Filters

**Electronic filters** are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an integrated circuit.

Electronic filters remove unwanted frequency components from the applied signal, enhance wanted ones, or both. They can be:

* passive or active
* analog or digital
* high-pass, low-pass, band-pass, band-stop (band-rejection; notch), or all-pass.
* discrete-time (sampled) or continuous-time
* linear or non-linear
* infinite impulse response (IIR type) or finite impulse response (FIR type)

The most common types of electronic filters are linear filters, regardless of other aspects of their design. See the article on linear filters for details on their design and analysis.

## Analogue filter

**Analogue filters** are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass, mid-range, and tweeter loudspeakers; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen radio station in a radio receiver and rejection of others.

Passive linear electronic analogue filters are those filters which can be described with linear differential equations (linear); they are composed of capacitors, inductors and, sometimes, resistors (passive) and are designed to operate on continuously varying (analogue) signals. There are many linear filters which are not analogue in implementation (digital filter), and there are many electronic filters which may not have a passive topology – both of which may have the same transfer function of the filters described in this article. Analogue filters are most often used in wave filtering applications, that is, where it is required to pass particular frequency components and to reject others from analogue (continuous-time) signals.

Analogue filters have played an important part in the development of electronics. Especially in the field of telecommunications, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with transmission lines. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of network synthesis techniques greatly enhanced the degree of control of the designer.

Today, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue filters are now implemented in a filter topology which is active in order to avoid the wound components (i.e. inductors, transformers, etc.) required by passive topology.

It is possible to design linear analogue mechanical filters using mechanical components which filter mechanical vibrations or acoustic waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of transducers to convert to and from the electrical domain. Indeed, some of the earliest ideas for filters were acoustic resonators because the electronics technology was poorly understood at the time. In principle, the design of such filters can be achieved entirely in terms of the electronic counterparts of mechanical quantities, with kinetic energy, potential energy and heat energy corresponding to the energy in inductors, capacitors and resistors respectively.

## Infinite impulse response

**Infinite impulse response** (**IIR**) is a property applying to many linear time-invariant systems. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as *IIR systems* or *IIR filters*, and are distinguished by having an impulse response which does not become exactly zero past a certain point, but continues indefinitely. This is in contrast to a finite impulse response (FIR) in which the impulse response *h*(*t*) *does* become exactly zero at times *t* > *T* for some finite *T*, thus being of finite duration.

In practice, the impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction. For instance, analog electronic filters composed of resistors, capacitors, and/or inductors (and perhaps linear amplifiers) are generally IIR filters. The capacitors (or inductors) in the analog filter have a "memory" and their internal state never completely relaxes following an impulse (assuming the classical model of capacitors and inductors where quantum effects are ignored).

On the other hand, discrete-time filters (usually digital filters) based on a tapped delay line *employing no feedback* are necessarily FIR filters. After an impulse has reached the end of the tapped delay line, the system has no further memory of that impulse and has returned to its initial state; its impulse response beyond that point is exactly zero.
## Linear phase

**Linear phase** is a property of a filter, where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount (the slope of the linear function), which is referred to as the group delay. And consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR) filter by having coefficients which are symmetric or anti-symmetric.  Approximations can be achieved with infinite impulse response (IIR) designs, which are more computationally efficient.  Several techniques are**:**

* a Bessel transfer function which has a maximally flat group delay approximation function
* a phase equalizer

A filter is called a linear phase filter if the phase component of the frequency response is a linear function of frequency. For a continuous-time application, the frequency response of the filter is the Fourier transform of the filter's impulse response, and a linear phase version has the form:

H
(
ω
# )
A
(
ω
)

e

−
j
ω
τ

,

{\displaystyle H(\omega )=A(\omega )\ e^{-j\omega \tau },}

where:

* A(ω) is a real-valued function.
* τ

{\displaystyle \tau }

 is the group delay.

For a discrete-time application, the discrete-time Fourier transform of the linear phase impulse response has the form:

H

2
π

(
ω
# )
A
(
ω
)

e

−
j
ω
k

/

2

,

{\displaystyle H\_{2\pi }(\omega )=A(\omega )\ e^{-j\omega k/2},}

where:

* A(ω) is a real-valued function with 2π periodicity.
* k is an integer, and k/2 is the group delay in units of samples.

H

2
π

(
ω
)

{\displaystyle H\_{2\pi }(\omega )}

 is a Fourier series that can also be expressed in terms of the Z-transform of the filter impulse response. I.e.:

H

2
π

(
ω
# )

H
^

(
z
)

# z

e

j
ω

=

H
^

(

e

j
ω

)
,

{\displaystyle H\_{2\pi }(\omega )=\left.{\widehat {H}}(z)\,\right|\_{z=e^{j\omega }}={\widehat {H}}(e^{j\omega }),}

where the

H
^

{\displaystyle {\widehat {H}}}

 notation distinguishes the Z-transform from the Fourier transform.

## Causal filter

In signal processing, a **causal filter** is a linear and time-invariant causal system. The word *causal* indicates that the filter output depends only on past and present inputs. A filter whose output also depends on future inputs is **non-causal**, whereas a filter whose output depends *only* on future inputs is **anti-causal**. Systems (including filters) that are *realizable* (i.e. that operate in real time) must be causal because such systems cannot act on a future input. In effect that means the output sample that best represents the input at time

t
,

{\displaystyle t,}

 comes out slightly later. A common design practice for digital filters is to create a realizable filter by shortening and/or time-shifting a non-causal impulse response. If shortening is necessary, it is often accomplished as the product of the impulse-response with a window function.

An example of an anti-causal filter is a maximum phase filter, which can be defined as a stable, anti-causal filter whose inverse is also stable and anti-causal.