When analyzing filters and calculating their parameters, some standard terms are always used and it makes sense to stick to them from the very beginning.

Let's assume that you want a low-pass filter that has a flat response in the passband and a sharp transition to the stopband. The final slope of the response in the stopband will always be 6n dB/octave, where n is the number of “poles”. One capacitor (or inductor) is needed per pole, so the final roll-off rate requirements of the filter roughly determine its complexity.

Now let's say you decide to use a 6-pole low-pass filter. You are guaranteed a final roll-off at high frequencies of 36 dB/octave. In turn, it is now possible to optimize the filter design in the sense of providing the most flat response in the passband by reducing the slope of the transition from the passband to the stopband. On the other hand, by allowing some ripple in the passband, a steeper transition from the passband to the stopband can be achieved. The third criterion, which may be important, describes the ability of the filter to pass signals with a spectrum lying within the passband without distorting their shape due to phase shifts. You can also be interested in rise time, overshoot, and settling time.

Filter design methods are known that are suitable for optimizing any of these characteristics or combinations thereof. Truly smart filter selection doesn't happen as described above; As a rule, the required uniformity of the characteristic in the passband and the required attenuation at a certain frequency outside the passband and other parameters are first set. After this, the most suitable circuit is selected with the number of poles sufficient to satisfy all these requirements. The next few sections will look at the three most popular types of filters, namely the Butterworth filter (the flattest passband response), the Chebyshev filter (the steepest transition from passband to stopband), and the Bessel filter (the flattest lag time response). Any of these filter types can be implemented using various filter circuits; We will discuss some of them later. All of them are equally suitable for constructing low- and high-pass filters and bandpass filters.

Butterworth and Chebyshev filters. The Butterworth filter provides the flattest response in the passband, which is achieved at the cost of smoothness in the transition region, i.e. between passbands and delaybands. As will be shown later, it also has a poor phase-frequency response. Its amplitude-frequency characteristic is given by the following formula:
U out /U in = 1/ 1/2,
where n defines the filter order (number of poles). Increasing the number of poles makes it possible to flatten the portion of the characteristic in the passband and increase the steepness of the roll-off from the passband to the suppression band, as shown in Fig. 5.10.

Rice. 5.10 Normalized characteristics of Butterworth low-pass filters. Note the increase in the steepness of the characteristic rolloff with increasing filter order.

When choosing a Butterworth filter, we sacrifice everything else for the sake of the flattest characteristics. Its characteristic goes horizontally, starting from zero frequency, its inflection begins at the cutoff frequency ƒ s - this frequency usually corresponds to the -3 dB point.

In most applications, the most important consideration is that the passband ripple should not exceed a certain amount, say 1 dB. The Chebyshev filter meets this requirement, while some unevenness of the characteristic is allowed throughout the entire passband, but at the same time the sharpness of its break greatly increases. For the Chebyshev filter, the number of poles and the unevenness in the passband are specified. Allowing for increased unevenness in the passband, we obtain a sharper kink. The amplitude-frequency response of this filter is given by the following relation
U out /U in = 1/ 1/2,
where C n is a Chebyshev polynomial of the first kind of degree n, and ε is a constant that determines the unevenness of the characteristic in the passband. The Chebyshev filter, like the Butterworth filter, has phase-frequency characteristics that are far from ideal. In Fig. Figure 5.11 compares the characteristics of the 6-pole Chebyshev and Butterworth low-pass filters. As you can easily see, both are much better than a 6-pole RC filter.

Rice. 5.11. Comparison of the characteristics of some commonly used 6-pole low-pass filters. The characteristics of the same filters are shown in both logarithmic (top) and linear (bottom) scales. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebyshev filter (ripple 0.5 dB).

In fact, a Butterworth filter with a very flat passband response is not as attractive as it might seem, since in any case you have to put up with some unevenness in the passband (for a Butterworth filter this will be a gradual decrease in response as the frequency approaches ƒ c, and for the Chebyshev filter - ripples distributed over the entire passband). In addition, active filters built from elements whose ratings have some tolerance will have a characteristic that differs from the calculated one, which means that in reality there will always be some unevenness in the passband in the Butterworth filter characteristic. In Fig. Figure 5.12 illustrates the effect of the most undesirable deviations in the values ​​of capacitor capacitance and resistor resistance on the filter characteristic.

