Lowpass filter
A lowpass filter (LPF) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a highcut filter, or treblecut filter in audio applications. A lowpass filter is the complement of a highpass filter.
In optics, highpass and lowpass may have different meanings, depending on whether referring to frequency or wavelength of light, since these variables are inversely related. Highpass frequency filters would act as lowpass wavelength filters, and vice versa. For this reason it is a good practice to refer to wavelength filters as "Shortpass" and "Longpass" to avoid confusion, which would correspond to "highpass" and "lowpass" frequencies ^{[1]}.
Lowpass filters exist in many different forms, including electronic circuits such as a hiss filter used in audio, antialiasing filters for conditioning signals prior to analogtodigital conversion, digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on. The moving average operation used in fields such as finance is a particular kind of lowpass filter, and can be analyzed with the same signal processing techniques as are used for other lowpass filters. Lowpass filters provide a smoother form of a signal, removing the shortterm fluctuations and leaving the longerterm trend.
Filter designers will often use the lowpass 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 lowpass, highpass, bandpass or bandstop).
Examples[edit]
Examples of lowpass filters occur in acoustics, optics and electronics.
A stiff physical barrier tends to reflect higher sound frequencies, and so acts as an acoustic lowpass 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 with the same function can correctly be called a lowpass filter, but conventionally is called a longpass filter (low frequency is long wavelength), to avoid confusion.^{[2]}
In an electronic lowpass 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.)
Electronic lowpass filters are used on inputs to subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently reproduce. Radio transmitters use lowpass filters to block harmonic emissions that might interfere with other communications. The tone knob on many electric guitars is a lowpass filter used to reduce the amount of treble in the sound. An integrator is another time constant lowpass filter.^{[3]}
Telephone lines fitted with DSL splitters use lowpass and highpass filters to separate DSL and POTS signals sharing the same pair of wires.^{[4]}^{[5]}
Lowpass filters also play a significant role in the sculpting of sound created by analogue and virtual analogue synthesisers. See subtractive synthesis.
A lowpass filter is used as an antialiasing filter prior to sampling and for reconstruction in digitaltoanalog conversion.
Ideal and real filters[edit]
An ideal lowpass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged; its frequency response is a rectangular function and is a brickwall filter. The transition region present in practical filters does not exist in an ideal filter. An ideal lowpass 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, in order to perform the convolution. It is effectively realizable for prerecorded 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 realtime 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. This delay is manifested as phase shift. Greater accuracy in approximation requires a longer delay.
An ideal lowpass filter results in ringing artifacts 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 [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."^{[6]}
The Whittaker–Shannon interpolation formula describes how to use a perfect lowpass filter to reconstruct a continuous signal from a sampled digital signal. Real digitaltoanalog converters use real filter approximations.
Time response[edit]
The time response of a lowpass filter is found by solving the response to the simple lowpass RC filter.
Using Kirchhoff's Laws we arrive at the differential equation^{[7]}
Step input response example[edit]
If we let be a step function of magnitude then the differential equation has the solution^{[8]}
Where is the cutoff frequency of the filter
Frequency response[edit]
The most common way to characterize the frequency response of a circuit is to find its Laplace transform^{[7]} transfer function, . Taking the Laplace transform of our differential equation and solving for we get
Difference equation through discrete time sampling[edit]
A discrete difference equation is easily obtained by sampling the step input response above at regular intervals of where and is the time between samples. Taking the difference between two consecutive samples we have
Solving for we get
Where
Using the notation and , and substituting our sampled value, , we get the difference equation
Error analysis[edit]
Comparing the reconstructed output signal from the difference equation, , to the step input response, , 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 , this model approximates the input signal as a series of step functions with duration producing an error in the reconstructed output signal. The error produced from time variant inputs is difficult to quantify^{[citation needed]} but decreases as .
Discretetime realization[edit]
Many digital filters are designed to give lowpass characteristics. Both infinite impulse response and finite impulse response low pass filters as well as filters using Fourier transforms are widely used.
Simple infinite impulse response filter[edit]
The effect of an infinite impulse response lowpass 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:

(V)


(Q)


(I)

