Linear filter
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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.
They are often used to eliminate unwanted frequencies from an input signal or to select a wanted frequency from amidst many others. There are a wide range of types of filter 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.
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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.
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.
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.
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.
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:
- Bessel filters
- Butterworth filters
- Chebyshev filters
- Cauer filters (elliptic filters)
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.
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
- Filter design
- Laplace transform
- Green's function
- Z-transform
- System theory
- Non-linear filter
- Wiener filter
External links and references
- Template:Book reference The Bible for practical electronic filter design.
- National Semiconductor AN-779 (http://www.national.com/an/AN/AN-779.pdf) application note describing analog filter theory