General mn-type image filter

Last updated

These filters are electrical wave filters designed using the image method. They are an invention of Otto Zobel at AT&T Corp.. [1] They are a generalisation of the m-type filter in that a transform is applied that modifies the transfer function while keeping the image impedance unchanged. For filters that have only one stopband there is no distinction with the m-type filter. However, for a filter that has multiple stopbands, there is the possibility that the form of the transfer function in each stopband can be different. For instance, it may be required to filter one band with the sharpest possible cut-off, but in another to minimise phase distortion while still achieving some attenuation. If the form is identical at each transition from passband to stopband the filter will be the same as an m-type filter (k-type filter in the limiting case of m=1). If they are different, then the general case described here pertains.

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 of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, Zi 1, seen looking into port 1 when port 2 is terminated with the image impedance, Zi 2, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical with respect to the ports.


A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level. Depending on application, the required attenuation within the stopband may typically be a value between 20 and 120 dB higher than the nominal passband attenuation, which often is 0 dB.


The k-type filter acts as a prototype for producing the general mn designs. For any given desired bandform there are two classes of mn transformation that can be applied, namely, the mid-series and mid-shunt derived sections; this terminology being more fully explained in the m-derived filter article. Another feature of m-type filters that also applies in the general case is that a half section will have the original k-type image impedance on one side only. The other port will present a new image impedance. The two transformations have equivalent transfer functions but different image impedances and circuit topology.

Prototype filter

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

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 an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

Bandform diagram showing frequency response of a general image filter. The oc are the critical frequencies (the frequency where cut-off begins) and the o[?] are the poles of attenuation in the stop bands. Image filter bandform terminology.svg
Bandform diagram showing frequency response of a general image filter. The ωc are the critical frequencies (the frequency where cut-off begins) and the ω are the poles of attenuation in the stop bands.
Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.

Mid-series multiple stopband

Image filter sections 1.svg

If Z and Y are the series impedance and shunt admittance of a constant k half section and;

where Z1, Z2 etc are a cascade of antiresonators,

the transformed series impedance for a mid-series derived filter becomes;

Where the mn are arbitrary positive coefficients. For an invariant image impedance ZiT and invariant bandform (that is, invariant cut-off frequencies ωc) the transformed shunt admittance, expressed in terms of Zmn, is given by;

where and is a constant by definition. When the mn are all equal this reduces to the expression for an m-type filter and where they are all equal to one it reduces further to the k-type filter.
2-bandstop mid-series derived m1, m2 filter half-section Series derived multi-stop mn type filter.svg
2-bandstop mid-series derived m1, m2 filter half-section

A result of this relationship is that the N antiresonators in Zmn will transform into 2N resonators in Ymn. The coefficients mn can be adjusted by the designer to set the frequency of one of the two poles of attenuation, ω, in each stopband. The second pole of attenuation is dependent and cannot be set separately.

Special cases

In the case of a filter with a stopband extending to zero frequency, one of the antiresonators in Z will reduce to a single inductor. In this case the resonators in Ymn are reduced by one to 2N-1. Similarly, for a filter with a stopband extending to infinity, one antiresonator will reduce to a single capacitor and the resonators will again be reduced by one. In a filter where both conditions pertain, the number of resonators will be 2N-2. For these end stopbands, there is only one pole of attenuation in each, as would be expected from the reduced number of resonators. These forms are the maximum allowable complexity while maintaining invariance of bandform and one image impedance.

Mid-shunt multiple stopband

2-bandstop mid-shunt derived m1, m2 filter half-section. Frequency response is equivalent to the corresponding mid-series derived filter Shunt derived multi-stop mn type filter.svg
2-bandstop mid-shunt derived m1, m2 filter half-section. Frequency response is equivalent to the corresponding mid-series derived filter

By dual analogy, the shunt derived filter starts from;

Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance is its reciprocal, or algebraic inverse . For this reason the dual impedance is also called the inverse impedance. Another way of stating this is that the dual of is the admittance .

For an invariant image admittance Y and invariant bandform the transformed series impedance is given by;

Simple bandpass section

General image bandpass filter, mid-shunt derived General image bandpass filter.svg
General image bandpass filter, mid-shunt derived

The bandpass filter can be characterised as a 2-bandstop filter with ωc = 0 for the lower critical frequency of the lower band and ωc = ∞ for the upper critical frequency of the upper band. The two resonators reduce to an inductor and a capacitor respectively. The number of antiresonators reduces to two.

