Lowpass filter
A lowpass filter is a filter that passes signals with a frequency lower than a certain cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The amount of attenuation for each frequency depends on the filter design. The filter is sometimes called a highcut filter, or treble cut filter in audio applications. A lowpass filter is the opposite of a highpass filter. A bandpass filter is a combination of a lowpass and a highpass filter.
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.
An optical filter can correctly be called a lowpass filter, but conventionally is called a longpass filter (low frequency is long wavelength), to avoid confusion.
Contents

Examples 1
 Acoustics 1.1
 Electronics 1.2
 Ideal and real filters 2

Continuoustime lowpass filters 3
 Laplace notation 3.1

Electronic lowpass filters 4
 Passive electronic realization 4.1
 Active electronic realization 4.2

Discretetime realization 4.3
 Simple infinite impulse response filter 4.3.1
 Finite impulse response 4.3.2
 Fourier 4.3.3
 See also 5
 References 6
 External links 7
Examples
Acoustics
A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a lowpass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated.
Electronics
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.^{[1]}
Telephone lines fitted with DSL splitters use lowpass and highpass filters to separate DSL and POTS signals sharing the same pair of wires.^{[2]}^{[3]}
Lowpass filters also play a significant role in the sculpting of sound created by analogue and virtual analogue synthesisers. See subtractive synthesis.
Ideal and real filters
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."^{[4]}
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.
Continuoustime lowpass filters
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 higher 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\scriptstyle n allpole filter is \scriptstyle 6n dB per octave (i.e., \scriptstyle 20n 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^{[5]} et al. For thirdorder filters, the peaking and its frequency of occurrence can too be predicted without calculus as recently shown by Cartwright^{[6]} 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
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:
 \frac{\text{Output}}{\text{Input}} = K \frac{1}{\tau s + 1}
where s is the Laplace transform variable, τ is the filter time constant, and K is the filter passband gain.
Electronic lowpass filters
Passive electronic realization
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 \scriptstyle \tau \;=\; RC (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:
 f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}
or equivalently (in radians per second):
 \omega_\mathrm{c} = {1 \over \tau} = {1 \over R C}
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 \scriptstyle V_\mathrm{out} (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.
Active electronic realization
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:
 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 firstorder filter.
Discretetime realization
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
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_{\text{in}}(t)  v_{\text{out}}(t) = R \; i(t) 

(V) 
Q_c(t) = C \, v_{\text{out}}(t) 

(Q) 
i(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t} 

(I) 
where Q_c(t) is the charge stored in the capacitor at time \scriptstyle t. Substituting equation Q into equation I gives \scriptstyle i(t) \;=\; C \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}, which can be substituted into equation V so that:
 v_{\text{in}}(t)  v_{\text{out}}(t) = RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}
This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenlyspaced points in time separated by \scriptstyle \Delta_T time. Let the samples of \scriptstyle v_{\text{in}} be represented by the sequence \scriptstyle (x_1,\, x_2,\, \ldots,\, x_n), and let \scriptstyle v_{\text{out}} be represented by the sequence \scriptstyle (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_{i1}}{\Delta_T}
And rearranging terms gives the recurrence relation
 y_i = \overbrace{x_i \left( \frac{\Delta_T}{RC + \Delta_T} \right)}^{\text{Input contribution}} + \overbrace{y_{i1} \left( \frac{RC}{RC + \Delta_T} \right)}^{\text{Inertia from previous output}}.
That is, this discretetime implementation of a simple RC lowpass filter is the exponentiallyweighted moving average
 y_i = \alpha x_i + (1  \alpha) y_{i1} \qquad \text{where} \qquad \alpha \triangleq \frac{\Delta_T}{RC + \Delta_T}
By definition, the smoothing factor \scriptstyle 0 \;\leq\; \alpha \;\leq\; 1. The expression for \scriptstyle \alpha yields the equivalent time constant \scriptstyle RC in terms of the sampling period \scriptstyle \Delta_T and smoothing factor \scriptstyle \alpha:
 RC = \Delta_T \left( \frac{1  \alpha}{\alpha} \right)
If \scriptstyle \alpha \;=\; 0.5, then the \scriptstyle RC time constant is equal to the sampling period. If \scriptstyle \alpha \;\ll\; 0.5, then \scriptstyle RC is significantly larger than the sampling interval, and \scriptstyle \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 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] := 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 \scriptstyle RC increases, the discretetime smoothing parameter \scriptstyle \alpha decreases, and the output samples \scriptstyle (y_1,\, y_2,\, \ldots,\, y_n) respond more slowly to a change in the input samples \scriptstyle (x_1,\, x_2,\, \ldots,\, x_n); the system has more inertia. This filter is an infiniteimpulseresponse (IIR) singlepole lowpass filter.
Finite impulse response
Finiteimpulseresponse filters can be built that approximate to the sinc function timedomain response of an ideal sharpcutoff lowpass filter. 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.^{[7]}
Fourier
For minimum distortion the finite impulse response filter has an unbounded number of coefficients.
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, and then the inverse Fourier transform is performed. This means only O(n log(n)) operations are required.
This can also sometimes be done in 'realtime', where the signal is delayed long enough to perform the fourier on shorter, overlapping blocks, which can substantially reduce the processing required.
See also
References
 ^
 ^ "ADSL filters explained". Epanorama.net. Retrieved 20130924.
 ^ "Home Networking – Local Area Network". Pcweenie.com. 20090412. Retrieved 20130924.
 ^ Mastering Windows: Improving Reconstruction
 ^ 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 and E. J. Kaminsky (2013). "Finding the maximum and minimum magnitude responses (gains) of thirdorder filters without calculus". Lat. Am. J. Phys. Educ. 7 (4): 582–587.
 ^ Signal recovery from noise in electronic instrumentation – T H Whilmshurst
External links
 Lowpass filter
 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.
