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| A '''low-pass filter''' is a [[Filter (signal processing)|filter]] that passes low-[[frequency]] [[signal (electrical engineering)|signals]] and [[attenuate]]s (reduces the [[amplitude]] of) signals with frequencies higher than the [[cutoff frequency]]. The actual amount of attenuation for each frequency varies depending on specific filter design. It is sometimes called a '''high-cut filter''', or '''treble cut filter''' in audio applications. A low-pass filter is the opposite of a [[high-pass filter]]. A [[band-pass filter]] is a combination of a low-pass and a high-pass.
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| Low-pass filters exist in many different forms, including electronic circuits (such as a ''hiss filter'' used in [[Sound recording|audio]]), [[anti-aliasing filter]]s for conditioning signals prior to [[analog-to-digital conversion]], [[digital filter]]s for smoothing sets of data, acoustic barriers, blurring of images, and so on. The [[moving average (finance)|moving average]] operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same [[signal processing]] techniques as are used for other low-pass filters. Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations, and leaving the longer-term trend.
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| An [[optical filter]] can correctly be called a low-pass filter, but conventionally is called a ''longpass'' filter (low frequency is long wavelength), to avoid confusion.
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| == Examples of low-pass filters==
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| ===Acoustic===
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| 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.
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| ===Electronic===
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| In an electronic low-pass [[RC filter]] for voltage signals, high frequencies in the input signal are attenuated, but the filter has little attenuation below the [[cutoff frequency]] determined by its [[RC time constant]]. For current signals, a similar circuit, using a resistor and capacitor in [[Parallel circuits#Parallel circuits|parallel]], works in a similar manner. (See [[current divider]] discussed in more detail [[#Electronic low-pass filters|below]].)
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| Electronic low-pass filters are used on inputs to [[subwoofer]]s and other types of [[loudspeaker]]s, to block high pitches that they can't efficiently reproduce. Radio transmitters use low-pass filters to block [[harmonic]] emissions that might interfere with other communications. The tone knob on many [[electric guitar]]s is a low-pass filter used to reduce the amount of treble in the sound. An [[integrator]] is another [[Time constant#Time constants in electrical circuits|time constant]] low-pass filter.<ref>{{cite book
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| | title = Microelectronic Circuits, 3 ed.
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| | page = 60
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| | first1 = Adel
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| | last1 = Sedra
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| | authorlink1 = Adel Sedra
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| | first2 = Kenneth C.
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| | last2 = Smith
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| | publisher = Saunders College Publishing
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| | year = 1991
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| | isbn = 0-03-051648-X
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| }}</ref>
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| Telephone lines fitted with [[DSL splitter]]s use low-pass and [[high-pass filter|high-pass]] filters to separate [[Digital Subscriber Line|DSL]] and [[Plain old telephone service|POTS]] signals sharing the same [[twisted pair|pair]] of wires. <ref>{{cite web|url=http://www.epanorama.net/documents/telecom/adsl_filter.html |title=ADSL filters explained |publisher=Epanorama.net |date= |accessdate=2013-09-24}}</ref><ref>{{cite web|url=http://www.pcweenie.com/hni/broadband/broad6.shtml |title=Home Networking - Local Area Network |publisher=Pcweenie.com |date=2009-04-12 |accessdate=2013-09-24}}</ref>
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| Low-pass filters also play a significant role in the sculpting of sound created by analogue and virtual analogue [[synthesiser]]s. ''See [[subtractive synthesis]].''
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| ==Ideal and real filters==
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| [[File:Sinc function (normalized).svg|thumb|The [[sinc function]], the [[impulse response]] of an ideal low-pass filter.]]
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| An [[sinc filter|ideal low-pass filter]] completely eliminates all frequencies above the [[cutoff frequency]] while passing those below unchanged; its [[frequency response]] is a [[rectangular function]] and is a [[brick-wall filter]]. 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 its [[impulse response]], a [[sinc function]], in the time domain.
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| However, the ideal filter is impossible to realize without also having signals of infinite extent in time, 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, or more typically by making the signal repetitive and using Fourier analysis.
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| Real filters for [[Real-time computing|real-time]] applications approximate the ideal filter by truncating and [[window function|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. This delay is manifested as [[Phase (waves)|phase shift]]. Greater accuracy in approximation requires a longer delay.
