Prototype filter
Encyclopedia
Prototype filters are electronic filter
designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised
design from which the desired filter can be scaled
or transformed. They are most often seen in regards to electronic filter
s 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.
Filters are required to operate at many different frequencies
, impedances
and bandwidths. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype. The filter design need thus only be carried out once in full, with other filters being obtained by simply applying a scaling factor.
Especially useful is the ability to transform from one bandform to another. In this case, the transform is more than a simple scale factor. Bandform here is meant to indicate the category of passband
that the filter possesses. The usual bandforms are lowpass, highpass, bandpass and bandstop, but others are possible. In particular, it is possible for a filter to have multiple passbands. In fact, in some treatments, the bandstop filter is considered to be a type of multiple passband filter having two passbands. Most commonly, the prototype filter is expressed as a lowpass filter, but other techniques are possible.
' = 1 rad/s
. Occasionally, frequency f' ' = 1 Hz
is used instead of ωc' = 1. Likewise, the nominal or characteristic impedance of the filter is set to R ' = 1 Ω.
In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design. For example, for filters with ripple in the passband the corner frequency is usually defined as the highest frequency at maximum ripple rather than 3dB. Another case is in image parameter filters
(an older design method than the more modern network synthesis filters
) which use the cut-off frequency rather than the 3dB point since cut-off is a well defined point in this types of filter.
The prototype filter can only be used to produce other filters of the same classThe class of a filter is the mathematical class of the polynomials in the rational function
that describe its transfer function
. Image parameter filters are not rational and hence do not have a polynomial class. Such filters are classified by type (k-type, m-type etc). Type serves as the class name for image filters and is based on the filter circuit topology. and order.The order of a filter is the order
of the filter's rational function. A rational function is a ratio of two polynomial
s and the order of the function is the order of the highest order polynomial. Any filter constructed from a finite number of discrete elements will be described by a rational function and in general, the order will be equal to the number of reactive elements that are used. For instance, a fifth order Bessel filter
prototype can be converted into any other fifth order Bessel filter, but it cannot be transformed into a third order Bessel filter or a fifth order Tchebyscheff filter.
where ωc' is the value of the frequency parameter (e.g. cut-off frequency) for the prototype and ωc is the desired value. So if ωc' = 1 then the transfer function of the filter is transformed as:
It can readily be seen that to achieve this, the non-resistive components of the filter must be transformed by:
and, 
It may be more convenient on some elements to scale the admittance instead:
It can readily be seen that to achieve this, the non-resistive components of the filter must be scaled as:
and, 
Impedance scaling by itself has no effect on the transfer function of the filter (providing that the terminating impedances have the same scaling applied to them). However, it is usual to combine the frequency and impedance scaling into a single step:
and, 
where ωc is the point on the highpass filter corresponding to ωc' on the prototype. The transfer function then transforms as:
Inductors are transformed into capacitors according to,
and capacitors are transformed into inductors,
the primed quantities being the component value in the prototype.
where Q is the Q-factor and is equal to the inverse of the fractional bandwidth:
If ω1 and ω2 are the lower and upper frequency points (respectively) of the bandpass response corresponding to ωc' of the prototype, then,
and 
Δω is the absolute bandwidth, and ω0 is the resonant frequency of the resonators in the filter. Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.
The transfer function of the filter is transformed according to:
Inductors are transformed into series resonator
s,
and capacitors are transformed into parallel resonators,
Inductors are transformed into parallel resonators,
and capacitors are transformed into series resonators,
The number of resonators in the expression corresponds to the number of passbands required. Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms becoming zero as appropriate. Bandstop filters can be regarded as a combination of a lowpass and a highpass filter. Multiple bandstop filters can always be expressed in terms of a multiple bandpass filter. In this way it, can be seen that this transformation represents the general case for any bandform, and all the other transformations are to be viewed as special cases of it.
