Accordingly, in the phase straight line, the resistance R ₁ R ₂ in the circuit of Fig. 2a is set as follows. In other words, R ₁ /R ₂ =2-1/Q...(13) From equation (12), Q ₁ of basic element 2 in the first stage is Q ₁ =
Since it is 0.5111703, R ₁ /R ₂ = 0.043705, and Q ₂ of the basic element 3 in the second stage is Q ₂ =
Since it is 0.7472383, R ₁ /R ₂ =0.661739.

In this case, the selectivity Qi when the amplitude is maximum and flat is
Q ₁ =0.541196, Q ₂ =1.306563. Further, the overall cutoff frequency fc is given by fc=1/2R _F C _F.

Therefore, as mentioned above, the cutoff constant C _F is set so that the cutoff frequency fci of the basic element of each stage matches the overall cutoff frequency fc of the filter.
A filter is configured by fixing the R _F product and setting the selectivity Qi of each basic element in each stage to the calculated predetermined value. As a result, the switching mechanism can be greatly simplified, and the number of parts can also be reduced, leading to a corresponding reduction in cost. Incidentally, when the cut-off frequency fc is defined as a 3 dB attenuation frequency, the group delay characteristic according to the present invention approximates the characteristic of an ideal Betz cell as shown by the dashed line b in the figure, as shown by the solid line a in the same figure, and its amplitude characteristic can also be approximated to the characteristics of the ideal Betz cell as shown by the solid line a in FIG. 4, as shown by the dashed-dotted line b in FIG.

It should be noted that the present invention is not limited to the above-mentioned embodiments, but can be implemented with appropriate modifications without changing the gist. For example, in the above-described embodiment, a 4-order variable low-pass filter was described, but a filter having an arbitrary order can be similarly implemented.

[Brief explanation of drawings]

FIG. 1 is a block diagram of a filter constructed according to an embodiment of the present invention, FIGS. 4 are characteristic diagrams for explaining the delay characteristics and amplitude characteristics of an ideal Beth cell, and the delay characteristics and amplitude characteristics of filters according to the embodiment and the conventional configuration, respectively. 1...Input terminal, 2, 3...Basic element, 4...Output terminal.

Claims (1)

[Claims] 1. The basic element has a cutoff frequency fci defined by the cutoff constant CR product and a selectivity Qi, and when the order n is an even number, n/2 stages of 2nd order basic elements are connected in cascade. , and when the degree n is an odd number, the second-order fundamental element is (n-1)/2 stages and the first-order fundamental element is 1
In a method for designing an n-th order variable frequency filter that is constructed by cascading stages and has a total cutoff frequency fc, the basic elements of each stage have the cutoff frequency fci defined by the product of cutoff constants CR. The overall cutoff frequency fc is the same as the total cutoff frequency fc, and for each of these basic elements, the n-th group delay function obtained from the group delay of the basic elements of each stage is expanded into a Matslaurin series, and when the order n is an even number, n/ Obtained by solving n/2 or (n+1)/2 simultaneous equations obtained by equating the differential value with respect to the frequency up to the second order, or (n+1)/second order when the order n is an odd number, to zero. A method for designing a filter, characterized in that it provides a selectivity Qi.