Rice. 5.12. The influence of changes in element parameters on the characteristics of the active filter.

In light of the above, a very rational structure is the Chebyshev filter. Sometimes it is called an equal-wave filter, since its characteristic in the transition region has a greater steepness due to the fact that several equal-sized pulsations are distributed over the passband, the number of which increases with the order of the filter. Even with relatively small ripples (about 0.1 dB), the Chebyshev filter provides a much greater slope in the transition region than the Butterworth filter. To quantify this difference, assume that a filter is required with a passband flatness of no more than 0.1 dB and an attenuation of 20 dB at a frequency that differs by 25% from the cutoff frequency of the passband. Calculation shows that in this case a 19-pole Butterworth filter or just an 8-pole Chebyshev filter is required.

The idea that one can tolerate ripple in the passband for the sake of increasing the steepness of the transition section is taken to its logical conclusion in the idea of ​​the so-called elliptic filter (or Cauer filter), in which ripple is allowed in both the passband and the delay in order to ensure the steepness of the transition section is even greater than that of the Chebyshev filter characteristic. With the help of a computer, elliptic filters can be designed as simply as the classical Chebyshev and Butterworth filters. In Fig. Figure 5.13 shows a graphical description of the amplitude-frequency response of the filter. In this case (low pass filter), the acceptable range of the filter gain (i.e. ripple) in the passband, the minimum frequency at which the characteristic leaves the passband, the maximum frequency where the characteristic enters the stopband, and the minimum attenuation in the band are defined. detention.

Rice. 5.13. Setting the filter frequency response parameters.

Bessel filters. As was established earlier, the amplitude-frequency response of a filter does not provide complete information about it. A filter with a flat amplitude-frequency response can have large phase shifts. As a result, the shape of the signal, the spectrum of which lies in the passband, will be distorted when passing through the filter. In situations where the waveform is of paramount importance, it is desirable to have a linear phase filter (constant delay time filter) available. Demanding a filter to ensure a linear change in the phase shift as a function of frequency is equivalent to requiring constant delay time for a signal whose spectrum is located in the passband, i.e., the absence of distortion of the signal shape. The Bessel filter (also called the Thomson filter) has the flattest part of the passband lag time curve, just as the Butterworth filter has the flattest frequency response. To understand the time domain improvement a Bessel filter provides, look at Fig. Figure 5.14 shows frequency-normalized lag time plots for 6-pole Bessel and Butterworth low-pass filters. The poor lag time characteristics of the Butterworth filter cause overshoot-type effects to occur when pulsed signals pass through the filter. On the other hand, you have to pay for the constancy of the delay times of the Bessel filter by the fact that its amplitude-frequency characteristic has an even flatter transition section between the passband and stopband than even the characteristic of the Butterworth filter.

Rice. 5.14. Comparison of time delays for 6-band Bessel (1) and Butterworth (2) low-pass filters. The Bessel filter, due to its excellent time domain properties, produces the least waveform distortion.

There are many different filter design techniques that attempt to improve the time-domain performance of a Bessel filter, partially sacrificing constant lag time in order to reduce rise time and improve frequency response. The Gaussian filter has almost as good phase characteristics as the Bessel filter, but with improved transient response. Another interesting class are filters that make it possible to achieve identical ripples in the delay time curve in the passband (similar to ripples in the amplitude-frequency characteristic of a Chebyshev filter) and provide approximately the same delay for signals with a spectrum up to the stopband. Another approach to creating filters with constant lag time is the use of all-pass filters, otherwise called time-domain equalizers. These filters have a constant amplitude-frequency response, and the phase shift can be changed according to specific requirements. Thus, they can be used to equalize the delay time of any filters, in particular Butterworth and Chebyshev filters.

Comparison of filters. Despite earlier comments about the transient response of Bessel filters, it still has very good time domain properties compared to Butterworth and Chebyshev filters. The Chebyshev filter itself, with its very suitable amplitude-frequency response, has the worst parameters in the time domain of all these three types of filters. The Butterworth filter makes a trade-off between frequencies and timing characteristics. In Fig. Figure 5.15 provides information on the performance characteristics of these three types of filters in the time domain, complementing the earlier graphs of amplitude-frequency characteristics. Based on these data, we can conclude that in cases where filter parameters in the time domain are important, it is advisable to use a Bessel filter.