The same response can equivalently be obtained, sometimes with a more convenient component topology, by transforming to multiple stopbands instead of multiple passbands. The required transformation in those cases is:
, Zobel provided an alternative basis for constructing a prototype which is not based in the frequency domain
. The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing; however, it is not in common use.
The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a two-port network
rather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series impedance
, Z, and the shunt admittance
Y of a filter half-section. See the article Image impedance
for a description of half-sections. This quantity is nondimensional
, adding to the prototype's generality. Generally, ZY is a complex quantity,
and as U and V are both, in general, functions of ω we should properly write,
With image filters, it is possible to obtain filters of different classes from the constant k filter
prototype by means of a different kind of transformation (see composite image filter
), constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of U(ω) for a constant k, which is notated as,
In the case of dissipationless networks, i.e. no resistors, the quantity V(ω) is zero and only U(ω) need be considered. Uk(ω) ranges from 0 at the centre of the passband
to -1 at the cut-off frequency and then continues to increase negatively into the stopband
regardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used:
For a lowpass constant k prototype that is scaled:
the independent variable of the response plot is,
The bandform transformations from this prototype are,
for lowpass,
for highpass,
and for bandpass,
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...
design from which the desired filter can be scaled
Scale factor
A scale factor is a number which scales, or multiplies, some quantity. In the equation y=Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x...
or transformed. They are most often seen in regards to electronic filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
s and especially linear analogue passive filters
Linear filter
Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....
. However, in principle, the method can be applied to any kind of linear filter or signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, including mechanical, acoustic and optical filters.
Filters are required to operate at many different frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
, impedances
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
and bandwidths. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype. The filter design need thus only be carried out once in full, with other filters being obtained by simply applying a scaling factor.
Especially useful is the ability to transform from one bandform to another. In this case, the transform is more than a simple scale factor. Bandform here is meant to indicate the category of passband
Passband
A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
that the filter possesses. The usual bandforms are lowpass, highpass, bandpass and bandstop, but others are possible. In particular, it is possible for a filter to have multiple passbands. In fact, in some treatments, the bandstop filter is considered to be a type of multiple passband filter having two passbands. Most commonly, the prototype filter is expressed as a lowpass filter, but other techniques are possible.
Low-pass prototype
The prototype is most often a low-pass filter with a 3dB corner frequency of angular frequency ωcAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
. Occasionally, frequency f
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
is used instead of ωc
In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design. For example, for filters with ripple in the passband the corner frequency is usually defined as the highest frequency at maximum ripple rather than 3dB. Another case is in image parameter filters
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 in an infinite chain of such sections. In this, the...
(an older design method than the more modern network synthesis filters
Network synthesis filters
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...
) which use the cut-off frequency rather than the 3dB point since cut-off is a well defined point in this types of filter.
The prototype filter can only be used to produce other filters of the same classThe class of a filter is the mathematical class of the polynomials in the rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
that describe its transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
. Image parameter filters are not rational and hence do not have a polynomial class. Such filters are classified by type (k-type, m-type etc). Type serves as the class name for image filters and is based on the filter circuit topology. and order.The order of a filter is the order
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
of the filter's rational function. A rational function is a ratio of two polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s and the order of the function is the order of the highest order polynomial. Any filter constructed from a finite number of discrete elements will be described by a rational function and in general, the order will be equal to the number of reactive elements that are used. For instance, a fifth order Bessel filter
Bessel filter
In electronics and signal processing, a Bessel filter is a type of linear filter with a maximally flat group delay . Bessel filters are often used in audio crossover systems...
prototype can be converted into any other fifth order Bessel filter, but it cannot be transformed into a third order Bessel filter or a fifth order Tchebyscheff filter.
