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Roll-off is the steepness of a
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
with
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
and a
stopband A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level. Depending on applic ...
. 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 and is usually expressed in decibels (dB). If the power transmitted to the load before insertion ...
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 frequency; consequently, the units of roll-off are either
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
s per
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...
(dB/decade), where a decade is a tenfold increase in frequency, or decibels per
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(dB/8ve), where an octave is a twofold 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 A cut-off, battle jacket, battle vest or kutte in heavy metal subcultures, is a type of vest or jacket which originated in the U.S. military, specifically the Army Air Corps, where pilots and other aviation personnel would collect patches or o ...
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 bandform of the filter being considered: for instance a
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
will roll-off with increasing frequency, but a
high-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
or the lower
stopband A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level. Depending on applic ...
of a
band-pass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-por ...
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 filter Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or tra ...
s; the same principles may be applied to high-pass filters by interchanging phrases such as "above cut-off frequency" and "below cut-off frequency".


First-order roll-off

A simple
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
network such as a
RC circuit A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
will have a roll-off of 20 dB/decade. This is approximately equal (to within normal engineering required accuracy) to 6 dB/octave and is the more usual description given for this roll-off. This can be shown to be so by considering the voltage
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
, ''A'', of the RC network:J. Michael Jacob, ''Advanced AC circuits and electronics: principles & applications'', pages 150-152, Cengage Learning 2003 . :A=\frac=\frac
Frequency scaling In computer architecture, frequency scaling (also known as frequency ramping) is the technique of increasing a processor's frequency so as to enhance the performance of the system containing the processor in question. Frequency ramping was the dom ...
this to ''ω''c = 1/''RC'' = 1 and forming the power ratio gives, :, A, ^2=\frac = \frac In decibels this becomes, :10\log \left(\right) or expressed as a loss, :L=10\log \left(\right) \ \mathrm At frequencies well above ''ω''=1, this simplifies to, :L \approx 10\log \left(\omega^2\right)= 20\log \omega \ \mathrm Roll-off is given by, :\Delta L = 20\log \left( \right) \ \mathrm For a decade this is; :\Delta L = 20\log 10 = 20 \ \mathrm and for an octave, :\Delta L = 20\log 2 \approx 20 \times 0.3 = 6 \ \mathrm


Higher order networks

A higher order network can be constructed by cascading first-order sections together. If a
unity gain buffer amplifier A buffer amplifier (sometimes simply called a buffer) is one that provides electrical impedance transformation from one circuit to another, with the aim of preventing the signal source from being affected by whatever currents (or voltages, for a cu ...
is placed between each section (or some other active topology 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; :A_=A^n \ consequently, the total roll-off is given by, :\Delta L_\text = n \, \Delta L = 6n \text A similar effect can be achieved in the
digital domain Digital Domain is an American visual effects and digital production company based in Playa Vista, Los Angeles, California. The company is known for creating digital imagery for feature films, advertising and games from its locations in Californ ...
by repeatedly applying the same filtering algorithm to the signal. The calculation of transfer function becomes somewhat more complicated when the sections are not all identical, or when the popular
ladder topology Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their t ...
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 Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the Br ...
, the insertion loss is still monotonically 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 one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
converges to a roll-off of 6''n'' dB/8ve, but in others, such as the
Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...
or
elliptic filter An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amo ...
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 6''n'' 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.


Applications

Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM systems. Roll-off is also significant on audio loudspeaker crossover filters: 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 (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocortex ...
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.Mayer et al, pp 104–105.


See also

* Bode plot


Notes


References

*J. William Helton, Orlando Merino, ''Classical control using H
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methods: an introduction to design'', pages 23–25, Society for Industrial and Applied Mathematics 1998 . *Todd C. Handy, ''Event-related potentials: a methods handbook'', pages 89–92, 107–109, MIT Press 2004 . *Fay S. Tyner, John Russell Knott, W. Brem Mayer (ed.), ''Fundamentals of EEG Technology: Basic concepts and methods'', pages 101–102, Lippincott Williams & Wilkins 1983 {{ISBN, 0-89004-385-X. Electronic design Tone, EQ and filter Filter frequency response