Low-pass filter - Citizendia

A low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency. Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted Frequency is a measure of the number of occurrences of a repeating event per unit Time. In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity In Physics, attenuation (in some context also called extinction) is the gradual loss in intensity of any kind of Flux through a medium Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications.

The concept of a low-pass filter exists in many different forms, including electronic circuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data, acoustic barriers, blurring of images, and so on. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Low-pass filters play the same role in signal processing that moving averages do in some other fields, such as finance; both tools provide a smoother form of a signal which removes the short-term oscillations, leaving only the long-term trend. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as In Statistics, a moving average, also called a rolling average and sometimes a running average, refers to a statistical technique used to analyze a

1 Examples of low pass filters
- 1.1 Acoustic
- 1.2 Electronic
2 Ideal and real filters
3 Electronic low-pass filters
4 Digital simulation
5 See also
6 External links

Examples of low pass filters

Figure 1: A low-pass electronic filter realized by an RC circuit

Figure 1 shows a low pass RC filter for voltage signals, discussed in more detail below. Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by A low-pass filter is a filter that passes low- Frequency signals but Attenuates (reduces the Amplitude of signals with frequencies Signal V_out retains unattenuated only frequencies below the cut-off frequency of the filter set by its RC time constant. In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency In Physics and Engineering, the time constant usually denoted by the Greek letter \tau, (tau characterizes the Frequency For current signals, a similar circuit using a resistor and capacitor in parallel works the same way. If two or more circuit components are connected end to end like a daisy chain it is said they are connected in series. See current divider. In Electronics, a current divider is a simple Linear Circuit that produces an output Current ( I X that is a fraction

Acoustic

A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated.

Electronic

Electronic low-pass filters are used to drive subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast. A subwoofer is a Woofer, or a complete Loudspeaker dedicated to the reproduction of bass audio frequencies, from perhaps 150 Hz down For the Marty Friedman album see Loudspeaker (album A loudspeaker, speaker, or speaker system is an electroacoustical

Radio transmitters use lowpass filters to block harmonic emissions which might cause interference with other communications. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer

An integrator is another example of a low-pass filter. An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in Calculus.

DSL splitters use low-pass and high-pass filters to separate DSL and POTS signals sharing the same pair of wires. A high-pass filter is a filter that passes high frequencies well but attenuates (reduces the amplitude of frequencies lower than the Cutoff frequency Twisted pair Cabling is a form of wiring in which two conductors (two halves of a single circuit) are wound together for the purposes of canceling out

Low-pass filters also play a significant role in the sculpting of sound for electronic music as created by analogue synthesisers. Electronic music is music that employs Electronic musical instruments and Electronic Music technology in its production See subtractive synthesis. Subtractive synthesis is a method of subtracting Harmonic content from a sound via Sound synthesis, characterised by the application of an Audio filter

Ideal and real filters

An ideal low-pass filter completely eliminates all frequencies above the cut-off frequency while passing those below unchanged. In Signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth leaves the low frequencies alone and has In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency The transition region present in practical filters does not exist in an ideal filter. An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with a sinc function in the time domain. The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized Boxcar function) In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes

However, the ideal filter is impossible to realize without also having signals of infinite extent, and so generally needs to be approximated for real ongoing signals, because the sinc function's support region extends to all past and future times. The filter would therefore need to have infinite delay, or knowledge of the infinite future and past, in order to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, but even that is not typically practical.

Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a finite impulse response; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future. See also Window function (SQL In Signal processing, a window function (also known as an apodization function or A finite impulse response (FIR filter is a type of a Digital filter. This delay is manifested as phase shift. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 Greater accuracy in approximation requires a longer delay.

The Whittaker–Shannon interpolation formula describes how to use a perfect low-pass filter to reconstruct a continuous signal from a sampled digital signal. The Whittaker–Shannon interpolation formula is a method to reconstruct a Continuous-time Bandlimited signal from a set of equally spaced samples A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain usually time The term digital signal is used to refer to more than one concept Real digital-to-analog converters use real filter approximations. In Electronics, a digital-to-analog converter ( DAC or D-to-A) is a device for converting a digital (usually binary code to an Analog signal

Electronic low-pass filters

The frequency response of a first-order filter

There are a great many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot. A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log