Frequency scaling
The prototype filter is scaled to the frequency required with the following transformation:where ωc
It can readily be seen that to achieve this, the non-resistive components of the filter must be transformed by:
and, Impedance scaling
Impedance scaling is invariably a scaling to a fixed resistance. This is because the terminations of the filter, at least nominally, are taken to be a fixed resistance. To carry out this scaling to a nominal impedance R, each impedance element of the filter is transformed by:It may be more convenient on some elements to scale the admittance instead:
and, Impedance scaling by itself has no effect on the transfer function of the filter (providing that the terminating impedances have the same scaling applied to them). However, it is usual to combine the frequency and impedance scaling into a single step:
and, Bandform transformation
In general, the bandform of a filter is transformed by replacing iω where it occurs in the transfer function with a function of iω. This in turn leads to the transformation of the impedance components of the filter into some other component(s). The frequency scaling above is a trivial case of bandform transformation corresponding to a lowpass to lowpass transformation.Lowpass to highpass
The frequency transformation required in this case is:where ωc is the point on the highpass filter corresponding to ωc
Inductors are transformed into capacitors according to,
and capacitors are transformed into inductors,
the primed quantities being the component value in the prototype.
Lowpass to bandpass
In this case, the required frequency transformation is:where Q is the Q-factor and is equal to the inverse of the fractional bandwidth:
If ω1 and ω2 are the lower and upper frequency points (respectively) of the bandpass response corresponding to ωc
and Δω is the absolute bandwidth, and ω0 is the resonant frequency of the resonators in the filter. Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.
The transfer function of the filter is transformed according to:
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...
s,
and capacitors are transformed into parallel resonators,
Lowpass to bandstop
The required frequency transformation for lowpass to bandstop is:Inductors are transformed into parallel resonators,
and capacitors are transformed into series resonators,
Lowpass to multi-band
Filters with multiple passbands may be obtained by applying the general transformation:The number of resonators in the expression corresponds to the number of passbands required. Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms becoming zero as appropriate. Bandstop filters can be regarded as a combination of a lowpass and a highpass filter. Multiple bandstop filters can always be expressed in terms of a multiple bandpass filter. In this way it, can be seen that this transformation represents the general case for any bandform, and all the other transformations are to be viewed as special cases of it.
The same response can equivalently be obtained, sometimes with a more convenient component topology, by transforming to multiple stopbands instead of multiple passbands. The required transformation in those cases is:
Alternative prototype
In his treatment of image filtersComposite 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 in an infinite chain of such sections. In this, the...
, Zobel provided an alternative basis for constructing a prototype which is not based in the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
. The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing; however, it is not in common use.
The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a two-port network
Two-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
rather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
, Z, and the shunt admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...
Y of a filter half-section. See the article Image impedance
Image impedance
Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking in to the ports of a network. Usually a two-port network is implied but the concept is capable of being extended to...
for a description of half-sections. This quantity is nondimensional
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...
, adding to the prototype's generality. Generally, ZY is a complex quantity,
and as U and V are both, in general, functions of ω we should properly write,With image filters, it is possible to obtain filters of different classes from the constant k filter
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 components...
prototype by means of a different kind of transformation (see composite image filter
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 in an infinite chain of such sections. In this, the...
), constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of U(ω) for a constant k, which is notated as,
In the case of dissipationless networks, i.e. no resistors, the quantity V(ω) is zero and only U(ω) need be considered. Uk(ω) ranges from 0 at the centre of the passband
Passband
A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
to -1 at the cut-off frequency and then continues to increase negatively into the stopband
Stopband
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...
regardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used:
For a lowpass constant k prototype that is scaled:
the independent variable of the response plot is,
The bandform transformations from this prototype are,
for lowpass,
for highpass,
and for bandpass,
See also
- Electronic filter topologyElectronic filter topologyElectronic 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....
- Electronic filterElectronic filterElectronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
- Linear filterLinear filterLinear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....
- Composite image filterComposite image filterA composite image filter is an electronic filter consisting of multiple image filter sections of two or more different types.The image method of filter design determines the properties of filter sections by calculating the properties they have in an infinite chain of such sections. In this, the...