Image bandpass filter with o[?]1 set to zero and o[?]2 set to correspond to o'c1. Image bandpass filter with one pole af attenuation.svg
Image bandpass filter with ω∞1 set to zero and ω∞2 set to correspond to ω'c1.

If, however, ω∞1 is set to zero (that is, there is no pole of attenuation in the lower stopband) and ω∞2 is set to correspond to the upper critical frequency ω'c1, then a particularly simple form of the bandpass filter is obtained consisting of just antiresonators coupled by capacitors. This was a popular topology for multi-section band-pass filters due its low component count, particularly of inductors. [2] [3] Many other such reduced forms are possible by setting one of the poles of attenuation to correspond to one of the critical frequencies for various classes of basic filter. [4]

See also

Filter bandforms: see, low-pass, high-pass, band-pass, band-stop. Bandform template.svg
Filter bandforms: see, low-pass, high-pass, band-pass, band-stop.

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 components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

mm'-type filters, also called double-m-derived filters, are a type of electronic filter designed using the image method. They were patented by Otto Zobel in 1932. Like the m-type filter from which it is derived, the mm'-type filter type was intended to provide an improved impedance match into the filter termination impedances and originally arose in connection with telephone frequency division multiplexing. The filter has a similar transfer function to the m-type, having the same advantage of rapid cut-off, but the input impedance remains much more nearly constant if suitable parameters are chosen. In fact, the cut-off performance is better for the mm'-type if like-for-like impedance matching are compared rather than like-for-like transfer function. It also has the same drawback of a rising response in the stopband as the m-type. However, its main disadvantage is its much increased complexity which is the chief reason its use never became widespread. It was only ever intended to be used as the end sections of composite filters, the rest of the filter being made up of other sections such as k-type and m-type sections.

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


  1. Zobel, 1923
  2. Matthaei, p425
  3. Bray, p63
  4. Zobel, pp42-43

Related Research Articles

Cutoff frequency frequency response boundary

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

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

Butterworth filter

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

Electronic filter 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.

Zobel network type of filter section based on the image-impedance design principle

Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

Lattice phase equaliser

A lattice phase equaliser or lattice filter is an example of an all-pass filter. That is, the attenuation of the filter is constant at all frequencies but the relative phase between input and output varies with frequency. The lattice filter topology has the particular property of being a constant-resistance network and for this reason is often used in combination with other constant resistance filters such as bridge-T equalisers. The topology of a lattice filter, also called an X-section is identical to bridge topology. The lattice phase equaliser was invented by Otto Zobel. using a filter topology proposed by George Campbell.

Filters designed using the image impedance methodology suffer from a peculiar flaw in the theory. The predicted characteristics of the filter are calculated assuming that the filter is terminated with its own image impedances at each end. This will not usually be the case; the filter will be terminated with fixed resistances. This causes the filter response to deviate from the theoretical. This article explains how the effects of image filter end terminations can be taken into account.

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

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

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

Distributed-element filter

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

RLC circuit Resistor Inductor Capacitor Circuit

An RLC circuit is an electrical circuit consisting of a 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 constituent components of this circuit, where the sequence of the components may vary from RLC.

Frequency selective surface

A frequency-selective surface (FSS) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

A symmetrical lattice is a two-port electrical wave filter in which diagonally-crossed shunt elements are present – a configuration which sets it apart from ladder networks. The component arrangement of the lattice is shown in the diagram below. The filter properties of this circuit were first developed using image impedance concepts, but later the more general techniques of network analysis were applied to it.

Lattice and bridged-T equalizers are circuits which are used to correct for the amplitude and/or phase errors of a network or transmission line. Usually, the aim is to achieve an overall system performance with a flat amplitude response and constant delay over a prescribed frequency range, by the addition of an equalizer. In the past, designers have used a variety of techniques to realize their equalizer circuits. These include the method of complementary networks; the method of straight line asymptotes; using a purpose built test-jig; the use of standard circuit building blocks,; or with the aid of computer programs. In addition, trial and error methods have been found to be surprisingly effective, when performed by an experienced designer.


  • Zobel, O. J.,Theory and Design of Uniform and Composite Electric Wave Filters, Bell System Technical Journal, Vol. 2 (1923), pp. 1-46.
  • Mathaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964.
  • Bray, J, Innovation and the Communications Revolution, Institution of Electrical Engineers, 2002 ISBN   0-85296-218-5
International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.