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Low-pass filter

A low-pass filter 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 high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter.

In optics, high-pass and low-pass may have different meanings, depending on whether referring to the frequency or wavelength of light, since these variables are inversely related. High-pass frequency filters would act as low-pass wavelength filters, and vice versa. For this reason, it is a good practice to refer to wavelength filters as short-pass and long-pass to avoid confusion, which would correspond to high-pass and low-pass frequencies.[1]


Low-pass filters exist in many different forms, including electronic circuits such as a hiss filter used in audio, anti-aliasing filters for conditioning signals before 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.


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[edit]

Examples of low-pass filters occur in acoustics, optics and electronics.


A stiff physical barrier tends to reflect higher sound frequencies, acting 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 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.[1]


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.)


Electronic low-pass filters are used on inputs to subwoofers and other types of loudspeakers, to block high pitches that they cannot 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.[2]


Telephone lines fitted with DSL splitters use low-pass filters to separate DSL from POTS signals (and high-pass vice versa), which share the same pair of wires (transmission channel).[3][4]


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 before sampling and for reconstruction in digital-to-analog conversion.

A first-order 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 ); 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, smoothly transitioning 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.

octave

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 reduces the signal amplitude to one-fourth of its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See RLC circuit.

Butterworth filter

Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order- n all-pole filter is 6n dB per octave (20n dB per decade).

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.

Baseband

Low Pass Filter java simulator

a short primer on the mathematical analysis of (electrical) LTI systems.

ECE 209: Review of Circuits as LTI Systems

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.

ECE 209: Sources of Phase Shift

for digital implementation of Butterworth, Bessel, and Chebyshev filters created by the late Dr. Tony Fisher of the University of York (York, England).

C code generator