Generally speaking, a filter is a device that receives as input a signal containing a certain set of frequencies, and returns as output another signal containing only frequencies belonging to a subset of the input frequencies. In particular, the filter is called
- high-pass if it lets through all frequencies above a given frequency, called the cut-off frequency
- low-pass if it lets through all frequencies below a given frequency, called the cut-off frequency
- band-pass if it lets through all and only the frequencies around a given frequency, called the resonance frequency
- band-stop if it blocks all and only the frequencies around a given frequency, called the resonance frequency
The concept becomes clear at once with a few examples:
- A coloured glass is an optical filter: it receives white light as input (that is, light made of all visible frequencies), and returns as output only the frequencies corresponding to the colour of the glass. The energy of the input signal corresponding to the frequencies "stopped" by the glass stays in the glass, and turns into heat.
- A mechanical filter can consist, in its simplest form, of a harmonic oscillator. The oscillator receives an external force and transmits it to another mechanical system. However, if the oscillator is forced to vibrate at a frequency far from its own resonance, it behaves essentially as a rigid connector, efficiently transmitting all the energy it receives to the final system, while near resonance it will itself tend to oscillate strongly, dissipating the work done by the external driving force as heat. Working at resonance, the oscillator can therefore filter external forces in a certain frequency range (for details see the page on resonance in buildings).
- A simple tube is an acoustic waveguide: the waves that travel along it bouncing between the walls undergo reflection and interference in the radial direction, while along the tube they can propagate freely. These boundary conditions determine a cut-off frequency for each of these modes, which therefore behave as high-pass filters; the plane wave travelling along the axis of the tube, instead, propagates at all frequencies.
The function of a filter, therefore, is to select and let through undisturbed some particular frequencies, attenuating all the others. We do not want to go into the description of the various possible filters here, but rather to show that, thanks to the use of physical analogy and of the concept of impedance, the operation of filters of any kind (electrical, mechanical, acoustic) can be understood in a unified way.
The band-pass filter
This type of filter essentially consists of a resonant system. Resonance, we recall, is the property of a system to absorb energy only in a particular range of frequencies, called the passband, while coupling very weakly to frequencies outside it.
This property is particularly useful when one wants to build devices that can isolate a physical system from an external disturbance, such as a building from earthquake shocks, or an electrical circuit from the influence of other circuits or from electromagnetic anomalies. Seen the other way round, the function of the filter is to couple strongly with a signal at a given frequency (or rather with all frequencies within the passband), as happens when tuning a radio. Tuning is in fact achieved by changing the resonance frequency of the receiver so that the centre of the band falls at the frequency of the station one wants to receive, eliminating the signals of all the other stations. These examples are better illustrated on the related pages resonance in buildings, oscillating circuits, radio.
Graphically, we can picture the process as follows:
Note that, as long as we reason in terms of frequencies (we say that we work in Fourier space), the action of the filter is mathematically described by a simple multiplication between the input frequencies and the impedance of the filter at each frequency.
In real space, that is, in the time domain, things are a little more complicated. Indeed, if the input signal contains several frequencies, it means that it does not have the shape of a sinusoid, and, since the different spectral components of the signal are attenuated differently, the output signal will no longer have the same shape it had at the input.
Even though it is mathematically simpler to describe the action of the filter in the frequency domain, we live in real space, and so the effect of a band-pass filter always shows itself to us as a change in the space-time shape of the signal. In acoustics, for example, this shape is connected to the property of perceived sound that we call timbre (see timbre perception). Here are some examples
Band-pass filters for sound
- A textbook example is given by Helmholtz resonators. They are resonant cavities with a rather narrow band, if they are of good quality, and indeed each of them has the property of selecting a particular frequency and thus "extracting" it from the overall sound that strikes them. A bank of resonators tuned to different frequencies is therefore a rudimentary instrument for the Fourier analysis of sound.
- Again, our ear apparently works like a bank of resonators (see physiology of the auditory system and pitch perception). The cochlea, that is, the last part of the inner ear in which sound still exists as a mechanical wave before being turned into an electrical impulse, is a waveguide with a conical cross-section and actually behaves like a series of band-pass filters with adjacent bands, and it is able to distribute the sound energy in real space according to the energy present in each band in Fourier space.
- A great many musical instruments can be described by a source-filter model: a source of oscillations produces a complex vibration, containing many frequencies. The oscillation of the source is forced to pass through a filter (generally very complex, made of many different band-passes), which determines the quality of the sound radiated by the instrument. For example:
- in bowed strings, such as the violin, the source is the strings. Excited by the bow, they produce a "sawtooth" wave, which is transmitted to the body of the instrument, which radiates it into the air. The filter (and impedance matcher) is made up of all the parts that form the wooden body of the instrument.
- in reed winds, such as the oboe or the clarinet, the player blows through a reed, which produces a rather harsh sound. The vibration of the reed is then filtered by the body of the instrument, made of a waveguide (cylindrical or conical, or of composite shape), which determines its pitch (through the holes) and its timbre.
- in other winds, such as the flute or the trumpet, the oscillation of the air is not produced by the vibration of a reed, but the filter principle applies in the same way. Indeed, the mouthpiece of the instrument alone is not able to change the pitch of the sound, nor does it produce a timbre comparable to that of the complete instrument.
For further details see the page sound and resonance, and the pages of the individual instruments
Links and further reading
The effect of filters can be explored interactively in the 2D Waves app. Several guided paths for the experiments are available in the following sections:
Many concepts only mentioned here are illustrated more thoroughly on the pages
For more detailed examples of the action of filters in various contexts see
