Complex signals can represent circuits fairly accurately. Complex signals consist of a real component along with an imaginary component. The imaginary component is represented by the letter j if your an engineer, a letter i for the rest of the mathematics literal world.

j=$ \sqrt{-1} $


$ j^2=-1 $


An important conversion:


$ x(t)=e^{j\omega t} $

$ e^{j\omega t}= cos\omega t + jsin\omega t $


An example of a complex signal/system would be x = 10 + 12j Complex signals can do a fairly good job of describing systems involving circuits and springs! As you can see above in the "Important conversion" $ e^{j\omega t} $ is essentially a complex number that represents typically oscillating mathematical symbols.

What we care about is that by altering the $ \omega $ we can represent periodic oscillating systems as well as damped and undamped, but we really care about the periodic ones.

We can test to see if our function\signal is periodic by $ {\omega/ 2\Pi = rational number!} $

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