For filters, you can fix the default parameters of the filter, the most important is the frequency response. We have already discussed at the statti how it will be possible to reduce the given filter to the low-pass filter prototype. The butt of the vimog to the amplitude-frequency characteristic of the low-pass filter prototype of the projected filter is aimed at the little 1.

Figure 1. Application of the normal amplitude-frequency characteristics of the low-pass filter

This graph shows the accumulation of the filter transmission rate up to the standard frequency ξ , de ξ = f / f v

On the hovering little 1 graph, you can see that the smoothness of the transmission can be set to the permissible unevenness of the transmission rate. Smoothies have a minimal effect of smothering the signal. A real filter can be a form. Smolder, you won’t overturn the boundaries of the assignments.

To reach a trivial hour of filtering the filter was carried out by the method of selecting the amplitude-frequency characteristics for the additional standard Lanka (M-Lanka or K-Lanka). A similar method is called the aplication method. Win buv to reach the folding i without giving optimal performance the quality of the crushed filter and the number of lanes. To that, the mathematical methods of approximation of the amplitude-frequency characteristics from the given characteristics have been broken down.

Approximation in mathematics is called the manifestation of folding fallowness as a function. The name of the function is simple. In the case of the filter design, it is important that the approximate function could easily be implemented circuitry. In addition, the function is implemented with the additional zero and pole of the transmission coefficient of the four-pole, in the case of the filter. The stench is easy to implement behind the additional LC-contour or the ringing sounds.

A broader type of approximation of the frequency response of the filter є approximation behind Butterworth. Some of the filters were called Butterworth Filter.

Filtree Butterworth

Due to the peculiarity of the amplitude-frequency characteristic of the Butterworth filter є the visibility of the minimum and maximum in the smoothie, the transmission and the shut-off. The decrease in the frequency response between the smog and the transmission of the filters to the doors is 3 dB. If the filter requires less than the value of the irregularity in the smoothness of the throughput, then the frequency of the filter is turned f to vibrate at a given upper frequency and transmission. The frequency response approximation function for the LPF prototype of the Butterworth filter is as follows:

(1),

de ξ - The frequency is normal;
n- Filter order.

At the same real amplitude-frequency characteristic of the filter can be trimmed by multiplying the standard frequency ξ filter frequency. For the Butterworth filter of low frequencies, the frequency response approximation function is shown as follows:

(2).

Infectious beastly respect, when the filtrating is widespread, the understanding of the complex area is widely victorious, a circular frequency is put on the yak along the ordinate axis. jω, and on the abscis axis - the value, turned around by the quality factor. In such a rank, it is possible by virtue of the main parameters of the LC-circuits, which enter before the warehouse of the filter circuit: the frequency of adjustment (resonant frequency) and the quality factor. Go to the s-area, go for help.

A detailed map of the position of the Butterworth filter poles on the complex s-area is placed in. For us, it’s smarter than the poles of the whole filter are roztasvani on a single number at a time from one place to one. The number of poles starts with the order of the filter.

On little 2, the poles for the first order Butterworth filter have been moved. The handrail is shown by the frequency response, which is related to the given rosetting of the poles on the complex s-area.

Malunok 2. Roztashuvannya poles and frequency response Butterworth filter first order

On little 2 it can be seen that for the filter of the first order, the pole is to blame for the adjustment to the zero frequency, and the quality factor is at fault for the single ones. The frequency response graph shows that the frequency of the adjustment of the pole is efficiently zero, and the quality of the pole is such, that at the frequency of the standard Butterworth filter, that is, the transmission efficiency is −3 dB.

So the poles for the Butterworth filter are of a different order. Once the frequency of the alignment of the pole is vibrated at the cross-retraction of a single pole with a straight line, then pass through the center of the pole at 45 °.

