deletions | additions       diff --git a/subsection_Quadrature_Noise_The_motion__.tex b/subsection_Quadrature_Noise_The_motion__.tex  index 8cd9151..15f39e4 100644  --- a/subsection_Quadrature_Noise_The_motion__.tex  +++ b/subsection_Quadrature_Noise_The_motion__.tex 
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The motion of a Harmonic Oscillator can be described in phase space, i.e. the coordinates that define the system are its position and momentum in time. In normalized phase space, the points only differ in phase and the curve follows a perfect circle.  A point in phase space, together with the time-evolution equations completely determines the motion of the oscillator, however, due to noise, it's impossible to determine this point exactly - there's an amount of uncertainty as to where exactly the oscillator is in phase space.  The same analysis can be applied to signals - suppose we have some signal $f(t)$, the signal can be written in the same form above, with a time-dependent "position"(in-phase component)  and momentum (quadrature component). "momentum" or "in-phase" and "quadrature" components, respectively.  \[f(t) = X(t) cos(\omega t) + Y(t) sin(\omega t)\]  As in the case of the harmonic oscillator, it's impossible to exactly determine $X(t)$ and $Y(t)$ for a given $t$ - there's an amount of uncertainty that can be due to classical noise or quantum uncertainty relations.