Editing Cutoff frequency (section)

==Electronics==
In [[electronics]], cutoff frequency or corner frequency is the [[frequency]] either above or below which the power output of a [[Electronic circuit|circuit]], such as a [[telephone line|line]], amplifier, or [[electronic filter]] has fallen to a given proportion of the power in the [[passband]].  Most frequently this proportion is one half the passband power, also referred to as the 3&nbsp;[[decibel|dB]] point since a fall of 3&nbsp;dB corresponds approximately to half power. As a voltage ratio this is a fall to <math display="inline"> \sqrt{1/2} \ \approx \ 0.707</math> of the passband voltage.<ref>
{{cite book
|title=Network Analysis
|edition=3rd
|last=Van Valkenburg
|first=M. E.
|year=1974
|pages=[https://archive.org/details/networkanalysis00vanv/page/383 383–384]
|isbn=0-13-611095-9
|url=https://archive.org/details/networkanalysis00vanv/page/383
|access-date=2008-06-22
|url-access=registration
}}</ref> Other ratios besides the 3&nbsp;dB point may also be relevant, for example see {{section link|#Chebyshev filters}} below. Far from the cutoff frequency in the transition band, the rate of increase of attenuation ([[roll-off]]) with logarithm of frequency is [[asymptotic]] to a constant. For a [[First-order linear differential equation|first-order]] network, the roll-off is −20&nbsp;dB per [[decade (log scale)|decade]] (approximately −6&nbsp;dB per [[octave]].)

===Single-pole transfer function example===
The [[transfer function]] for the simplest [[low-pass filter]],
<math display="block">H(s) = \frac {1}{1+\alpha s},</math>
has a single [[Pole (complex analysis)|pole]] at {{math|1=''s'' = −1/''α''}}. The magnitude of this function in the {{math|''jω''}} plane is
<math display="block">\left | H(j\omega) \right | = \left |  \frac {1}{1+\alpha j \omega} \right | =\sqrt{ \frac {1}{1 + \alpha^2\omega^2}}.</math>

At cutoff
<math display="block">\left | H(j\omega_ \mathrm c) \right | = \frac {1}{\sqrt{2}} = \sqrt{ \frac {1}{1 + \alpha^2\omega_\mathrm c ^2}}.</math>

Hence, the cutoff frequency is given by
<math display="block">\omega_ \mathrm c = \frac {1}{\alpha}.</math>

Where {{mvar|s}} is the [[s-plane]] variable, {{math|ω}} is [[angular frequency]] and {{math|''j''}} is the [[imaginary unit]].

===Chebyshev filters===
Sometimes other ratios are more convenient than the 3&nbsp;dB point. For instance, in the case of the [[Chebyshev filter]] it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple.  The amount of ripple in this class of filter can be set by the designer to any desired value, hence the ratio used could be any value.<ref>Mathaei, Young, Jones ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', pp.85-86, McGraw-Hill 1964.</ref>