Malunok 3. Roztashuvannya poles and frequency response of the Butterworth filter in a different order

If the resonant frequency of the pole is different, it is not close to the frequency of the normal filter. Won the door 0.707. The quality of the pole behind the graph of the rosetting of the poles at the roots of two times is for the quality of the Butterworth filter pole of the first order, so the steepness of the decline in the amplitude-frequency characteristic is greater. (I respect the numbers in the right part of the graph. When the frequency is set to 2, it is adjusted to the level of 13 dB). The left part of the amplitude-frequency characteristics of the pole is flat. Tse z injected poles, scho in the zones of negative frequencies.

The rotation of the poles and the amplitude-frequency characteristic of the Butterworth filter of the third order is shown in small 4.

Malunok 4. Raztashuvannya poles of Butterworth filter of the third order

It can be seen from the graphs shown on the little ones 2 ... 5, with an increase in the order of the Butterworth filter, the steepness of the fall of the amplitude-frequency characteristic increases and the increase requires the quality of the lantern of a different order of the characteristic (contour), more The very growth of the required quality factor is interconnected with the maximum order of the filter, which can be realized. It is possible to realize the Butterworth filter up to the eighth - tenth order.

Filtri Chebisheva

In Chebishev filters, the approximation of the amplitude-frequency characteristic is carried out by the offensive rank:

(3),

With a large amplitude-frequency characteristic of a real Chebishev filter, so it is possible to correct the Butterworth filter by multiplying the standard frequency ξ to the frequency of the filter, so that it breaks down. For the Chebishev filter of low frequencies, the amplitude-frequency characteristic can be as follows:

(4).

Amplitude-frequency response of the Chebishev filter low frequencies characterized by a greater steep drop in the frequency range above the upper transmission frequency. Tsey vigrash reach for the rakhunok, show the uneven frequency response of the smooth flow. The inconsistency of the function of approximation of the frequency response of the Chebishev filter is to be determined by the higher quality of the poles.

A detailed diagram of the position of the poles of the approximate function of the Chebishev filter on the s-area is pointed at. It is important for us that the poles of the Chebishev filter roztashovani on elipsi, there is a great deal of how to get out of all the standard frequencies. On the central axis, the elips pass through the frequency point and the lower frequency filter.

The standard version has a single point. Another way is to start with an uneven function of approximation of the frequency response of the smoothie throughput. Chim more is permissible unevenness in smoothies passing, tim mensha tsya. To see how to "flatten" a single stake of Butterworth filter. The poles approach the frequency axis. The increase in the quality factor of the filter poles. Something more unevenness in smoothies, the more solidity of poles, more quickness of growth in smoothies, non-skipping filter of Chebishev. The number of poles in the AFC approximation function starts with the order of the Chebishev filter.

Slide means that Chebishev's first filter is dumb. The rotation of the poles and the frequency response of the Chebishev filter has a different order. The stench reflects the maxima of the frequency response of the smoothie throughput. The filter has a different order, the frequency of the pole is ξ =0.707.

Plan:

Entry

• 1 Look around
  • 1.1 Norman Butterworth Polynomy
  • 1.2 Maximum smoothness
  • 1.3 Decay of characteristics on high frequencies
• 2 Filter design
  • 2.1 Topology Kauer
  • 2.2 Sullen-Cay topology
• 3 Matching with the largest line filters
• 4 Butt
Literature

Entry

Filter Butterworth- one of the types of electronic filters. Filters of the class are derived from the design method. The Butterworth filter is designed so that the amplitude-frequency characteristic of the boole is as smooth as possible at the frequencies of the smog throughput.

Some of the filters were first described by the British engineer Stephen Butterworth in the article "About the theory of filter drives" (eng. On Theory of Filter Amplifiers ), in the journal Wireless Engineer 1930 rock.