where is the charge stored in the capacitor at time . Substituting equation Q into equation I gives , which can be substituted into equation V so that:
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 time. Let the samples of be represented by the sequence , and let be represented by the sequence , which correspond to the same points in time. Making these substitutions:
And rearranging terms gives the recurrence relation
That is, this discretetime implementation of a simple RC lowpass filter is the exponentially weighted moving average
By definition, the smoothing factor . The expression for yields the equivalent time constant in terms of the sampling period and smoothing factor :
Recalling that
 so
then and are related by:
and
 .
If , then the time constant is equal to the sampling period. If , then is significantly larger than the sampling interval, and .
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 lowpass filter on a series of digital samples:
// Return RC lowpass 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 α := dt / (RC + dt) y[0] := α * x[0] for i from 1 to n y[i] := α * x[i] + (1α) * y[i1] return y
The loop that calculates each of the n outputs can be refactored into the equivalent:
for i from 1 to n y[i] := y[i1] + α * (x[i]  y[i1])
That is, the change from one filter output to the next is proportional to the difference between the previous output and the next input. This exponential smoothing property matches the exponential decay seen in the continuoustime system. As expected, as the time constant increases, the discretetime smoothing parameter decreases, and the output samples respond more slowly to a change in the input samples ; the system has more inertia. This filter is an infiniteimpulseresponse (IIR) singlepole lowpass filter.
Finite impulse response[edit]
Finiteimpulseresponse filters can be built that approximate to the sinc function timedomain response of an ideal sharpcutoff lowpass filter. For minimum distortion the finite impulse response filter has an unbounded number of coefficients operating on an unbounded signal. In practice, the timedomain 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.^{[9]}
Fourier transform[edit]
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For nonrealtime 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^{2}) for the time domain filtering algorithm.
This can also sometimes be done in realtime, where the signal is delayed long enough to perform the Fourier transformation on shorter, overlapping blocks.
Continuoustime realization[edit]
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 firstorder filter, for example, reduces the signal amplitude by half (so power reduces by a factor of 4, or 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 firstorder 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 firstorder lowpass filter has a zero as well as a 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 onepole filter; this onepole–onezero filter is still a firstorder lowpass. See Pole–zero plot and RC circuit.
 A secondorder filter attenuates high frequencies more steeply. The Bode plot for this type of filter resembles that of a firstorder filter, except that it falls off more quickly. For example, a secondorder 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 allpole secondorder 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 firstorder filters, zeroes in the transfer function can change the highfrequency asymptote. See RLC circuit.
 Third and higherorder filters are defined similarly. In general, the final rate of power rolloff for an order allpole filter is dB per octave (i.e., dB per decade).
On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upperleft (the asymptotes of the function), they intersect at exactly the cutoff frequency. The frequency response at the cutoff frequency in a firstorder filter is 3 dB below the horizontal line. The various types of filters (Butterworth filter, Chebyshev filter, Bessel filter, etc.) all have differentlooking knee curves. Many secondorder filters have "peaking" or 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^{[10]} et al. For thirdorder filters, the peaking and its frequency of occurrence can also be predicted without calculus as shown by Cartwright^{[11]} et al. See electronic filter for other types.
The meanings of 'low' and 'high'—that is, the cutoff frequency—depend on the characteristics of the filter. The term "lowpass filter" merely refers to the shape of the filter's response; a highpass filter could be built that cuts off at a lower frequency than any lowpass 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[edit]
Continuoustime 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 Ztransform of the impulse response.)
For example, a firstorder lowpass filter can be described in Laplace notation as:
where s is the Laplace transform variable, τ is the filter time constant, and K is the gain of the filter in the passband.
Electronic lowpass filters[edit]
First order[edit]
RC filter[edit]
One simple lowpass filter circuit consists of a resistor in series with a load, and a capacitor in parallel with the load. The capacitor exhibits reactance, and blocks lowfrequency 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 (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:
or equivalently (in radians per second):
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 (DC) cannot flow through the capacitor, DC input must flow out the path marked (analogous to removing the capacitor).
 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 circuiting 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 that show this variability.
RL filter[edit]
A resistor–inductor circuit or RL filter is an electric circuit composed of resistors and inductors driven by a voltage 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 filters. It consists of a resistor and an inductor, either in series driven by a voltage source or in parallel driven by a current source.
Second order[edit]
RLC filter[edit]
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 resistance, inductance and capacitance respectively. The circuit forms a harmonic oscillator 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 sets, 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 bandpass filter, bandstop filter, lowpass filter or highpass filter. The RLC filter is described as a secondorder circuit, meaning that any voltage or current in the circuit can be described by a secondorder differential equation in circuit analysis.
Higher order passive filters[edit]
Higher order passive filters can also be constructed (see diagram for a third order example).
Active electronic realization[edit]
Another type of electrical circuit is an active lowpass filter.
In the operational amplifier circuit shown in the figure, the cutoff frequency (in hertz) is defined as:
or equivalently (in radians per second):
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 firstorder filter.
See also[edit]
References[edit]
 ^ Long Pass Filters and Short Pass Filters Information, retrieved 20171004
 ^ Long Pass Filters and Short Pass Filters Information, retrieved 20171004
 ^ Sedra, Adel; Smith, Kenneth C. (1991). Microelectronic Circuits, 3 ed. Saunders College Publishing. p. 60. ISBN 003051648X.
 ^ "ADSL filters explained". Epanorama.net. Retrieved 20130924.
 ^ "Home Networking – Local Area Network". Pcweenie.com. 20090412. Archived from the original on 20130927. Retrieved 20130924.
 ^ Mastering Windows: Improving Reconstruction
 ^ ^{a} ^{b} Hayt, William H., Jr. and Kemmerly, Jack E. (1978). Engineering Circuit Analysis. New York: McGRAWHILL BOOK COMPANY. pp. 211–224, 684–729.CS1 maint: multiple names: authors list (link)
 ^ Boyce, William and DiPrima, Richard (1965). Elementary Differential Equations and Boundary Value Problems. New York: JOHN WILEY & SONS. pp. 11–24.CS1 maint: multiple names: authors list (link)
 ^ Whilmshurst, T H (1990) Signal recovery from noise in electronic instrumentation. ISBN 9780750300582
 ^ K. V. Cartwright, P. Russell and E. J. Kaminsky,"Finding the maximum magnitude response (gain) of secondorder filters without calculus," Lat. Am. J. Phys. Educ. Vol. 6, No. 4, pp. 559565, 2012.
 ^ Cartwright, K. V.; P. Russell; E. J. Kaminsky (2013). "Finding the maximum and minimum magnitude responses (gains) of thirdorder filters without calculus" (PDF). Lat. Am. J. Phys. Educ. 7 (4): 582–587.
External links[edit]
Wikimedia Commons has media related to Lowpass filters. 
 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 lowpass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.