Chebyshev filter

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Abstract

In the first section describes the General immunity of radio, the passage of noise through the path of the receiving device. And also made the choice and justification of the structural scheme and the circuit diagram. The second section consists of part of the settlement, where calculated the necessary data transmission speed and the duration of single elements, bandwidth filters transmission and reception, the effective value of interference in the channel, time of entry into synchronism and time synchronization reference fluctuations.

The characteristic of Bessel filters that makes them valuable to digital designers. Very few filters are designed with square waves in mind. Most of the time, the signals filtered are sine waves, or close enough that the effect of harmonics can be ignored. If a waveform with high harmonic content is filtered, such as a square wave, the harmonics can be delayed with respect to the fundamental frequency.

The content

Normative references

Conventions and abbreviations

Introduction

1. Classification of the filter

1.1 The filter on the concentrated elements

2. The review of similar schemes

3. The choice and justification of the filter circuit

4. Topological model of the filter and the transfer function of the voltage

4.1 Implementation of the normalization HPF

4.2 Definition of the polynomial Bessel

4.3 the Reverse transition from a fixed to a planned HPF

5. Calculation of elements of the circuit

5.1 Preparation of a complete scheme of filter

Conclusion

List of the used literature

Normative references

In this course work used references to the following documents:

Table №1 - Used normative references:

Symbol	Name
ГОСТ 2.701-84	The scheme. The kinds and types. General requirements to the implementation
ГОСТ 2.702-75	Rules of execution of electrical circuits
ГОСТ 2.708-81	Rules of execution of electrical circuits and digital computer technology
ГОСТ 2.709-72	The system of marking of circuits in electrical circuits
ГОСТ 2.752-71	Conventions document in graphical diagrams. Device telemechanics
ANSI	American national standard notation
DIN	The European standard notation
ГОСТ 2.201-80 ЕСКД	Designation of products and design documents
ГОСТ 2.601-95	Operational documentation
Ф.7.14-2008 СМК	Rules for the registration of training documentation. General requirements to graphic documents

radio noise reception channel

Conventions and abbreviations

RC - filters of lower

Op - operational amplifier

BPF - band-pass filter

LPF - low pass filters

BSP - band stop filter

HPF- the high pass filter

SV - source voltage,

VC- voltage-controlled

NF - negative-feedback

Introduction

Since 1926, for the first time have been measured by non-linear distortion, proposed more than 20 different measurement methods. All of these methods can be divided into 5 main groups: one tone, two tones, with discrete spectrum, with continuous spectrum, with the working signal. Each of these groups has its varieties, which differ according to the method of registration and allocation of products of non-linear distortions.

In General, the methods of measurement of nonlinear distortions in paths in the STI is a great variety. Widely spread method of harmonics as the most simple in experiments and convenient for calculations. Less common other methods: the difference of tone modulated tone intermodulation (two-tone).

For the mentioned methods has its own field of application. In this case, each of them uses a special signals, ensuring the most efficient detection products distortion.

Visibility of non-linear distortions of the real input signal is connected with the fact, how often, if we consider the process in time, or with what probability, if you apply to him statistical measure of the instantaneous values fall within the essential nonlinearity tract STI. Many, probably, had to observe, as the decrease of the signal level in congested channel disappears less of the hoarse sound. It the less, the less emissions signal fall within the overload.

In this work is discussed in detail strip Chebyshev filter, included in the composition of the installation for measurement of nonlinear distortion by the method of bands noise.

Filters with the characteristic of Chebyshev have a relatively high ratio cost/efficiency.

1. Classification of the filter

The filter this frequency-selective device which passes signals of certain frequencies and detains or weakens signals of other frequencies. The filters finding application in processing of signals happen: analog or digital, passive or active, linear and nonlinear, recursive and not recursive.

Filters can be classified by a number of signs:

1) by the form they are divided on: filters of the lower frequencies (FLF); filters of the top frequencies (FTF); strip filters (SF); the projector (blocking) filters (RZh). In separate group the phase filters (PF) can be allocated;

2) depending on the polynoms used at approximation of transfer function distinguish filters: critical attenuation, Bessel, Butterworth, Chebyshev;

3) on element base filters are divided on: passive and active filters. Active filters include in the scheme RLC - the filter of an active element as which operational amplifiers are often used.

