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{{short description|Type of signal filter}}
{{more citations needed|date=May 2023}}
A '''low-pass filter'''
In optics, '''high-pass''' and '''low-pass''' may have different meanings, depending on whether referring to the frequency or wavelength of light, since these variables are inversely related. High-pass frequency filters would act as low-pass wavelength filters, and vice versa. For this reason, it is a good practice to refer to wavelength filters as
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
Filter designers will often use the low-pass form as a [[prototype filter]]. That is
==Examples==
Examples of low-pass filters occur in [[acoustics]], [[optics]] and [[electronics]].
A stiff physical barrier tends to reflect higher sound frequencies,
An [[optical filter]] with the same function can correctly be called a low-pass filter, but conventionally is called a ''longpass'' filter (low frequency is long wavelength), to avoid confusion.<ref
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]].)
Electronic low-pass filters are used on inputs to [[subwoofer]]s and other types of [[loudspeaker]]s, to block high pitches that they
|page = [https://archive.org/details/microelectronicc00sedr_0/page/60 60]
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}}</ref>
Telephone lines fitted with [[DSL splitter]]s use low-pass
Low-pass filters also play a significant role in the sculpting of sound created by analogue and virtual analogue [[synthesiser]]s. ''See [[subtractive synthesis]].''
A low-pass filter is used as an [[anti-aliasing filter]]
==Ideal and real filters==
[[File:Sinc function (normalized).svg|thumb|The [[sinc function]], the time-domain [[impulse response]] of an ideal low-pass filter. The ripples of a true sinc extend infinitely to the left and right while getting smaller and smaller, but this particular graph is truncated.]]
[[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.]]
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.
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,
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.
The [[Whittaker–Shannon interpolation formula]] describes how to use a perfect low-pass filter to reconstruct a [[continuous signal]] from a sampled [[Digital signal (signal processing)|digital signal]]. Real [[digital-to-analog converter]]s
==Time response==
The time response of a low-pass filter is found by solving the response to the simple low-pass RC filter.
[[File:1st Order Lowpass Filter RC.svg|right|framed|A simple low-pass [[RC circuit|RC filter]]]]Using [[Kirchhoff's circuit laws|Kirchhoff's Laws]] we arrive at the differential equation<ref name=":0">{{Cite book|last=Hayt, William H.
:<math>v_{\text{out}}(t) = v_{\text{in}}(t) - RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}</math>
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The most common way to characterize the frequency response of a circuit is to find its Laplace transform<ref name=":0" /> transfer function, <math>H(s) = {V_{\rm out}(s) \over V_{\rm in}(s)}</math>. Taking the Laplace transform of our differential equation and solving for <math>H(s)</math> we get
:<math>H(s) = {V_{\rm out}(s) \over V_{\rm in}(s)} = {\
== Difference equation through discrete time sampling ==
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=== Error analysis ===
Comparing the reconstructed output signal from the difference equation, <math>V_n = \beta V_{n-1} + (1-\beta)v_n</math>, to the step input response, <math>v_{\text{out}}(t) = V_i (1 - e^{-\omega_0 t})</math>, we find that there is an exact reconstruction (0% error). This is the reconstructed output for a time
==Discrete-time realization==
{{For|another method of conversion from continuous- to discrete-time|Bilinear transform}}
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.
===Simple infinite impulse response filter===
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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>
That is, this discrete-time implementation of a simple ''RC'' low-pass filter is the [[Exponential smoothing|exponentially weighted moving average]]
:<math>y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha := \frac{\Delta_T}{RC + \Delta_T} .</math>
By definition, the ''smoothing factor'' is within the range <math> 0 \;\leq\; \alpha \;\leq\; 1</math>. The expression for {{mvar| α}} yields the equivalent [[time constant]] {{math|''RC''}} in terms of the sampling period <math>\Delta_T</math> and smoothing factor {{mvar| α}},
:<math>RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right).</math>
Recalling that
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:<math>f_c=\frac{\alpha}{(1 - \alpha)2\pi \Delta_T}.</math>
If {{mvar| α}}=0.5, then the ''RC'' time constant
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,
// time interval ''dt'', and time constant ''RC''
'''function''' lowpass(''real[
'''var''' ''real[
'''var''' ''real'' α := dt / (RC + dt)
y[
'''for''' i '''from'''
y[i] := α * x[i] + (1-α) * y[i-1]
'''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:
'''for''' i '''from'''
y[i] := y[i-1] + α * (x[i] - y[i-1])
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===Finite impulse response===
Finite-impulse-response filters can be built that approximate
===Fourier transform===
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For non-realtime filtering, to achieve a low pass filter, the entire signal is usually taken as a looped signal, the Fourier transform is taken, filtered in the frequency domain, followed by an inverse Fourier transform. Only O(n log(n)) operations are required compared to O(n<sup>2</sup>) for the time domain filtering algorithm.
This can also sometimes be done in real
==Continuous-time realization==
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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.
* 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 (log scale)|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,
* A '''second-order filter''' attenuates high 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
* Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-{{mvar| n}} all-pole filter is 6{{mvar|n}} dB per octave (20{{mvar|n}} dB per decade).▼
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'', 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 ''above'' the horizontal line at this peak.