Rice. 5.15. Transient comparison of 6-pole low-pass filters. The curves are normalized by reducing the attenuation value of 3 dB to a frequency of 1 Hz. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebyshev filter (ripple 0.5 dB).

Plan:

Introduction

• 1 Review
  • 1.1 Normalized Butterworth polynomials
  • 1.2 Maximum smoothness
  • 1.3 High-frequency roll-off
• 2 Filter design
  • 2.1 Cauer topology
  • 2.2 Sallen-Kay topology
• 3 Comparison with other linear filters
• 4 Example
Literature

Introduction

Butterworth filter- one of the types of electronic filters. Filters of this class differ from others in the design method. The Butterworth filter is designed so that its amplitude-frequency response is as smooth as possible at passband frequencies.

Such filters were first described by British engineer Stefan Butterworth in the article “On the Theory of Filter Amplifiers.” On the Theory of Filter Amplifiers ), In the magazine Wireless Engineer in 1930.

1. Review

The frequency response of the Butterworth filter is maximally smooth at passband frequencies and decreases to almost zero at stopband frequencies. When plotting the frequency response of a Butterworth filter on a logarithmic phase response, the amplitude decreases toward minus infinity at the stopband frequencies. In the case of a first-order filter, the frequency response attenuates at a rate of −6 decibels per octave (-20 decibels per decade) (in fact, all first-order filters, regardless of type, are identical and have the same frequency response). For a second-order Butterworth filter, the frequency response attenuates by −12 dB per octave, for a third-order filter - by −18 dB, and so on. The frequency response of the Butterworth filter is a monotonically decreasing function of frequency. The Butterworth filter is the only filter that retains the shape of the frequency response for higher orders (with the exception of a steeper roll-off of the characteristic at the suppression band), while many other types of filters (Bessel filter, Chebyshev filter, elliptic filter) have different shapes of the frequency response at different orders.

Compared to Chebyshev types I and II filters or the elliptical filter, the Butterworth filter has a flatter rolloff and therefore must be of higher order (which is more difficult to implement) in order to provide the desired performance at stopband frequencies. However, the Butterworth filter has a more linear phase-frequency response at passband frequencies.

The frequency response for low-pass Butterworth filters is of the order of 1 to 5. The slope of the characteristic is 20 n dB/decade, where n- filter order.

As with all filters, when considering frequency characteristics, a low-pass filter is used, from which you can easily obtain a high-pass filter, and by connecting several such filters in series, a band-pass filter or notch filter.

The frequency response of a th-order Butterworth filter can be obtained from the transfer function:

It is easy to see that for infinite values, the frequency response becomes a rectangular function, and frequencies below the cutoff frequency will be passed with a gain, and frequencies above the cutoff frequency will be completely suppressed. For finite values, the decline in the characteristic will be gentle.

Using a formal replacement, we present the expression as:

The poles of the transfer function are located on a circle of radius equidistant from each other in the left half-plane. That is, the transfer function of a Butterworth filter can only be determined by determining the poles of its transfer function in the left half-plane of the s-plane. The th pole is determined from the following expression:

The transfer function can be written as:

Similar reasoning applies to digital Butterworth filters, with the only difference being that the relationships are not written for s-plane, and for z-plane.

The denominator of this transfer function is called the Butterworth polynomial.

1.1. Normalized Butterworth polynomials

Butterworth polynomials can be written in complex form, as shown above, but they are usually written as relations with real coefficients (complex conjugate pairs are combined using multiplication). The polynomials are normalized by the cutoff frequency: . The normalized Butterworth polynomials thus have the following canonical form:

Below are the Butterworth polynomial coefficients for the first eight orders:

1.2. Maximum smoothness

Taking and , the derivative of the amplitude characteristic with respect to frequency will look like this:

It decreases monotonically for everyone since the gain is always positive. Thus, the frequency response of the Butterworth filter has no ripple. When expanding the amplitude characteristic into a series, we obtain:

In other words, all derivatives of the amplitude-frequency characteristic with respect to frequency up to 2 n- are equal to zero, which implies “maximum smoothness”.