The Amplitude-frequency Characteristics (AFC) of various filters are submitted in fig. 2.23. The filter of the lower frequencies - passes low frequencies and detains high (fig. 2.23, a), the filter of the top frequencies - detains low frequencies and passes high (fig. 2.23, b), the strip filter - passes a strip of frequencies from щ1 to щ2 and detains those frequencies which are located above or below this strip of frequencies (fig. 2.23, c), the rezhektorny filter - detains a strip of frequencies from щ1 to щ2 and passes the frequencies located above or below this strip of frequencies (fig. 2.23, d). In the specified filters the coefficient of transfer and phase shift depend on the frequency of an entrance signal. Filters at which the coefficient of transfer remains to constants, and phase shift depends on frequency, are called as phase filters.



Figure 1 Amplitude-frequency characteristic

As it was mentioned earlier, depending on the approximating polynom filters are divided into filters of critical attenuation, Bessel, Butterworth, Chebyshev. At a statement of the principle of creation of the approximating functions of filters as a basis usually use the FILTER of the LOWER FREQUENCIES. In fig. 2.24 AMPLITUDE-frequency characteristics of the specified filters of the lower frequencies are shown.

Figure 2 Filter passes low frequencies and detains the high

Amplitude-frequency characteristics the filter of the lower frequencies of Butterworth has quite long horizontal site and sharply falls down behind cut frequency. The transitional characteristic of such filter at a step entrance signal has oscillatory character. With increase in an order of the filter of fluctuation amplify. The characteristic of the filter of Chebyshev falls down more coolly behind cut frequency. In a pass-band it has wavy character with a constant amplitude. Fluctuations of transition process at a step entrance signal are stronger, than at Butterworth's filter. Bessel's filter is characterized by the smaller length of a horizontal site, than Battevort's filter and more flat recession AMPLITUDE-FREQUENCY CHARACTERISTICS behind cut frequency, than Butterworth and Chebyshev's filters. This filter possesses the optimum transitional characteristic (transition process has practically no fluctuations). The filter of critical attenuation possesses considerably the worst amplitude-frequency characteristic in comparison with Bessel's filter, but has no reregulation. Generally the filter of critical attenuation is inferior to Bessel's filter concerning quality of working off of an entrance step signal.

Bessel's filter - in electronics and processing of signals one of the most widespread types of linear filters which distinctive feature is the maksimalnogladky group delay (the straight-line phase-frequency characteristic). Bessel's filters vsegoispolzut for audio-crossovers more often. Their group delay practically doesn't change on polosypropuskaniye frequencies owing to what the form of the filtered signal at the exit of such filter in polosepropuskaniye remains the almost invariable.

1.1 The filter on the concentrated elements

In drawing the example of the elementary LC filter of the lower frequencies is shown: when giving a signal of a certain frequency on a filter entrance (at the left), tension at the filter exit (on the right) is defined by the relation of jet resistance of the coil of inductance (X_L = \omega L) and the condenser (X_C = 1/\omega C)

The transfer coefficient can be calculated the FILTER of the LOWER FREQUENCIES, considering tension divider formed by frequency-dependent resistance. Complex (taking into account shift of phases between tension and current) resistance of the coil of inductance is Z_L = j\omega L = jX_L and the Z_C condenser = 1 / (j\omega C) = - j X_C, where {j} ^ {2} =-1, therefore, for not loaded LC filter

K = {Z_C Z_C Z_C} { Z_L + Z_C}\\, \!. (1.1)

Substituting values of resistance, we will receive for frequency-dependent coefficient of transfer:

K (\omega) = \frac {1} { 1-\omega^2 \, LC } = \frac {1} { 1-(\omega/\omega_0) ^2 }\\, \!. (1.2)

Figure 3 Simple LC filter of the lower frequencies

Apparently, the coefficient of transfer not loaded ideal the FILTER of the LOWER FREQUENCIES beyond all bounds grows with approach to the resonant frequency \omega_0=1/\sqrt {LC}, and then decreases. At very low frequencies transfer coefficient the FILTER of the LOWER FREQUENCIES is close to unit, on very high -- to zero. It is accepted to call dependence of the module of complex coefficient of transfer of the filter on frequency the amlitudno-frequency characteristic (AMPLITUDE-FREQUENCY CHARACTERISTICS), and dependence of a phase on frequency -- the phase-frequency characteristic.