1. Look around

The frequency response of the Butterworth filter є is as smooth as possible at the frequencies of the smog, the transmission is practically reduced to zero at the frequencies of the smog, the smothering. When the frequency response of the Butterworth filter is displayed on the logarithmic AFC, the amplitude decreases to a minimum at the frequencies of the smog. If the filter is of the first order, the frequency response is reduced to -6 decibels per octave (-20 decibels per decade) (for the sake of all the filters of the first order, they are of the same type). For the Butterworth filter of a different order, the frequency response is dimmed by -12 dB per octave, for the third order filter - by -18 dB, so far. The frequency response of the Butterworth filter is a monotonically falling function of the frequency. Butterworth filter - single iz filter, which takes the form of frequency response for higher orders (behind the winyat a larger steep decline in the characteristics for smothering smoothies) to some of the most sophisticated types of filter різні form AFC for different orders.

Accordingly, with the Chebishev filters I and II types, or with an elegant filter, the Butterworth filter has a slightly sloping drop in characteristics and that is due to the mother of a larger order (more folding in realizatsiya) in order to forget required characteristics at the frequencies of the smoggy. However, the Butterworth filter has a larger linear phase-frequency characteristic at the frequencies of the smog transmission.

Frequency response for Butterworth filters of low frequencies from 1 to 5. Nahil characteristics - 20 n dB / decade, de n- Filter order.

Yak and for all filters for an hour to look frequency characteristics low frequency filter, from which you can easily reject the high frequency filter, and, having removed the number of such filters last, - dark brown filter or notch filter.

The amplitude-frequency response of the Butterworth filter of the order can be adjusted from the transmission function:

It is easy to note that, for those who are not, the frequency response becomes a direct-current function, and the frequency and lower frequency will pass through the efficiency, and the frequency will increase the frequency and the frequency will increase. For Kintsevs, the decline in the indicator will be flat.

Behind the additional formal substitution, one can imagine a viraz at the viglyad:

The poles of the transmission function are rotted on the number of radii equal to one from one to one in the left. The transfer function of the Butterworth filter can only be achieved by depriving the assigned poles of its transfer function at the left side of the square. -th pole of start from offensive viraz:

The transfer of the function can be recorded at the viewer:

Analogous intercourse stays up to digital Butterworth filters, with this difference, s-area, and for z-area.

The banner of the central transmission function is called the Butterworth polynomial.

1.1. Norman Butterworth Polynomy

Butterworth's polynomies can be recorded in a complex form, like it is shown, if you want to get a stench, you can be registered with the help of speech functions (in a complex way, you can make a bet for additional help). Polynomies are normal for the frequency of vision:. Norman Butterworth polynomies, in such rank, mayut taku canonical form:

, - paired, - unpaired

Below are the parameters of Butterworth polynomials for the first eight orders:

Kofіtsієnti polіnomіv 1 2 3 4 5 6 7 8

1.2. Maximum smoothness

Accepted and lost amplitude characteristics in terms of frequency will be seen as an offensive rank:

There is a monotonous change in all the scores, the performance is always positive. In such a rank, the frequency response of the Butterworth filter is not pulsation. When the amplitude characteristics are laid out in a row, we can measure:

In other words, all the same amplitude-frequency characteristics in terms of frequency up to 2 n-ї equal to zero, which means "maximum smoothness".

1.3. Slope at high frequencies

Having accepted, it is known to have nahil the logarithm of the frequency response at high frequencies:

In decibels, the high-frequency asymptote is ma nahil -20 n dB / decade.

2. Filter design

There is a low low filter topology, in addition to realizing linear analog filters. Tsi schemes become deprived of significant elements, the structure becomes invisible.

2.1. Topology Kauer

The topology of Kauer is a vicious passive element (a lot and an inductance). Filter of Butteworth due to the given function of transmission can be found in the form of Kauer type 1. kth element filter to ask for children:

; k is unpaired; k paired

2.2. Sullen-Cay topology

The topology of Sullen-Kay is victorious for the passive as well as active elements (operative pidsilyuvachi and amnosti). The skin cascade of the Sallen-Kay scheme is a part of the filter, which is mathematically described by a pair of complex-derived poles. The entire filter goes to the last day of the cascades. At the vipadku, where the pole is traversing, the fault is to blame for the implementation of okremos, call in the viglyadi RC-lance, and the inclusions to the out-of-the-box scheme.

