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A linear filter applies a linear operator to a time-varying input signal. Linear filters are very common in electronics and digital signal processing (see the article on electronic filters), but they can also be found in mechanical engineering and other technologies. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
The field of electronics is the study and use of systems that operate by controlling the flow of electrons (or other charge carriers) in devices such as thermionic valves and semiconductors. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
Electronic filters are electronic circuits which perform signal processing functions. ...
The W16 engine from a Bugatti Veyron Mechanical engineering is a very broad field of engineering that involves the application of physical principles for analysis, design, manufacturing, and maintenance of mechanical systems. ...
They are often used to eliminate unwanted frequencies from an input signal or to select a wanted frequency amongst many others. There are a wide range of types of filter and filter technologies, of which this article will present an overview. Sine waves of various frequencies; the lower waves have higher frequencies than those above. ...
In information theory, a signal is the sequence of states of a communications channel that encodes a message. ...
Regardless of whether they are electronic, electrical, or mechanical, or what frequency ranges or timescales they work on, the mathematical theory of linear filters is universal.
Classification by transfer function
Impulse response Linear filters can be divided into two classes: infinite impulse response (IIR), and finite impulse response (FIR) filters. In general, a filter with a compact frequency response will have an infinite impulse response and a filter with a compact impulse response will have an infinite frequency response. Until recently, only analog IIR filters were practical to construct. However, technologies such as analog delay lines and digital filters have made the construction of FIR filters practical. IIR (Infinite impulse response) Filters have an impulse response function which is non-zero over an infinite length of time. ...
A finite impulse response (FIR) filter is a type of a digital filter. ...
An FIR filter In electronics, a digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. ...
Frequency response There are several common kinds of linear filters: - A low-pass filter passes low frequencies.
- A high-pass filter passes high frequencies.
- A band-pass filter passes a limited range of frequencies.
- A band-stop filter passes all frequencies except a limited range.
- An all-pass filter passes all frequencies, but alters the phase relationship among them.
- A notch filter is a specific type of band-stop filter that acts on a particularly narrow range of frequencies.
- some filters are not designed to stop any frequencies, but instead to gently vary the amplitude response at different frequencies: filters used as pre-emphasis filters, equalizers, or tone controls are good examples of this
Band-stop and band-pass filters can be constructed by combining low-pass and high-pass filters. A popular form of 2 pole filter is the Sallen-Key type. This is able to provide low-pass, band-pass, and high pass versions. A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. ...
A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. ...
The frequency axis of this symbolic diagram would be logarithmically scaled. ...
In electronics, a band-stop filter is a filter that attenuates, usually to very low levels, all frequencies between two non-zero, finite limits and passes all frequencies not within the limits. ...
A notch filter, also called a band-stop filter, sometimes a narrow band-pass filter, or T-notch filter, is an electronic filter typically used when the high frequency and the low frequency are less than 1 to 2 decades apart (that is, the high frequency is less than 10...
In audio processing, equalization (EQ) is the process of modifying the frequency envelope of a sound. ...
A Sallen and Key filter is a type of active filter, particularly valued for its simplicity. ...
Mathematics of filter design Linear filters of all types can be completely described by their frequency response and phase response, the specification of which uniquely defines their impulse response, and vice versa. From a mathematical viewpoint, continuous-time IIR filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation. Continuous-time filters can also be described in terms of the Laplace transform of their impulse response in a way which allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of their Laplace transform in the complex plane (and in can discrete time, one can similarly consider the Z-transform of the impulse response). Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...
Phase response is the relationship between the phase of a periodic input and an ouput signal passing through any device which accepts an input and produces an output signal such as an amplifier or a filter. ...
In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ...
Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...
In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...
In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...
Before the advent of computer filter synthesis tools, graphical tools such as Bode plots and Nyquist plots were extensively used as design tools. Even today, they are invaluable tools to understanding filter behavior. A Bode plot, named for Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show the transfer function or frequency response of an LTI...
A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...
Many different analog filter designs have been developed, each trying to optimise some feature of the system response. For practical filters, a custom design is sometimes desirable, that can offer the best tradeoff between different design criteria, which may include component count and cost, as well as filter response characteristics.
Some classic IIR filter types include the following:
These descriptions refer to the mathematical properties of the filter (that is, the frequency and phase response). These can be implemented as analog circuits (for instance, using a Sallen Key filter topology, a type of active filter), or as algorithms in digital signal processing systems. In electronics and signal processing, a Bessel filter is a variety of linear filter with a maximally flat group delay (linear phase response). ...
The Butterworth filter is one type of electronic filter designs. ...
The frequency response of a fourth-order type I Chebyshev low-pass filter Chebyshev filters, are analog or digital filters having a steeper roll-off and more passband ripple than Butterworth filters. ...
The frequency response of a fourth-order elliptic low-pass filter An elliptic filter (also known as a Cauer filter) is a filter with equiripple behavior in both the passband and the stopband. ...
The frequency response of a fourth-order elliptic low-pass filter An elliptic filter (also known as a Cauer filter) is a filter with equiripple behavior in both the passband and the stopband. ...
A Sallen and Key filter is a type of active filter, particularly valued for its simplicity. ...
Active filters are a special kind of electronic filters. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
Digital filters are much more flexible to synthesize and use than analog filters, where the constraints of the design permits their use. Notably, there is no need to consider component tolerances, and very high Q levels may be obtained.
FIR digital filters may be implemented by the direct convolution of the desired impulse response with the input signal. For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...
IIR digital filters are also easy to design. However, IIR digital filters do have their own mathematical design problems, in particular relating to dynamic range and roundoff nonlinearity problems.
See also The design of a frequency domain filter must begin by deciding what kind of filter is required for the task at hand. ...
In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...
In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...
In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...
Yo man This page is a candidate for speedy deletion. ...
In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ...
A non-linear filter is a signal-processing device whose output is not a linear function of its input. ...
The Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published [1]. // Description Unlike the typical filtering theory of designing a filter for a desired frequency response the Wiener filter approaches filtering from a different angle. ...
External links and references - Williams, Arthur B & Taylor, Fred J (1995). Electronic Filter Design Handbook. McGraw-Hill. ISBN 0-07-070441-4. The Bible for practical electronic filter design.
- National Semiconductor AN-779 application note describing analog filter theory
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