In real schemes active loading which lowers good quality of the filter is connected to an exit of the filter and prevents a sharp resonance AMPLITUDE-FREQUENCY CHARACTERISTICS near frequency \omega_0. Sizes \rho = \call sqrt {L/C} the characteristic resistance of the filter or wave resistance of the filter. The FILTER of the LOWER FREQUENCIES loaded on the active resistance equal characteristic, has not resonant AMPLITUDE-FREQUENCY CHARACTERISTICS, approximately constant for frequencies \omega <\omega_0, and decreasing as 1/\omega^2 at frequencies above \omega_0. Therefore, frequency \omega_0 is called cut frequency.

2. The review of similar schemes

The term Bessel refers to a type of filter response, not a type of filter. It features flat group delay in the passband:

Figure 4- the flat group delay in the pass band

This is the characteristic of Bessel filters that makes them valuable to digital designers. Very few filters are designed with square waves in mind. Most of the time, the signals filtered are sine waves, or close enough that the effect of harmonics can be ignored. If a waveform with high harmonic content is filtered, such as a square wave, the harmonics can be delayed with respect to the fundamental frequency if a Butterworth or Chebyshev response is used. The Fourier series of a square wave is:

(2.1)

This means that a square wave is an infinite series of odd harmonics, or sine waves, summed together to create the square shape. Obviously, if a square wave is to be transmitted without distortion, all of the harmonics - out to infinity - must be transmitted. This means that the square wave can be high pass filtered without distortion, if the 3 dB point of the filter is significantly lower than the fundamental. If the square wave is low pass filtered, however, the situation changes dramatically. Harmonics will be eliminated, producing distortion in the square wave. It is the job of the designer to decide just how many harmonics must be passed and what can be eliminated. Suppose that the designer wants to keep five harmonics. The resulting waveform looks something like this:

Figure 5 - Form of a wave of five harmonicas

This may be acceptable to the designer - it depends on the timing of the leading and trailing edge of the waveform. The elimination of harmonics will result in rounding of the edges, and therefore delay in the leading and trailing edges of the digital signal. Of more importance, the harmonics that are passed will not be delayed.

The Bessel approximation has a smooth passband and stop band response, like the Butterworth. For the same filter order, the stop band attenuation of the Bessel approximation is much lower that that of the Butterworth approximation.

The following figures are representative of a low pass filter. The response characteristics are mirror imaged for high pass filters.



Figure 6- Pass-bands of the filter of Bessel

The designer can see that there is no ripple in the passband of a Bessel filter, and that it does not have as much rejection in the stop band as a Butterworth filter.

The phase response of the three filter types is shown below. The Bessel response has the slowest rate of change of phase.