The transfer function of the skin cascade in the Sallen-Kay ma viglyad circuit:

It is necessary that the banner will be one of the multipliers in the Butterworth polynomial. Having accepted, otrimaєmo:

The rest of the business is given to two unavailable ones, which can be found pretty well.

3. Adjustment with the largest line filters

The little one below shows the frequency response of the Butterworth filter, in proportion to the most popular linear filters of the same (fifth) order:

From the small it is seen that the drop in the frequency response of the Butterworth filter is the most common from the chotiroh, protein low and high frequency response at the frequencies of the smog throughput.

4. Butt

Analogue Butterworth filter of low frequencies (Cauer topology) due to the frequency of the upcoming nominal elements: farad, ohm and genry.

Gustini's logarithmic graph of the transfer function H (s) on the area of ​​the complex argument for the Butterworth filter of the third order with frequency. Three poles lie on the number of a single radius at the left side of the square.

A clear analog low-frequency Butterworth filter of the third order with farad, ohm, and genre. Having determined the new definition of the capacities C yak 1 / Cs the same op_r of inductances L yak Ls, de - complex change, that vikorystovyuchi іvnyannya for rozrahunku electrical circuits, I will immediately step on the transmission function for such a filter:

The frequency response is set to equal:

and the PFC should be set equal to:

Group coverage starts as the minus is lost to the phase of the circular frequency and to the world with the signal in phase at the lower frequencies. The logarithmic frequency response of such a filter is not pulsating, neither smoothing, nor smoothing.

The graph of the module of the transfer function on the complex area clearly points to three poles at the sides of the area. The transfer function is to increase the visibility of the rosetting of these poles on a single number symmetrically to the active axis.

Replacing the skin inductance with the ominousness, and the multiplicity with the inductances, we can recognize the high-frequency Butterworth filter.

1st group coverage of Butterworth filter of the third order due to frequency of vision

Literature

• V.A. Lucas Theory of automatic keruvannya. - M .: Nadra, 1990.
• B.Kh. Krivitsky Dovidnik s theoretical foundations radioelectronics. - M .: Energia, 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. - 4rd Edition. - Paramus, NJ: Prentice-Hall, 2001 .-- ISBN 0-13-090126-1
• Michael L. Honig, David G. Messerschmitt Adaptive Filters - Structures, Algorithms, i 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

Topic 28: Classification of electric filters.

28.1 Business value.

An electric frequency filter is called a chotiripolus, which is a stream of some pass frequencies for good extinguishing (3 dB attenuation), and a stream of lower frequencies for great extinguishing (30 dB).

The range of frequencies, in some weakened ones, is not very much called a dark transmission.

The range of frequencies, for which weakening is great, is called smog overshadowed.

Introduce the swamp to the transition with crimp braids.

The main characteristic of electric filters is the presence of the working frequency extinguishing.

The qia characteristic is called the frequency characteristic of the dissipation.

- The frequency is 3 dB.

- Suppose fading, set the mechanical parameters of the filter.

- the permissible frequency is based on the permissible extinguishing.

PP - smuga transmission - frequency range, in
dB.

PZ - smuga zatrimuvannya - the frequency range, for those who work more extinguishing is greater than the permissible.

28.2 Classification

1
Passage for smuga roasting:

a) LPF - low frequency filter - skipping low frequencies and overriding high frequencies.

Get stuck in the equipment (TV reception).

b
) HPF - filter high frequencies- skipping high frequencies and overriding low ones.

v
) PF - smugovі filtri - skip the less frequent smogy frequencies.

G
) ZF - rejection chi zhisnі filters - do not pass the frequency that is too singular, and do not pass.

