Insignal processing, afilter is a device or process that removes some unwanted components or features from asignal. Filtering is a class ofsignal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing somefrequencies or frequency bands. However, filters do not exclusively act in thefrequency domain; especially in the field ofimage 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 inelectronics andtelecommunication, inradio,television,audio recording,radar,control systems,music synthesis,image processing,computer graphics, andstructural dynamics.

There are many different bases of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Filters may be:

• non-linear orlinear
• time-variant ortime-invariant, also known as shift invariance. If the filter operates in a spatial domain then the characterization is space invariance.
• causal or non-causal: A filter is non-causal if its present output depends on future input. Filters processing time-domain signals inreal time must be causal, but not filters acting onspatial domain signals or deferred-time processing of time-domain signals.
• analog ordigital
• discrete-time (sampled) orcontinuous-time
• passive oractive type of continuous-time filter
• infinite impulse response (IIR) orfinite impulse response (FIR) type of discrete-time or digital filter.

Linear continuous-time circuit is perhaps the most common meaning for filter in the analog signal processing world, and simply "filter" is often taken to be synonymous. These circuits are generallydesigned to remove certainfrequencies and allow others to pass. Circuits that perform this function are generallylinear in their response, or at least approximately so. Any nonlinearity would potentially result in the output signal containing frequency components not present in the input signal.

The modern design methodology for linear continuous-time filters is callednetwork synthesis. Some important filter families designed in this way are:

• Chebyshev filter, has the best approximation to the ideal response of any filter for a specified order and ripple.
• Butterworth filter, has a maximally flat frequency response.
• Bessel filter, has a maximally flatphase delay.
• Elliptic filter, has the steepest cutoff of any filter for a specified order and ripple.

The difference between these filter families is that they all use a differentpolynomial function to approximate to theideal filter response. This results in each having a differenttransfer function.

Another older, less-used methodology is theimage parameter method. Filters designed by this methodology are archaically called "wave filters". Some important filters designed by this method are:

• Constant k filter, the original and simplest form of wave filter.
• m-derived filter, a modification of the constant k with improved cutoff steepness andimpedance matching.

Some terms used to describe and classify linear filters:

• The frequency response can be classified into a number of different bandforms describing which frequencybands the filter passes (thepassband) and which it rejects (thestopband):
  • 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 or band-reject filter – only frequencies in a frequency band are attenuated.
  • Notch filter – rejects just one specific frequency - an extreme band-stop filter.
  • Comb filter – has multiple regularly spaced narrow passbands giving the bandform the appearance of a comb.
  • All-pass filter – all frequencies are passed, but the phase of the output is modified.
• Cutoff frequency is the frequency beyond which the filter will not pass signals. It is usually measured at a specific attenuation such as 3 dB.
• Roll-off is the rate at which attenuation increases beyond the cut-off frequency.
• Transition band, the (usually narrow) band of frequencies between a passband and stopband.
• Ripple is the variation of the filter'sinsertion loss in the passband.
• The order of a filter is thedegree of the approximating polynomial and in passive filters corresponds to the number of elements required to build it. Increasing order increases roll-off and brings the filter closer to the ideal response.

One important application of filters is intelecommunication.Many telecommunication systems usefrequency-division multiplexing, where the system designers divide a wide frequency band into many narrower frequency bands called "slots" or "channels", and each stream of information is allocated one of those channels.The people who design the filters at each transmitter and each receiver try to balance passing the desired signal through as accurately as possible, keeping interference to and from other cooperating transmitters and noise sources outside the system as low as possible, at reasonable cost.

Multilevel andmultiphasedigital modulation systems require filters that have flat phase delay—arelinear phase in the passband—to preserve pulse integrity in the time domain,^[1]giving lessintersymbol interference than other kinds of filters.

On the other hand,analog audio systems usinganalog transmission can tolerate much larger ripples inphase delay, and so designers of such systems often deliberately sacrifice linear phase to get filters that are better in other ways—better stop-band rejection, lower passband amplitude ripple, lower cost, etc.

Filters can be built in a number of different technologies. The same transfer function can be realised in several different ways, that is the mathematical properties of the filter are the same but the physical properties are quite different. Often the components in different technologies are directly analogous to each other and fulfill the same role in their respective filters. For instance, the resistors, inductors and capacitors of electronics correspond respectively to dampers, masses and springs in mechanics. Likewise, there are corresponding components indistributed-element filters.