1.3. High-frequency roll-off

Having accepted, we find the slope of the logarithm of the frequency response at high frequencies:

In decibels, the high-frequency asymptote has a slope of −20 n dB/decade.

2. Filter design

There are a number of different filter topologies with which linear analog filters are implemented. These schemes differ only in the values ​​of the elements, but the structure remains unchanged.

2.1. Cauer topology

Cauer's topology uses passive elements (capacitance and inductance). A Butteworth filter with a given transfer function can be constructed in the form of a type 1 Cowher. The kth filter element is given by the relation:

2.2. Sallen-Kay topology

The Sallen-Kay topology uses, in addition to passive ones, also active elements (operational amplifiers and capacitors). Each stage of the Sallen-Kay circuit is a part of the filter, mathematically described by a pair of complex conjugate poles. The entire filter is obtained by connecting all stages in series. If a valid pole is found, it must be implemented separately, usually as an RC circuit, and included in the overall circuit.

The transfer function of each stage in the Sallen-Kay circuit has the form:

The denominator must be one of the factors of the Butterworth polynomial. Having accepted, we get:

The last relation gives two unknowns that can be chosen arbitrarily.

3. Comparison with other linear filters

The figure below shows the frequency response of the Butterworth filter in comparison with other popular linear filters of the same (fifth) order:

It can be seen from the figure that the Butterworth filter roll-off is the slowest of the four, but it also has the smoothest frequency response at passband frequencies.

4. Example

Analog low-pass Butterworth filter (Cauer topology) with cutoff frequency with the following element values: farad, ohm, and henry.

Logarithmic density plot of the transfer function H(s) on the complex argument plane for a third-order Butterworth filter with cutoff frequency . The three poles lie on a circle of unit radius in the left half-plane.

Consider a third-order analog low-pass Butterworth filter with farad, ohm, and henry. Indicating the total resistance of the capacitors C How 1/Cs and impedance of inductances L How Ls, where is a complex variable, and using equations for calculating electrical circuits, we obtain the following transfer function for such a filter:

The frequency response is given by the equation:

and the phase response is given by the equation:

Group delay is defined as minus the derivative of phase with respect to circular frequency and is a measure of the phase distortion of a signal at various frequencies. The logarithmic frequency response of such a filter has no ripples either in the passband or in the suppression band.

The graph of the modulus of the transfer function in the complex plane clearly indicates three poles in the left half-plane. The transfer function is completely determined by the location of these poles on the unit circle symmetrically about the real axis.

By replacing each inductance with a capacitance, and the capacitances with inductances, we obtain a high-pass Butterworth filter.

And the group delay of a third order Butterworth filter with cutoff frequency

Literature

• V.A. Lucas Theory of automatic control. - M.: Nedra, 1990.
• B.H. Krivitsky Handbook on the theoretical foundations of radio electronics. - M.: Energy, 1977.
• Miroslav D. Lutovac Filter Design for Signal Processing using MATLAB© and Mathematica©. - New Jersey, USA.: Prentice Hall, 2001. - ISBN 0-201-36130-2
• Richard W. Daniels Approximation Methods for Electronic Filter Design. - New York: McGraw-Hill, 1974. - ISBN 0-07-015308-6
• Steven W. Smith The Scientist and Engineer's Guide to Digital Signal Processing. - Second Edition. - San-Diego: California Technical Publishing, 1999. - ISBN 0-9660176-4-1
• Britton C. Rorabaugh Approximation Methods for Electronic Filter Design. - New York: McGraw-Hill, 1999. - ISBN 0-07-054004-7
• B. Widrow, S.D. Stearns Adaptive Signal Processing. - Paramus, NJ: Prentice-Hall, 1985. - ISBN 0-13-004029-0
• S. Haykin Adaptive Filter Theory. - 4th Edition. - Paramus, NJ: Prentice-Hall, 2001. - ISBN 0-13-090126-1
• Michael L. Honig, David G. Messerschmitt Adaptive Filters - Structures, Algorithms, and Applications. - Hingham, MA: Kluwer Academic Publishers, 1984. - ISBN 0-89838-163-0
• J.D. Markel, A.H. Gray, Jr. Linear Prediction of Speech. - New York: Springer-Verlag, 1982. - ISBN 0-387-07563-1
• L.R. Rabiner, R.W. Schafer Digital Processing of Speech Signals. - Paramus, NJ: Prentice-Hall, 1978. - ISBN 0-13-213603-1
• Richard J. Higgins Digital Signal Processing in VLSI. - Paramus, NJ: Prentice-Hall, 1990. - ISBN 0-13-212887-X
• A. V. Oppenheim, R. W. Schafer Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1975. - ISBN 0-13-214635-5
• L. R. Rabiner, B. Gold Theory and Application of Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1986. - ISBN 0-13-914101-4
• John G. Proakis, Dimitris G. Manolakis Introduction to Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1988. - ISBN 0-02-396815-X