2 Behind the element base:

a) filters LC (passive)

b) filter RC (passive)

c) active filter ARC

d) special types of filters:

P'єzoelectric

Magnetostriction

3 3 Mathematical security:

a
) Filter Butterworth. Characteristics of the working extinguishing
low at frequency f = 0 value 0 and then monotonically increase. Smoothies have a flat characteristic - the price is good, but the smoothies have a cool one - they are short.

b) Chebishev's filter. I’ll try to correct the cool characteristic of Chebishev’s vikoristovuyt filter, but they have a smooth flow of “boastfulness”, but it’s not enough.

c) Zolotarov's filters. Characteristics of the working extinguishing
in smoothies, there is a lack of vigor, and in smoothies, the failure of characteristics is overridden.

Busy Topic 29: Filter LF and HF Butterworth.

29.1 Fnch Butterworth.

Butterworth, having proponated the following formula for extinction:

dB

de
- Butterworth function (normalized frequency)

n-filter order

For LPF
, de - If you need a frequency

- Frequency zrіzu, yaka dorіvnyuє

In order to realize such a characteristic, it is possible to use the filter LіC.

І

Inductiveness to put lastly,
and for growth get better
It is easy to pass through the low frequency streams through the inductance, while the high frequency streams can be overridden and not consumed during the installation.

The capacitor should be installed parallel to the mounting, splinters
To that, the capacitor is good for passing the high frequencies and badly for the lower ones. The streams of high frequencies are closed through the capacitor, and the streams of low frequencies pass at the navantazhennya.

The scheme of the filter is stored in LіC, which can be swiped.

Butterworth LPF 3rd order T-sub

LPF Butterworth. 3rd order P-podibny.

When analyzing filters and racking of these parameters, depending on the size of the ear, it is necessary to determine the standard terms and the senses of the process from the ear itself.

Admittedly, a filter of low frequencies is required for the flat characteristics of the smog of transmission and a sharp transition to smog of smothering. Residual nahil characteristics in smoothies overshadowed by 6n dB / octave, de n - the number of "poles". One capacitor is required on the skin pole (aka inductance coil), so, until the residual fluidity, the frequency response of the filter decreases, roughly apparently, because of its foldability.

Now, it is permissible that you have shown a 6-pole low-frequency filter by the way. You are guaranteed a residual rolloff of 36 dB / octave at the higher frequencies. At your side, you can now optimize the filter circuit in the sense of ensuring the most flat characteristics of the smog, the flow for the rakhunok, the decrease in the steepness of the transition from the smog, the flow to the swamp, is tamped. On the other side, admitting an unevenness of the characteristics of the smoothie throughput, it is possible to reach a steep transition from the swamp through to the swamp throughtrimmed. The third criterion, which can be important, is the quality of the filter to pass the signal with the spectrum, to lie near the smoothie through, without forming it, and to experience phase changes. It is also possible to ts_kavitisya hour of growth, wikid that hour of establishment.

We provide methods for designing filters, which are available for optimizing either of these characteristics or of combinations. It is true that the intelligent vibration of the filter is not displayed as it is described in the food; As a rule, it is necessary to set the necessary parameters for the characteristics of the smog, the transmission and the need to be saved for the day of the frequency, the posture of the smog, the transmission and the parameters. For the sake of choosing the most suitable scheme with a number of poles, sufficient to make all people happy. At the next few times, there will be three most popular types of filters, and the Butterworth filter itself (the flattest characteristic in smooth flow), Chebishev filter (the most wide range of the air from the smog to the maximum flow rate) Whether from a number of types of filters can be realized for additional help other schemes filter; All sorts of them are negotiable for all sorts of smells can also be used to induce low- and high-frequency filters and dark colors.

