A low-pass filter is a filter that passes low-frequency signals and attenuates signals
with frequencies higher than the cutoff frequency. The actual amount of attenuation for each
frequency varies depending on specific filter design. It is sometimes called a high-cut
filter, or treble cut filter in audio applications. A low-pass filter is the opposite of a high-pass
filter. A band-pass filter is a combination of a low-pass and a high-pass.
Low-pass filters exist in many different forms, including electronic circuits, anti-aliasing
filters 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 operation used in fields such as finance is a particular kind of low-pass filter, and
can be analyzed with the same signal processing techniques as are used for other low-pass
filters. Low-pass filters provide a smoother form of a signal, removing the short-term
fluctuations, and leaving the longer-term trend.
An optical filter can correctly be called a low-pass filter, but conventionally is called
a longpass filter, to avoid confusion.
Examples of low-pass filters Acoustic
A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a
low-pass filter for transmitting sound. When music is playing in another room, the low
notes are easily heard, while the high notes are attenuated.
Electronic 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.
Electronic low-pass 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 low-pass
filters to block harmonic emissions that might interfere with other communications. The tone
knob on many electric guitars is a low-pass filter used to reduce the amount of treble
in the sound. An integrator is another time constant low-pass filter.
Telephone lines fitted with DSL splitters use low-pass and high-pass filters to separate
DSL and POTS 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.
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 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 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 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; 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 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." 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 converters use real filter approximations.
Continuous-time low-pass 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 first-order filter, for example, reduces
the signal amplitude by half, or 6 dB, every time the frequency doubles; more precisely,
the power rolloff approaches 20 dB per decade in the limit of high frequency. The magnitude
Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency,
and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary
between the two, which smoothly transitions between the two straight line regions. If
the transfer function of a first-order low-pass filter has a zero 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 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 higher frequencies more steeply. The Bode plot for this type
of filter resembles that of a first-order filter, except that it falls off more quickly.
For example, a second-order Butterworth filter reduces the signal amplitude to one fourth
its original level every time the frequency doubles. 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- all-pole filter is dB per octave.
On any Butterworth filter, if one extends the horizontal line to the right and the diagonal
line to the upper-left, they intersect at exactly the cutoff frequency. The frequency
response at the cutoff frequency in a first-order filter is 3 dB below the horizontal line.
The various types of filters all have different-looking knee curves. Many second-order 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 et al. For third-order filters, the peaking and
its frequency of occurrence can too be predicted without calculus as recently shown by Cartwright
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 "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 and higher. Laplace notation
Continuous-time filters can also be described in terms of the Laplace transform of their
impulse response, in a way that lets all characteristics of the filter be easily analyzed by considering
the pattern of poles and zeros of the Laplace transform in the complex plane.
For example, a first-order low-pass 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
filter passband gain. Electronic low-pass filters
Passive electronic realization
One simple low-pass 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 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 . The break frequency, also called
the turnover frequency or cutoff frequency, is determined by the time constant:
or equivalently:
One way to understand this circuit is to focus on the time the capacitor takes to charge.
It takes time to charge or discharge the capacitor through that resistor:
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 with the idea of reactance at a particular
frequency: Since DC cannot flow through the capacitor,
DC input must "flow out" the path marked . Since 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.
The capacitor is not an "on/off" object. 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 low-pass filter.
In the operational amplifier circuit shown in the figure, the cutoff frequency is defined
as:
or equivalently:
The gain in the passband is −R2/R1, and the stopband drops off at −6 dB per octave
as it is a first-order filter. Discrete-time realization
Many digital filters are designed to give low-pass 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 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:
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 discrete-time implementation of a simple RC low-pass 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 :
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 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[0..n] x, real dt, real
RC) var real[0..n] y
var real α := dt / y[0] := x[0]
for i from 1 to n y[i] := α * x[i] + * y[i-1]
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[i-1] + α *
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 continuous-time system. As expected, as the time constant
increases, the discrete-time 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 infinite-impulse-response single-pole low-pass filter.
Finite impulse response Finite-impulse-response filters can be built
that approximate to the sinc function time-domain response of an ideal sharp-cutoff low-pass
filter. In practice, the time-domain response must be time truncated and is often of a simplified
shape; in the simplest case, a running average can be used, giving a square time response.
See also
Baseband DSL filter
References
External links Low-pass filter
Low Pass Filter java simulator ECE 209: Review of Circuits as LTI Systems,
a short primer on the mathematical analysis of 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 by means of trigonometric
identity.