Something interesting from class - HW2

Max Woesner

Homework #2 - Something interesting from class

The Linear Time Invariant System Game can be used to help us understand the impulse response of a linear time invariant system.

Input$-----⟶$	Linear Time Invariant System	$⟶$Output	Reason
$\delta (t)$	$-------⟶$	$h(t)$	Given
$\delta (t-t_0)$	$-------⟶$	$h(t-t_0)$	Time Invariance
$x(t_0)\delta (t-t_0)$	$-------⟶$	$x(t_0)h(t-t_0)$	Proportionality
$$\begin{gathered} \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t_0)\delta \\ (t-t_0)d{t}_0 \end{gathered}$$	$-------⟶$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t_0)h(t-t_0)d{t}_0$	Superposition

where $$\begin{gathered} \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t_0)\delta \\ (t-t_0)d{t}_0=x(t) \end{gathered}$$ for any $x(t)$ and $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t_0)h(t-t_0)d{t}_0$ is the convolution integral.
We can expand the game further.

Input$-----⟶$	Linear Time Invariant System	$⟶$Output	Reason
$\delta (t)$	$-------⟶$	$h(t)$	Given
$\delta (t-t_0)$	$-------⟶$	$h(t-t_0)$	Time Invariance
$$\begin{gathered} x(t_0)\delta \\ (t-t_0) \end{gathered}$$	$-------⟶$	$x(t_0)h(t-t_0)$	Proportionality
$x(t)$	$-------⟶$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t_0)h(t-t_0)d{t}_0$	Superposition
$e^{j2\pi ft}$	$-------⟶$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f{t}_0}h(t-t_0)d{t}_0$	Superposition

Let $$\begin{gathered} \lambda \\ =t-t_0 \end{gathered}$$, so $t_0=t-\lambda$ and $d{t}_0=-d\lambda$
Therefore $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f{t}_0}h(t-t_0)d{t}_0=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(\lambda )e^{j2\pi f(t-\lambda )}(-d\lambda )=e^{j2\pi ft}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(\lambda )e^{-j2\pi f\lambda }d\lambda$
This tells us that $e^{j2\pi ft}$ is the eigenfunction and $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}h(\lambda )e^{-j2\pi f\lambda }d\lambda$ is the eigenvalue of all linear time invariant systems.
This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.