• Electronic filters were originally entirely passive consisting of resistance, inductance and capacitance. Active technology makes design easier and opens up new possibilities in filter specifications.
• Digital filters operate on signals represented in digital form. The essence of a digital filter is that it directly implements a mathematical algorithm, corresponding to the desired filter transfer function, in its programming or microcode.
• Mechanical filters are built out of mechanical components. In the vast majority of cases they are used to process an electronic signal andtransducers are provided to convert this to and from a mechanical vibration. However, examples do exist of filters that have been designed for operation entirely in the mechanical domain.
• Distributed-element filters are constructed out of components made from small pieces oftransmission line or otherdistributed elements. There are structures in distributed-element filters that directly correspond to thelumped elements of electronic filters, and others that are unique to this class of technology.
• Waveguide filters consist of waveguide components or components inserted in the waveguide. Waveguides are a class of transmission line and many structures of distributed-element filters, for instance thestub, can also be implemented in waveguides.
• Optical filters were originally developed for purposes other than signal processing such as lighting and photography. With the rise ofoptical fiber technology, however, optical filters increasingly find signal processing applications and signal processing filter terminology, such aslongpass andshortpass, are entering the field.
• Transversal filter, or delay line filter, works by summing copies of the input after various time delays. This can be implemented with various technologies includinganalog delay lines, active circuitry,CCD delay lines, or entirely in the digital domain.

Digital signal processing allows the inexpensive construction of a wide variety of filters. The signal is sampled and ananalog-to-digital converter turns the signal into a stream of numbers. A computer program running on aCPU or a specializedDSP (or less often running on a hardware implementation of thealgorithm) calculates an output number stream. This output can be converted to a signal by passing it through adigital-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 oraliasing will occur.

In the late 1930s, engineers realized that small mechanical systems made of rigid materials such asquartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up to several hundred megahertz. Some early resonators were made ofsteel, but quartz quickly became favored. The biggest advantage of quartz is that it ispiezoelectric. 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.

A large number of crystals can be collapsed into a single component, by mounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tappeddelay 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 (surface acoustic wave) filters areelectromechanical devices commonly used inradio frequency applications. Electrical signals are converted to a mechanical wave in a device constructed of apiezoelectric crystal or ceramic; this wave is delayed as it propagates across the device, before being converted back to an electrical signal by furtherelectrodes. The delayed outputs are recombined to produce a direct analog implementation of afinite impulse response filter. This hybrid filtering technique is also found in ananalog sampled filter.SAW filters are limited to frequencies up to 3 GHz. The filters were developed by ProfessorTed Paige and others.^[2]

BAW (bulk acoustic wave) filters areelectromechanical devices. BAW filters can implement ladder or lattice filters. BAW filters typically operate at frequencies from around 2 to around 16 GHz, and may be smaller or thinner than equivalent SAW filters. Two main variants of BAW filters are making their way into devices:thin-film bulk acoustic resonator or FBAR and solid mounted bulk acoustic resonators (SMRs).

Another method of filtering, atmicrowave frequencies from 800 MHz to about 5 GHz, is to use a syntheticsingle crystalyttrium iron garnet sphere made of a chemical combination ofyttrium andiron (YIGF, or yttrium iron garnet filter). The garnet sits on a strip of metal driven by atransistor, and a small loopantenna touches the top of the sphere. Anelectromagnet 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 themagnetic field.

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

Thetransfer function of a filter is most often defined in the domain of the complex frequencies. The back and forth passage to/from this domain is operated by theLaplace transform and its inverse (therefore, here below, the term "input signal" shall be understood as "the Laplace transform of" the time representation of the input signal, and so on).

Thetransfer function${\displaystyle H(s)}$ of a filter is the ratio of the output signal${\displaystyle Y(s)}$ to the input signal${\displaystyle X(s)}$ as a function of the complex frequency${\displaystyle s}$:

${\displaystyle H(s)=\frac{Y(s)}{X(s)}}$

with${\displaystyle s=\sigma +j\omega }$.

