Analogue
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
are a basic building block of
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
much used in
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
. Amongst their many applications are the separation of an audio signal before application to
bass,
mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
:M=\frac.
The mid-range is closely related to the range, a measure of ...
, and
tweeter
A tweeter or treble speaker is a special type of loudspeaker (usually dome, inverse dome or horn-type) that is designed to produce high audio frequencies, typically deliver high frequencies up to 100 kHz. The name is derived from the high ...
loudspeaker
A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...
s; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen
radio station
Radio broadcasting is transmission of audio (sound), sometimes with related metadata, by radio waves to radio receivers belonging to a public audience. In terrestrial radio broadcasting the radio waves are broadcast by a land-based radio ...
in a
radio receiver
In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. Th ...
and rejection of others.
Passive linear electronic analogue filters are those filters which can be described with
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
s (linear); they are composed of
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s,
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and, sometimes,
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s (
passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of on ...
) and are designed to operate on continuously varying
analogue signal
An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies c ...
s. There are many
linear filter
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 (or shift invariant) in which case they can be analyzed exactly using ...
s which are not analogue in implementation (
digital filter
In signal processing, 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, t ...
), and there are many
electronic filter
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 ...
s which may not have a passive topology – both of which may have the same
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
of the filters described in this article. Analogue filters are most often used in wave filtering applications, that is, where it is required to pass particular frequency components and to reject others from analogue (
continuous-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
) signals.
Analogue filters have played an important part in the development of electronics. Especially in the field of
telecommunication
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
s, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of
network synthesis
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. ...
techniques greatly enhanced the degree of control of the designer.
Today, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue filters are now implemented in a
filter topology which is
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
in order to avoid the wound components (i.e. inductors, transformers, etc.) required by
passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of on ...
topology.
It is possible to design linear analogue
mechanical filter
A mechanical filter is a signal processing filter usually used in place of an electronic filter at radio frequencies. Its purpose is the same as that of a normal electronic filter: to pass a range of signal frequencies, but to block others. T ...
s using mechanical components which filter mechanical vibrations or
acoustic waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of
transducer
A transducer is a device that converts energy from one form to another. Usually a transducer converts a signal in one form of energy to a signal in another.
Transducers are often employed at the boundaries of automation, measurement, and contr ...
s to convert to and from the electrical domain. Indeed, some of the earliest ideas for filters were acoustic resonators because the electronics technology was poorly understood at the time. In principle, the design of such filters can be achieved entirely in terms of the electronic counterparts of mechanical quantities, with
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
,
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
and
heat energy
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
corresponding to the energy in inductors, capacitors and resistors respectively.
Historical overview
There are three main stages in the history of passive analogue filter development:
#Simple filters. The frequency dependence of electrical response was known for capacitors and inductors from very early on. The resonance phenomenon was also familiar from an early date and it was possible to produce simple, single-branch filters with these components. Although attempts were made in the 1880s to apply them to
telegraphy
Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas p ...
, these designs proved inadequate for successful
frequency-division multiplexing
In telecommunications, frequency-division multiplexing (FDM) is a technique by which the total bandwidth available in a communication medium is divided into a series of non-overlapping frequency bands, each of which is used to carry a separate ...
. Network analysis was not yet powerful enough to provide the theory for more complex filters and progress was further hampered by a general failure to understand the
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
nature of signals.
#
Image filters. Image filter theory grew out of transmission line theory and the design proceeded in a similar manner to transmission line analysis. For the first time filters could be produced that had precisely controllable
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
s and other parameters. These developments took place in the 1920s and filters produced to these designs were still in widespread use in the 1980s, only declining as the use of analogue telecommunications has declined. Their immediate application was the economically important development of frequency division multiplexing for use on
intercity and international lines.
#
Network synthesis filters
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
. The mathematical bases of network synthesis were laid in the 1930s and 1940s. After World War II, network synthesis became the primary tool of
filter design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...
. Network synthesis put filter design on a firm mathematical foundation, freeing it from the mathematically sloppy techniques of image design and severing the connection with physical lines. The essence of network synthesis is that it produces a design that will (at least if implemented with ideal components) accurately reproduce the response originally specified in
black box
In science, computing, and engineering, a black box is a system which can be viewed in terms of its inputs and outputs (or transfer characteristics), without any knowledge of its internal workings. Its implementation is "opaque" (black). The te ...
terms.
Throughout this article the letters R, L, and C are used with their usual meanings to represent
resistance,
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
, and
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
, respectively. In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for
electrical impedance
In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the comp ...
, any 2-terminal
[A terminal of a network is a connection point where current can enter or leave the network from the world outside. This is often called a ''pole'' in the literature, especially the more mathematical, but is not to be confused with a ]pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
of the transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
which is a meaning also used in this article. A 2-terminal network amounts to a single impedance (although it may consist of many elements connected in a complicated set of meshes
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, ex ...
) and can also be described as a one-port network. For networks of more than two terminals it is not necessarily possible to identify terminal pairs as ports. combination of RLC elements and in some sections D is used for the rarely seen quantity
elastance
Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of c ...
, which is the inverse of capacitance.
Resonance
Early filters utilised the phenomenon of
resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
to filter signals. Although
electrical resonance
Electrical resonance occurs in an electric circuit at a particular '' resonant frequency'' when the impedances or admittances of circuit elements cancel each other. In some circuits, this happens when the impedance between the input and output of ...
had been investigated by researchers from a very early stage, it was at first not widely understood by electrical engineers. Consequently, the much more familiar concept of
acoustic resonance
Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration (its ''resonance frequencies'').
The term "acoustic resonance" is sometimes used to nar ...
(which in turn, can be explained in terms of the even more familiar
mechanical resonance
Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its ''resonance frequency'' or ''resonant frequency'') close ...
) found its way into filter design ahead of electrical resonance.
[Lundheim, p.24] Resonance can be used to achieve a filtering effect because the resonant device will respond to frequencies at, or near, to the resonant frequency but will not respond to frequencies far from resonance. Hence frequencies far from resonance are filtered out from the output of the device.
Electrical resonance
![Oudin coil](https://upload.wikimedia.org/wikipedia/commons/2/20/Oudin_coil.png)
Resonance was noticed early on in experiments with the
Leyden jar
A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typi ...
