Editing Linear filter (section)

== Impulse response and transfer function ==
A [[LTI system theory|linear time-invariant]] (LTI) filter can be uniquely specified by its [[impulse response]] ''h'', and the output of any filter is mathematically expressed as the [[convolution]] of the input with that impulse response. The [[frequency response]], given by the filter's [[transfer function]] <math>H(\omega)</math>, is an alternative characterization of the filter. Typical filter design goals are to realize a particular frequency response, that is, the magnitude of the transfer function <math>|H(\omega)|</math>; the importance of the [[Phase (waves)|phase]] of the transfer function varies according to the application, inasmuch as the shape of a waveform can be distorted to a greater or lesser extent in the process of achieving a desired (amplitude) response in the frequency domain. The frequency response may be tailored to, for instance, eliminate unwanted frequency components from an input [[Signal (information theory)|signal]], or to limit an amplifier to signals within a particular band of frequencies.

The [[impulse response]] ''h'' of a linear time-invariant causal filter specifies the output that the filter would produce if it were to receive an input consisting of a single impulse at time 0. An "impulse" in a continuous time filter means a [[Dirac delta function]]; in a [[discrete time]] filter the [[Kronecker delta function]] would apply. The impulse response completely characterizes the response of any such filter, inasmuch as any possible input signal can be expressed as a (possibly infinite) combination of weighted delta functions. Multiplying the impulse response shifted in time according to the arrival of each of these delta functions by the amplitude of each delta function, and summing these responses together (according to the [[superposition principle]], applicable to all linear systems) yields the output waveform.

Mathematically this is described as the [[convolution]] of a time-varying input signal ''x(t)'' with the filter's [[impulse response]] ''h'', defined as:

:<math>y(t) =  \int_{0}^{T} x(t-\tau)\, h(\tau)\, d\tau</math> or
:<math>y_k =  \sum_{i=0}^{N} x_{k-i}\, h_i </math>.

The first form is the continuous-time form, which describes mechanical and analog electronic systems, for instance. The second equation is a discrete-time version used, for example, by digital filters implemented in software, so-called ''[[digital signal processing]]''. The impulse response ''h'' completely characterizes any linear time-invariant (or shift-invariant in the discrete-time case) filter. The input ''x'' is said to be "[[convolved]]" with the impulse response ''h'' having a (possibly infinite) duration of time ''T'' (or of ''N'' [[sampling period]]s).

Filter design consists of finding a possible transfer function that can be implemented within certain practical constraints dictated by the technology or desired complexity of the system, followed by a practical design that realizes that transfer function using the chosen technology. The complexity of a filter may be specified according to the [[degree of a polynomial|order]] of the filter.

Among the time-domain filters we here consider, there are two general classes of filter transfer functions that can approximate a desired frequency response. Very different mathematical treatments apply to the design of filters termed [[infinite impulse response]] (IIR) filters, characteristic of mechanical and analog electronics systems, and [[finite impulse response]] (FIR) filters, which can be implemented by [[discrete time]] systems such as computers (then termed ''[[digital signal processing]]'').

=== Implementation issues ===

Classical analog filters are IIR filters, and classical filter theory centers on the determination of transfer functions given by low order [[rational functions]], which can be synthesized using the same small number of reactive components.<ref>However, there are a few cases in which FIR filters directly process analog signals, involving non-feedback topologies and analog delay elements. An example is the discrete-time ''[[analog sampled filter]]'', implemented using a so-called [[bucket-brigade device]] clocked at a certain sampling rate, outputting copies of the input signal at different delays that can be combined with some weighting to realize an FIR filter. Electromechanical filters such as [[Electronic filter#SAW filters|SAW filters]] can likewise implement FIR filter responses; these operate in continuous time and can thus be designed for higher frequencies.</ref> Using digital computers, on the other hand, both FIR and IIR filters are straightforward to implement in software.

A digital IIR filter can generally approximate a desired filter response using less computing power than a FIR filter, however this advantage is more often unneeded given the increasing power of digital processors. The ease of designing and characterizing FIR filters makes them preferable to the filter designer (programmer) when ample computing power is available. Another advantage of  FIR filters is that their impulse response can be made symmetric, which implies a response in the frequency domain that has [[linear phase|zero phase at all frequencies]] (not considering a finite delay), which is absolutely impossible with any IIR filter.<ref>Outside of trivial cases, stable IIR filters with zero phase response are possible if they are not causal (and thus are unusable in real-time applications) or implementing transfer functions classified as unstable or "marginally stable" such as a [[double integrator]].</ref>