Linear filter: Information from Answers.com

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.

1 Classification by transfer function
- 1.1 Impulse response
- 1.2 Frequency response
2 Mathematics of filter design
3 See also
4 Notes and references
5 Further reading

Classification by transfer function

Impulse response

Linear filters can be divided into two classes: infinite impulse response (IIR) and finite impulse response (FIR) filters.

An FIR filter (which may only be implemented in discrete time) 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, and this is the reason for the name finite impulse response. A discrete-time transfer function of such a filter contains only poles at the origin (i.e., delays) and zeros; it cannot have off-origin poles.^{[citation needed]}

For an IIR filter, by contrast, if the input is set to 0 and the initial conditions are non-zero, then the set of time where the output is non-zero will be unbounded; the filter's energy will decay but will be ever present. Therefore, the impulse response extends to infinity, and the filter is said to have an infinite impulse response. There are no special restrictions on the transfer function of an IIR filter; it can have arbitrary poles and zeros, and it need not be expressible as a rational transfer function (for example, a sinc filter).^{[citation needed]}

Until about the 1970s, only analog IIR filters were practical to construct. The distinction between FIR and IIR filters is generally applied only in the discrete-time domain. Because digital systems necessarily have discrete-time domains, both FIR and IIR filters are straightforward to implement digitally. Analog FIR filters can be built with analog delay lines.^{[citation needed]}

Frequency response

Here is an image comparing the Fourier transform (i.e., "frequency response") of several popular continuous-time IIR filters: Butterworth, Chebyshev, and elliptic filters. The filters in this illustration are all fifth-order low-pass filters.

As is clear from the image, elliptic filters are sharper than the others, but they show ripples in their passband.

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 bandpass 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

Linear analog electronic filters
Network synthesis filters Butterworth filter Chebyshev filter Elliptic (Cauer) filter Bessel filter Gaussian filter Optimum "L" (Legendre) filter Linkwitz-Riley filter Image impedance filters Constant k filter m-derived filter General image filters Zobel network (constant R) filter Lattice filter (all-pass) Bridged T delay equaliser (all-pass) Composite image filter mm'-type filter Simple filters RL filter RC filter RLC filter LC filter
edit

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^[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^[2] 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

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

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

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

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
$ω 0$ is the center frequency
$s = σ + j ω$ is the complex frequency

Notes and references

^ A. Zverev, Handbook of Filter Synthesis, John Wiley and Sons, 1967, ISBN 0-471-98680-1
^ 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.

	Which is better in terms of sound a Sony CD changer that has 4 or 8 times oversampling filter or one that has high density linear converter? Read answer...
	What is linear law? Read answer...
	What is linear mime? Read answer...

	Second order linear time invariant system acts as which filter?
	Design linear phase low pass filter with cut off frequency 0.5pi using frequency sampling technique and M equals 17?
	What are linear equations and linear functions?

Linear filter

Contents

Classification by transfer function

Impulse response

Frequency response

Mathematics of filter design

See also

Notes and references

Further reading