# Electronic filters in loudspeakers

## Transfer function

There are several common kinds of electronic filters:

- A low-pass filter passes low frequencies
- A high-pass filter passes high frequencies
- A band-pass filter passes a limited range of frequencies
- A band-stop filter passes all frequencies except a limited range
- A notch filter is a type of band-stop filter that acts on a particularly narrow range of frequencies
- some filters are not designed to stop any frequencies, but instead to gently vary the amplitude response at different frequencies: filters used as pre-emphasis filters, equalizers or tone controls are good examples of this

Band-stop and band-pass filters can be constructed by combining low-pass and high-pass filters. A popular form of 2 pole filter is the Sallen-Key type. This is able to provide low-pass, band-pass, and high pass versions.

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## Technology

### Passive filters

The simplest electronic filters are based on combinations of resistors, inductors and capacitors. Since resistance has the symbol R, inductance the symbol L and capacitance the symbol C, these filters exist in so-called RC, RL, LC and RLC varieties. All these types are collectively known as passive filters, because they are activated by the power in the signal and not by an external power supply.

Here's how passive filters work: inductors block high-frequency signals and conduct low-frequency signals, while capacitors do the reverse. A filter in which the signal passes through an inductor, or in which a capacitor provides a path to earth, therefore transmits low-frequency signals more strongly than high-frequency signals and is a low-pass filter. If the signal passes through a capacitor, or has a path to ground through an inductor, then the filter transmits high-frequency signals more strongly than low-frequency signals and is a high-pass filter. Resistors on their own have no frequency-selective properties, but are added to inductors and capacitors to determine the time-constants of the circuit, and therefore the frequencies to which it responds.

At very high frequencies (above about 100 megahertz), sometimes the inductors consist of single loops or strips of sheet metal, and the capacitors consist of adjacent strips of metal. These are called stubs. Other components can be added to LC filters to make them more precise.

Filters are measured by their quality or "Q" factor. A filter is said to have a high Q if it selects or rejects a narrow range of frequencies compared with the absolute frequency at which it operates. Quality can be measured by the precision of a harmonic oscillator implemented with that type of device.

### Active filters

Active filters are implemented using a combination of passive and active (amplifying) components. Operational amplifiers are frequently used in active filter designs. These can have high Q, and achieve resonance without the use of inductors. However, their upper frequency limit is limited by the bandwidth of the amplifiers used.

### Digital filters

A finite impulse response filterDigital signal processing allows the inexpensive construction of a wide variety of filters. The signal is sampled and an analog to digital converter turns the signal into a stream of numbers. A computer program running on a CPU or a specialized DSP, less often a hardware implementation of the algorithm, calculates an output number stream. This output is converted to a signal by passing it through a digital to analog converter. There are problems with noise introduced by the conversions, but these can be controlled and limited for many useful filters. Due to the sampling involved, the input signal must be of limited frequency content or aliasing will occur. See also: Digital filter.

## Other filter technologies

### Quartz filters and piezoelectrics

In the late 1930s, engineers realized that small mechanical systems made of rigid materials such as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up to several hundred megahertz.

Some early resonators were made of steel, but quartz quickly became favored. The biggest advantage of quartz is that it is piezoelectric. This means that quartz resonators can directly convert their own mechanical motion into electrical signals. Quartz also has a very low coefficient of thermal expansion. This means that quartz resonators produce stable frequencies over a wide temperature range.

Quartz crystal filters have much higher quality factors than LCR filters. When higher stabilities are required, the crystals and their driving circuits may be mounted in a "crystal oven" to control the temperature. For very narrow filters, sometimes several crystals are operated in series.

Engineers realized that a large number of crystals could be collapsed into a single component, by mounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tapped delay line" reinforces the desired frequencies as the sound waves flow across the surface of the quartz crystal. The tapped delay line has become a general scheme of making high-Q filters in many different ways.

### Garnet filters

Another method of filtering, at frequencies from 800MHz to about 5GHz, is to use a synthetic single-crystal garnet sphere made of a chemical combination of titanium, iron and nitrogen. The garnet sits on a strip of metal driven by a transistor, and a small loop antenna touches the top of the sphere. An electromagnet changes the frequency that the garnet will pass. The advantage of this method is that the garnet can be tuned over a very wide frequency by varying the strength of the magnetic field.

### Atomic filters

For even higher frequencies and greater precision, the electrons of atoms must be used. Atomic clocks use caesium masers as ultra-high Q filters to stabilize their primary oscillators. Another method, used at high, fixed frequencies with very weak radio signals, is to use a ruby maser tapped delay line.

## Mathematics of filter design

Filters of all types can be described by their frequency response and phase response, the specification of which uniquely defines their impulse response, and vice versa. From a mathematical viewpoint, continuous-time IIR filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation. Continuous-time filters can also be described in terms of the Laplace transform of their impulse response in a way which allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of their Laplace transform in the complex plane.

Before the advent of computer filter synthesis tools, graphical tools such as Bode plots and Nyquist plots were extensively used as design tools. Even today, they are invaluable tools to understanding filter behavior.

Many different analog filter designs have been developed, each trying to optimise some feature of the system response. For practical filters, a custom design is sometimes desirable, that can offer the best tradeoff between different design criteria, which may include component count and cost, as well as filter response characteristics.

Some classic IIR filter designs include the following:

- Bessel filters
- Butterworth filters
- Chebyshev filters
- elliptic filters
- Sallen-Key filters

Analagous discrete-time designs for IIR digital filters can be analyzed and designed using the Z-transform, which is closely related to the Laplace transform. Digital filters are much more flexible to synthesize and use than analog filters, where the constraints of the design permits their use. In particular, FIR digital filters may be implemented by the direct convolution of the desired impulse response with the input signal.

IIR digital filters are also easy to design. Notably, there is no need to consider component tolerances, and very high Q levels may be obtained. However, IIR digital filters do have their own mathematical design problems, in particular relating to dynamic range and roundoff nonlinearity problems.

## Bibliography

Description |
ISDN | Available |

Williams, Arthur B & Taylor, Fred J (1995). Electronic Filter Design Handbook. McGraw-Hill. The Bible for practical electronic filter design. |
0070704414 |