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A low-pass filter is a filter that passes signals with a
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a
high-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequenc ...
. In optics, high-pass and low-pass may have different meanings, depending on whether referring to 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 ''short-pass'' and ''long-pass'' to avoid confusion, which would correspond to ''high-pass'' and ''low-pass'' frequencies. Low-pass filters exist in many different forms, including electronic circuits such as a hiss filter used in audio,
anti-aliasing filter An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruc ...
s for conditioning signals prior to analog-to-digital conversion, digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on. The
moving average In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
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. Filter designers will often use the low-pass form as a prototype filter. That is, a filter with unity bandwidth and impedance. The desired filter is obtained from the prototype by scaling for the desired bandwidth and impedance and transforming into the desired bandform (that is low-pass, high-pass, band-pass or band-stop).


Examples

Examples of low-pass filters occur in acoustics, optics and electronics. A stiff physical barrier tends to reflect higher sound frequencies, and so acts as an acoustic 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. An
optical filter An optical filter is a device that selectively transmits light of different wavelengths, usually implemented as a glass plane or plastic device in the optical path, which are either dyed in the bulk or have interference coatings. The optic ...
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. 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, works in a similar manner. (See current divider discussed in more detail
below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...
.) Electronic low-pass filters are used on inputs to subwoofers and other types of
loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or ...
s, to block high pitches that they cannot efficiently reproduce. Radio transmitters use low-pass filters to block
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
emissions that might interfere with other communications. The tone knob on many
electric guitar An electric guitar is a guitar that requires external amplification in order to be heard at typical performance volumes, unlike a standard acoustic guitar (however combinations of the two - a semi-acoustic guitar and an electric acoustic gu ...
s is a low-pass filter used to reduce the amount of treble in the sound. An
integrator An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an importan ...
is another
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
low-pass filter. Telephone lines fitted with DSL splitters use low-pass and high-pass filters to separate DSL and
POTS Pot may refer to: Containers * Flowerpot, a container in which plants are cultivated * Pottery, ceramic ware made by potters * A type of cookware Places * Ken Jones Aerodrome, IATA airport code POT * Palestinian Occupied Territories, the We ...
signals sharing the same pair of wires. Low-pass filters also play a significant role in the sculpting of sound created by analogue and virtual analogue synthesisers. ''See subtractive synthesis.'' A low-pass filter is used as an
anti-aliasing filter An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruc ...
prior to sampling and for reconstruction in
digital-to-analog conversion In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archi ...
.


Ideal and real filters

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged; its
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
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 In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
with its
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
, a
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the ...
, 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, 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. Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
; 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 shift. Greater accuracy in approximation requires a longer delay. An ideal low-pass filter results in
ringing artifacts In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "ec ...
via the Gibbs phenomenon. These can be reduced or worsened by choice of windowing function, and the design and choice of real filters involves understanding and minimizing these artifacts. For example, "simple truncation
f sinc F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''. His ...
causes severe ringing artifacts," in signal reconstruction, and to reduce these artifacts one uses window functions "which drop off more smoothly at the edges." 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 In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archi ...
s use real filter approximations.


Time response

The time response of a low-pass filter is found by solving the response to the simple low-pass RC filter. Using Kirchhoff's Laws we arrive at the differential equation :v_(t) = v_(t) - RC \frac


Step input response example

If we let v_(t) be a step function of magnitude V_i then the differential equation has the solution :v_(t) = V_i (1 - e^), where \omega_0 = is the cutoff frequency of the filter.


Frequency response

The most common way to characterize the frequency response of a circuit is to find its Laplace transform transfer function, H(s) = . Taking the Laplace transform of our differential equation and solving for H(s) we get :H(s) = =


Difference equation through discrete time sampling

A discrete
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
is easily obtained by sampling the step input response above at regular intervals of nT where n = 0, 1, ... and T is the time between samples. Taking the difference between two consecutive samples we have :v_(nT) - v_((n-1)T) = V_i (1 - e^) - V_i (1 - e^) Solving for v_(nT) we get :v_(nT) = \beta v_((n-1)T) + (1-\beta)V_i Where \beta = e^ Using the notation V_n = v_(nT) and v_n = v_(nT), and substituting our sampled value, v_n = V_i, we get the difference equation :V_n = \beta V_ + (1-\beta)v_n


