# General mn-type image filter

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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.

## Contents

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 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.

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

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

${\displaystyle Z=Z_{1}+Z_{2}+Z_{3}+\dots +Z_{N}}$
where Z1, Z2 etc are a cascade of antiresonators,

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

${\displaystyle Z_{m_{n}}=m_{1}Z_{1}+m_{2}Z_{2}+m_{3}Z_{3}+\dots +m_{N}Z_{N}}$

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;

${\displaystyle Y_{m_{n}}={\frac {Z_{m_{n}}}{k^{2}+Z^{2}-Z_{m_{n}}^{2}}}}$
where ${\displaystyle k={\sqrt {\frac {Z}{Y}}}}$ 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.

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

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 .

${\displaystyle Y_{m_{n}}=m_{1}Y_{1}+m_{2}Y_{2}+m_{3}Y_{3}+\dots +m_{N}Y_{N}}$

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

${\displaystyle Z_{m_{n}}={\frac {Y_{m_{n}}}{k^{2}+Y^{2}-Y_{m_{n}}^{2}}}}$

## Simple bandpass section

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.

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]

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.

## Notes

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

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## References

• 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

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