Figure 7 - The phase response of the three filter types

3. The choice and justification of the filter circuit

Input voltage, Uin	9 мВ	0,009
Amplification factor, КU	1000	1000
Input resistance, Rin	15 kОм	15000
The range of amplifi frequency, fН .. fВ	1000..10000hz	1000
The recession of the amplitude-frequency characteristics
on the bottom, fН	40 дб/дек
the upper frequency, fВ	40 дб/дек
Type of filter	the Bessel filter
Output current Iout,	12 мА	0,012
Power supply voltage Ups,	± 15 В
Solution
Proceeding from the formula of determining the coefficient of the gain:
found R2 = 100Ч100000 = 10 МОм.	1500000
To reduce the dependence of the offset of the input current is found resistance R3:
	14851,48515
Selected resistance R3 = 100 кОм.
The calculation of the filter high frequency.
As the HPF chosen the filter on ИНУН. This scheme allows you to build a filter in the minimum number of elements. It has low-impedance output, the small dispersion of the values of elements and the possibility to receive relatively high values of the coefficient of amplification.
For the Butterworth filter of the second order, from the application A[1], selected coefficients:В = 1,414214; С = 1,000000;
The gain for the HPF took KU = 10;
Choose the value of capacitance of C1 and C2 according to the formula:
С1 = С2 = 10 / fC = 10 / 1 = 10 мкФ	0,01
Took C1 = C2 =10 мкФ.
Calculated the value of the resistors
where:
wc = 2ЧpЧf = 2Ч3,14Ч1 = 6,283 рад/с.	6280
Chosen R5 = 6.2 кОм.
кОм
Chosen R4 = 39 кОм.
Chosen R7 = 6.8 кОм и R6 = 62 кОм.
The gain level shall be determined by the formula [1]:
The resulting gain HPF KU = 10;
3.4 . The calculation of the low-frequency filter.
To obtain the total gain to the voltage, satisfying job, from the condition of the KU preamplifier =100, and the KU HPF= 10, calculated the required HP-KU
	1
Ratios: В = 1,414214; С = 1,000000;
The gain for the LPF took KU = 10;
Choose the value of capacitance C3 according to the formula:	C3=
С3 = 10 / fC = 10 / 1000 = 0,01 мкФ
Took С3 = 10 нФ.
Found C4, satisfies the condition:
Ф	9,5E-08	C4
Took С4 = 91 нФ.
Calculated value сопротивленй by the formulas
Ом
where:
wc = 2ЧpЧf = 2Ч3,14Ч1000 = 6,283Ч103 рад/с.	62800
Chosen R8 = 12 кОм.
Ом
Chosen R9 = 2.4 кОм.
Ом		1357,497938
Ом
The gain level shall be determined by the formula
Chosen R10 = 150 кОм R11 = 16 кОм.
The resulting gain LPF KU = 10;
To select a chip, you want to find current and voltage on the elements of the scheme.
To simplify the calculations take some assumption - operational amplifier will be considered ideal.
Take marginal input parameters: Uin = 0,2 •10-3 В и f = 1000 Hz.
The voltage at the output of the first stage is:
Uout1 = КU1 * Uin = 100 • 0,2•10-3 = 0,02 V.	0,9	Eoy1
The output voltage of the second stage as well:
Uout2 = КU2 * Uout1 = 10 • 0,02 = 0,2 V.	9	Eoy2
The output voltage of the third cascade as well:
Uout3 = КU3 * Uout2 = 10 • 0,2 = 2 V.	9	Eoy3
Load resistance is defined from the following expression:
Ом
For determining the loads were scheme of replacement (Annex 2). For this input voltage presented voltage source Евх = 0,2 •10-3 In, OA replaced voltage sources Eoy1 = 0.02 V Eoy2 = 0.2 V, Еоу3 = 2 V equal to voltage on their output, the input current of the op neglected, but capacitors replaced capacitive resistance
,	1,59236E-11
	15923,56688
	15923566,88
	1676164,88
where f is the upper frequency range bandwidth.
According to the scheme of substitution, using Kirchhoff's laws, amounted to a system of equations:
I1 + Ioy1 - I2 = 0	I1	5,88119E-07
I2 + I3 - I4 = 0	I2	59518801981
I4 + Ioy2 - I5 = 0	I3	0,000003
I5+ I6 - I7 = 0	I4	59518801981
I7 + Ioy3 = Iн	I5	0,037029252
I1·(R1 + R2) = Eвх - Eoy1	I6	0,0000001
I2·(- j·Xc1) - I3·(R5 - j·Xc2) = Eoy1	I7	0,037029352
I3·(R5 - j·Xc2) + I4·R4 = -Eoy2	Ioy1	59518801981
I5·R8 - I6·(R9 - j·Xc4) = Eoy2	Ioy2	-59518801981
I6·(R9 - j·Xc4) + I7·(- j·Xc3) = -Eoy3	Ioy3	-0,025029352
As a result of solution of the system were the currents in the branches of the system:
I1 = 2 нА, I2 = 3,77 мкА, I3 = 1,81 мкА, I4 = 5,58 мкА, I5 = 1,07 нА,
I6 = 0,114 мкА, I7 = 0,113 мкА, Ioy1 = 3,77 мкА, Ioy2 = 5,58 мкА,Ioy3 = 10мА.
Select from the directory of the op type К140УД6А with the following parameters:
The nominal value of the supply voltage Ups = ±15 В.
Amplification factor КU = 70000
Input offset voltage UВХ.СМ. = 5 мВ.
Input bias current IВХ.СМ. = 30 нА.
Differential input resistance RВХ.ДИФ = 3 MОм.
Maximum output current IВЫХ.MAX. = 25 мА.
Slew rate output voltage VUВЫХ = 2,5 В/мкс.		2,5
That is, the maximum output current of the circuit does not exceed the permissible values for the op-amp, the gain to the voltage of the op-much more than the gain of the cascades and the input resistance many times more than the resistance of a resistor R1.
The dependence of the rate of rise of the output voltage has the form:
The maximum slew rate output voltage is equal to the amplitude:
В/мкс	0,157

