A linear filter applies a linear operator to a time-varying input signal. Mathematically, it is a combination
of delays S and advances S-1.
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
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.
- 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).
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.
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