Editing Bode plot (section)

==Frequency response ==
This section illustrates that a Bode plot is a visualization of the frequency response of a system.

Consider a [[LTI system theory|linear, time-invariant]] system with transfer function <math>H(s)</math>. Assume that the system is subject to a sinusoidal input with frequency <math>\omega</math>,

:<math>u(t) = \sin (\omega t),</math>

that is applied persistently, i.e. from a time <math>-\infty</math> to a time <math>t</math>. The response will be of the form

:<math>y(t) = y_0 \sin (\omega t + \varphi),</math>

i.e., also a sinusoidal signal with amplitude <math>y_0</math> shifted by a phase <math>\varphi</math> with respect to the input.

It can be shown<ref name=multivar_fb_control>{{cite book|last1=Skogestad|first1=Sigurd|last2=Postlewaite|first2=Ian|title=Multivariable Feedback Control|date=2005|publisher=John Wiley & Sons, Ltd.|location=Chichester, West Sussex, England|isbn=0-470-01167-X}}</ref> that the magnitude of the response is
{{NumBlk|:|<math>y_0 = |H(\mathrm{j} \omega)|</math>|{{EquationRef|1}}}}
and that the phase shift is
{{NumBlk|:|<math>\varphi = \arg H(\mathrm{j} \omega).</math>|{{EquationRef|2}}}}

In summary, subjected to an input with frequency <math>\omega</math>, the system responds at the same frequency with an output that is amplified by a factor <math>|H(\mathrm{j} \omega)|</math> and phase-shifted by <math>\arg H(\mathrm{j} \omega)</math>. These quantities, thus, characterize the frequency response and are shown in the Bode plot.