Lesson topic 28: Classification of electrical filters.

28.1 Definitions.

An electric frequency filter is a four-port network that passes currents of some frequencies well with low attenuation (3 dB attenuation), and currents of other frequencies poorly with high attenuation (30 dB).

The range of frequencies in which there is little attenuation is called the passband.

The range of frequencies in which the attenuation is large is called the stopband.

A transition strip is introduced between these stripes.

The main characteristic of electric filters is the dependence of the operating attenuation on frequency.

This characteristic is called the frequency attenuation characteristic.

- cutoff frequency at which the operating attenuation is 3 dB.

- permissible attenuation, set by the mechanical parameters of the filter.

- permissible frequency corresponding to permissible attenuation.

PP passband – the frequency range in which
dB.

PB - stopband - frequency range in which the operating attenuation is greater than permissible.

28.2 Classification

1
By bandwidth location:

a) LPF - low pass filter - passes low frequencies and delays high ones.

It is used in communication equipment (TV receivers).

b
) HPF - high pass filter - passes high frequencies and delays low ones.

V
) PF - bandpass filters - pass only a certain frequency band.

G
) SF - notch or blocking filters - do not pass only a certain frequency band, and let the rest pass.

2 According to the element base:

a) LC filters (passive)

b) RC filters (passive)

c) active ARC filters

d) special types of filters:

Piezoelectric

Magnetostrictive

3 For mathematical support:

A
) Butterworth filters. Operating attenuation characteristic
has a value of 0 at frequency f=0 and then increases monotonically. In the passband it has a flat characteristic - this is an advantage, but in the stopband it is not steep - this is a disadvantage.

b) Chebyshev filters. To obtain a steeper characteristic, Chebyshev filters are used, but they have a “waviness” in the passband, which is a disadvantage.

c) Zolotarev filters. Operating attenuation characteristic
in the passband it has undulations, and in the stopband there is a dip in characteristics.

Lesson topic 29: Low-pass and high-pass Butterworth filters.

29.1 Butterworth LF.

Butterworth proposed the following attenuation formula:

,dB

Where
- Butterworth function (normalized frequency)

n – filter order

For low pass filter
, Where - any desired frequency

- cutoff frequency, which is equal to

To implement this characteristic, L and C filters are used.

AND

The inductance is placed in series with the load, since
and with growth increases
Therefore, low-frequency currents will easily pass through the inductance resistance, and high-frequency currents will be delayed and will not reach the load.

The capacitor is placed in parallel with the load, since
, therefore the capacitor passes high-frequency currents well and poorly lower ones. High frequency currents will be closed through the capacitor, and low frequency currents will pass to the load.

The filter circuit consists of alternating L and C.

Butterworth low-pass filter 3rd order T-shaped

Butterworth low-pass filter. 3rd order U-shaped.

The frequency response of the Butterworth filter is described by the equation

Features of the Butterworth filter: nonlinear phase response; cutoff frequency independent of the number of poles; oscillatory nature of the transient response with a step input signal. As the filter order increases, the oscillatory nature increases.

Chebyshev filter

The frequency response of the Chebyshev filter is described by the equation

,

Where T n 2 (ω/ω n ) – Chebyshev polynomial n-th order.

The Chebyshev polynomial is calculated using the recurrent formula

Features of the Chebyshev filter: increased unevenness of the phase response; wave-like characteristic in the passband. The higher the coefficient of unevenness of the frequency response of the filter in the passband, the sharper the decline in the transition region at the same order. The transient oscillation of a stepped input signal is greater than that of a Butterworth filter. The quality factor of the Chebyshev filter poles is higher than that of the Butterworth filter.