A first-order filter, for example, will reduce the signal amplitude by half (about –6 dB) every time the frequency doubles (goes up one octave). The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to In Music, an octave ( is the the use of which is "common in most musical systems The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. See RC circuit. A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by

A second-order filter does a better job of attenuating higher frequencies. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles (–12 dB per octave). The Butterworth filter is one type of Electronic filter design The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to Other second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of –12 dB per octave. For other uses of the terms Q and Q factor see Q value. In Physics and Engineering the quality See RLC circuit. An RLC circuit (also known as a Resonant circuit tuned circuit or LCR circuit is an Electrical circuit consisting of a Resistor (R an

Third- and higher-order filters are defined similarly. In general, the final rate of rolloff for an n-order filter is 6n dB per octave. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they will intersect at exactly the "cutoff frequency". An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity The frequency response at the cutoff frequency in a first-order filter is –3 dB below the horizontal line. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to The various types of filters — Butterworth filter, Chebyshev filter, Bessel filter, etc. The Butterworth filter is one type of Electronic filter design Chebyshev filters are analog or Digital filters having a steeper Roll-off and more Passband ripple (type I or In Electronics and Signal processing, a Bessel filter is a variety of Linear filter with a maximally flat Group delay (linear Phase response — all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. Electrical resonance occurs in an electric circuit at a particular resonance frequency when the impedance between the input and output of the See electronic filter for other types. Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted

The meanings of 'low' and 'high' — that is, the cutoff frequency — depend on the characteristics of the filter. In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency The term "low-pass filter" merely refers to the shape of the filter's response; a high-pass filter could be built that cuts off at a lower frequency than any low-pass filter – it is their responses that set them apart. Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1000 MHz) and higher.

Passive electronic realization

Passive, first order low-pass RC filter

One simple electrical circuit that will serve as a low-pass filter consists of a resistor in series with a load, and a capacitor in parallel with the load. An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | If an electric circuit has a well-defined output terminal the circuit connected to this terminal (or its Input impedance) is the load. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors The capacitor exhibits reactance, and blocks low-frequency signals, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The combination of resistance and capacitance gives you the time constant of the filter $τ = R C$ (represented by the Greek letter tau). In Physics and Engineering, the time constant usually denoted by the Greek letter \tau, (tau characterizes the Frequency Tau (uppercase Τ, lowercase τ; Ταυ) is the 19th letter of the Greek alphabet. The break frequency, also called the turnover frequency or cutoff frequency (in hertz), is determined by the time constant:

$f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}$

or equivalently (in radians per second):

$\omega_\mathrm{c} = {1 \over \tau} = { 1 \over R C}.$

One way to understand this circuit is to focus on the time the capacitor takes to charge. In Physics and Electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's Frequency The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 It takes time to charge or discharge the capacitor through that resistor:

At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.
At high frequencies, the capacitor only has time to charge up a small amount before the input switches direction. The output goes up and down only a small fraction of the amount the input goes up and down. At double the frequency, there's only time for it to charge up half the amount.

Another way to understand this circuit is with the idea of reactance at a particular frequency:

Since DC cannot flow through the capacitor, DC input must "flow out" the path marked $V out$ (analogous to removing the capacitor). Direct current ( DC) is the unidirectional flow of Electric charge.
Since AC flows very well through the capacitor — almost as well as it flows through solid wire — AC input "flows out" through the capacitor, effectively short circuiting to ground (analogous to replacing the capacitor with just a wire). An alternating current ( AC) is an Electric current whose direction reverses cyclically as opposed to Direct current, whose direction remains constant Short Circuit is a 1986 comedy Science fiction film starring Ally Sheedy and Steve Guttenberg and directed by

It should be noted that the capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the Bode plot and frequency response that show this variability. A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log Frequency response is the measure of any system's spectrum response at the output to a signal of varying Frequency (but constant amplitude at its input

Active electronic realization

An active low-pass filter

Another type of electrical circuit is an active low-pass filter.

In the operational amplifier circuit shown in the figure, the cutoff frequency (in hertz) is defined as:

$f_\mathrm{c} = {1 \over 2 \pi R_2 C }$

or equivalently (in radians per second):

$\omega_\mathrm{c} = \frac{1}{R_2 C}$

The gain in the passband is $\frac{-R_2}{R_1}$ , and the stopband drops off at −6 dB per octave, as it is a first-order filter. An operational amplifier, often called an op-amp, is a DC - coupled high- Gain electronic voltage amplifier with differential The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second.

