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| {{About|roll-off in electrical network analysis|the dumpster|Roll-off (dumpster)}}
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| '''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.
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| 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".
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| ==First order roll-off==
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| [[File:First order RC circuit.svg|thumb|250px|First order RC filter [[low-pass filter]] circuit.]]
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| [[File:Roll-off graph 6dB.svg|thumb|250px|Roll-off of a first order low-pass filter at 6 dB/8ve (20 dB/decade)]]
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| 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>
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| :<math>A=\frac{V_o}{V_i}=\frac{1}{1+i\omega RC}</math>
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| [[Prototype filter#Frequency scaling|Frequency scaling]] this to ''ω''<sub>c</sub>=1/''RC''=1 and forming the power ratio gives,
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| :<math>|A|^2=\frac{1}{1+\left( {\omega \over \omega_c} \right)^2} = \frac{1}{1+\omega^2}</math>
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| In decibels this becomes,
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| :<math>10\log \left({\frac{1}{1+\omega^2}}\right)</math>
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| or expressed as a loss,
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| :<math>L=10\log \left({1+\omega^2}\right) \ \mathrm{dB}</math>
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| At frequencies well above ''ω''=1, this simplifies to,
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| :<math>L \approx 10\log \left(\omega^2\right)= 20\log \omega \ \mathrm{dB}</math>
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| Roll-off is given by,
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| :<math>\Delta L = 20\log \left( {\omega_2 \over \omega_1} \right) \ \mathrm{dB/interval_{2,1}}</math> | |
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| For a decade this is;
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| :<math>\Delta L = 20\log 10 = 20 \ \mathrm{dB/decade}</math> | |
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| and for an octave,
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| :<math>\Delta L = 20\log 2 \approx 20 \times 0.3 = 6 \ \mathrm{dB/8ve}</math>
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| ==Higher order networks==
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| [[File:High order buffered RC circuit.svg|thumb|500px|left|Multiple order RC filter buffered between stages.]]
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| [[File:Roll-off graph multiple.svg|thumb|250px|none|Roll-off graph of higher-order low-pass filters showing various rates of roll-off]]
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| 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/>
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| :<math>A_{\mathrm T}=A^n \ </math>
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| consequently, the total roll-off is given by,
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| :<math>\Delta L_{\mathrm T} = n \Delta L = 6n \ \mathrm{dB/8ve}</math>
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| 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>
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| [[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]].]]
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| 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>
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| ==Applications==
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| 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>
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| ==See also==
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| *[[Bode plot]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *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.
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| *Todd C. Handy, ''Event-related potentials: a methods handbook'', pages 89-92, 107-109, MIT Press 2004 ISBN 0-262-08333-7.
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| *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.
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| [[Category:Electronic design]]
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| [[Category:Tone, EQ and filter]]
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| [[Category:Filter frequency response]]
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Why should you select a vacation rental the next time youre planning for a trip away from home? Because theres nothing better than getting up to the sounds of the spectacular views and seagulls found only across the white, powdery shores of Floridas Gulf of Mexico coast-line. Until youd somewhat dance to the early hours of the day at Miamis hottest clubs and see some of the finest physiology youve ever seen (even some that money can find), that"s, needless to say! In either case, the view youll get of the popular tourist destination is guaranteed in full to be better if you see it from one of many Florida vacation rentals that exist year-round.
Whether you visit Panama City or Destin in the panhandle, Miami or the Florida Keys in the south, or somewhere in between like the pristine beaches of Clear-water, you need not accept a when you can appreciate all the comforts of home plus a view more spectacular than youve probably ever seen before by just choosing to stay in a Florida vacation rental. Get extra info on our favorite related encyclopedia - Click this website: url.
Perhaps you choose the roominess and isolation of a home. Or even you like to be surrounded by all the amenities that only house life offers. To read more, we recommend you check out: clicky. And even though you choose something in-between like a apartment, town house, duplex, or triplex youll have not a problem choosing the hotels. Look Into Rental Property includes further about why to study it. And with charges that are comparable to what youll buy a typical hotel room, selecting a Florida vacation rental makes financial sense, too.
Yes, theres a vacation rental thats just right for you. The difficulty is, youll likely be so comfortable that you wont want to depart. And thats ok. Texas may be the one state that will beckon you back with open arms every year!.
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