Bessel filter

The frequency response of the Bessel filter is described by the equation

,

Where
;B n 2 (ω/ω cp h ) – Bessel polynomial n-th order.

The Bessel polynomial is calculated using the recurrent formula

Features of the Bessel filter: fairly uniform frequency response and phase response, approximated by the Gaussian function; the phase shift of the filter is proportional to the frequency, i.e. the filter has a frequency-independent group delay time. The cutoff frequency changes as the number of filter poles changes. The filter's frequency response is usually flatter than that of Butterworth and Chebyshev. This filter is especially suitable for pulse circuits and phase-sensitive signal processing.

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

.

Features of the Cauer filter: uneven frequency response in the passband and stopband; the sharpest drop in frequency response of all the above filters; implements the required transfer functions with a lower filter order than when using other types of filters.

Determining the filter order

The required filter order is determined by the formulas below and rounded to the nearest integer value. Butterworth filter order

.

Chebyshev filter order

.

For the Bessel filter, there is no formula for calculating the order; instead, tables are provided that correspond the filter order to the minimum required deviation of the delay time from unity at a given frequency and the loss level in dB).

When calculating the Bessel filter order, the following parameters are specified:

Permissible percentage deviation of group delay time at a given frequency ω ω cp h ;

The filter gain attenuation level can be set in dB at frequency ω , normalized relative to ω cp h .

Based on these data, the required order of the Bessel filter is determined.

Circuits of cascades of low-pass filters of the 1st and 2nd order

In Fig. 12.4, 12.5 show typical circuits of low-pass filter cascades.

A) b)

Rice. 12.4. Low-pass filter cascades of Butterworth, Chebyshev and Bessel: A - 1st order; b – 2nd order

A) b)

Rice. 12.5. Cauer low-pass filter cascades: A - 1st order; b – 2nd order

General view of the transfer functions of the Butterworth, Chebyshev and Bessel low-pass filters of the 1st and 2nd order

,
.

General view of the transfer functions of the Cauer low-pass filter of the 1st and 2nd order

,
.

The key difference between a 2nd order Cauer filter and a bandstop filter is that in the Cauer filter transfer function the frequency ratio Ω s ≠ 1.

Calculation method for Butterworth, Chebyshev and Bessel low-pass filters

This technique is based on the coefficients given in the tables and is valid for Butterworth, Chebyshev and Bessel filters. The method for calculating Cauer filters is given separately. The calculation of Butterworth, Chebyshev and Bessel low-pass filters begins with determining their order. For all filters, the minimum and maximum attenuation parameters and cutoff frequency are set. For Chebyshev filters, the coefficient of frequency response unevenness in the passband is additionally determined, and for Bessel filters, the group delay time is determined. Next, the transfer function of the filter is determined, which can be taken from the tables, and its 1st and 2nd order cascades are calculated, the following calculation procedure is observed:

Depending on the order and type of the filter, the circuits of its cascades are selected, while an even-order filter consists of n/2 2nd order cascades, and an odd order filter - from one 1st order cascade and ( n– 1)/2 cascades of the 2nd order;

To calculate a 1st order cascade:

The selected filter type and order determines the value b 1 1st order cascade;

By reducing the occupied area, the capacity rating is selected C and calculated R according to the formula (you can also choose R, but it is recommended to choose C, for reasons of accuracy)

;

The gain is calculated TO at U 1 1st order cascade, which is determined from the relation

,

Where TO at U– gain of the filter as a whole; TO at U 2 , …, TO at Un– gain factors of 2nd order cascades;

To realize gain TO at U 1 it is necessary to set resistors based on the following relationship

R B = R A ּ (TO at U1 –1) .

To calculate a 2nd order cascade:

By reducing the occupied area, the nominal values ​​of the containers are selected C 1 = C 2 = C;

Coefficients are selected from tables b 1 i And Q pi for 2nd order cascades;

According to a given capacitor rating C resistors are calculated R according to the formula

;

For the selected filter type, you must set the appropriate gain TO at Ui = 3 – (1/Q pi) of each 2nd order stage, by setting resistors based on the following relationship

R B = R A ּ (TO at Ui –1) ;

For Bessel filters, it is necessary to multiply the ratings of all capacitors by the required group delay time.