Error analysis

Comparing the reconstructed output signal from the difference equation, V_n = \beta V_ + (1-\beta)v_n, to the step input response, v_(t) = V_i (1 - e^), we find that there is an exact reconstruction (0% error). This is the reconstructed output for a time invariant input. However, if the input is ''time variant'', such as v_(t) = V_i \sin(\omega t), this model approximates the input signal as a series of step functions with duration T producing an error in the reconstructed output signal. The error produced from ''time variant'' inputs is difficult to quantify but decreases as T\rightarrow0.


Discrete-time realization

Many digital filters are designed to give low-pass characteristics. Both infinite impulse response and
finite impulse response In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse ...
low pass filters as well as filters using
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s are widely used.


Simple infinite impulse response filter

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 discretizing the model. From the circuit diagram to the right, according to Kirchhoff's Laws and the definition of
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized a ...
: where Q_c(t) is the charge stored in the capacitor at time . Substituting equation into equation gives i(t) \;=\; C \frac, which can be substituted into equation so that :v_(t) - v_(t) = RC \frac. 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 \Delta_T time. Let the samples of v_ be represented by the sequence (x_1,\, x_2,\, \ldots,\, x_n), and let v_ be represented by the sequence (y_1,\, y_2,\, \ldots,\, y_n), which correspond to the same points in time. Making these substitutions, :x_i - y_i = RC \, \frac{y_{i}-y_{i-1{\Delta_T}. Rearranging terms gives the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...
: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. That is, this discrete-time implementation of a simple ''RC'' low-pass filter is the exponentially weighted moving average :y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha := \frac{\Delta_T}{RC + \Delta_T} . By definition, the ''smoothing factor'' is within the range 0 \;\leq\; \alpha \;\leq\; 1. The expression for yields the equivalent
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
in terms of the sampling period \Delta_T and smoothing factor , :RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right). Recalling that :f_c=\frac{1}{2\pi RC} so RC=\frac{1}{2\pi f_c}, note and f_c are related by, :\alpha = \frac{2\pi \Delta_T f_c}{2\pi \Delta_T f_c + 1} and :f_c=\frac{\alpha}{(1 - \alpha)2\pi \Delta_T}. If =0.5, then the ''RC'' time constant is equal to the sampling period. If \alpha \;\ll\; 0.5, then ''RC'' is significantly larger than the sampling interval, and \Delta_T \;\approx\; \alpha RC. 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 In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
algorithm simulates 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 ..n' x, ''real'' dt, ''real'' RC) var ''real ..n' y var ''real'' α := dt / (RC + dt) y := α * x for i from 2 to n y := α * x + (1-α) * y -1 return y The
loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, ...
that calculates each of the ''n'' outputs can be refactored into the equivalent: for i from 2 to n y := y -1+ α * (x - y -1 That is, the change from one filter output to the next is
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
to the difference between the previous output and the next input. This
exponential smoothing Exponential smoothing is a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average the past observations are weighted equally, exponential functions are used to assign e ...
property matches the exponential decay seen in the continuous-time system. As expected, as the
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
''RC'' increases, the discrete-time smoothing parameter \alpha decreases, and the output samples (y_1,\, y_2,\, \ldots,\, y_n) respond more slowly to a change in the input samples (x_1,\, x_2,\, \ldots,\, x_n); the system has more ''
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law o ...
''. This filter is an infinite-impulse-response (IIR) single-pole low-pass filter.


Finite impulse response

Finite-impulse-response filters can be built that approximate to the
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the ...
time-domain response of an ideal sharp-cutoff low-pass filter. For minimum distortion the finite impulse response filter has an unbounded number of coefficients operating on an unbounded signal. In practice, the time-domain response must be time truncated and is often of a simplified shape; in the simplest case, a
running average In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
can be used, giving a square time response.Whilmshurst, T H (1990) ''Signal recovery from noise in electronic instrumentation.''


Fourier transform

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(n2) for the time domain filtering algorithm. This can also sometimes be done in real-time, where the signal is delayed long enough to perform the Fourier transformation on shorter, overlapping blocks.