, invented in 1746. The Leyden jar stores electricity due to its
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
, and is, in fact, an early form of capacitor. When a Leyden jar is discharged by allowing a spark to jump between the electrodes, the discharge is oscillatory. This was not suspected until 1826, when
Felix Savary
Felix may refer to:
* Felix (name), people and fictional characters with the name
Places
* Arabia Felix is the ancient Latin name of Yemen
* Felix, Spain, a municipality of the province Almería, in the autonomous community of Andalusia, S ...
in France, and later (1842)
Joseph Henry
Joseph Henry (December 17, 1797– May 13, 1878) was an American scientist who served as the first Secretary of the Smithsonian Institution. He was the secretary for the National Institute for the Promotion of Science, a precursor of the Smith ...
in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction. They both independently drew the conclusion that there was a transient oscillation dying with time.
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
in 1847 published his important work on conservation of energy in part of which he used those principles to explain why the oscillation dies away, that it is the resistance of the circuit which dissipates the energy of the oscillation on each successive cycle. Helmholtz also noted that there was evidence of oscillation from the
electrolysis
In chemistry and manufacturing, electrolysis is a technique that uses direct electric current (DC) to drive an otherwise non-spontaneous chemical reaction. Electrolysis is commercially important as a stage in the separation of elements from n ...
experiments of
William Hyde Wollaston
William Hyde Wollaston (; 6 August 1766 – 22 December 1828) was an English chemist and physicist who is famous for discovering the chemical elements palladium and rhodium. He also developed a way to process platinum ore into malleable ingo ...
. Wollaston was attempting to decompose water by electric shock but found that both hydrogen and oxygen were present at both electrodes. In normal electrolysis they would separate, one to each electrode.
Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to
Sir William Thomson
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
(Lord Kelvin) who, in 1853, postulated that there was inductance present in the circuit as well as the capacitance of the jar and the resistance of the load. This established the physical basis for the phenomenon – the energy supplied by the jar was partly dissipated in the load but also partly stored in the magnetic field of the inductor.
So far, the investigation had been on the natural frequency of transient oscillation of a resonant circuit resulting from a sudden stimulus. More important from the point of view of filter theory is the behaviour of a resonant circuit when driven by an external
AC signal: there is a sudden peak in the circuit's response when the driving signal frequency is at the resonant frequency of the circuit.
[The resonant frequency is very close to, but usually not exactly equal to, the natural frequency of oscillation of the circuit] James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
heard of the phenomenon from
Sir William Grove in 1868 in connection with experiments on
dynamo
file:DynamoElectricMachinesEndViewPartlySection USP284110.png, "Dynamo Electric Machine" (end view, partly section, )
A dynamo is an electrical generator that creates direct current using a commutator (electric), commutator. Dynamos were the f ...
s, and was also aware of the earlier work of
Henry Wilde in 1866. Maxwell explained resonance
Oliver Lodge
Sir Oliver Joseph Lodge, (12 June 1851 – 22 August 1940) was a British physicist and writer involved in the development of, and holder of key patents for, radio. He identified electromagnetic radiation independent of Heinrich Rudolf Hertz, H ...
and some other English scientists tried to keep acoustic and electric terminology separate and promoted the term "syntony". However it was "resonance" that was to win the day. Blanchard, p.422 mathematically, with a set of differential equations, in much the same terms that an
RLC circuit
An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the ...
is described today.
[
]Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's Maxwell's equations, equations of electrom ...
(1887) experimentally demonstrated the resonance phenomena by building two resonant circuits, one of which was driven by a generator and the other was tunable and only coupled to the first electromagnetically (i.e., no circuit connection). Hertz showed that the response of the second circuit was at a maximum when it was in tune with the first. The diagrams produced by Hertz in this paper were the first published plots of an electrical resonant response.[
]
Acoustic resonance
As mentioned earlier, it was acoustic resonance that inspired filtering applications, the first of these being a telegraph system known as the "harmonic telegraph
Acoustic telegraphy (also known as harmonic telegraphy) was a name for various methods of multiplexing (transmitting more than one) telegraph messages simultaneously over a single telegraph wire by using different audio frequencies or channels for ...
". Versions are due to Elisha Gray
Elisha Gray (August 2, 1835 – January 21, 1901) was an American electrical engineer who co-founded the Western Electric Manufacturing Company. Gray is best known for his development of a telephone prototype in 1876 in Highland Park, Illinois. ...
, Alexander Graham Bell
Alexander Graham Bell (, born Alexander Bell; March 3, 1847 – August 2, 1922) was a Scottish-born inventor, scientist and engineer who is credited with patenting the first practical telephone. He also co-founded the American Telephone and Te ...
(1870s),[ ]Ernest Mercadier
Ernest is a given name derived from Germanic word ''ernst'', meaning "serious". Notable people and fictional characters with the name include:
People
*Archduke Ernest of Austria (1553–1595), son of Maximilian II, Holy Roman Emperor
*Ernest, M ...
and others. Its purpose was to simultaneously transmit a number of telegraph messages over the same line and represents an early form of frequency division multiplexing
In telecommunications, frequency-division multiplexing (FDM) is a technique by which the total bandwidth available in a communication medium is divided into a series of non-overlapping frequency bands, each of which is used to carry a separate ...
(FDM). FDM requires the sending end to be transmitting at different frequencies for each individual communication channel. This demands individual tuned resonators, as well as filters to separate out the signals at the receiving end. The harmonic telegraph achieved this with electromagnetically driven tuned reeds at the transmitting end which would vibrate similar reeds at the receiving end. Only the reed with the same resonant frequency as the transmitter would vibrate to any appreciable extent at the receiving end.[Blanchard, p.425]
Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as transducer
A transducer is a device that converts energy from one form to another. Usually a transducer converts a signal in one form of energy to a signal in another.
Transducers are often employed at the boundaries of automation, measurement, and contr ...
s converting sound to and from an electrical signal. It is no great leap from this view of the harmonic telegraph to the idea that speech can be converted to and from an electrical signal.[
]
Early multiplexing
By the 1890s electrical resonance was much more widely understood and had become a normal part of the engineer's toolkit. In 1891 Hutin and Leblanc patented an FDM scheme for telephone circuits using resonant circuit filters. Rival patents were filed in 1892 by Michael Pupin and John Stone Stone
John Stone Stone (September 24, 1869 – May 20, 1943) was an American mathematician, physicist and inventor. He initially worked in telephone research, followed by influential work developing early radio technology, where he was especially ...
with similar ideas, priority eventually being awarded to Pupin. However, no scheme using just simple resonant circuit filters can successfully multiplex
Multiplex may refer to:
* Multiplex (automobile), a former American car make
* Multiplex (comics), a DC comic book supervillain
* Multiplex (company), a global contracting and development company
* Multiplex (assay), a biological assay which measu ...
