Low-pass filter

From Academic Kids

A low-pass filter passes low frequencies fairly well, but attenuates, or blocks, 'high' frequencies. (The low frequencies are relative to the unwanted higher frequencies and therefore do not have a definitive range. The actual frequencies that are cut vary from filter to filter.) Therefore it is also called a high-cut filter or treble cut filter. A high-pass filter is the opposite, and a bandpass filter is a combination of a high- and low-pass.

The concept of a low-pass filter exists in many different forms, including electronic circuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data, acoustic barriers, blurring of images, and so on. Low-pass filters play the same 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.

Contents

Examples of low-pass filters

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Low_pass_filter.png
A low-pass electronic filter realized by 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, attenuating all of the treble.

Electronic low-pass filters are 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. See subtractive synthesis.

Ideal and real filters

An ideal low-pass filter completely eliminates all frequencies above the cut-off frequency while passing those below unchanged. The transition region is infinitesimal. It can be realized mathematically (theoretically) 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 for practical, real, signals because the sinc function extends to infinity. The filter would therefore need to predict the future and have infinite knowledge of the past in order to perform the convolution. It is effectively realizable for pre-recorded digital signals (by padding the ends of the signal with zeros to the point that the error after filtering is less than the quantization error), or perfectly cyclic signals that repeat for infinity.

Real filters for real-time applications approximate the ideal filter by delaying the signal for a small period of time, allowing them to "see" a little bit into the future. Greater accuracy in approximation requires a longer delay.

The Nyquist-Shannon sampling theorem describes how to use a perfect low-pass filter and the Nyquist-Shannon interpolation formula to reconstruct a continuous signal from a sampled digital signal. Real digital-to-analog converters use real filter approximations.

Electronic low-pass filters

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Butterworth_response.png
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 (about −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 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. See RC circuit.
  • A second-order filter (Butterworth) filter 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. See RLC circuit.
  • Third and higher order filters are defined similarly.

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left, they will 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 — Butterworth filter, Chebyshev filter, etc. — all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. See electronic filter for other types.

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.) Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1000 MHz) and higher.

Passive electronic realization

A passive low-pass filter showing impedance values
A passive low-pass filter showing impedance values

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 or cutoff frequency (in hertz), is determined by the choice of resistance and capacitance:

<math> f_c = {1 \over 2 \pi R C } <math>

or equivalently (in radians per second):

<math> \omega_c = \frac{1}{RC} <math>

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 <math>V_{out}<math> (analogous to removing the capacitor).
  • 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 (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 electronic realization

An active low-pass filter
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:

<math> f_c = {1 \over 2 \pi R_2 C } <math>

or equivalently (in radians per second):

<math> \omega_c = \frac{1}{R_2 C} <math>

The gain in the passband is <math>\frac{-R_2}{R_1}<math>, 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 (See operational 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 (what we can hear) only goes up to 20 kHz or so, 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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