Network synthesis filters

Last updated

Network synthesis filters are signal processing filters designed by the network synthesis method. The method has produced several important classes of filter including the Butterworth filter, the Chebyshev filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but its results can also be applied to implementations in active filters and digital filters. The essence of the method is to obtain the component values of the filter from a given rational function representing the desired transfer function.

Contents

Description of method

The method can be viewed as the inverse problem of network analysis. Network analysis starts with a network and by applying the various electric circuit theorems predicts the response of the network. Network synthesis on the other hand, starts with a desired response and its methods produce a network that outputs, or approximates to, that response. [1]

Network synthesis was originally intended to produce filters of the kind formerly described as wave filters but now usually just called filters. That is, filters whose purpose is to pass waves of certain frequencies while rejecting waves of other frequencies. Network synthesis starts out with a specification for the transfer function of the filter, H(s), as a function of complex frequency, s. This is used to generate an expression for the input impedance of the filter (the driving point impedance) which then, by a process of continued fraction or partial fraction expansions results in the required values of the filter components. In a digital implementation of a filter, H(s) can be implemented directly. [2]

The advantages of the method are best understood by comparing it to the filter design methodology that was used before it, the image method. The image method considers the characteristics of an individual filter section in an infinite chain (ladder topology) of identical sections. The filters produced by this method suffer from inaccuracies due to the theoretical termination impedance, the image impedance, not generally being equal to the actual termination impedance. With network synthesis filters, the terminations are included in the design from the start. The image method also requires a certain amount of experience on the part of the designer. The designer must first decide how many sections and of what type should be used, and then after calculation, will obtain the transfer function of the filter. This may not be what is required and there can be a number of iterations. The network synthesis method, on the other hand, starts out with the required function and generates as output the sections needed to build the corresponding filter. [2]

In general, the sections of a network synthesis filter are of identical topology (usually the simplest ladder type) but different component values are used in each section. By contrast, the structure of an image filter has identical values at each section, as a consequence of the infinite chain approach, but may vary the topology from section to section to achieve various desirable characteristics. Both methods make use of low-pass prototype filters followed by frequency transformations and impedance scaling to arrive at the final desired filter. [2]

Important filter classes

The class of a filter refers to the class of polynomials from which the filter is mathematically derived. The order of the filter is the number of filter elements present in the filter's ladder implementation. Generally speaking, the higher the order of the filter, the steeper the cut-off transition between passband and stopband. Filters are often named after the mathematician or mathematics on which they are based rather than the discoverer or inventor of the filter.

Butterworth filter

Butterworth filters are described as maximally flat, meaning that the response in the frequency domain is the smoothest possible curve of any class of filter of the equivalent order. [3]

The Butterworth class of filter was first described in a 1930 paper by the British engineer Stephen Butterworth after whom it is named. The filter response is described by Butterworth polynomials, also due to Butterworth. [4]

Chebyshev filter

A Chebyshev filter has a faster cut-off transition than a Butterworth, but at the expense of there being ripples in the frequency response of the passband. There is a compromise to be had between the maximum allowed attenuation in the passband and the steepness of the cut-off response. This is also sometimes called a type I Chebyshev, the type 2 being a filter with no ripple in the passband but ripples in the stopband. The filter is named after Pafnuty Chebyshev whose Chebyshev polynomials are used in the derivation of the transfer function. [3]

Cauer filter

Cauer filters have equal maximum ripple in the passband and the stopband. The Cauer filter has a faster transition from the passband to the stopband than any other class of network synthesis filter. The term Cauer filter can be used interchangeably with elliptical filter, but the general case of elliptical filters can have unequal ripples in the passband and stopband. An elliptical filter in the limit of zero ripple in the passband is identical to a Chebyshev Type 2 filter. An elliptical filter in the limit of zero ripple in the stopband is identical to a Chebyshev Type 1 filter. An elliptical filter in the limit of zero ripple in both passbands is identical to a Butterworth filter. The filter is named after Wilhelm Cauer and the transfer function is based on elliptic rational functions. [5] Cauer-type filters use generalized continued fractions. [6] [7] [8]

Bessel filter

The Bessel filter has a maximally flat time-delay (group delay) over its passband. This gives the filter a linear phase response and results in it passing waveforms with minimal distortion. The Bessel filter has minimal distortion in the time domain due to the phase response with frequency as opposed to the Butterworth filter which has minimal distortion in the frequency domain due to the attenuation response with frequency. The Bessel filter is named after Friedrich Bessel and the transfer function is based on Bessel polynomials. [9]