Filtry Butterworth and Chebishev. The Butterworth filter will preserve the best flat characteristics of the smooth flow, so that it can reach the same smoothness of the characteristics of the transitional area to tobto. mіzh swears passing that zatrimuvannya. As it will be shown far away, the phase-frequency characteristic is also nasty. Yogo amplitude-frequency characteristic is set by the following formula:
U out / U in = 1 / 1/2,
de n is the first order of the filter (number of poles). The increase in the number of poles gives the possibility of turning into a greater flatness of the characteristics of the smoothie throughput and the steepness of the decline from the smogy throughput to the smoggy smothering, as shown in Fig. 5.10.

Small. 5.10 Normal characteristics of low frequency Butterworth filters. Worst respect for the increase in the coolness of the decline in the characteristics with the increase in the order of the filter.

Vibrating Butterworth filter, in order to maximize the flat characteristics of the actions of all the people. Its characteristics go horizontally, adjusting from zero frequency, bending and repairing at the frequency of the rise - the frequency is adjusted to the point of -3 dB.

In most cases, the most common furnishings and those, where the unevenness of the characteristics of the smoothies, the transmission of a higher singing value, say 1 dB. Filtr Chebisheva promises, at the same time, the deyak is allowed to have an uneven character in all smoothies, but at the same time there is a lot of hospitality and evil. For the Chebishev filter, set the number of poles and the unevenness in the smoothie. Allowed to increase the nervousness in smoothies, pass, accept hostility to evil. Amplitude-frequency characteristic of the whole filter should start with the onset of sports
U out / U in = 1 / 1/2,
de З n - the field of Chebishev of the first kind of step n, and - a constant, which is the origin of the irregularity of the characteristics in smoothies. Filter of Chebishev, yak and filter of Butterworth, low phase-frequency characteristics, far from ideal. In fig. 5.11 shows the characteristics of 6-pole low-frequency filters for Chebishev and Butterworth. The yak is easy to pick, and the one that can be easily shrunk by the 6-pole RC filter.

Small. 5.11. Adjustment of the characteristics of the 6-pole low frequency filters, so that you can start to freeze. The characteristics of the images themselves are quiet in the logarithmic (in the mountains), and in the linear (below) scales. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebishev filter (pulsation 0.5 dB).

For the sake of the Butterworth filter with a maximally flat characteristic in smoothies, the transmission is not too prickly, as it is possible to get up, some kind of problem can be reconciled with the fact that the smoothness of the transmission filter of Chebishev-pulsation, distribution of all smoothies through). In addition, the active filters, prompted by the elements, the nominal values ​​of such tolerance, refer to the characteristic, which is based on the rosary, and this means that, for the sake of the characteristics of the filter, the Butterworth's mother deserves In fig. 5.12 illustrated by the injection of the most irrelevant value of the capacity of the capacitor and the support of the resistor on the characteristic of the filter.

Small. 5.12. Injection of change parameters of elements on the characteristic of the active filter.

In the light of the vischevikladenogo with a more rational structure є the filter of Chebishev. One of them is called an equal-sized filter, since its characteristic in the area of ​​transition is very steep for the sake of the fact that, according to the smoothness of the transmission, a number of equal-sized pulsations are generated, a number of bright growths at once. Navigate at randomly small pulsations (close to 0.1 dB) the Chebishev filter will not provide more steepness in the transition area, lower Butterworth filter. However, it is possible that a filter is required for an inadequate characteristic of the smog, the transmission is not more than 0.1 dB and the extinguishing of 20 dB at a frequency that appears at 25% of the boundary frequency. Rozrakhunok will show that a 19-pole Butterworth filter is required in the whole range, or an 8-pole Chebishev filter.

A thought about those, how you can put up with the pulsations of the characteristics of a smoothie, passing for an increase in the coolness of a transitional design, is brought to its logical conclusion in the idea of ​​the so-called elegant filter (or a filter of Causer), in such a way tweaks for securing the coolness of the transitional file to see more, lower at the characteristics of the Chebishev filter. With the help of EOM, you can construct an elegant filter as simple as the classic Chebishev and Butterworth filter. In fig. 5.13 the graphical setting of the amplitude-frequency characteristic of the filter is induced. In a wide range of filters (low frequency filter), there is an acceptable range of the filter transmission efficiency (to be unreasonable) in smoothness, the minimum frequency, the characteristic is too dark, the transmission is overshadowed, the maximum frequency

Small. 5.13. The value of the parameters of the frequency characteristics of the filter.