Continuous-time realization

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 rolloff. In all cases, at the ''cutoff frequency,'' the filter attenuates 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 , every time the frequency doubles (goes up one
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
); more precisely, the power rolloff approaches 20 dB per
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "d ...
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 In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
of a first-order low-pass filter has a
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
as well as a
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets * Pole star, a visible star that is approximately aligned with th ...
, 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 resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ...
.'' * 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 In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
, 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- all-pole filter is 6 dB per octave (20 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
that puts their frequency response ''above'' the horizontal line at this peak. 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.


Laplace notation

Continuous-time filters can also be described in terms of the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
of their
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
, 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 In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-t ...
of the impulse response.) For example, a first-order low-pass filter can be described in Laplace notation as: : \frac{\text{Output{\text{Input = K \frac{1}{\tau s + 1} where ''s'' is the Laplace transform variable, ''τ'' is the filter
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
, and ''K'' is the
gain Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in d ...
of the filter in the
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its anten ...
.


Electronic low-pass filters


First order


RC filter

One simple low-pass filter
circuit Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circu ...
consists of a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
in series with a load, and a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...
in parallel with the load. The capacitor exhibits 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 In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a s ...
of the filter \tau \;=\; RC (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: : f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C} or equivalently (in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s per second): : \omega_\mathrm{c} = {1 \over \tau} = {1 \over R C} This circuit may be understood by considering the time the capacitor needs to charge or discharge through the 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 through the concept of reactance at a particular frequency: * Since
direct current Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or eve ...
(DC) cannot flow through the capacitor, DC input must flow out the path marked V_\mathrm{out} (analogous to removing the capacitor). * Since
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
(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 A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circuit ...
ing to 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 In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
that show this variability.


RL filter

A resistor–inductor circuit or RL filter is an electric circuit composed of
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
s and
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...
s driven by a
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge t ...
or current source. A first order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit. A first order RL circuit is one of the simplest analogue infinite impulse response
electronic filter 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 component ...
s. It consists of a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
and an
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...
, either in series driven by a voltage source or in parallel driven by a current source.


Second order


RLC filter

An RLC circuit (the letters R, L and C can be in a different sequence) is an
electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sour ...
consisting of a
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
, an
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...
, and a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...
, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''T ...
,
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of t ...
and
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized a ...
respectively. The circuit forms a
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
for current and will 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 damping. The presence of the resistance also reduces the peak resonant frequency somewhat. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a component. An ideal, pure LC circuit is an abstraction for the purpose of theory. There are many applications for this circuit. They are used in many different types of oscillator circuits. Another important application is for tuning, such as in radio receivers or
television set A television set or television receiver, more commonly called the television, TV, TV set, telly, tele, or tube, is a device that combines a tuner, display, and loudspeakers, for the purpose of viewing and hearing television broadcasts, or using ...
s, where they are used to select a narrow range of frequencies from the ambient radio waves. In this role the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a
band-pass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two- ...
, band-stop filter, low-pass filter or
high-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequenc ...
. The RLC filter is described as a ''second-order'' circuit, meaning that any voltage or current in the circuit can be described by a second-order
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
in circuit analysis.


Higher order passive filters

Higher order passive filters can also be constructed (see diagram for a third order example).


Active electronic realization

Another type of electrical circuit is an ''active'' low-pass filter. In the
operational amplifier An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to ...
circuit shown in the figure, the cutoff frequency (in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one her ...
) is defined as: :f_{\text{c = \frac{1}{2 \pi R_2 C} or equivalently (in radians per second): :\omega_{\text{c = \frac{1}{R_2 C} The gain in the passband is −''R''2/''R''1, and the stopband drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.


See also

*
Baseband In telecommunications and signal processing, baseband is the range of frequencies occupied by a signal that has not been modulated to higher frequencies. Baseband signals typically originate from transducers, converting some other variable into ...


References


External links


Low Pass Filter java simulator

ECE 209: Review of Circuits as LTI Systems
a short primer on the mathematical analysis of (electrical) LTI systems.
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 In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
by means of trigonometric identity. {{DEFAULTSORT:Low-Pass Filter Signal processing Linear filters Synthesiser modules Filter frequency response Acoustics Sound