4. Topological model of the filter and the transfer function of the voltage

Filter Topology

The most common filter topologies are Sallen-Key and Multiple Feedback filters. The major differences are:

Sallen-Key	Multiple Feedback
Non-inverting	Inverting
Very precise DC-gain of 1	Any gain is dependent on the resistor precision
Less components for gain = 1	Less components for gain > 1 or < 1
Op-amp input capacity must possibly be taken into account	Op-amp input capacity has almost no effect
Resistive load for sources even in high-pass filters	Capacitive loads can become very high for sources in high-pass filters

4.1 Implementation of the normalization HPF

In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay (maximally linear phase response). Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband.

The filter's name is a reference to Friedrich Bessel, a German mathematician (1784-1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel-Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design.

The transfer function



Figure 8 - The schedule of the amplitude-frequency characteristic and group delay for the low-frequency filter of Bessel of the fourth order.

A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the pass band. The Bessel filter maximizes the flatness of the group delay curve at zero frequency.

A Bessel low-pass filter is characterized by its transfer function:

(4.1)

where is a reverse Bessel polynomial from which the filter gets its name and is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of .

4.2 Definition of the polynomial Bessel

The roots of the third-order Bessel polynomial are the poles of filter transfer function in the s plane, here plotted as crosses.



Figure 9 - Bessel polynomials

The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial, such as the following:

The reverse Bessel polynomials are given by:

(4.2)

Where

(4.3)

Example

(a) (b)

Figure 7 (a,b) - Gain plot of the third-order Bessel filter, versus normalized frequency

Group delay plot of the third-order Bessel filter, illustrating flat unit delay in the pass-band

4.3 the Reverse transition from a fixed to a planned HPF

The transfer function for a third-order (three-pole) Bessel low-pass filter, normalized to have unit group delay, is

The roots of the denominator polynomial, the filter's poles, include a real pole at s = ?2.3222, and a complex-conjugate pair of poles at s = ?1.8389 ± j1.7544, plotted above. The numerator 15 is chosen to give a gain of 1 at DC (at s = 0).

The gain is then

 (4.4)

The phase is

(4.5)

The group delay is

(4.6)

The Taylor series expansion of the group delay is

(4.7)

Note that the two terms in щ2 and щ4 are zero, resulting in a very flat group delay at щ = 0. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at щ = 0 and a second specifies that the gain be zero at щ = ?, leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order n: the first n ? 1 terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at щ = 0.

5. Calculation of elements of the circuit

From the task it is visible, that it is necessary to provide recession AFC on the bottom and top frequency. Hence, it is necessary to u^e filters of the botton and top frequencies (FBF and FTF). As recession AFC both on bottom and on the top frequency is equal a db/deck it is clear that it is necessary to use filters of the first order. Also it is impossible to forget that is necessary to provide the fixed input impedance, size in 75 kOhm. On the basis of these reasons the block diagramme of the amplifier of an alternate current is made. It is presented on fig. 1

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Figure 10. The Block diagramme of the amplifier of an alternate current where P - a preamplifier, FBF - the filter of the bottom frequencies, FTF - the filter of the top

Frequencies

The recommended design procedure is shown on an example. Proceeding from the block diagramme of amplifiers of an alternate current it is necessary to develop the basic circuit.

The preamplifier is represented by not inverting scale amplifier captured by an individual feedback. That is its amplification factor Kl=10. In parallel to its input

terminal there is resistor Rl, 75kOhm. By means of 1 his resistor the set fixed input impedance is provided. The preamplifier exit is jointed to input terminal FBF. FTF free passes a signal with frequency, below 1MHz and carries out recession AFC on limit inferiors of frequencies a -15db/deck on frequencies, above 100 Hz. Exit FBF is jointed to input terminal FBF. FTF free passes a signal with frequency, above certain frequencies in 100Hz and carries out recession ACF on a complete limit a + 15db/deck on frequencies, below limit inferiors of frequencies of 1MHz.

In the given course work it is necessary to apply active filters on operating amplifiers (OA). Active filters usually have one or several active components (transistors, etc.) and, unlike passive, have the big complexity and can use on low frequencies (more low 0,5 mHz) owing to absence in them of reactors.