(i.e. combine) the wider bandwidth of telephone channels (as opposed to telegraph) without either an unacceptable restriction of speech bandwidth or a channel spacing so wide as to make the benefits of multiplexing uneconomic.[
The basic technical reason for this difficulty is that the frequency response of a simple filter approaches a fall of 6 dB/octave far from the point of resonance. This means that if telephone channels are squeezed in side by side into the frequency spectrum, there will be ]crosstalk
In electronics, crosstalk is any phenomenon by which a signal transmitted on one circuit or channel of a transmission system creates an undesired effect in another circuit or channel. Crosstalk is usually caused by undesired capacitive, induc ...
from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
like a low- Q resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to stopband like a high-Q resonant circuit.Q factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
is a dimensionless quantity enumerating the ''q''uality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled). One definition of Q factor, the most relevant one in this context, is the ratio of resonant frequency to bandwidth of a circuit. It arose as a measure of selectivity
Selectivity may refer to:
Psychology and behaviour
* Choice, making a selection among options
* Discrimination, the ability to recognize differences
* Socioemotional selectivity theory, in social psychology
Engineering
* Selectivity (radio), a ...
in radio receivers Obviously, these are contradictory requirements to be met with a single resonant circuit. The solution to these needs was founded in the theory of transmission lines and consequently the necessary filters did not become available until this theory was fully developed. At this early stage the idea of signal bandwidth, and hence the need for filters to match to it, was not fully understood; indeed, it was as late as 1920 before the concept of bandwidth was fully established. For early radio, the concepts of Q-factor, selectivity
Selectivity may refer to:
Psychology and behaviour
* Choice, making a selection among options
* Discrimination, the ability to recognize differences
* Socioemotional selectivity theory, in social psychology
Engineering
* Selectivity (radio), a ...
and tuning sufficed. This was all to change with the developing theory of transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s on which image filter
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
s are based, as explained in the next section.[
At the turn of the century as telephone lines became available, it became popular to add telegraph onto telephone lines with an earth return ]phantom circuit In telecommunication and electrical engineering, a phantom circuit is an electrical circuit derived from suitably arranged wires with one or more conductive paths being a circuit in itself and at the same time acting as one conductor of another circ ...
.[Telegraph lines are typically unbalanced with only a single conductor provided, the return path is achieved through an ]earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
connection which is common to all the telegraph lines on a route. Telephone lines are typically balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
with two conductors per circuit. A telegraph signal connected common-mode to both conductors of the telephone line will not be heard at the telephone receiver which can only detect voltage differences between the conductors. The telegraph signal is typically recovered at the far end by connection to the center tap
In electronics, a center tap (CT) is a contact made to a point halfway along a winding of a transformer or inductor, or along the element of a resistor or a potentiometer.
Taps are sometimes used on inductors for the coupling of signals, and ...
of a line transformer. The return path is via an earth connection as usual. This is a form of phantom circuit In telecommunication and electrical engineering, a phantom circuit is an electrical circuit derived from suitably arranged wires with one or more conductive paths being a circuit in itself and at the same time acting as one conductor of another circ ...
An LC filter
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
was required to prevent telegraph clicks being heard on the telephone line. From the 1920s onwards, telephone lines, or balanced lines dedicated to the purpose, were used for FDM telegraph at audio frequencies. The first of these systems in the UK was a Siemens and Halske installation between London and Manchester. GEC and AT&T
AT&T Inc. is an American multinational telecommunications holding company headquartered at Whitacre Tower in Downtown Dallas, Texas. It is the world's largest telecommunications company by revenue and the third largest provider of mobile tel ...
also had FDM systems. Separate pairs were used for the send and receive signals. The Siemens and GEC systems had six channels of telegraph in each direction, the AT&T system had twelve. All of these systems used electronic oscillators to generate a different carrier for each telegraph signal and required a bank of band-pass filters to separate out the multiplexed signal at the receiving end.
Transmission line theory
The earliest model of the transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
was probably described by Georg Ohm
Georg Simon Ohm (, ; 16 March 1789 – 6 July 1854) was a German physicist and mathematician. As a school teacher, Ohm began his research with the new electrochemical cell, invented by Italian scientist Alessandro Volta. Using equipment of his o ...
(1827) who established that resistance in a wire is proportional to its length.[At least, Ohm described the first model that was in any way correct. Earlier ideas such as Barlow's law from Peter Barlow were either incorrect, or inadequately described. See, for example. p.603 of; ]
*John C. Shedd, Mayo D. Hershey, "The history of Ohm's law", ''The Popular Science Monthly'', pp.599–614, December 1913 ISSN 0161-7370. The Ohm model thus included only resistance. Latimer Clark
Josiah Latimer Clark FRS FRAS (10 March 1822 – 30 October 1898), was an English electrical engineer, born in Great Marlow, Buckinghamshire.
Biography
Josiah Latimer Clark was born in Great Marlow, Buckinghamshire, and was younger brother ...
noted that signals were delayed and elongated along a cable, an undesirable form of distortion now called dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
but then called retardation, and Michael Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
(1853) established that this was due to the capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
present in the transmission line.Werner von Siemens
Ernst Werner Siemens (von Siemens from 1888; ; ; 13 December 1816 – 6 December 1892) was a German electrical engineer, inventor and industrialist. Siemens's name has been adopted as the SI unit of electrical conductance, the siemens. He foun ...
had also noted the retardation effect a few years earlier in 1849 and came to a similar conclusion as Faraday. However, there was not so much interest in Germany in underwater and underground cables as there was in Britain, the German overhead cables did not noticeably suffer from retardation and Siemen's ideas were not accepted. (Hunt, p.65.) Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
(1854) found the correct mathematical description needed in his work on early transatlantic cables; he arrived at an equation identical to the conduction of a heat pulse along a metal bar. This model incorporates only resistance and capacitance, but that is all that was needed in undersea cables dominated by capacitance effects. Kelvin's model predicts a limit on the telegraph signalling speed of a cable but Kelvin still did not use the concept of bandwidth, the limit was entirely explained in terms of the dispersion of the telegraph symbols
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
.[ The mathematical model of the transmission line reached its fullest development with ]Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
. Heaviside (1881) introduced series inductance and shunt conductance into the model making four distributed elements
: ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.''