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| An ideal low-pass filter results in [[ringing artifacts]] via the [[Gibbs phenomenon]]. These can be reduced or worsened by choice of windowing function, and the [[Window function#Filter design|design and choice of real filters]] involves understanding and minimizing these artifacts. For example, "simple truncation [of sinc] causes severe ringing artifacts," in signal reconstruction, and to reduce these artifacts one uses window functions "which drop off more smoothly at the edges."<ref>[http://www.cg.tuwien.ac.at/research/vis/vismed/Windows/MasteringWindows.pdf Mastering Windows: Improving Reconstruction]</ref>
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| The [[Whittaker–Shannon interpolation formula]] describes how to use a perfect low-pass filter to reconstruct a [[continuous signal]] from a sampled [[digital signal]]. Real [[digital-to-analog converter]]s use real filter approximations. | |
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| == Continuous-time low-pass filters ==
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| [[File:Butterworth response.svg|thumb|350px|The gain-magnitude frequency response of a first-order (one-pole) low-pass filter. ''Power gain'' is shown in decibels (i.e., a 3 [[Decibel|dB]] decline reflects an additional half-power attenuation). [[Angular frequency]] is shown on a logarithmic scale in units of radians per second.]]
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| There are 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]], and the filter is characterized by its [[cutoff frequency]] and rate of frequency [[roll-off|rolloff]]. In all cases, at the ''cutoff frequency,'' the filter [[attenuate]]s the input power by half or 3 dB. So the '''order''' of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.
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| * A '''first-order filter''', for example, reduces the signal amplitude by half (so power reduces by a factor of 4), or {{nowrap|6 dB}}, every time the frequency doubles (goes up one [[octave]]); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency. 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. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the [[transfer function]] of a first-order low-pass filter has a [[zero (complex analysis)|zero]] as well as a [[pole (complex analysis)|pole]], the Bode plot flattens out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this one-pole–one-zero filter is still a first-order low-pass. ''See [[Pole–zero plot]] and [[RC circuit]].''
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| * A '''second-order filter''' attenuates higher frequencies more steeply. 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]] reduces the signal amplitude to one fourth its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole 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; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See [[RLC circuit]].
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| * Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-<math>\scriptstyle n</math> all-pole filter is <math>\scriptstyle 6n</math> dB per octave (i.e., <math>\scriptstyle 20n</math> dB per decade).
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| On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the [[asymptote]]s of the function), they intersect at exactly the ''cutoff frequency''. The frequency response at the cutoff frequency in a first-order filter is 3 dB below the horizontal line. The various types of filters ([[Butterworth filter]], [[Chebyshev filter]], [[Bessel filter]], etc.) all have different-looking ''knee curves''. Many second-order filters have "peaking" or [[Electrical resonance|resonance]] that puts their frequency response at the cutoff frequency ''above'' the horizontal line. Furthermore, the actual frequency where this peaking occurs can be predicted without calculus, as shown by Cartwright<ref>K. V. Cartwright, P. Russell and E. J. Kaminsky,"[http://www.journal.lapen.org.mx/dec2012/8_LAJPE_703_Kenneth_Cartwright_preprint_corr_f.pdf Finding the maximum magnitude response (gain) of second-order filters without calculus]," Lat. Am. J. Phys. Educ. Vol. 6, No. 4, pp. 559-565, 2012.</ref> et al. ''See [[electronic filter]] for other types.''
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| The meanings of 'low' and 'high'—that is, the [[cutoff frequency]]—depend on the characteristics of the filter. 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 1 GHz) and higher.
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| ===Laplace notation===
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| Continuous-time filters can also be described in terms of the [[Laplace transform]] of their [[impulse response]], in a way that lets all characteristics of the filter 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.)
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| For example, a first-order low-pass filter can be described in Laplace notation as:
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| :<math>
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| \frac{\text{Output}}{\text{Input}} = K \frac{1}{1 + s \tau}
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| </math>
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| where ''s'' is the Laplace transform variable, ''τ'' is the filter [[time constant]], and ''K'' is the filter [[passband]] [[gain]].