Driving point impedance

Low-pass filter implemented as a ladder (Cauer) topology Cauer lowpass.svg
Low-pass filter implemented as a ladder (Cauer) topology

The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform (s-domain) or Fourier transform (jω-domain). Treating it as a one-port network, the expression is expanded using continued fraction or partial fraction expansions. The resulting expansion is transformed into a network (usually a ladder network) of electrical elements. Taking an output from the end of this network, so realised, will transform it into a two-port network filter with the desired transfer function. [1]

Not every possible mathematical function for driving point impedance can be realised using real electrical components. Wilhelm Cauer (following on from R. M. Foster [10] ) did much of the early work on what mathematical functions could be realised and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer. [11]

There are a number of canonical forms of driving point impedance that can be used to express all (except the simplest) realisable impedances. The most well known ones are; [12]

Further theoretical work on realizable filters in terms of a given rational function as transfer function was done by Otto Brune in 1931 [13] and Richard Duffin with Raoul Bott in 1949. [14] The work was summarized in 2010 by John H. Hubbard. [15] When a transfer function is specified as a positive-real function (the set of positive real numbers is invariant under the transfer function), then a network of passive components (resistors, inductors, and capacitors) can be designed with that transfer function.

Prototype filters

Prototype filters are used to make the process of filter design less labour-intensive. The prototype is usually designed to be a low-pass filter of unity nominal impedance and unity cut-off frequency, although other schemes are possible. The full design calculations from the relevant mathematical functions and polynomials are carried out only once. The actual filter required is obtained by a process of scaling and transforming the prototype. [16]

Values of prototype elements are published in tables, one of the first being due to Sidney Darlington. [17] Both modern computing power and the practice of directly implementing filter transfer functions in the digital domain have largely rendered this practice obsolete.

A different prototype is required for each order of filter in each class. For those classes in which there is attenuation ripple, a different prototype is required for each value of ripple. The same prototype may be used to produce filters which have a different bandform from the prototype. For instance low-pass, high-pass, band-pass and band-stop filters can all be produced from the same prototype. [18]

See also

Notes

  1. 1 2 E. Cauer, p4
  2. 1 2 3 Matthaei, pp83-84
  3. 1 2 Matthaei et al., pp85-108
  4. Butterworth, S, "On the Theory of Filter Amplifiers", Wireless Engineer, vol. 7, 1930, pp. 536-541.
  5. Mathaei, p95
  6. Fry, T. C. (1929). "The use of continued fractions in the design of electrical networks". Bull. Amer. Math. Soc. 35 (4): 463–498. doi: 10.1090/s0002-9904-1929-04747-5 . MR   1561770.
  7. Milton. G. W. (1987). "Multicomponent composites of networks and new types of continued fraction. I". Comm. Math. Physics. 111 (2): 281–327. Bibcode:1987CMaPh.111..281M. doi:10.1007/bf01217763. MR   0899853.
  8. Milton. G. W. (1987). "Multicomponent composites of networks and new types of continued fraction. II". Comm. Math. Physics. 111 (3): 329–372. Bibcode:1987CMaPh.111..329M. doi:10.1007/bf01238903. MR   0900499.
  9. Matthaei, pp108-113
  10. Foster, R M, "A Reactance Theorem", Bell System Technical Journal, vol 3, pp259-267, 1924.
  11. E. Cauer, p1
  12. Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", IEEE Trans. Circuits and Systems, vol 31, p6, 1984.
  13. Otto Brune (1931) "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", MIT Journal of Mathematics and Physics, Vol 10, pp 191–236
  14. Richard Duffin & Raoul Bott, "Impedance synthesis without the use of transformers", Journal of Applied Physics 20:816
  15. John H. Hubbard (2010) "The Bott-Duffin Synthesis of Electrical Circuits", pp 33 to 40 in A Celebration of the Mathematical Legacy of Raoul Bott, P. Robert Kotiuga editor, CRM Proceedings and Lecture Notes #50, American Mathematical Society
  16. Matthaei, p83
  17. Darlington, S, "Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics", Jour. Math. and Phys., Vol 18, pp257-353, September 1939.
  18. See Matthaei for examples.

Related Research Articles

Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant in which case they can be analyzed exactly using LTI system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real-time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions. An analog electronic circuit consisting only of linear components will necessarily fall in this category, as will comparable mechanical systems or digital signal processing systems containing only linear elements. Since linear time-invariant filters can be completely characterized by their response to sinusoids of different frequencies, they are sometimes known as frequency filters.