In electrical engineering, the distributed-element model or transmission-line model of electrical E ...
in all. This model is now known as the telegrapher's equation
The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver ...
and the distributed-element parameters are called the primary line constants
The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevan ...
.
From the work of Heaviside (1887) it had become clear that the performance of telegraph lines, and most especially telephone lines, could be improved by the addition of inductance to the line. George Campbell at AT&T
AT&T Inc. is an American multinational telecommunications holding company headquartered at Whitacre Tower in Downtown Dallas, Texas. It is the world's largest telecommunications company by revenue and the third largest provider of mobile tel ...
implemented this idea (1899) by inserting loading coil
A loading coil or load coil is an inductor that is inserted into an electronic circuit to increase its inductance. The term originated in the 19th century for inductors used to prevent signal distortion in long-distance telegraph transmission c ...
s at intervals along the line. Campbell found that as well as the desired improvements to the line's characteristics in the passband there was also a definite frequency beyond which signals could not be passed without great attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable att ...
. This was a result of the loading coils and the line capacitance forming a low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
, an effect that is only apparent on lines incorporating lumped components such as the loading coils. This naturally led Campbell (1910) to produce a filter with ladder 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.
Filter design characterises filter circuits primarily by their ...
, a glance at the circuit diagram of this filter is enough to see its relationship to a loaded transmission line. The cut-off phenomenon is an undesirable side-effect as far as loaded lines are concerned but for telephone FDM filters it is precisely what is required. For this application, Campbell produced band-pass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-por ...
s to the same ladder topology by replacing the inductors and capacitors with resonator
A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator ...
s and anti-resonators respectively.[The exact date Campbell produced each variety of filter is not clear. The work started in 1910, initially patented in 1917 (US1227113) and the full theory published in 1922, but it is known that Campbell's filters were in use by AT&T long before the 1922 date (Bray, p.62, Darlington, p.5)] Both the loaded line and FDM were of great benefit economically to AT&T and this led to fast development of filtering from this point onwards.
Image filters
The filters designed by Campbell[Campbell has publishing priority for this invention but it is worth noting that ]Karl Willy Wagner
Karl Willy Wagner (22 February 1883 – 4 September 1953) was a German pioneer in the theory of electronic filters. He is noted by Hendrik Bode as being one of two Germans whose;
The other German being referred to is Wilhelm Cauer. Wagner wa ...
independently made a similar discovery which he was not allowed to publish immediately because World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
was still ongoing. (Thomas H. Lee, ''Planar microwave engineering'', p.725, Cambridge University Press 2004 .) were named wave filters because of their property of passing some waves and strongly rejecting others. The method by which they were designed was called the image parameter method[The term "image parameter method" was coined by Darlington (1939) in order to distinguish this earlier technique from his later "insertion-loss method"]["History of Filter Theory"]
Quadrivium, retrieved 26 June 2009[S. Darlington,]
Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics
, ''Journal of Mathematics and Physics'', vol 18, pp.257–353, September 1939 and filters designed to this method are called image filters.[The terms wave filter and image filter are not synonymous, it is possible for a wave filter to not be designed by the image method, but in the 1920s the distinction was moot as the image method was the only one available] The image method essentially consists of developing the transmission constants of an infinite chain of identical filter sections and then terminating the desired finite number of filter sections in the 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 into a Port (circuit theory), port of a network. Usually a two-port ...
. This exactly corresponds to the way the properties of a finite length of transmission line are derived from the theoretical properties of an infinite line, the image impedance corresponding to the characteristic impedance
The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
of the line.
From 1920 John Carson, also working for AT&T, began to develop a new way of looking at signals using the operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.
History
Th ...
of Heaviside which in essence is working in the frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
. This gave the AT&T engineers a new insight into the way their filters were working and led Otto Zobel
Otto Julius Zobel (October 20, 1887 – January 1970) was an electrical engineer who worked for the American Telephone & Telegraph Company (AT&T) in the early part of the 20th century. Zobel's work on filter design was revolutionary and led ...
to invent many improved forms. Carson and Zobel steadily demolished many of the old ideas. For instance the old telegraph engineers thought of the signal as being a single frequency and this idea persisted into the age of radio with some still believing that frequency modulation
Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The technology is used in telecommunications, radio broadcasting, signal processing, and Run-length limited#FM: .280. ...
(FM) transmission could be achieved with a smaller bandwidth than the baseband
In telecommunications and signal processing, baseband is the range of frequencies occupied by a signal that has not been modulated to higher frequencies. Baseband signals typically originate from transducers, converting some other variable into a ...
signal right up until the publication of Carson's 1922 paper. Another advance concerned the nature of noise, Carson and Zobel (1923) treated noise as a random process with a continuous bandwidth, an idea that was well ahead of its time, and thus limited the amount of noise that it was possible to remove by filtering to that part of the noise spectrum which fell outside the passband. This too, was not generally accepted at first, notably being opposed by Edwin Armstrong
Edwin Howard Armstrong (December 18, 1890 – February 1, 1954) was an American electrical engineer and inventor, who developed FM (frequency modulation) radio and the superheterodyne receiver system. He held 42 patents and received numerous awa ...
(who ironically, actually succeeded in reducing noise with wide-band FM) and was only finally settled with the work of Harry Nyquist
Harry Nyquist (, ; February 7, 1889 – April 4, 1976) was a Swedish-American physicist and electronic engineer who made important contributions to communication theory.
Personal life
Nyquist was born in the village Nilsby of the parish Stora Ki ...
whose thermal noise power formula is well known today.
Several improvements were made to image filters and their theory of operation by Otto Zobel
Otto Julius Zobel (October 20, 1887 – January 1970) was an electrical engineer who worked for the American Telephone & Telegraph Company (AT&T) in the early part of the 20th century. Zobel's work on filter design was revolutionary and led ...
. Zobel coined the term constant k filter
Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive co ...
(or k-type filter) to distinguish Campbell's filter from later types, notably Zobel's m-derived filter
m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was a ...
