Filter (signal processing)

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, computer graphics, and structural 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:

has the best approximation to the ideal response of any filter for a specified order and ripple.

Chebyshev filter

has a maximally flat frequency response.

Butterworth filter

has a maximally flat phase delay.

Bessel filter

has the steepest cutoff of any filter for a specified order and ripple.

Elliptic filter

were originally entirely passive consisting of resistance, inductance and capacitance. Active technology makes design easier and opens up new possibilities in filter specifications.

Electronic 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.

Digital filters

are built out of mechanical components. In the vast majority of cases they are used to process an electronic signal and transducers 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.

Mechanical filters

are constructed out of components made from small pieces of transmission line or other distributed elements. There are structures in distributed-element filters that directly correspond to the lumped elements of electronic filters, and others that are unique to this class of technology.

Distributed-element 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 the stub, can also be implemented in waveguides.

Waveguide filters

were originally developed for purposes other than signal processing such as lighting and photography. With the rise of optical fiber technology, however, optical filters increasingly find signal processing applications and signal processing filter terminology, such as longpass and shortpass, are entering the field.

Optical filters

or delay line filter, works by summing copies of the input after various time delays. This can be implemented with various technologies including analog delay lines, active circuitry, CCD delay lines, or entirely in the digital domain.

Transversal filter

Their transfer function will be the ratio of polynomials in $s$ , i.e. a of $s$ . The order of the transfer function will be the highest power of $s$ encountered in either the numerator or the denominator polynomial.

rational function

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.

Impedance matching[edit]

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 a ladder 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 regarding power dividers and directional couplers. When implemented in a distributed-element format, these devices can take the form of a distributed-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.

Audio filter

Line filter

high-pass filter for correlations

Scaled correlation

Texture filtering

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, 2001 0201361302.

ISBN

B. A. Shenoi, Introduction to Digital Signal Processing and Filter Design, John Wiley & Sons, 2005 0471656380.

ISBN

L. D. Paarmann, Design and Analysis of Analog Filters: A Signal Processing Perspective, Springer, 2001 0792373731.

ISBN

J.S.Chitode, Digital Signal Processing, Technical Publications, 2009 8184316461.

ISBN

Leland B. Jackson, Digital Filters and Signal Processing, Springer, 1996 079239559X.