# Filter

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**Electronic filters** are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signal components and/or enhance wanted ones. Electronic filters can be:

- passive or active
- analog or digital
- discrete-time (sampled) or continuous-time
- linear or non-linear
- infinite impulse response (IIR type) or finite impulse response (FIR type)

The most common types of electronic filters are linear filters, regardless of other aspects of their design. See the article on linear filters for details on their design and analysis.

## History Edit

The oldest forms of electronic filters are passive analog linear filters, constructed using only resistors and capacitors or resistors and inductors. These are known as RC and RL single pole filters respectively.

Hybrid filters have also been made, typically involving combinations of analog amplifiers with mechanical resonators or delay lines.
Other devices such as CCD delay lines have also been used as discrete-time filters. With the availability of digital signal processing, active digital filters have become common.

## Classification by technology Edit

### Passive filtersEdit

#### Single pole typesEdit

The simplest electronic implementations of linear filters are based on combinations of resistors, inductors and capacitors. These filters exist in so-called RC, RL, LC and RLC varieties. All these types are collectively known as *passive filters*, because they do not depend upon an external power supply.
Inductors block high-frequency signals and conduct low-frequency signals, while capacitors do the reverse. A filter in which the signal passes through an inductor, or in which a capacitor provides a path to earth, presents less attenuation to low-frequency signals than high-frequency signals and is a *low-pass filter*. If the signal passes through a capacitor, or has a path to ground through an inductor, then the filter presents less attenuation to high-frequency signals than low-frequency signals and is a *high-pass filter*. Resistors on their own have no frequency-selective properties, but are added to inductors and capacitors to determine the *time-constants* of the circuit, and therefore the frequencies to which it responds.

At very high frequencies (above about 100 megahertz), sometimes the inductors consist of single loops or strips of sheet metal, and the capacitors consist of adjacent strips of metal. These are called stubs.

#### Multipole typesEdit

Higher order filters are measured by their quality or "*Q*" factor. A filter is said to have a high *Q* if it selects or rejects a range of frequencies that is narrow in comparison to the centre frequency. *Q* may be defined as the ratio of centre frequency divided by 3dB bandwidth.

### Active filtersEdit

Active filters are implemented using a combination of passive and active (amplifying) components, and require an outside power source. Operational amplifiers are frequently used in active filter designs. These can have high *Q*, and can achieve resonance without the use of inductors. However, their upper frequency limit is limited by the bandwidth of the amplifiers used.

### Digital filtersEdit

Digital signal processing allows the inexpensive construction of a wide variety of filters. The signal is sampled and an analog to digital converter turns the signal into a stream of numbers. A computer program running on a CPU or a specialized DSP (or less often running on a hardware implementation of the algorithm) calculates an output number stream. This output can be converted to a signal by passing it through a digital to analog converter. There are problems with noise introduced by the conversions, but these can be controlled and limited for many useful filters. Due to the sampling involved, the input signal must be of limited frequency content or aliasing will occur. See also: Digital filter.

### Other filter technologiesEdit

#### Quartz filters and piezoelectricsEdit

In the late 1930s, engineers realized that small mechanical systems made of rigid materials such as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up to several hundred megahertz. Some early resonators were made of steel, but quartz quickly became favored. The biggest advantage of quartz is that it is piezoelectric. This means that quartz resonators can directly convert their own mechanical motion into electrical signals. Quartz also has a very low coefficient of thermal expansion which means that quartz resonators can produce stable frequencies over a wide temperature range. Quartz crystal filters have much higher quality factors than LCR filters. When higher stabilities are required, the crystals and their driving circuits may be mounted in a "crystal oven" to control the temperature. For very narrow band filters, sometimes several crystals are operated in series.

Engineers realized that a large number of crystals could be collapsed into a single component, by mounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tapped delay line" reinforces the desired frequencies as the sound waves flow across the surface of the quartz crystal.
The tapped delay line has become a general scheme of making high-*Q* filters in many different ways.

