Linear filter

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.

They are often used to eliminate unwanted frequencies from an input signal or to select a desired frequency among many others. There are a wide range of types of filters and filter technologies, of which this article will present an overview.

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.

An FIR filter may be described as a weighted sum of delayed inputs. For such a filter, if the input becomes zero at any time, then the output will eventually become zero as well, as soon as enough time has passed so that all the delayed inputs are zero, too. Therefore the impulse response lasts only a finite time, hence the name "finite impulse response". The transfer function of such a filter contains only zeros, and no poles.

In an IIR filter, by contrast, if the input is set to 0, then the output will decay exponentially, but never become precisely zero. Therefore the impulse response extends to infinity, and the filter is said to have an "Infinite Impulse Response". The transfer function of such a filter will contain poles as well as zeros.

Until about the 1970s, only analog IIR filters were practical to construct. However, in digital logic, both FIR and IIR filters are straightforward, and both techniques are commonly used in digital filters. Analog FIR filters are still uncommon, though they can be built with analog delay lines.

Frequency response

Here is an image comparing the most popular IIR filters: 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 little difference [A sampled data linear filter, such as most digital filters, will have a slightly different response since it is described by the Z transform, not the Laplace transform. The frequency response will repeat if the frequency becomes high enough, and the curves will be slightly different, but the main features are unchanged.] ; the output from any implementation would still match this image.

As is clear from the image, elliptic filters are sharper than the others, but they show ripples on the whole bandwidth.

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 particular bandform of filter can be obtained by transformation of a prototype filter of that class.

Mathematics of filter design

LTI system theory describes linear "time-invariant" (LTI) filters of all types. LTI filters 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 LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation. Continuous-time LTI filters may also be described in terms of the Laplace transform of their impulse response, which allows all of the characteristics of the filter to be analyzed by considering the pattern of poles and zeros of their Laplace transform in the complex plane. Similarly, discrete-time LTI filters may be analyzed via the Z-transform of their impulse response.

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. Reference books [A. Zverev, "Handbook of Filter Synthesis", John Wiley and Sons, 1967, ISBN 0-471-98680-1] had extensive plots of frequency response, phase response, group delay, and impulse response for various types of filters, of various orders. They also contained tables of values showing how to implement such filters as RLC ladders - very useful when amplifying elements were expensive compared to passive components. Such a ladder can also be designed to have minimal sensitivity to component variation [Normally, computing sensitivities is a very laborious operation. But in the special case of an LC ladder driven by an impedance and terminated by a resistor, there is a neat argument showing the sensitivities are small. In such as case, the transmission at the maximum frequency(s) transfers the maximal possible energy to the output load, as determined by the physics of the source and load impedances. Since this point is a maximum, "all" derivatives with respect to "all" component values must be zero, since the result of changing "any" component value in "any" direction can only result in a reduction. This result only strictly holds true at the peaks of the response, but is roughly true at nearby points as well.] a property hard to evaluate without computer tools.

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.

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.

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.They can easily be designed to give a matched filter for any arbitrary pulse shape.

IIR digital filters are often more difficult to design, due to problems including dynamic range issues, quantization noise and instability.Typically digital IIR filters are designed as a series of digital biquad filters.

All low-pass second-order continuous-time filters have a transfer function given by

: $H\left(s\right)=frac\left\{K omega^\left\{2\right\}_\left\{0\left\{s^\left\{2\right\}+frac\left\{omega_\left\{0\left\{Q\right\}s+omega^\left\{2\right\}_\left\{0.$

All band-pass second-order continuous-time have a transfer function given by

: $H\left(s\right)=frac\left\{K frac\left\{omega_\left\{0\left\{Q\right\}s\right\}\left\{s^\left\{2\right\}+frac\left\{omega_\left\{0\left\{Q\right\}s+omega^\left\{2\right\}_\left\{0.$

where
* "K" is the gain (low-pass DC gain, or band-pass mid-band gain) ("K" is 1 for passive filters)
* "Q" is the Q factor
* $omega_\left\{0\right\}$ is the center frequency
* $s=sigma+jomega$ is the complex frequency

ee also

* Filter design
* Laplace transform
* Green's function
* Prototype filter
* Z-transform
* System theory
** LTI system theory
* Nonlinear filter
* Wiener filter

*

* [http://www.national.com/an/AN/AN-779.pdf National Semiconductor AN-779] application note describing analog filter theory
* [http://www.latticesemi.com/lit/docs/appnotes/pac/an6017.pdf Lattice AN6017] application note comparing and contrasting filters (in order of damping coefficient, from lower to higher values): Gaussian, Bessel, linear phase, Butterworth, Chebyshev, Legendre, elliptic. (with graphs).
* [http://books.google.com/books?id=l7oC-LJwyegC&pg=PA267&lpg=PA267&dq=%22legendre+filter%22&source=web&ots=xRLtCLfslz&sig=0Nw2zhb8Y7FSrylN3wDaoIMkekQ#PPA238,M1 "Design and Analysis of Analog Filters: A Signal Processing Perspective"] by L. D. Paarmann

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