Sometimes, a simple gain amplifier (as opposed to the very-high-gain operation amplifier) is turned into a low-pass filter by simply adding a feedback capacitor C. This feedback decreases the frequency response at high frequencies via the Miller effect, and helps to avoid oscillation in the amplifier. In Electronics, the Miller effect accounts for an increase in the equivalent input Capacitance of an inverting voltage Amplifier due to amplification of For example, an audio amplifier can be made into a low-pass filter with cutoff frequency 100 kHz to reduce gain at frequencies which would otherwise oscillate. Since the audio band (what we can hear) only goes up to 20 kHz or so, the frequencies of interest fall entirely in the passband, and the amplifier behaves the same way as far as audio is concerned. In brief the Passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated

Laplace notation

Continuous-time filters can also be described in terms of the Laplace transform of their impulse response in a way that allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of the Laplace transform in the complex plane (in discrete time, one can similarly consider the Z-transform of the impulse response). In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real

A first-order low-pass filter can be described in Laplace notation as

$\frac{\mathrm{Output}}{\mathrm{Input}} = \frac{1}{1 + s \tau}$

where s is the Laplace transform variable and τ is the filter time constant. In Physics and Engineering, the time constant usually denoted by the Greek letter \tau, (tau characterizes the Frequency

Digital simulation

For a more general algorithm to convert an analog filter into a digital filter, see Bilinear transform. The bilinear transform (also known as Tustin's method) is used in Digital signal processing and discrete-time Control theory to transform continuous-time

The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model. A discrete signal or discrete-time signal is a Time series, perhaps a signal that has been sampled from a continuous-time signal.

A simple low-pass RC filter

From the circuit diagram to the right, according to Kirchoff's Laws and the definition of capacitance:

V i n (t) - V o u t (t) = I (t) R

Q c (t) = C V o u t (t)

Taking the time derivative of the second equation, $I(t) = C \frac{dV_{out}}{dt}$ . A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by Capacitance is a measure of the amount of Electric charge stored (or separated for a given Electric potential. Combining this with the first equation:

$V_{in}(t) - V_{out}(t) = C \left[\frac{dV_{out}}{dt}\right] R$

Now we may discretize the equation. Let us represent $V i n$ by a series of samples $x 1. . . n$ . We will likewise represent $V o u t$ by a series of sample $y 1. . . n$ at the same points in time. For simplicity we assume that the samples are taken at evenly-spaced points in time separated by $Δ t$ . Making these substitutions:

$x_i - y_i = C \left[ \frac{y_{i}-y_{i-1}}{\Delta t} \right] R$

And rearranging terms:

$y_i = x_i \left( \frac{\Delta t}{RC + \Delta t} \right) + y_{i-1} \left( \frac{RC}{RC + \Delta t} \right)$

or more succinctly,

$y_n = \alpha x_n + (1 - \alpha) y_{n-1}\,$

where $\alpha = \frac{\Delta t}{RC + \Delta t}$

This gives us a way to determine the output samples in terms of the input samples and the preceding output. The following algorithm will simulate the effect of a low-pass filter on a series of digital samples:

 // Return RC low-pass filter output samples, given input samples,
 // time interval dt, and time constant RC
 function lowpass(real[0. . n] x, real dt, real RC)
   var real[0. . n] y
   var real alpha := dt / (RC + dt)
       y[0] := x[0]
   for i from 1 to n
       y[i] := alpha * x[i] + (1-alpha) * y[i-1]
   
   return y

Equivalently, more efficiently, and somewhat more intuitively (the change in filter output is proportional to the difference between the last output and the current input, which is the essence of exponential decay):

     for i from 1 to n
       y[i] := y[i-1] + alpha * (x[i] - y[i-1])

External links

In Signal processing, baseband is an adjective that describes signals and systems whose range of Frequencies is measured from zero to a maximum bandwidth In Electronics, Computer science and Mathematics, a digital filter is a system that performs mathematical operations on a sampling, A high-pass filter is a filter that passes high frequencies well but attenuates (reduces the amplitude of frequencies lower than the Cutoff frequency In Signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered but Attenuates A band-pass filter is a device that passes frequencies within a certain range and rejects ( Attenuates frequencies outside that range

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