#### SAW filtersEdit

SAW (surface acoustic wave) filters are electromechanical devices commonly used in radio frequency applications. Electrical signals are converted to a mechanical wave in a piezoelectric crystal; this wave is delayed as it propagates across the crystal, before being converted back to an electrical signal by further electrodes. The delayed outputs are recombined to produce a direct analog implementation of a finite impulse response filter. This hybrid filtering technique is also found in an analog sampled filter. SAW filters are limited to frequencies up to 3GHz.

#### BAW filtersEdit

BAW (Bulk Acoustic Wave) filters are electromechanical devices. These filters are in the research state for the moment. BAW filters can implement ladder or lattice filters. BAW filters seem to be smaller than SAW filters, and can operate at frequencies up to 16 GHz.

#### Garnet filtersEdit

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Another method of filtering, at microwave frequencies from 800MHz to about 5 GHz, is to use a synthetic single crystal yttrium iron garnet sphere made of a chemical combination of yttrium and iron (**YIGF**, or **yttrium iron garnet filter**). The garnet sits on a strip of metal driven by a transistor, and a small loop antenna touches the top of the sphere. An electromagnet changes the frequency that the garnet will pass. The advantage of this method is that the garnet can be tuned over a very wide frequency by varying the strength of the magnetic field.

#### Atomic filtersEdit

For even higher frequencies and greater precision, the vibrations of atoms must be used. Atomic clocks use caesium masers as ultra-high *Q* filters to stabilize their primary oscillators. Another method, used at high, fixed frequencies with very weak radio signals, is to use a ruby maser tapped delay line.

## The transfer function Edit

The transfer function of a filter is the ratio of the output signal to that of the input signal as a function of the complex frequency :

with .

The transfer function of all linear time-invariant filters generally share certain characteristics:

- Since the filters are constructed of discrete components, their transfer function will be the ratio of two polynomials in , i.e. a rational function of . The order of the transfer function will be the highest power of encountered in either the numerator or the denominator.
- The polynomials of the transfer function will all have real coefficients. Therefore, the poles and zeroes of the transfer function will either be real or occur in complex conjugate pairs.
- Since the filters are assumed to be stable, the real part of all poles (i.e. zeroes of the denominator) will be negative, i.e. they will lie in the left half-plane in complex frequency space.

## Classification by transfer function Edit

Filters may be specified by family and passband. A filter's family is specified by certain design criteria which give general rules for specifying the transfer function of the filter. Some common filter families and their particular design criteria are:

- Butterworth filter - no gain ripple in pass band and stop band, slow cutoff
- Chebyshev filter(Type I) - no gain ripple in stop band, moderate cutoff
- Chebyshev filter(Type II) - no gain ripple in pass band, moderate cutoff
- Bessel filter - no group delay ripple, no gain ripple in both bands, slow gain cutoff
- Elliptic filter - gain ripple in pass and stop band, fast cutoff
- Optimum "L" filter
- Gaussian filter - no ripple in response to step function
- Hourglass filter
- Raised-cosine filter

Generally, each family of filters can be specified to a particular order. The higher the order, the more the filter will approach the "ideal" filter. The ideal filter has full transmission in the pass band, and complete attenuation in the stop band, and the transition between the two bands is abrupt (often called brick-wall).

Each family can be used to specify a particular pass band in which frequencies are transmitted, while frequencies in the stop band (i.e. outside the pass band) are more or less attenuated.

- Low-pass filter - Low frequencies are passed, high frequencies are attenuated.
- High-pass filter - High frequencies are passed, Low frequencies are attenuated.
- Band-pass filter - Only frequencies in a frequency band are passed.
- Band-stop filter - Only frequencies in a frequency band are attenuated.
- All-pass filter - All frequencies are passed, but the phase of the output is modified.

## Classification by topology Edit

The above classifications will specify completely the transfer function of the filter (i.e. its electronic behavior), but it remains to choose the particular circuit topology to implement the filter. In other words, there are a number of different ways of achieving a particular transfer function when designing a circuit. These topologies may be further subdivided into passive filters and active filters. Some common circuit topologies are:

- Cauer topology - Passive
- Sallen Key topology - Active
- Multiple Feedback topology - Active
- State Variable Topology - Active
- Biquadratic topology - Active