Differential privacy

2014-01-20T20:51:48Z

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{{About|roll-off in electrical network analysis|the dumpster|Roll-off (dumpster)}}

'''Roll-off''' is a term commonly used to describe the steepness of a [[Transfer function|transmission function]] with [[frequency]], particularly in [[network analysis (electrical circuits)|electrical network analysis]], and most especially in connection with [[filter (signal processing)|filter circuits]] in the transition between a [[passband]] and a [[stopband]]. It is most typically applied to the [[insertion loss]] 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 scale|logarithmic]] frequency, consequently, the units of roll-off are either [[decibel]]s per [[decade (log scale)|decade]] (dB/decade), where a decade is a 10-times increase in frequency, or decibels per [[octave (electronics)|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 frequency|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 [[:File:Bandform template.svg|bandform]] of the filter being considered: for instance a [[low-pass filter]] will roll-off with increasing frequency, but a [[high-pass filter]] or the lower [[stopband]] of a [[band-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 filter]]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==
[[File:First order RC circuit.svg|thumb|250px|First order RC filter [[low-pass filter]] circuit.]]
[[File:Roll-off graph 6dB.svg|thumb|250px|Roll-off of a first order low-pass filter at 6 dB/8ve (20 dB/decade)]]
A simple [[First order linear differential equation|first order]] network such as a [[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'', of the RC network:<ref name=Jacob>J. Michael Jacob, ''Advanced AC circuits and electronics: principles & applications'', pages 150-152, Cengage Learning 2003 ISBN 0-7668-2330-X.</ref>

:<math>A=\frac{V_o}{V_i}=\frac{1}{1+i\omega RC}</math>

[[Prototype filter#Frequency scaling|Frequency scaling]] this to ''ω''<sub>c</sub>=1/''RC''=1 and forming the power ratio gives,

:<math>|A|^2=\frac{1}{1+\left( {\omega \over \omega_c} \right)^2} = \frac{1}{1+\omega^2}</math>

In decibels this becomes,

:<math>10\log \left({\frac{1}{1+\omega^2}}\right)</math>

or expressed as a loss,

:<math>L=10\log \left({1+\omega^2}\right) \ \mathrm{dB}</math>

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

:<math>L \approx 10\log \left(\omega^2\right)= 20\log \omega \ \mathrm{dB}</math>

Roll-off is given by,

:<math>\Delta L = 20\log \left( {\omega_2 \over \omega_1} \right) \ \mathrm{dB/interval_{2,1}}</math>

For a decade this is;

:<math>\Delta L = 20\log 10 = 20 \ \mathrm{dB/decade}</math>

and for an octave,

:<math>\Delta L = 20\log 2 \approx 20 \times 0.3 = 6 \ \mathrm{dB/8ve}</math>

==Higher order networks==
[[File:High order buffered RC circuit.svg|thumb|500px|left|Multiple order RC filter buffered between stages.]]
[[File:Roll-off graph multiple.svg|thumb|250px|none|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 filter|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;<ref name=Jacob/>

:<math>A_{\mathrm T}=A^n \ </math>

consequently, the total roll-off is given by,

:<math>\Delta L_{\mathrm T} = n \Delta L = 6n \ \mathrm{dB/8ve}</math>

A similar effect can be achieved in the [[digital filter|digital domain]] by repeatedly applying the same filtering algorithm to the signal.<ref>Todd, pp107-108</ref>

[[File:LC ladder circuit.svg|thumb|500px|LC low-pass ladder circuit. Each element (that is L or C) adds an order to the filter and a [[pole (complex analysis)|pole]] 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 ''A<sup>n</sup>'' even when all the sections are identical. For some filter classes, such as the [[Butterworth filter]], the insertion loss is still [[monotonic function|monotonically]] increasing with frequency and quickly [[Asymptote|asymptotically]] converges to a roll-off of 6''n'' dB/8ve, but in others, such as the [[Chebyshev filter|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 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 capacitance|parasitic effects]] may well start to dominate long before this happens.<ref>Giovanni Bianchi, Roberto Sorrentino, ''Electronic filter simulation & design'', pages 129-130, McGraw-Hill Professional 2007 ISBN 0-07-149467-7.</ref>

==Applications==
Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone [[Frequency division multiplexing|FDM]] systems.<ref>Lundheim, L, "On Shannon and "Shannon's Formula", ''Telektronikk'', '''vol. 98''', no. 1, 2002, pp. 24-25.</ref> Roll-off is also significant on audio loudspeaker [[audio crossover|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]] 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.<ref>Mayer et al, pp104-105.</ref>

==See also==
*[[Bode plot]]

==Notes==
{{reflist}}

==References==
*J. William Helton, Orlando Merino, ''Classical control using H [infinity] methods: an introduction to design'', pages 23-25, Society for Industrial and Applied Mathematics 1998 ISBN 0-89871-424-9.
*Todd C. Handy, ''Event-related potentials: a methods handbook'', pages 89-92, 107-109, MIT Press 2004 ISBN 0-262-08333-7.
*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.

[[Category:Electronic design]]
[[Category:Tone, EQ and filter]]
[[Category:Filter frequency response]]

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