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| == Electronic low-pass filters ==
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| ===Passive electronic realization===
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| [[File:RC Divider.svg|thumb|200px|Passive, first order low-pass RC filter]]
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| One simple low-pass filter [[electrical circuit|circuit]] consists of a [[resistor]] in series with a [[External electric load|load]], and a [[capacitor]] in parallel with the load. The capacitor exhibits [[Reactance (electronics)|reactance]], and blocks low-frequency signals, forcing them 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 the [[time constant]] of the filter <math>\scriptstyle \tau \;=\; RC </math> (represented by the Greek letter [[tau]]). The break frequency, also called the turnover frequency or [[cutoff frequency]] (in hertz), is determined by the time constant:
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| :<math>
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| f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}
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| </math>
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| or equivalently (in [[radian]]s per second):
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| :<math>
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| \omega_\mathrm{c} = {1 \over \tau} = {1 \over R C}
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| </math>
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| One way to understand this circuit is to focus on the time the capacitor takes to charge. It takes time to charge or discharge the capacitor through that resistor:
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| * At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.
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| * 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.
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| Another way to understand this circuit is with the idea of [[Reactance (electronics)|reactance]] at a particular frequency:
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| * Since [[Direct current|DC]] cannot flow through the capacitor, DC input must "flow out" the path marked <math>\scriptstyle V_\mathrm{out}</math> (analogous to removing the capacitor).
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| * Since [[Alternating current|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 circuit]]ing to ground (analogous to replacing the capacitor with just a wire).
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| The capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor variably acts between these two extremes. It is the [[Bode plot]] and [[frequency response]] that show this variability.
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| ===Active electronic realization===
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| [[File:Active Lowpass Filter RC.svg|thumb|right|300px|An active low-pass filter]]
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| Another type of electrical circuit is an ''active'' low-pass filter.
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| In the [[operational amplifier]] circuit shown in the figure, the cutoff frequency (in [[hertz]]) is defined as:
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| :<math>f_{\text{c}} = \frac{1}{2 \pi R_2 C}</math>
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| or equivalently (in radians per second):
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| :<math>\omega_{\text{c}} = \frac{1}{R_2 C}</math>
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| The gain in the passband is −''R''<sub>2</sub>/''R''<sub>1</sub>, and the [[stopband]] drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.
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| === Discrete-time realization ===
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| {{For|another method of conversion from continuous- to discrete-time|Bilinear transform}}
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| Many [[digital filter]]s are designed to give low-pass characteristics. Both [[infinite impulse response]] and [[finite impulse response]] low pass filters as well as filters using [[fourier transform]]s are widely used.
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| ====Simple infinite impulse response filter====
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| The effect of an infinite impulse response low-pass filter can be simulated on a computer by analyzing an RC filter's behavior in the time domain, and then [[discrete signal|discretizing]] the model.
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| [[File:1st Order Lowpass Filter RC.svg|right|framed|A simple low-pass [[RC circuit|RC filter]]]]
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| From the circuit diagram to the right, according to [[Kirchhoff's circuit laws|Kirchhoff's Laws]] and the definition of [[capacitance]]:
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| :{{NumBlk|:|<math>v_{\text{in}}(t) - v_{\text{out}}(t) = R \; i(t)</math>|{{EquationRef|V}}}}
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| :{{NumBlk|:|<math>Q_c(t) = C \, v_{\text{out}}(t)</math>|{{EquationRef|Q}}}}
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| :{{NumBlk|:|<math>i(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t}</math>|{{EquationRef|I}}}}
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| where <math>Q_c(t)</math> is the charge stored in the capacitor at time <math>\scriptstyle t</math>. Substituting equation {{EquationNote|Q}} into equation {{EquationNote|I}} gives <math>\scriptstyle i(t) \;=\; C \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}</math>, which can be substituted into equation {{EquationNote|V}} so that:
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| :<math>v_{\text{in}}(t) - v_{\text{out}}(t) = RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}</math>
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| This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by <math>\scriptstyle \Delta_T</math> time. Let the samples of <math>\scriptstyle v_{\text{in}}</math> be represented by the sequence <math>\scriptstyle (x_1,\, x_2,\, \ldots,\, x_n)</math>, and let <math>\scriptstyle v_{\text{out}}</math> be represented by the sequence <math>\scriptstyle (y_1,\, y_2,\, \ldots,\, y_n)</math>, which correspond to the same points in time. <!--
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| :Replace <math>V_{\text{in}(t)</math> with <math>x_i</math>
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| :Replace <math>V_{\text{out}}(t)</math> with <math>y_i</math>