If in the task kind ACF of the amplifier filters Battervort are applied to realisation FBF and FTF in other cases, or filters Bessell or Chebishev is not specified. As already it was spoken earlier filters of the first order us?.

The amplification factor of consistently jointed links thus accordingly is multiplied.

5.1 Preparation of a complete scheme of filter

Full scheme shall be drawn up in accordance with the results of calculations of the circuit elements HPF, and LPF. When this is taken into account, that for creation of the band-pass filter (PF), filters of high and low frequencies should be connected in series. As shown by engineering practice, to get a good range filter, it is advisable to first place the HP-filter to suppress the low-frequency components of the signal and noise at the output of the LPF. In figure 7, the full scheme of band pass filter with the indication of values of parameters of elements of the circuit, received in the result of calculations. On the basis of this scheme after incorporating additional elements will be developed drawing of the basic electric circuit of the filter.



Figure 11 Complete the calculation scheme of the band--pass filter

5.2 Development of the basic electric circuit

Development of the basic electric circuit is the most important point of work. The basic electric circuit of the block of the band-pass filter determines the full composition of its elements and links between them and gives a detailed presentation on the work of the filters. The diagram depicts all the electrical elements, links between them, as well as elements, which ends the input and the output circuits. Grounds for execution of the basic electric circuit is the scheme of band pass filter (figure 11).

Conclusion

In fact Bessel filter with a maximally flat response in the pass band is not so attractive as it may seem, because in any case have to put up with some uneven performance in bandwidth (for the filter Butterworth it will be a gradual lowering of the characteristics of the approach to frequency fc, and for the Chebyshev filter - pulsation, distributed throughout the bandwidth). In addition, active filters, built from elements, ratings of which have a certain tolerance, will possess the characteristic different from the target, and this means that, in reality, the characteristics of the Butterworth filter will always have a place some unevenness in bandwidth.

In the light of very rational structure is the Chebyshev filter. Sometimes it is called filter, because of its characteristics in the field of transition has a large slope due to the fact that the amount of bandwidth allocated several equal ripple, the number of which increases with the order of the filter. Even with a relatively small ripple (order 0, 1dB) Chebyshev filter provides much greater slope of the characteristics of the area than the Butterworth filter. To Express this difference quantitatively, suppose that you want to filter with the uneven performance in a bandwidth of not more than 0.1 dB attenuation at the frequency by 25% from the limit of the frequency bandwidth. The calculation shows that in this case it is required 19-pole Butterworth filter, or just a 8-pole Bessel filter.

The idea that you can put up with the pulsations of the characteristics of the bandwidth for the sake of the steepness of the transition area characteristics, be brought to its logical conclusion in the idea of the so-called filter (or filter Cauer), which allowed pulsation characteristics as the amount of bandwidth, so, and in the store for the sake of the security of the slope of the transition area even more than the characteristics of the Chebyshev filter. With the help of computer it is possible to construct elliptic filters so as easy as the classical Chebyshev, Butterworth filters and Bessel.

List of the used literature

1. Opadchy Yu.F., Gludkin O. P. Analog and digital electronics: The textbook for higher education institutions. Under the editorship of O. P. Gludkin. - M.:goryachy liniya-Telkom. 2005. - 768s.

2. Gorbachev G. N., Chaplygin E.E. Industrial electronics. M.: Energoatomizdat, 1988.

3. Zhigalov A.A. Electronics bases. M:energoatomizdat, 1985.

4. Tokheym R. Microprocessors. Courses and exercises. М: World of 1988.

5. Gusev of V. G. Elektronik and microprocessor equipment: The textbook for higher education institutions. - M of a.:vyssh.shkol., 2006. - 799s.

6. Apsemetov A.T., Ismailov S. U. Development and calculation of relaxation generators on operational usilitelyakh./Methodical instructions to a term paper on discipline of "Electronic engineer": Shymkent, 2000.Steven W. Smith The Scientist and Engineer's Guide to Digital Signal Processing. -- Second Edition. -- San-Diego: California Technical Publishing, 1999.

7. Britton C. Rorabaugh Approximation Methods for Electronic Filter Design. -- New York: McGraw-Hill, 1999.

8. B. Widrow, S.D. Stearns Adaptive Signal Processing. -- Paramus, NJ: Prentice-Hall, 1985.

9. S. Haykin Adaptive Filter Theory. -- 4rd Edition. -- Paramus, NJ: Prentice-Hall, 2001.

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