Filtri Bezsel. Since the bulo was set earlier, the amplitude-frequency characteristic of the filter does not give any more information. Filter from a flat amplitude-frequency characteristic can cause great phase failure. In addition to the form of the signal, the spectrum of which lies in the smoothness of the transmission, will be created when passing through the filter. In a situation, if the form of the signal is of the highest value, the mother has a separate line-phase filter (filter with a permanent hour of recording). Before the filter, before the filter, there is no change in the phase in the depletion of the frequency, which is equivalent to that of the hour when the signal is lost, the spectrum of bright changes in the smoothie is transmitted, so that the signal is formed during the daytime. Bessel's filter (also called Thomson's filter) is the best flat for a crooked hour stored in smoothies, possibly before the Butterworth filter has the best flat amplitude-frequency response. The mind of the eye, like a shrinking in the clock area, gives the Bessel filter, wonder at the fig. 5.14 Display rates are standard for the hour of recording for 6-pole Bessel and Butterworth low frequency filters. Shit characteristic of the hour of recording the filter Butterworth will zoom in on the effect of the wikid type for an hour passing through the filter of impulse signals. On the other hand, for the steel of hours spent at the Bessel filter, the payment is due to the fact that the amplitude-frequency characteristic is less likely to cross the distance between the smogs and the transmission and control characteristics of the filter.

Small. 5.14. Adjustment of team-hour recordings for 6-dark Bessel low frequency filters (1) and Butterworth (2). Bessel's Filter is the best way to form a signal for his miraculous powers in the watch area.

isnu bagato different ways Designing filters, for those who try to polish the working parameters of the Bessel filter in the watch area, often donating an hour to save the hour of increasing and improving the amplitude-frequency characteristics. The Gaus filter can be as good as phase-frequency characteristics, like the Bessel filter, even if the transition characteristics are reduced. The second tsikaviy class is a filter, which allows the signal to reach the same for the value of the pulsation crooked hour stored in the smoothie (similar to the pulsations of the amplitude-frequency characteristics of the Chebishev's filter) and One more time before the start of the filters, after the last hour of the time, there is no more filter, which is called all-passable, called by the correctors in the time zone. Cycle filters can change the amplitude-frequency response, and the phase can be changed depending on the specific parameters. With such a rank, it is possible to store for the hour of remembrance of any filters, the seed of the filters of Butterworth and Chebishev.

Filters Unimportant at the earliest, I was concerned about the transfer of the characteristics of Bessel's filters, but all the same, there is even less good authority in the watch area in the context of Butterworth's and Chebishev's filters. The Chebishev filter itself for the even more suitable amplitude-frequency characteristics of the most important parameters in the clock area with three types of filters. The Butterworth filter gives a compromise between frequencies and clock characteristics. In fig. 5.15 Information about the working characteristics of the three types of filters in the watch area is given, with additional guidance earlier than the graph of the amplitude-frequency characteristics. For the tsim of the data, you can go unimportantly, but in the fall, if the parameters of the filter in the watch area are important, there is a lot of Bessel filter.

Small. 5.15. Adjustment of transition processes of 6-pole low-frequency filters. The curves are normalized to the reduced attenuation values ​​of 3 dB up to a frequency of 1 Hz. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebishev filter (pulsation 0.5 dB).

