Ornstein–Uhlenbeck process

2013-12-28T21:14:28Z

176.27.97.189: /* Scaling limit interpretation */ small addition for clarity

[[File:Three_decades.png|thumb|Three decades: 0.01, 0.1, 1, 10 (10-2, 10-1, 100, 101).]]
[[File:Three_decades_x10.png|thumb|Three decades: One-thousand 0.01's, one-hundred 0.1's, ten 1's, one 10.]]

One '''decade''' is a [[factorization|factor]] of 10 difference between two numbers (an [[order of magnitude]] difference) measured on a [[logarithmic scale]]. Along with the [[octave (electronics)|octave]], it is a [[units of measurement|unit]] used to describe [[frequency|frequency bands]] or [[interval ratio|frequency ratios]].<ref name="Levine">Levine, William S. (2010). ''The Control Handbook: Control System Fundamentals'', p.9-29. ISBN 9781420073621.</ref><ref name="Perdikaris">Perdikaris, G. (1991). ''Computer Controlled Systems: Theory and Applications'', p.117. ISBN 9780792314226.</ref> It is especially useful when referring to frequencies and when describing [[frequency response]] of [[electronics|electronic systems]], such as [[audio amplifier]]s and [[electronic filter|filters]].

==Calculations==
The factor-of-ten in a decade can be in either direction: so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, so 3.14 rad/s is one decade down from 31.4 rad/s.

To determine the number of decades between two frequencies (<math>f_1</math> & <math>f_2</math>), use the [[logarithm]] of the ratio of the two values:
*<math>\log_{10} (f_2/f_1)</math> decades<ref name="Levine"/><ref name="Perdikaris"/>
or, using [[natural logarithm]]s:
*<math>\ln f_2 - \ln f_1\over\ln 10</math> decades<ref>Davis, Don and Patronis, Eugene (2012). ''Sound System Engineering'', p.13. ISBN 9780240808307.</ref>

:How many decades is it from 15 rad/s to 150,000 rad/s?
::<math>\log_{10} (150000/15) = 4</math> decades
:How many decades is it from 3.2 GHz to 4.7 MHz?
::<math>\log_{10} (4.7\times10^6 / 3.2\times10^9 ) = -2.83</math> decades
:How many decades is one octave?
::One octave is a factor of 2, so <math>\log_{10} (2) = 0.301</math> decades per octave (decade = [[just major third]] + three octaves, 10/1 = 5/4)

To find out what frequency is a certain number of decades from the original frequency, multiply by appropriate powers of 10:
:What is 3 decades down from 220 Hz?
::<math>220 \times 10^{-3} = 0.22</math> Hz
:What is 1.5 decades up from 10?
::<math>10 \times 10^{1.5} = 316.23</math>

To find out the size of a step for a certain number of frequencies per decade, raise 10 to the power of the inverse of the number of steps:
:What is the step size for 30 steps per decade?
::<math> 10^{1/30} = 1.079775</math> - or each step is 7.9775% larger than the last.

[[Image:Butterworth filter bode plot.png|350px|thumb|right|[[Bode plot]] showing the concept of a decade: each major division on the horizontal axis is one decade]]

==Graphical representation and analysis==
Decades on a logarithmic scale, rather than unit steps (steps of 1) or other [[linear]] scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in [[Bode plot]]s, since depicting large frequency ranges on a linear scale is often not practical. For example, an [[audio amplifier]] will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (100) and go up to perhaps 100 kHz (105), to comfortably include the full audio band in a standard-sized [[graph paper]], as shown below. Whereas in the same distance on a linear scale, with 10 as the major step-size, you might only get from 0 to 50.

[[Image:Decade vs Linear.svg|400px|1,10,100,1k,10k,100k using decades vs. 0,10,20,30,40,50 using linear scale]]

Electronic frequency responses are often described in terms of "per decade". The example Bode plot shows a slope of -20 [[Decibel|dB]]/decade in the stopband, which means that for every factor-of-ten increase in frequency (going from 10 rad/s to 100 rad/s in the figure), the gain decreases by 20 dB.

==See also==
* [[Savart]]

==Sources==
{{reflist}}

{{DEFAULTSORT:Decade (Log Scale)}}
[[Category:Charts]]
[[Category:Logarithmic scales of measurement]]

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2012-04-22T13:15:21Z

176.27.239.4:

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Ornstein–Uhlenbeck process

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