Definitions

# 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; 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:

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 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 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

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

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

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

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