Filter Butterworth

Butterworth Low Pass Filter Transmit Function n-th order is characterized by viraz:

The amplitude-frequency response of the Butterworth filter can be as follows:

1) For some order n frequency response value

2) at frequency

The frequency response of the low-pass filter monotonously changes with increasing frequencies. Tom Butterworth filters are called filters with maximally flat characteristics. On little 3, graphs of amplitude-frequency characteristics of Butterworth LPF of 1-5 orders are shown. Obviously, if the order of the filter is larger, the frequency response of the ideal low-frequency filter is more accurately approximated.

Malunok 3 - frequency response for low frequency Butterworth filter from 1 to 5

On little 4, a schematic implementation of the Butterworth HPF is presented.

Malunok 4 - HPF-II Butterworth

By the passable Butterworth filter є the smoothest AFC at the smog frequencies, the transmission is practically reduced to zero at the smog frequencies. Butterworth filter - single from filter, which takes the form of frequency response for higher orders (due to a higher drop in characteristics for smoothies), which is also very rich in quality of filter

However, in the case of Chebishev filters of I and II types, or with an elegant filter, the Butterworth filter has a slightly sloping drop in characteristics, and the mother is responsible for a greater order (more foldable in real life for the loss of frequencies) in order to

Filtr Chebishev

The square of the module of the transmission function of the Chebishev filter is defined by the viraz:

de - polіnom Chebishev. The module for the transmission function of the Chebishev filter is set to zero at the same frequency.

Filter Chebisheva invite you to be there, because it is necessary for an additional filter of a small order to preserve the necessary characteristics of the frequency response, winter, it is good to adjust the frequencies from the smog to suppress, and when the frequency is not so important, the frequency is not so important.

Razrіznyayut filters Chebishev I and II canopy.

Filtr Chebishev of the 1st kind. The most frequent development is the modification of Chebishev's filters. In smoothies, the passage of such a filter shows pulsations, the amplitude of which appears to be an indicator of pulsations. At the same time, the Chebishev analogue electronic filter has an order of magnitude for the number of reactive components, which are victorious during its implementation. Greater steep drop in characteristics can be removed, so as to allow pulsation in smoothies, letting go, and in smoothies, stifling, adding a filter to the forward function of zero on the clear axis of jsh in a complex area. Tse, however, is the result of the least effective smuggling of smog. Otrimaniy filter є an elegant filter is also seen as a filter Kauer.

The frequency response for the Chebishev filter of the lower frequencies of the first kind of the fourth order is presented by malunka 5.

Figure 5 - AFC for the Chebishev filter of low frequencies of the first order of the fourth order

The filter of the Chebishev of the II kind (the inverted filter of the Chebishev) grows faster, the lower the filter of the Chebishev of the I kind through the lesser of the slopes the decrease in the amplitude characteristics, should be produced to an increase in the number of components. In new daytime pulsation, smoothness in smoothies, protea є in smoothies, stifling.

The frequency response for the Chebishev filter of low frequencies of the II kind of the fourth order is represented by a small 6.

Malunok 6 - AFC for the Chebishev filter of low frequencies of the II kind

On little 7, the schematic realizations of the Chebishev HPF are presented in the I and II order.

Malyunok 7 - high-frequency filter Chebishev: a) I order; b) II order

The power of the frequency characteristics of the Chebishev filters:

1) In smoothies, the frequency response has a significant character. On the interval (-1? N? 1) є n points, in which the functions of the reach of the maximum value, the road 1, or the minimum value, the road. Where n is unpaired, where n is pair;

2) the value of the frequency response of the Chebishev filter at the frequency of the road

3) When the function is monotonically changing and not zero.

4) The parameter is the initial value of the frequency response of the Chebishev filter in the smoothie throughput:

Adjusting the frequency response of Butterworth's filters and Chebishev's will show, but Chebishev's filter will be more weakened in the smoothie, but not the Butterworth's filter in the same order. There is a lack of filters in Chebishev's polarity in the fact that the phase-frequency characteristics of the smoothies are significantly transformed from the linear ones.

For Butterworth and Chebishev filters є report tables, for which the coordinates of the poles and the performance of the transmission functions of the different orders are set.