Chebyshev filters are analog or digital filters having a steeper roll-off than Butterworth filters, and have passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. The type I Chebyshev filters are called usually as just "Chebyshev filters", the type II ones are usually called "inverse Chebyshev filters".

Active filter active filter

An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a filter.

Butterworth filter type of signal processing filter designed to have a frequency response as flat as possible in the passband

The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

Electronic filter electronic circuit that removes unwanted components from the signal, or enhances wanted ones, or both

Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an integrated circuit.

An elliptic filter is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple. Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.

In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat group/phase delay, which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems.

Optimum "L" filter

The Optimum "L" filter was proposed by Athanasios Papoulis in 1958. It has the maximum roll off rate for a given filter order while maintaining a monotonic frequency response. It provides a compromise between the Butterworth filter which is monotonic but has a slower roll off and the Chebyshev filter which has a faster roll off but has ripple in either the pass band or stop band. The filter design is based on Legendre polynomials which is the reason for its alternate name and the "L" in Optimum "L".

Parks–McClellan filter design algorithm

The Parks–McClellan algorithm, published by James McClellan and Thomas Parks in 1972, is an iterative algorithm for finding the optimal Chebyshev finite impulse response (FIR) filter. The Parks–McClellan algorithm is utilized to design and implement efficient and optimal FIR filters. It uses an indirect method for finding the optimal filter coefficients.

Electronic filter topology electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected

Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.

A composite image filter is an electronic filter consisting of multiple image filter sections of two or more different types.

Prototype filter electronic filter designs that are used as a template to produce a modified filter design for a particular application

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

Wilhelm Cauer German mathematician and scientist

Wilhelm Cauer was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prior to his work, electronic filter design used techniques which accurately predicted filter behaviour only under unrealistic conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.

Analogue filters are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass, mid-range, and tweeter loudspeakers; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen radio station in a radio receiver and rejection of others.

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, and computer graphics.

Distributed-element filter type of electronic filter circuit

A distributed-element filter is an electronic filter in which capacitance, inductance, and resistance are not localised in discrete capacitors, inductors, and resistors as they are in conventional filters. Its purpose is to allow a range of signal frequencies to pass, but to block others. Conventional filters are constructed from inductors and capacitors, and the circuits so built are described by the lumped element model, which considers each element to be "lumped together" at one place. That model is conceptually simple, but it becomes increasingly unreliable as the frequency of the signal increases, or equivalently as the wavelength decreases. The distributed-element model applies at all frequencies, and is used in transmission-line theory; many distributed-element components are made of short lengths of transmission line. In the distributed view of circuits, the elements are distributed along the length of conductors and are inextricably mixed together. The filter design is usually concerned only with inductance and capacitance, but because of this mixing of elements they cannot be treated as separate "lumped" capacitors and inductors. There is no precise frequency above which distributed element filters must be used but they are especially associated with the microwave band.

Waveguide filter electronic filter that is constructed with waveguide technology

A waveguide filter is an electronic filter that is constructed with waveguide technology. Waveguides are hollow metal tubes inside which an electromagnetic wave may be transmitted. Filters are devices used to allow signals at some frequencies to pass, while others are rejected. Filters are a basic component of electronic engineering designs and have numerous applications. These include selection of signals and limitation of noise. Waveguide filters are most useful in the microwave band of frequencies, where they are a convenient size and have low loss. Examples of microwave filter use are found in satellite communications, telephone networks, and television broadcasting.

Waffle-iron filter

A waffle-iron filter is a type of waveguide filter used at microwave frequencies for signal filtering. It is a variation of the corrugated-waveguide filter but with longitudinal slots cut through the corrugations resulting in an internal structure that has the appearance of a waffle-iron.

Lattice delay network

Lattice delay networks are an important subgroup of lattice networks. They are all-pass filters, so they have a flat amplitude response, but a phase response which varies linearly with frequency. All lattice circuits, regardless of their complexity, are based on the schematic shown below, which contains two series impedances, Za, and two shunt impedances, Zb. Although there is duplication of impedances in this arrangement, it offers great flexibility to the circuit designer so that, in addition to its use as delay network it can be configured to be a phase corrector, a dispersive network, an amplitude equalizer, or a low pass filter, according to the choice of components for the lattice elements.

Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response of a given circuit is calculated. Network synthesis was a great leap forward in circuit design. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits.

References