(or m-type filter). The particular problems Zobel was trying to address with these new forms were impedance matching into the end terminations and improved steepness of roll-off. These were achieved at the cost of an increase in filter circuit complexity.[Darlington, p.5]
A more systematic method of producing image filters was introduced by Hendrik Bode
Hendrik Wade Bode ( ; ;Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", IEEE Transactions on Automatic Control, Vol. AC-29, No 3., March 1984, pp. 193–194. Quote: "Something should be ...
(1930), and further developed by several other investigators including Piloty (1937–1939) and Wilhelm Cauer
Wilhelm Cauer (24 June 1900 – 22 April 1945) 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. Prio ...
(1934–1937). Rather than enumerate the behaviour (transfer function, attenuation function, delay function and so on) of a specific circuit, instead a requirement for the image impedance itself was developed. The image impedance can be expressed in terms of the open-circuit and short-circuit impedances[The open-circuit impedance of a two-port network is the impedance looking into one port when the other port is open circuit. Similarly, the short-circuit impedance is the impedance looking into one port when the other is terminated in a short circuit. The open-circuit impedance of the first port in general (except for symmetrical networks) is not equal to the open-circuit impedance of the second and likewise for short-circuit impedances] of the filter as . Since the image impedance must be real in the passbands and imaginary in the stopbands according to image theory, there is a requirement that the poles and zeroes
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
of ''Zo'' and ''Zs'' cancel in the passband and correspond in the stopband. The behaviour of the filter can be entirely defined in terms of the positions in the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
of these pairs of poles and zeroes. Any circuit which has the requisite poles and zeroes will also have the requisite response. Cauer pursued two related questions arising from this technique: what specification of poles and zeroes are realisable as passive filters; and what realisations are equivalent to each other. The results of this work led Cauer to develop a new approach, now called network synthesis.[Belevitch, p.851][Cauer et al., p.6]
This "poles and zeroes" view of filter design was particularly useful where a bank of filters, each operating at different frequencies, are all connected across the same transmission line. The earlier approach was unable to deal properly with this situation, but the poles and zeroes approach could embrace it by specifying a constant impedance for the combined filter. This problem was originally related to FDM telephony but frequently now arises in loudspeaker crossover filters.[
]
Network synthesis filters
The essence of network synthesis
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. ...
is to start with a required filter response and produce a network that delivers that response, or approximates to it within a specified boundary. This is the inverse of network analysis Network analysis can refer to:
* Network theory, the analysis of relations through mathematical graphs
** Social network analysis, network theory applied to social relations
* Network analysis (electrical circuits)
See also
*Network planning and ...
which starts with a given network and by applying the various electric circuit theorems predicts the response of the network.[Cauer et al., p.4] The term was first used with this meaning in the doctoral thesis of Yuk-Wing Lee
Lee Yuk-wing (; April 14, 1904 – November 8, 1989) was a Professor of Electrical Engineering at the Massachusetts Institute of Technology. He is best known for adapting and popularizing the pioneering work of Norbert Wiener and for his own resea ...
(1930) and apparently arose out of a conversation with Vannevar Bush
Vannevar Bush ( ; March 11, 1890 – June 28, 1974) was an American engineer, inventor and science administrator, who during World War II headed the U.S. Office of Scientific Research and Development (OSRD), through which almost all wartime ...
. The advantage of network synthesis over previous methods is that it provides a solution which precisely meets the design specification. This is not the case with image filters, a degree of experience is required in their design since the image filter only meets the design specification in the unrealistic case of being terminated in its own image impedance, to produce which would require the exact circuit being sought. Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed.
The development of network analysis needed to take place before network synthesis was possible. The theorems of Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
He coine ...
and others and the ideas of Charles Steinmetz
Charles Proteus Steinmetz (born Karl August Rudolph Steinmetz, April 9, 1865 – October 26, 1923) was a German-born American mathematician and electrical engineer and professor at Union College. He fostered the development of alternating ...
( phasors) and Arthur Kennelly (complex impedance
In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the compl ...
) laid the groundwork. The concept of a port
A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Ham ...
also played a part in the development of the theory, and proved to be a more useful idea than network terminals.[ The first milestone on the way to network synthesis was an important paper by ]Ronald M. Foster
Ronald Martin Foster (3 October 1896 – 2 February 1998), was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines. He published an important paper, ''A Reactance Theorem'', (see Foster ...
(1924), ''A Reactance Theorem'', in which Foster introduces the idea of a driving point impedance
Driving is the controlled operation and movement of a vehicle, including cars, motorcycles, trucks, buses, and bicycles. Permission to drive on public highways is granted based on a set of conditions being met and drivers are required to fol ...
, that is, the impedance that is connected to the generator. The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. Foster's reactance theorem
Foster's reactance theorem is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always strictly monotonically increases ...
stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.[Cauer et al., p.1]
Wilhelm Cauer
Wilhelm Cauer (24 June 1900 – 22 April 1945) 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. Prio ...
expanded on the work of Foster (1926) and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them. He also found ladder realisations[which is the best known of the filter topologies. It is for this reason that ladder topology is often referred to as Cauer topology (the forms used earlier by Foster are quite different) even though ladder topology had long since been in use in image filter design] of the network using Thomas Stieltjes
Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
' continued fraction expansion. This work was the basis on which network synthesis was built, although Cauer's work was not at first used much by engineers, partly because of the intervention of World War II, partly for reasons explained in the next section and partly because Cauer presented his results using topologies that required mutually coupled inductors and ideal transformers. Designers tend to avoid the complication of mutual inductances and transformers where possible, although transformer-coupled double-tuned amplifier
A double-tuned amplifier is a tuned amplifier with transformer coupling between the amplifier stages in which the inductances of both the primary and secondary windings are tuned separately with a capacitor across each. The scheme results in a w ...
s are a common way of widening bandwidth without sacrificing selectivity.[Belevitch, p.850]
Image method versus synthesis
Image filters continued to be used by designers long after the superior network synthesis techniques were available. Part of the reason for this may have been simply inertia, but it was largely due to the greater computation required for network synthesis filters, often needing a mathematical iterative process. Image filters, in their simplest form, consist of a chain of repeated, identical sections. The design can be improved simply by adding more sections and the computation required to produce the initial section is on the level of "back of an envelope" designing. In the case of network synthesis filters, on the other hand, the filter is designed as a whole, single entity and to add more sections (i.e., increase the order)[ the designer would have no option but to go back to the beginning and start over. The advantages of synthesised designs are real, but they are not overwhelming compared to what a skilled image designer could achieve, and in many cases it was more cost effective to dispense with time-consuming calculations.][Darlington, p.9] This is simply not an issue with the modern availability of computing power, but in the 1950s it was non-existent, in the 1960s and 1970s available only at cost, and not finally becoming widely available to all designers until the 1980s with the advent of the desktop personal computer. Image filters continued to be designed up to that point and many remained in service into the 21st century.
