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Sinusoids are Eigenfunctions of CT LTI Systems

Module by: Richard Baraniuk. E-mail the author

Figure 1
Figure 1 (f1.png)
y⁢t=∫−∞∞ V ω ⁢t−τ⁢h⁢τd τ =∫−∞∞h⁢τ⁢ei⁢ω⁢(t−τ)d τ =ei⁢ω⁢t⁢∫−∞∞h⁢τ⁢e−(i⁢ω⁢τ)d τ =H⁢ω⁢ei⁢ω⁢t y t τ V ω t τ h τ τ h τ ω t τ ω t τ h τ ω τ H ω ω t
(1)
where H⁢ω H ω is the eigenvalue (scale factor) and ei⁢ω⁢t ω t is V ω ⁢t V ω t , which is the eigenfunction.

Summary: Figure 2.

Figure 2: H⁢ω H ω is the eigenvalue and ei⁢ω⁢t ω t is the eigenfunction.
Figure 2 (f2.png)
H⁢ω=|H⁢ω|⁢ei⁢∡⁢H⁢ω H ω H ω ∡ H ω
(2)
where |H⁢ω| H ω changes the amplitude of ei⁢ω⁢t ω t and ei⁢∡⁢H⁢ω ∡ H ω changes the phase of ei⁢ω⁢t ω t .
output=|H⁢ω|⁢ei⁢(ω⁢t+∡⁢H⁢ω) output H ω ω t ∡ H ω
(3)

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