Roll-off is a term commonly used to describe the steepness of a transmission function with frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. Loosely speaking, 1 year is the period of the Earth's orbit around the Sun, and the Earth's rotation on its axis has, particularly in electrical network analysis A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this. However, for the most part, they assume that the components of the network, and most especially in connection with filter circuits In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies and not others in order to suppress in the transition between a passband A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated. A passband signal is a bandpass filtered signal , as opposed to a baseband signal and a stopband. It is most typically applied to the insertion loss In telecommunications, insertion loss is the loss of signal power resulting from the insertion of a device in a transmission line or optical fiber. As it is usually expressed as a ratio in dB relative to the transmitted signal power, it can also be referred to as attenuation of the network, but can in principle be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of logarithmic A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself frequency, consequently, the units of roll-off are either decibels The decibel is a logarithmic unit for the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. Being a ratio of two measurements of a physical quantity in the same units, it is a dimensionless per decade (dB/decade), where a decade is a 10-times increase in frequency, or decibels per octave (dB/8ve), where an octave is 2-times increase in frequency.

The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the cut-off point of the frequency curve. Roll-off enables the cut-off performance of such a filter network to be reduced to a single number. Note that roll-off can occur with decreasing frequency as well as increasing frequency, depending on the of the filter being considered: for instance a low-pass filter A low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications. A low-pass filter is the opposite will roll-off with increasing frequency, but a high-pass filter or the lower stopband of a band-pass filter A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range. An example of an analogue electronic band-pass filter is an RLC circuit (a resistor–inductor–capacitor circuit). These filters can also be created by combining a low-pass filter with a high-pass filter will roll-off with decreasing frequency. For brevity, this article describes only low-pass filters. This is to be taken in the spirit of prototype filters; the same principles may be applied to high-pass filters by interchanging phrases such as "above cut-off frequency" and "below cut-off frequency".

Contents

First order roll-off

First order RC filter low-pass filter A low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications. A low-pass filter is the opposite circuit. Roll-off of a first order low-pass filter at 6 dB/8ve (20 dB/decade)

A simple first order network such as a RC circuit A resistor–capacitor circuit , or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. The 1st order RC circuit composed of one resistor and one capacitor, is the simplest example of an RC circuit will have a roll-off of 20 dB/decade. This is approximately equal (to within normal engineering required accuracy) to 6 dB/8ve and is the more usual description given for this roll-off. This can be shown to be so by considering the voltage transfer function A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity, A, of the RC network:[1]

Frequency scaling this to ωc=1/RC=1 and forming the power ratio gives,

In decibels this becomes,

or expressed as a loss,

At frequencies well above ω=1, this simplifies to,

Roll-off is given by,

For a decade this is;

and for an octave,

Higher order networks

Multiple order RC filter buffered between stages. Roll-off graph of higher-order low-pass filters showing various rates of roll-off

A higher order network can be constructed by cascading first-order sections together. If a unity gain buffer amplifier is placed between each section (or some other active topology An active filter is a type of analog electronic filter, distinguished by the use of one or more active components i.e. voltage amplifiers or buffer amplifiers. Typically this will be a vacuum tube, or solid-state is used) there is no interaction between the stages. In that circumstance, for n identical first-order sections in cascade, the voltage transfer function of the complete network is given by;[1]

AT = An

consequently, the total roll-off is given by,

A similar effect can be achieved in the digital domain In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is an electronic circuit operating on continuous-time by repeatedly applying the same filtering algorithm to the signal.[2]

LC low-pass ladder circuit. Each element (that is L or C) adds an order to the filter and a pole In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at z = 0. This means that, in particular, a pole of the function f is a point z = a such that f(z) approaches infinity uniformly as z approaches a to the driving point impedance.

The calculation of transfer function becomes somewhat more complicated when the sections are not all identical, or when the popular ladder topology construction is used to realise the filter. In a ladder filter each section of the filter has an effect on its immediate neighbours and a lesser effect on more remote sections so the response is not a simple An even when all the sections are identical. For some filter classes, such as the Butterworth filter, the insertion loss is still monotonically In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory increasing with frequency and quickly asymptotically In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In some contexts, such as algebraic geometry, an asymptote is converges to a roll-off of 6n dB/8ve, but in others, such as the Chebyshev or elliptic filter the roll-off near the cut-off frequency is much faster and elsewhere the response is anything but monotonic. Nevertheless, all filter classes eventually converge to a roll-off of 6n dB/8ve theoretically at some arbitrarily high frequency, but in many applications this will occur in a frequency band of no interest to the application and parasitic effects may well start to dominate long before this happens.[3]

Applications

Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM Frequency-division multiplexing is a form of signal multiplexing which involves assigning non-overlapping frequency ranges to different signals or to each "user" of a medium systems.[4] Roll-off is also significant on audio loudspeaker crossover filters Audio crossovers are a class of electronic filters designed specifically for use in audio applications, especially hi-fi. Individual loudspeaker drivers are incapable of covering the entire audio spectrum, from low frequencies to high frequencies, with acceptable volume and lack of distortion by themselves. Therefore a combination of multiple: here the need is not so much for a high roll-off but that the roll-offs of the high frequency and low-frequency sections are symmetrical and complementary. An interesting need for high roll-off arises in EEG Electroencephalography is the recording of electrical activity along the scalp produced by the firing of neurons within the brain. In clinical contexts, EEG refers to the recording of the brain's spontaneous electrical activity over a short period of time, usually 20–40 minutes, as recorded from multiple electrodes placed on the scalp. In machines. Here the filters mostly make do with a basic 6 dB/8ve roll-off, however, some instruments provide a switchable 35 Hz filter at the high frequency end with a faster roll-off to help filter out noise generated by muscle activity.[5]

See also

Notes

  1. ^ a b J. Michael Jacob, Advanced AC circuits and electronics: principles & applications, pages 150-152, Cengage Learning 2003 ISBN 076682330X.
  2. ^ Todd, pp107-108
  3. ^ Giovanni Bianchi, Roberto Sorrentino, Electronic filter simulation & design, pages 129-130, McGraw-Hill Professional 2007 ISBN 0071494677.
  4. ^ Lundheim, L, "On Shannon and "Shannon's Formula", Telektronikk, vol. 98, no. 1, 2002, pp. 24-25.
  5. ^ Mayer et al, pp104-105.

References

Categories: Electronics terms | Electronic design | Tone, EQ and filter | Filter frequency response

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