The computational difficulty of the network synthesis method was addressed by tabulating the component values of a 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 tra ...
and then scaling the frequency and impedance and transforming the bandform to those actually required. This kind of approach, or similar, was already in use with image filters, for instance by Zobel,[Zobel, O. J.,''Theory and Design of Uniform and Composite Electric Wave Filters'', Bell System Technical Journal, Vol. 2 (1923), pp. 1–46.
] but the concept of a "reference filter" is due to Sidney Darlington
Sidney Darlington (July 18, 1906 – October 31, 1997) was an American electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertion-loss synth ...
. Darlington (1939),[ was also the first to tabulate values for network synthesis prototype filters, nevertheless it had to wait until the 1950s before the Cauer-Darlington ]elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
first came into use.
Once computational power was readily available, it became possible to easily design filters to minimise any arbitrary parameter, for example time delay or tolerance to component variation. The difficulties of the image method were firmly put in the past, and even the need for prototypes became largely superfluous.[Darlington, p.12] Furthermore, the advent of 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.
...
s eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.[
]
Realisability and equivalence
Realisability (that is, which functions are realisable as real impedance networks) and equivalence (which networks equivalently have the same function) are two important questions in network synthesis. Following an analogy with Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
, Cauer formed the matrix equation,
:
where ''Z ''R ''Land ''Dare the ''n''x''n'' matrices of, respectively, impedance, resistance, inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
and elastance
Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of c ...
of an ''n''-mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, ex ...
network and ''s'' is the complex frequency
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the compl ...
operator . Here ''R ''Land ''Dhave associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the driving point impedance
Driving is the controlled operation and movement of a vehicle, including cars, motorcycles, trucks, buses, and bicycles. Permission to drive on public highways is granted based on a set of conditions being met and drivers are required to fol ...
by the method of Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...
;
:
where ''a11'' is the complement of the element ''A11'' to which the one-port is to be connected. From stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
Cauer found that ''R ''Land ''Dmust all be positive-definite matrices
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
for ''Z''p(''s'') to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology.[ This work is also partly due to ]Otto Brune
Otto Walter Heinrich Oscar Brune (10 January 1901 – 1982) undertook some key investigations into network synthesis at the Massachusetts Institute of Technology (MIT) where he graduated in 1929. His doctoral thesis was supervised by Wilhelm Ca ...
(1931), who worked with Cauer in the US prior to Cauer returning to Germany.[ A well known condition for realisability of a one-port rational][A rational impedance is one expressed as a ratio of two finite polynomials in ''s'', that is, a ]rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in ''s''. The implication of finite polynomials is that the impedance, when realised, will consist of a finite number of meshes with a finite number of elements impedance due to Cauer (1929) is that it must be a function of ''s'' that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the Poisson integral
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
representation of these functions. Brune coined the term positive-real for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to Sidney Darlington
Sidney Darlington (July 18, 1906 – October 31, 1997) was an American electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertion-loss synth ...
states that any positive-real function ''Z''(''s'') can be realised as a lossless two-port
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sat ...
terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.[Darlington, p.7]
As for equivalence, Cauer found that the group of real affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s,
:
:where,
:
is invariant in ''Z''p(''s''), that is, all the transformed networks are equivalents of the original.[
]
Approximation
The approximation problem in network synthesis is to find functions which will produce realisable networks approximating to a prescribed function of frequency within limits arbitrarily set. The approximation problem is an important issue since the ideal function of frequency required will commonly be unachievable with rational networks. For instance, the ideal prescribed function is often taken to be the unachievable lossless transmission in the passband, infinite attenuation in the stopband and a vertical transition between the two. However, the ideal function can be approximated with a rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was Stephen Butterworth (1930) using his Butterworth polynomials. Independently, Cauer (1931) used Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshe ...
, initially applied to image filters, and not to the now well-known ladder realisation of this filter.[
]
Butterworth filter
Butterworth filters are an important class[A class of filters is a collection of filters which are all described by the same class of mathematical function, for instance, the class of Chebyshev filters are all described by the class of ]Chebyshev polynomial
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshe ...
s. For realisable linear passive networks, the transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
must be a ratio of polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s. The order of a filter is the order of the highest order polynomial of the two and will equal the number of elements (or resonators) required to build it. Usually, the higher the order of a filter, the steeper the roll-off of the filter will be. In general, the values of the elements in each section of the filter will not be the same if the order is increased and will need to be recalculated. This is in contrast to the image method of design which simply adds on more identical sections of filters due to Stephen Butterworth (1930) which are now recognised as being a special case of Cauer's elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
s. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a valve amplifier
A valve amplifier or tube amplifier is a type of electronic amplifier that uses vacuum tubes to increase the amplitude or power of a signal. Low to medium power valve amplifiers for frequencies below the microwaves were largely replaced by sol ...
which made calculation of component values easy since the filter sections could not interact with each other and each section represented one term in the Butterworth polynomials. This gives Butterworth the credit for being both the first to deviate from image parameter theory and the first to design active filters. It was later shown that Butterworth filters could be implemented in ladder topology without the need for amplifiers. Possibly the first to do so was William Bennett (1932) in a patent which presents formulae for component values identical to the modern ones. Bennett, at this stage though, is still discussing the design as an artificial transmission line and so is adopting an image parameter approach despite having produced what would now be considered a network synthesis design. He also does not appear to be aware of the work of Butterworth or the connection between them.[Matthaei et al., pp.85–108]
Insertion-loss method
The insertion-loss method of designing filters is, in essence, to prescribe a desired function of frequency for the filter as an attenuation of the signal when the filter is inserted between the terminations relative to the level that would have been received were the terminations connected to each other via an ideal transformer perfectly matching them. Versions of this theory are due to Sidney Darlington
Sidney Darlington (July 18, 1906 – October 31, 1997) was an American electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertion-loss synth ...
, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis. Butterworth's filter implementation is, in those terms, an insertion-loss filter, but it is a relatively trivial one mathematically since the active amplifiers used by Butterworth ensured that each stage individually worked into a resistive load. Butterworth's filter becomes a non-trivial example when it is implemented entirely with passive components. An even earlier filter which influenced the insertion-loss method was Norton's dual-band filter where the input of two filters are connected in parallel and designed so that the combined input presents a constant resistance. Norton's design method, together with Cauer's canonical LC networks and Darlington's theorem that only LC components were required in the body of the filter resulted in the insertion-loss method. However, ladder topology proved to be more practical than Cauer's canonical forms.[Darlington, p.8]
Darlington's insertion-loss method is a generalisation of the procedure used by Norton. In Norton's filter it can be shown that each filter is equivalent to a separate filter unterminated at the common end. Darlington's method applies to the more straightforward and general case of a 2-port LC network terminated at both ends. The procedure consists of the following steps:
#determine the poles of the prescribed insertion-loss function,
#from that find the complex transmission function,
#from that find the complex reflection coefficients at the terminating resistors,
#find the driving point impedance from the short-circuit and open-circuit impedances,[
#expand the driving point impedance into an LC (usually ladder) network.
Darlington additionally used a transformation found by ]Hendrik Bode
Hendrik Wade Bode ( ; ;Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", IEEE Transactions on Automatic Control, Vol. AC-29, No 3., March 1984, pp. 193–194. Quote: "Something should be ...
that predicted the response of a filter using non-ideal components but all with the same ''Q''. Darlington used this transformation in reverse to produce filters with a prescribed insertion-loss with non-ideal components. Such filters have the ideal insertion-loss response plus a flat attenuation across all frequencies.[
]
Elliptic filters
Elliptic filters are filters produced by the insertion-loss method which use elliptic rational functions
In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic filter, elliptic electronic filters. (These functions are someti ...
in their transfer function as an approximation to the ideal filter response and the result is called a Chebyshev approximation. This is the same Chebyshev approximation technique used by Cauer on image filters but follows the Darlington insertion-loss design method and uses slightly different elliptic functions. Cauer had some contact with Darlington and Bell Labs before WWII (for a time he worked in the US) but during the war they worked independently, in some cases making the same discoveries. Cauer had disclosed the Chebyshev approximation to Bell Labs but had not left them with the proof. Sergei Schelkunoff
Sergei Alexander Schelkunoff (russian: Сергей Александрович Щелкунов; January 27, 1897 – May 2, 1992), who published as S. A. Schelkunoff, was a distinguished mathematician, engineer and electromagnetism theorist who ...
provided this and a generalisation to all equal ripple problems. Elliptic filters are a general class of filter which incorporate several other important classes as special cases: Cauer filter (equal ripple
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
in passband and stopband), Chebyshev filter (ripple only in passband), reverse Chebyshev filter (ripple only in stopband) and Butterworth filter (no ripple in either band).[
Generally, for insertion-loss filters where the transmission zeroes and infinite losses are all on the real axis of the complex frequency plane (which they usually are for minimum component count), the insertion-loss function can be written as;
:
where ''F'' is either an even (resulting in an antimetric filter) or an odd (resulting in an symmetric filter) function of frequency. Zeroes of ''F'' correspond to zero loss and the poles of ''F'' correspond to transmission zeroes. ''J'' sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of ''F'' and ''J'' can be set arbitrarily. The nature of ''F'' determines the class of the filter;
*if ''F'' is a Chebyshev approximation the result is a Chebyshev filter,
*if ''F'' is a maximally flat approximation the result is a passband maximally flat filter,
*if 1/''F'' is a Chebyshev approximation the result is a reverse Chebyshev filter,
*if 1/''F'' is a maximally flat approximation the result is a stopband maximally flat filter,
A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.][
Darlington relates that he found in the New York City library Carl Jacobi's original paper on elliptic functions, published in Latin in 1829. In this paper Darlington was surprised to find foldout tables of the exact elliptic function transformations needed for Chebyshev approximations of both Cauer's image parameter, and Darlington's insertion-loss filters.][
]
Other methods
Darlington considers the topology of coupled tuned circuits to involve a separate approximation technique to the insertion-loss method, but also producing nominally flat passbands and high attenuation stopbands. The most common topology for these is shunt anti-resonators coupled by series capacitors, less commonly, by inductors, or in the case of a two-section filter, by mutual inductance. These are most useful where the design requirement is not too stringent, that is, moderate bandwidth, roll-off and passband ripple.[
]
Other notable developments and applications
Mechanical filters
Edward Norton
Edward Harrison Norton (born August 18, 1969) is an American actor and filmmaker. He has received numerous awards and nominations, including a Golden Globe Award and three Academy Award nominations.
Born in Boston, Massachusetts and raised ...
, around 1930, designed a mechanical filter for use on phonograph
A phonograph, in its later forms also called a gramophone (as a trademark since 1887, as a generic name in the UK since 1910) or since the 1940s called a record player, or more recently a turntable, is a device for the mechanical and analogu ...
recorders and players. Norton designed the filter in the electrical domain and then used the correspondence of mechanical quantities to electrical quantities to realise the filter using mechanical components. Mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
corresponds to inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
, stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
to elastance
Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of c ...
and damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
to resistance. The filter was designed to have a maximally flat frequency response.
In modern designs it is common to use quartz crystal filter
A crystal filter allows some frequencies to 'pass' through an electrical circuit while attenuating undesired frequencies. An electronic filter can use quartz crystals as resonator components of a filter circuit. Quartz crystals are piezoelectric ...
s, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by transducer
A transducer is a device that converts energy from one form to another. Usually a transducer converts a signal in one form of energy to a signal in another.
Transducers are often employed at the boundaries of automation, measurement, and contr ...
s between the electrical and mechanical domains at the terminals of the crystal.
Distributed-element filters
Distributed-element filters are composed of lengths of transmission line that are at least a significant fraction of a wavelength long. The earliest non-electrical filters were all of this type. William Herschel
Frederick William Herschel (; german: Friedrich Wilhelm Herschel; 15 November 1738 – 25 August 1822) was a German-born British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Caroline H ...
