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Encyclopedia > Low pass filter

A low-pass filter passes low frequencies fairly well, but attenuates 'high' frequencies. Therefore it is better called a high-cut filter or treble cut filter. Also the term hiss filter sometimes used. See also: high-pass filter and bandpass filter.

Low-pass filters are used to block unwanted high-frequency signals, whilst passing the lower frequencies. The low frequencies to be filtered out are relative to the unwanted higher frequencies and therefore do not have a definitive range. The frequencies that are cut vary from filter to filter. A low-pass filter is the opposite of a high-pass filter.

Low-pass filters play a similar role in signal processing that moving averages do in some other fields, such as finance; both tools provide a smoother form of a signal which removes the short-term oscillations, leaving only the long-term trend.

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A perfect low-pass filter can be realized by multiplying with the rectangular function in the frequency domain or, equivalently, convolution with a sinc function in the time domain. However, this filter is not realizable because the sinc function requires infinite time in both past and future, which means the filter would need to predict the future and know infinite knowledge of the past.

The perfect low-pass filter is used in the Nyquist-Shannon interpolation formula in conjunction with the Nyquist-Shannon sampling theorem to reconstruct a digital signal from a continuous signal.

Real filters approximate the ideal filter by delaying the signal for a small period of time, allowing them to "see" a little bit into the future.

## Examples of low-pass filters

An example low-pass filter realized in an RC circuit

A physical barrier acts as a low-pass filter for waves. When music is playing in another room, the low notes are easily heard, while the high notes are largely filtered out. Similarly, very loud music played in one car is heard as a low throbbing by occupants of other cars, because the closed vehicles (and air gap) function as a very low-pass filter.

Low-pass filters are also used to drive subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast.

Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications.

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 for electronic music as created by analogue synthesisers, for example the TB-303 created by the Roland corporation.

## Types of low-pass filters

The frequency response of a first-order Butterworth filter

There are a great many different filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot.

• A first-order filter, for example, will reduce the signal strength by half (-6 dB) every time the frequency doubles (goes up one octave). The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a straight line approaching zero above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two regions. (See RC circuit.)
• A second-order filter (with a Butterworth response) will reduce the signal strength to one fourth its original level every time the frequency doubles (-12 dB per octave). The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. Third and higher order filters are defined similarly. (See RLC circuit.)

The meanings of 'low' and 'high'--i.e. 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.) A physical barrier acts as a filter at audio frequencies (between about 20 and 20000 Hz). Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1000 MHz) and higher.

### Passive electric circuit realization

A passive low-pass filter

One simple electrical circuit that will serve as a low-pass filter 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, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The break frequency, also called the turnover frequency (in radians per second), is determined by the choice of resistance and capacitance, .

Capacitors naturally resist changes in voltage. It is this natural resistance (not to be confused with Ohmic resistance) that the functionality of the low-pass filter is realized.

• With low-frequencies the voltage to the capacitor changes slowly and provides sufficient time for the capacitor to change voltage through the current-voltage relationship .
• For high-frequencies the voltage to the capacitor changes too fast for sufficient charge to build up in the capacitor to change the voltage.

This understanding is rooted in the concept of reactance where the capacitor will naturally block DC but pass AC. Taking a more fluidic vision of this passive circuit, then if the capacitor blocks DC then it must "flow out" the path marked Vout (analogous to removing the capacitor). If the capacitor passes AC then it "flows out" the path where the capacitor effectively short circuiting Vout with ground (analogous to replacing the capacitor with just a wire).

It should be noted that the capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the bode plot and frequency response that show this variability

### Active electric circuit realization

An active low-pass filter

Another type of electrical circuit is an active low-pass filter.

In this example, the cutoff frequency (in Hertz) is defined as:

The gain in the passband is , and the stopband drops off at −6 dB per octave, as it is a first-order filter.

Many times, a simple gain or attenuation amplifier is turned into a lowpass filter by adding the capacitor C. This decreases the frequency response at high frequencies and helps to avoid oscillation in the amplifier. For example, an audio amplifier can be made into a lowpass filter with cutoff frequency 100 kHz to reduce gain at frequencies which would otherwise oscillate. Since the audio band only goes up to 20 kHz, the frequencies of interest fall entirely in the passband, and the amplifier behaves the same way as far as audio is concerned.

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