▲* 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 (log scale)|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]].''
▲* A '''second-order filter''' attenuates high 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]].
▲* Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-{{mvar| n}} all-pole filter is 6{{mvar|n}} dB per octave
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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==== RC filter ====
{{Main|RC circuit#Series circuit}}
[[File:RC Divider.svg|thumb|200px|Passive, first order low-pass RC filter]]
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> \tau \;=\; RC </math> (represented by the Greek letter [[tau]]). The break frequency, also called the turnover frequency, corner frequency, or [[cutoff frequency]] (in hertz), is determined by the time constant:
:<math>
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Another way to understand this circuit is through the concept of [[Reactance (electronics)|reactance]] at a particular frequency:
* Since [[direct current]] (DC) cannot flow through the capacitor, DC input must flow out the path marked <math> V_\mathrm{out}</math> (analogous to removing the capacitor).
* Since [[alternating current]] (AC) flows very well through the capacitor, almost as well as it flows through a solid wire, AC input flows out through the capacitor, effectively [[short circuit]]ing to the ground (analogous to replacing the capacitor with just a wire).
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.
====
{{Main|RL circuit#Series circuit}}
A resistor–inductor circuit or [[RL filter]] is an [[electric circuit]] composed of [[
A first
=== Second order ===
====RLC filter====
[[File:
An [[RLC circuit]] (the letters R, L, and C can be in a different sequence) is an [[electrical circuit]] consisting of a [[resistor]], an [[inductor]], and a [[capacitor]], connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for [[electrical resistance|resistance]], [[inductance]], and [[capacitance]], respectively. The circuit forms a [[harmonic oscillator]] for current and will [[resonance|resonate]] in a similar way as an [[LC circuit]] will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source. This effect of the resistor is called [[
There are many applications for this circuit. They are used in many different types of [[electronic oscillator|oscillator circuit]]s. Another important application is for [[tuner (electronics)|tuning]], such as in [[receiver (radio)|radio receivers]] or [[television set]]s, where they are used to select a narrow range of frequencies from the ambient radio waves. In this role, the circuit is often
== Second-Order Low-Pass Filter Standard Form ==
The transfer function <math>H_{LP}(f)</math> of a second-order low-pass filter can be expressed as a function of frequency <math>f</math> as shown in Equation 1, the Second-Order Low-Pass Filter Standard Form.
<math>
H_{LP}(f) = -\frac{K}{f_{FSF} \cdot f_c^2 + \frac{1}{Q} \cdot jf_{FSF} \cdot f_c + 1} \quad (1)
</math>
In this equation, <math>f</math> is the frequency variable, <math>f_c</math> is the cutoff frequency, <math>f_{FSF}</math> is the frequency scaling factor, and <math>Q</math> is the quality factor. Equation 1 describes three regions of operation: below cutoff, in the area of cutoff, and above cutoff. For each area, Equation 1 reduces to:
* <math>f \ll f_c</math>: <math>H_{LP}(f) \approx K</math> - The circuit passes signals multiplied by the gain factor <math>K</math>.
* <math>\frac{f}{f_c} = f_{FSF}</math>: <math>H_{LP}(f) = jKQ</math> - Signals are phase-shifted 90° and modified by the quality factor <math>Q</math>.
* <math>f \gg f_c</math>: <math>H_{LP}(f) \approx -\frac{K}{f_{FSF} \cdot f^2}</math> - Signals are phase-shifted 180° and attenuated by the square of the frequency ratio. This behavior is detailed by Jim Karki in "Active Low-Pass Filter Design" (Texas Instruments, 2023).<ref>[https://www.ti.com/lit/an/sloa049d/sloa049d.pdf Active Low-Pass Filter Design" (Texas Instruments, 2023)]</ref>
With attenuation at frequencies above <math>f_c</math> increasing by a power of two, the last formula describes a second-order low-pass filter. The frequency scaling factor <math>f_{FSF}</math> is used to scale the cutoff frequency of the filter so that it follows the definitions given before.
=== Higher order passive filters ===
Higher
[[File:LowPass3poleICauer.svg|300px|centre|thumb|A third-order low-pass filter ([[Cauer topology]]). The filter becomes a Butterworth filter with [[cutoff frequency]]
{{clear}}
===Active electronic realization===
[[File:Active Lowpass Filter RC.svg|thumb|right|300px|An active low-pass filter]]{{See also|operational amplifier applications#Inverting integrator|Op amp integrator}}
An ''active'' low-pass filter adds an [[active device]] to create an [[active filter]] that allows for [[Gain (electronics)|gain]] in the passband.
In the [[operational amplifier]] circuit shown in the figure, the cutoff frequency (in [[hertz]]) is defined as:
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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.
* [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.
* [https://relayman.org/fisher/trad.html C code generator] for digital implementation of Butterworth, Bessel, and Chebyshev filters created by the late Dr. Tony Fisher of the University of York (York, England).
{{Electronic filters}}
{{Authority control}}
{{DEFAULTSORT:Low-Pass Filter}}
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