Analogue filter

1. 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, these designs proved inadequate for successful frequency-division multiplexing. 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 nature of signals.
2. 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 passbands 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.
3. Network synthesis filters. 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. 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 terms.

Throughout this article the letters R, L, and C are used with their usual meanings to represent

Early filters utilised the phenomenon of resonance to filter signals. Although electrical resonance 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 (which in turn, can be explained in terms of the even more familiar mechanical resonance) found its way into filter design ahead of electrical resonance.^[1] 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.^[2]

Electrical resonance

Resonance was noticed early on in experiments with the

Hermann von Helmholtz in 1847 published his important work on conservation of energy^[5] 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 experiments of William Hyde Wollaston. 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.^[6]

Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to

Heinrich Hertz (1887) experimentally demonstrated the resonance phenomena^[12] 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.^[1][13]

Acoustic resonance

As mentioned earlier, it was acoustic resonance that inspired filtering applications, the first of these being a telegraph system known as the "

Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as transducers 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.^[1][14]

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.

Transmission line theory

The earliest model of the

The filters designed by Campbell

of the filter as ${\displaystyle \scriptstyle Z_{i}={\sqrt {Z_{o}Z_{s}}}}$. 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

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.^[37]

Network synthesis filters

The essence of

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)[note 15] 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.^[51] 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.^[52]

The computational difficulty of the network synthesis method was addressed by tabulating the component values of a prototype filter 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,^[35] but the concept of a "reference filter" is due to Sidney Darlington.^[53] Darlington (1939),^[30] was also the first to tabulate values for network synthesis prototype filters,^[54] nevertheless it had to wait until the 1950s before the Cauer-Darlington elliptic filter first came into use.^[55]

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.^[56][57] Furthermore, the advent of active filters eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.^[51]

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, Cauer formed the matrix equation,

${\displaystyle \mathbf {[A]} =s^{2}\mathbf {[L]} +s\mathbf {[R]} +\mathbf {[D]} =s\mathbf {[Z]} }$

where [Z],[R],[L] and [D] are the nxn matrices of, respectively,

${\displaystyle \scriptstyle s=\sigma +i\omega }$

${\displaystyle Z_{\mathrm {p} }(s)={\frac {\det \mathbf {[A]} }{s\,a_{11}}}}$

where a₁₁ is the complement of the element A₁₁ to which the one-port is to be connected. From

As for equivalence, Cauer found that the group of real affine transformations,

${\displaystyle \mathbf {[T]} ^{T}\mathbf {[A]} \mathbf {[T]} }$

where,

${\displaystyle \mathbf {[T]} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}}$

is invariant in Z_p(s), that is, all the transformed networks are equivalents of the original.^[39]

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

Butterworth filters are an important class

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, 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.^[64]

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:

1. determine the poles of the prescribed insertion-loss function,
2. from that find the complex transmission function,
3. from that find the complex reflection coefficients at the terminating resistors,
4. find the driving point impedance from the short-circuit and open-circuit impedances,^{[note 13]}
5. expand the driving point impedance into an LC (usually ladder) network.

Darlington additionally used a transformation found by

Elliptic filters are filters produced by the insertion-loss method which use elliptic rational functions 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 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 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).^[64][66]

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;

${\displaystyle {\frac {1}{1+JF^{2}}}}$

where F is either an even (resulting in an

• 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.^[64]

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.^[51]

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.^[57]

Other notable developments and applications

Mechanical filters

In modern designs it is common to use quartz crystal filters, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by transducers between the electrical and mechanical domains at the terminals of the crystal.^[68]

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 (1738–1822), for instance, constructed an apparatus with two tubes of different lengths which attenuated some frequencies but not others. Joseph-Louis Lagrange (1736–1813) studied waves on a string periodically loaded with weights. The device was never studied or used as a filter by either Lagrange or later investigators such as Charles Godfrey. However, Campbell used Godfrey's results by analogy 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.^[69] The first electrical filters that were truly designed by distributed-element principles are due to Warren P. Mason starting in 1927.^[70]

Transversal filters

The purpose of matched filters is to maximise the signal-to-noise ratio (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)^[74] by Dwight North and are often eponymously referred to as "North filters".^[72][75]

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

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 filters that can have complex poles and zeros. In the 1950s, Sallen–Key active RC filters were made with vacuum tube amplifiers; these filters replaced the bulky inductors with bulky and hot vacuum tubes. Transistors offered more power-efficient active filter designs. Later, inexpensive operational amplifiers 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

• Audio filter
• Composite image filter
• Digital filter
• Electronic filter
• Linear filter
• Network synthesis filters

Footnotes