Sinusoids at the Same Frequency
An important property of sinusoids at a particular frequency is that they are closed with respect to addition. In other words, if you take a sinusoid, make many copies of it, scale them all by different gains, delay them all by different time intervals, and add them up, you always get a sinusoid at the same original frequency. This is a nontrivial property. It obviously holds for any constant signal


Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling, delaying, and summing its input signal(s) to create its output signal(s), it follows that when a sinusoid at a particular frequency is input to an LTI system, a sinusoid at that same frequency always appears at the output. Only the amplitude and phase can be changed by the system. We say that sinusoids are eigenfunctions of LTI systems. Conversely, if the system is nonlinear or time-varying, new frequencies are created at the system output. To prove this important invariance property of sinusoids, we may simply express all scaled and delayed sinusoids in the ``mix'' in terms of their in-phase and quadrature components and then add them up. Here are the details in the case of adding two sinusoids having the same frequency. Let





![\begin{eqnarray*}
y(t) &\isdef & g_1 x(t-t_1) + g_2 x(t-t_2) \\
&=& g_1 A \sin[\omega (t-t_1) + \phi]
+ g_2 A \sin[\omega (t-t_2) + \phi]
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img411.png)
![\begin{eqnarray*}
g_1 A \sin[\omega (t-t_1) + \phi]
&=&
g_1 A \sin[\omega t + (...
...omega t) \\
&\isdef & A_1 \cos(\omega t) + B_1 \sin(\omega t).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img412.png)
![$\displaystyle g_2 A \sin[\omega (t-t_2) + \phi]
=
A_2 \cos(\omega t) + B_2 \sin(\omega t)
$](http://www.dsprelated.com/josimages_new/mdft/img413.png)


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Constructive and Destructive Interference
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In-Phase & Quadrature Sinusoidal Components