10/3,6 - The Game

The Game

The idea behind the game is to use linearity (superposition and proportionality) and time invariance to find an output for a given input. An initial input and output are given.

Input	LTI System	Output	Reason
$\delta (t)$	$⟹$	$h(t)$	Given
$\delta (t-\lambda )$	$⟹$	$h(t-\lambda )$	Time Invarience
$x(\lambda )\delta (t-\lambda )$	$⟹$	$x(\lambda )h(t-\lambda )$	Proportionality
$x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(\lambda )\delta (t-\lambda )dx$	$⟹$	$\underset{ConvolutionIntegral}{\underbrace{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(\lambda )h(t-\lambda )dx}}$	Superposition

With the derived equation, note that you can put in any $x(t)$ to find the given output. Just change your t for a lambda and plug n chug.

Example 1

Let $x(t)=e^{j2\pi nt/T}=e^{j{\omega }_{n}t}$

$e^{j{\omega }_{n}t}$	$=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j{\omega }_n\lambda }h(t-\lambda )d\lambda$	Let $t-\lambda =u$ thus $du=-d\lambda$
	$=-\underset{\mathrm{\infty }}{\overset{-\mathrm{\infty }}{\int }}{e}^{j{\omega }_n(t-u)}h(u)du$	The order of integration switched due to changing from $-\lambda =u$
	$=\underset{eigenvalue}{\underbrace{(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-j{\omega }_{n}u}h(u)du)}}\underset{eigenfunction}{\underbrace{e^{j2\pi {\omega }_{n}t}}}$
	$=⟨h(u)\mid e^{j{\omega }_{n}u}⟩e^{j{\omega }_{n}t}$	Different notation
	$=H({\omega }_n)e^{j{\omega }_{n}t}$	Different notation

Example 2

Let $x(t)=x(t+T)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j2\pi nt/T}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j{\omega }_{n}t}$

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_n{e}^{j{\omega }_{n}t}$	$=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_{n}H({\omega }_n)e^{j{\omega }_{n}t}$	From Example 1
	$=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{T}⟨x(t)\mid e^{j{\omega }_{n}t}⟩⟨h(u)\mid e^{j{\omega }_{n}u}⟩e^{j{\omega }_{n}t}$	Different notation

Questions

• How do eigenfunction and basisfunctions differ?
• Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however. Very convenient.