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| :Replace <math>\operatorname{d}V_{\text{out}}</math> with <math>y_{i} - y_{i-1}</math>
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| :Replace <math>\operatorname{d}t</math> with <math>\Delta_T</math>
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| --> Making these substitutions:
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| :<math>x_i - y_i = RC \, \frac{y_{i}-y_{i-1}}{\Delta_T}</math>
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| And rearranging terms gives the [[recurrence relation]]
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| :<math>y_i = \overbrace{x_i \left( \frac{\Delta_T}{RC + \Delta_T} \right)}^{\text{Input contribution}} + \overbrace{y_{i-1} \left( \frac{RC}{RC + \Delta_T} \right)}^{\text{Inertia from previous output}}.</math>
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| That is, this discrete-time implementation of a simple RC low-pass filter is the [[Exponential smoothing|exponentially-weighted moving average]]
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| :<math>y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha \triangleq \frac{\Delta_T}{RC + \Delta_T}</math>
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| By definition, the ''smoothing factor'' <math>\scriptstyle 0 \;\leq\; \alpha \;\leq\; 1</math>. The expression for <math>\scriptstyle \alpha</math> yields the equivalent [[time constant]] <math>\scriptstyle RC</math> in terms of the sampling period <math>\scriptstyle \Delta_T</math> and smoothing factor <math>\scriptstyle \alpha</math>:
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| :<math>RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right)</math>
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| If <math>\scriptstyle \alpha \;=\; 0.5</math>, then the <math>\scriptstyle RC</math> time constant is equal to the sampling period. If <math>\scriptstyle \alpha \;\ll\; 0.5</math>, then <math>\scriptstyle RC</math> is significantly larger than the sampling interval, and <math>\scriptstyle \Delta_T \;\approx\; \alpha RC</math>.
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| The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following [[pseudocode]] algorithm simulates the effect of a low-pass filter on a series of digital samples:
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| // Return RC low-pass filter output samples, given input samples,
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| // time interval ''dt'', and time constant ''RC''
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| '''function''' lowpass(''real[0..n]'' x, ''real'' dt, ''real'' RC)
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| '''var''' ''real[0..n]'' y
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| '''var''' ''real'' α := dt / (RC + dt)
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| y[0] := x[0]
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| '''for''' i '''from''' 1 '''to''' n
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| y[i] := α * x[i] + (1-α) * y[i-1]
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| '''return''' y
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| The [[programming loop|loop]] that calculates each of the ''n'' outputs can be [[code refactoring|refactored]] into the equivalent:
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| '''for''' i '''from''' 1 '''to''' n
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| y[i] := y[i-1] + α * (x[i] - y[i-1])
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| That is, the change from one filter output to the next is [[proportionality (mathematics)|proportional]] to the difference between the previous output and the next input. This [[exponential smoothing]] property matches the [[exponential function|exponential]] decay seen in the continuous-time system. As expected, as the [[time constant]] <math>\scriptstyle RC</math> increases, the discrete-time smoothing parameter <math>\scriptstyle \alpha</math> decreases, and the output samples <math>\scriptstyle (y_1,\, y_2,\, \ldots,\, y_n)</math> respond more slowly to a change in the input samples <math>\scriptstyle (x_1,\, x_2,\, \ldots,\, x_n)</math>; the system has more ''[[inertia]]''. This filter is an [[infinite-impulse-response]] (IIR) single-pole lowpass filter.
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| ====Finite impulse response====
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| Finite impulse response filters can be built that approximate to the ideal sinc time domain response. In practice the time domain response must be time truncated and is often of a simplified shape; in the simplest case, a [[running average]] can be used giving a square time response.<ref>Signal recovery from noise in electronic instrumentation - T H Whilmshurst</ref>
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| ==See also==
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| {{Portal|Electronics}}
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| * [[Baseband]]
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| * [[DSL filter]]
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| ==References==
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| {{Reflist}}
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| == External links ==
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| * [http://www.allaboutcircuits.com/vol_2/chpt_8/2.html Low-pass filter]
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| * [http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/experiment/lowpass/lpf.html Low Pass Filter java simulator]
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| * [http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems], a short primer on the mathematical analysis of (electrical) LTI systems.
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| * [http://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf ECE 209: Sources of Phase Shift], an intuitive explanation of the source of phase shift in a low-pass filter. Also verifies simple passive LPF [[transfer function]] by means of trigonometric identity.
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| {{Electronic filters}}
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| {{DEFAULTSORT:Low-Pass Filter}}
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| [[Category:Signal processing]]
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| [[Category:Linear filters]]
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| [[Category:Synthesiser modules]]
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| [[Category:Filter frequency response]]
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