(1738–1822), for instance, constructed an apparatus with two tubes of different lengths which attenuated some frequencies but not others. Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ( ...](_blank)
to calculate the number of loading coils needed on his loaded lines, the device that led to his electrical filter development. Lagrange, Godfrey, and Campbell all made simplifying assumptions in their calculations that ignored the distributed nature of their apparatus. Consequently, their models did not show the multiple passbands that are a characteristic of all distributed-element filters. The first electrical filters that were truly designed by distributed-element principles are due to Warren P. Mason
Warren Perry Mason (September 28, 1900 – August 23, 1986) was an American electrical engineer and physicist at Bell Labs. A graduate of Columbia University, he had a prolific output, publishing four books and nearly a hundred papers. He was is ...
starting in 1927.
Transversal filters
Transversal filter
Transversal may refer to:
* Transversal (combinatorics), a set containing exactly one member of each of several other sets
* Transversal (geometry), a line that intersects two or more lines at different points
* Transversal (instrument making), a t ...
s are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from 1935 which describes a filter consisting of a cascade of all-pass sections. The outputs of the various sections are summed in the proportions needed to result in the required frequency function. This works by the principle that certain frequencies will be in, or close to antiphase, at different sections and will tend to cancel when added. These are the frequencies rejected by the filter and can produce filters with very sharp cut-offs. This approach did not find any immediate applications, and is not common in passive filters. However, the principle finds many applications as an active delay line implementation for wide band discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
filter applications such as television, radar and high-speed data transmission.[
]
Matched filter
The purpose of matched filters is to maximise the signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
(S/N) at the expense of pulse shape. Pulse shape, unlike many other applications, is unimportant in radar while S/N is the primary limitation on performance. The filters were introduced during WWII (described 1943) by Dwight North and are often eponymously referred to as " North filters".[Darlington, p.11]
Filters for control systems
Control systems have a need for smoothing filters in their feedback loops with criteria to maximise the speed of movement of a mechanical system to the prescribed mark and at the same time minimise overshoot and noise induced motions. A key problem here is the extraction of Gaussian signals from a noisy background. An early paper on this was published during WWII by Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (Kalman filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimat ...
) later reformulated this in terms of state-space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
smoothing and prediction where it is known as the linear-quadratic-Gaussian control problem. Kalman started an interest in state-space solutions, but according to Darlington this approach can also be found in the work of Heaviside and earlier.[
]
Modern practice
LC filters at low frequencies become awkward; the components, especially the inductors, become expensive, bulky, heavy, and non-ideal. Practical 1 H inductors require many turns on a high-permeability core; that material will have high losses and stability issues (e.g., a large temperature coefficient). For applications such as a mains filters, the awkwardness must be tolerated. For low-level, low-frequency, applications, RC filters are possible, but they cannot implement filters with complex poles or zeros. If the application can use power, then amplifiers can be used to make RC 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.
...
s that can have complex poles and zeros. In the 1950s, Sallen–Key active RC filters were made with vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied.
The type kn ...
amplifiers; these filters replaced the bulky inductors with bulky and hot vacuum tubes. Transistors offered more power-efficient active filter designs. Later, inexpensive operational amplifier
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
s enabled other active RC filter design topologies. Although active filter designs were commonplace at low frequencies, they were impractical at high frequencies where the amplifiers were not ideal; LC (and transmission line) filters were still used at radio frequencies.
Gradually, the low frequency active RC filter was supplanted by the switched-capacitor filter that operated in the discrete time domain rather than the continuous time domain. All of these filter technologies require precision components for high performance filtering, and that often requires that the filters be tuned. Adjustable components are expensive, and the labor to do the tuning can be significant. Tuning the poles and zeros of a 7th-order elliptic filter is not a simple exercise. Integrated circuits have made digital computation inexpensive, so now low frequency filtering is done with digital signal processors. Such digital filter
In signal processing, 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, t ...
s have no problem implementing ultra-precise (and stable) values, so no tuning or adjustment is required. Digital filters also don't have to worry about stray coupling paths and shielding the individual filter sections from one another. One downside is the digital signal processing may consume much more power than an equivalent LC filter. Inexpensive digital technology has largely supplanted analogue implementations of filters. However, there is still an occasional place for them in the simpler applications such as coupling where sophisticated functions of frequency are not needed. Passive filters are still the technology of choice at microwave frequencies.[ Lars Wanhammar, ''Analog Filters using MATLAB'', pp. 10–11, Springer, 2009 .]
See also
*Audio filter
An audio filter is a frequency dependent circuit, working in the audio frequency range, 0 Hz to 20 kHz. Audio filters can amplify (boost), pass or attenuate (cut) some frequency ranges. Many types of filters exist for different audio a ...
*Composite image filter
A 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 ...
*Digital filter
In signal processing, 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, t ...
*Electronic filter
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 ...
*Linear filter
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 (or shift invariant) in which case they can be analyzed exactly using ...
*Network synthesis filters
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
Footnotes
References
Bibliography
* Belevitch, V, "Summary of the history of circuit theory", ''Proceedings of the IRE'', vol. 50, iss. 5, pp. 848–855, May 1962 .
* Blanchard, J, "The History of Electrical Resonance", ''Bell System Technical Journal'', vol. 23, pp. 415–433, 1944.
* Cauer, E; Mathis, W; Pauli, R
"Life and work of Wilhelm Cauer (1900–1945)"
''Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)'', Perpignan, June, 2000.
*Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", ''IEEE Transactions on Circuits and Systems'', vol. 31, pp. 3–13, 1984 .
* Fagen, M D; Millman, S, ''A History of Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980)'', AT&T Bell Laboratories, 1984 .
* Godfrey, Charles
"On discontinuities connected with the propagation of wave-motion along a periodically loaded string"
''Philosophical Magazine'', ser. 5, vol. 45, no. 275, pp. 356–363, April 1898.
* Hunt, Bruce J
''The Maxwellians''
Cornell University Press, 2005 .
* Lundheim, L
"On Shannon and Shannon's formula"
''Telektronikk'', vol. 98, no. 1, pp. 20–29, 2002.
* Mason, Warren P
"Electrical and mechanical analogies"
''Bell System Technical Journal'', vol. 20, no. 4, pp. 405–414, October 1941.
* Matthaei, Young, Jones, ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', McGraw-Hill 1964.
Further reading
*Fry, T C
"The use of continued fractions in the design of electrical networks"
''Bulletin of the American Mathematical Society'', volume 35, pages 463–498, 1929 (full text available).
{{DEFAULTSORT:Analogue Filter
Linear filters
Filter theory
Analog circuits
History of electronic engineering
Electronic design