Linear Time Invariant System

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Linear Time Invariant Systems (LTI Systems)

A linear time invariant system is one that is linear (superposition and proportionality apply) and one that doesn't change with time. For example a circuit with fixed capacitors, resistors, and inductors having an input and an output is linear and time invariant. If a capacitor changed value with time, then it would not be time invariant.

Eigenfunctions and Eigenvalues of an LTI Systems

It is an interesting exercise to show that e^{j\omega t} are eigenfunctions of any LTI system. The eigenvalues are H(\omega).

LTI System
Input Output Reason
\delta (t) h(t) Given
\delta (t- \lambda ) h(t-\lambda ) Time invariance
x(\lambda) \delta (t- \lambda ) x(\lambda) h(t-\lambda ) Proportionality
x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda \int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t) Superposition
e^{j\omega t} \int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega  \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(\omega) Applying the line above to e^{j \omega t}

Note that the last line is obtained by doing a change of variables, then recognizing the Fourier Transform.