Network synthesis filters
Encyclopedia
Network synthesis is a method of designing signal processing filter
s. It 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 filter
s and digital filter
s. The essence of the method is to obtain the component values of the filter from a given mathematical polynomial
ratio expression representing the desired transfer function.
. 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.
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 wavelength
s while rejecting waves of other wavelengths. 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.
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 innaccuracies due to the theoretical termination impedance, the image impedance
, not generally being equal to the actual termination impedance. This is not the case 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 outputs the sections needed to build the corresponding filter.
In general, the sections of a network synthesis filter are 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 - this is 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 filter
s followed by frequency transformations and impedance scaling to arrive at the final desired filter.
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.
whose Chebyshev polynomials are used in the derivation of the transfer function.
and the transfer function is based on elliptic rational functions.
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.
Not every possible mathematical function for driving point impedance can be realised using real electrical components. Wilhelm Cauer
(following on from R. M. Foster
) 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.
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;
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.
Values of prototype elements are published in tables, one of the first being due to Sidney Darlington
. 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.
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
s. It has produced several important classes of filter including the Butterworth filter
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
, the Chebyshev filter
Chebyshev filter
Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple or stopband ripple than Butterworth filters...
and the Elliptic filter
Elliptic filter
An elliptic filter is a signal processing filter with equalized ripple behavior in both the passband and the stopband...
. 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 filter
Active filter
An active filter is a type of analog electronic filter that uses an amplifier stage. Amplifiers included in a filter design can be used to improve the performance, stability and predictability of a filter. An amplifier prevents the impedance of source or load stages from affecting the...
s and digital filter
Digital filter
In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
s. The essence of the method is to obtain the component values of the filter from a given mathematical polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
ratio expression representing the desired transfer function.
Description of method
The method can be viewed as the inverse problem of network analysisNetwork analysis
Network analysis can refer to:* Analysis of general networks: see Network theory.* Electrical network analysis see Network analysis .* Social network analysis.You may also be interested in Network planning and design...
. 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.
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 wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s while rejecting waves of other wavelengths. Network synthesis starts out with a specification for the transfer function of the filter, H(s), as a function of complex frequency
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
, 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
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
or partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
expansions results in the required values of the filter components. In a digital implementation of a filter, H(s) can be implemented directly.
The advantages of the method are best understood by comparing it to the filter design
Filter design
Filter design is the process of designing a filter , often a linear shift-invariant filter, that satisfies a set of requirements, some of which are contradictory...
methodology that was used before it, the image method
Image impedance
Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking in to the ports of a network. Usually a two-port network is implied but the concept is capable of being extended to...
. 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 innaccuracies due to the theoretical termination impedance, the image impedance
Image impedance
Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking in to the ports of a network. Usually a two-port network is implied but the concept is capable of being extended to...
, not generally being equal to the actual termination impedance. This is not the case 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 outputs the sections needed to build the corresponding filter.
In general, the sections of a network synthesis filter are 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 - this is 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 filter
Prototype filter
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 regards to...
s followed by frequency transformations and impedance scaling to arrive at the final desired filter.
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.The Butterworth class of filter was first described in a 1930 paper by the British engineer Stephen Butterworth
Stephen Butterworth
Stephen Butterworth was a British physicist who invented the Butterworth filter, a class of electrical circuits that are used to separate different frequencies of electrical signals....
after whom it is named. The filter response is described by Butterworth polynomials, also due to Butterworth.
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 ChebyshevPafnuty Chebyshev
Pafnuty Lvovich Chebyshev was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff .-Early years:One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk,...
whose Chebyshev polynomials are used in the derivation of the transfer function.
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 1 filter. An elliptical filter in the limit of zero ripple in the stopband is identical to a Chebyshev Type 2 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 CauerWilhelm Cauer
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...
and the transfer function is based on elliptic rational functions.
Bessel filter
- The Bessel filter has a maximally flat time-delay (group delayGroup delayGroup delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component...
) 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 BesselFriedrich Bessel-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...
and the transfer function is based on Bessel polynomialsBessel polynomialsIn mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions...
.
Driving point impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
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
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
(jω-domain). Treating it as a one-port network, the expression is expanded using continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
or partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
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
Two-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
filter with the desired transfer function.
Not every possible mathematical function for driving point impedance can be realised using real electrical components. Wilhelm Cauer
Wilhelm Cauer
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...
(following on from R. M. Foster
R. M. Foster
Ronald Martin Foster , was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines...
) did much of the early work on what mathematical functions could be realised and in which filter topologies
Electronic filter topology
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....
. The ubiquitous ladder topology of filter design is named after Cauer.
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;
- Cauer's first form of driving point impedance consists of a ladder of shunt capacitors and series inductors and is most useful for low-pass filterLow-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
s. - Cauer's second form of driving point impedance consists of a ladder of series capacitors and shunt inductors and is most useful for high-pass filterHigh-pass filterA high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...
s. - Foster's first form of driving point impedance consists of parallel connected LC resonators and is most useful for band-pass filterBand-pass filterA band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
s. - Foster's second form of driving point impedance consists of series connected LC anti-resonators and is most useful for band-stop filterBand-stop filterIn signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter...
s.
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
Nominal impedance
Nominal impedance in electrical engineering and audio engineering refers to the approximate designed impedance of an electrical circuit or device...
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.
Values of prototype elements are published in tables, one of the first being due to Sidney Darlington
Sidney Darlington
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair...
. 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.
See also
- Electronic filterElectronic filterElectronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
- Analogue filter
- Linear filterLinear filterLinear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....
- Composite image filterComposite image filterA composite image filter is an electronic filter consisting of multiple image filter sections of two or more different types.The image method of filter design determines the properties of filter sections by calculating the properties they have in an infinite chain of such sections. In this, the...