For filters that are constructed of discrete components (lumped elements):

• Their transfer function will be the ratio of polynomials in${\displaystyle s}$, i.e. arational function of${\displaystyle s}$. The order of the transfer function will be the highest power of${\displaystyle s}$ encountered in either the numerator or the denominator polynomial.
• 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.

Distributed-element filters do not, in general, have rational-function transfer functions, but can approximate them.

The construction of a transfer function involves theLaplace transform, and therefore it is needed to assume null initial conditions, because

${\displaystyle \mathcal{L}\left\{\frac{df}{dt}\right\}=s\cdot \mathcal{L}\left\{f(t)\right\}-f(0),}$

And whenf(0) = 0 we can get rid of the constants and use the usual expression

${\displaystyle \mathcal{L}\left\{\frac{df}{dt}\right\}=s\cdot \mathcal{L}\left\{f(t)\right\}}$

An alternative to transfer functions is to give the behavior of the filter as aconvolution of the time-domain input with the filter'simpulse response. Theconvolution theorem, which holds for Laplace transforms, guarantees equivalence with transfer functions.

Certain filters may be specified by family and bandform. A filter's family is specified by the approximating polynomial used, and each leads to certain characteristics of the transfer function of the filter. Some common filter families and their particular characteristics are:

• Butterworth filter – no gainripple 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 – nogroup 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
• Raised-cosine filter

Each family of filters can be specified to a particular order. The higher the order, the more the filter will approach the "ideal" filter; but also the longer the impulse response is and the longer the latency will be. An ideal filter has full transmission in the pass band, complete attenuation in the stop band, and an abrupt transition between the two bands, but this filter has infinite order (i.e., the response cannot be expressed as alinear differential equation with a finite sum) and infinite latency (i.e., itscompact support in theFourier transform forces its time response to be ever lasting).

Here is an image comparing Butterworth, Chebyshev, and elliptic filters. The filters in this illustration are all fifth-order low-pass filters. The particular implementation – analog or digital, passive or active – makes no difference; their output would be the same. As is clear from the image, elliptic filters are sharper than the others, but they show ripples on the whole bandwidth.

Any family can be used to implement a particular bandform of which frequencies are transmitted, and which, outside the passband, are more or less attenuated. The transfer function completely specifies the behavior of a linear filter, but not the particular technology used to implement it. In other words, there are a number of different ways of achieving a particular transfer function when designing a circuit. A particular bandform of filter can be obtained bytransformation of aprototype filter of that family.

Impedance matching structures invariably take on the form of a filter, that is, a network of non-dissipative elements. For instance, in a passive electronics implementation, it would likely take the form of aladder topology of inductors and capacitors. The design of matching networks shares much in common with filters and the design invariably will have a filtering action as an incidental consequence. Although the prime purpose of a matching network is not to filter, it is often the case that both functions are combined in the same circuit. The need for impedance matching does not arise while signals are in the digital domain.

Similar comments can be made regardingpower dividers and directional couplers. When implemented in a distributed-element format, these devices can take the form of adistributed-element filter. There are four ports to be matched and widening the bandwidth requires filter-like structures to achieve this. The inverse is also true: distributed-element filters can take the form of coupled lines.^[3]

• Audio filter
• Line filter
• Scaled correlation, high-pass filter for correlations
• Texture filtering

• Wiener filter
• Kalman filter
• Savitzky–Golay smoothing filter

• Electronic filter topology
• Lifter (signal processing)
• Noise reduction
• Sallen–Key topology
• Smoothing
• Multiplier (Fourier analysis)

• Miroslav D. Lutovac, Dejan V. Tošić, Brian Lawrence Evans,Filter Design for Signal Processing Using MATLAB and Mathematica, Miroslav Lutovac, 2001ISBN 0201361302.
• B. A. Shenoi,Introduction to Digital Signal Processing and Filter Design, John Wiley & Sons, 2005ISBN 0471656380.
• L. D. Paarmann,Design and Analysis of Analog Filters: A Signal Processing Perspective, Springer, 2001ISBN 0792373731.
• J.S.Chitode,Digital Signal Processing, Technical Publications, 2009ISBN 8184316461.
• Leland B. Jackson,Digital Filters and Signal Processing, Springer, 1996ISBN 079239559X.

• Signal processing
• Telecommunication theory
• Filter theory

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