# 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 $\delta (t) \,\!$ $\Longrightarrow$ $h(t) \,\!$ Given $\delta (t-\lambda)\,\!$ $\Longrightarrow$ $h(t-\lambda) \,\!$ Time Invarience $x(\lambda) \delta (t-\lambda)\,\!$ $\Longrightarrow$ $x(\lambda)h(t-\lambda) \,\!$ Proportionality $x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t-\lambda)\, dx$ $\Longrightarrow$ $\underbrace{\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx}_{Convolution Integral}$ 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 $x(t) = e^{j2\pi nt/T} = e^{j\omega_n t}$

 $e^{j\omega_n t}$ $= \int_{-\infty}^{\infty} e^{j \omega_n \lambda} h(t-\lambda)\, d\lambda$ Let $t-\lambda = u \,\!$ thus $du = -d\lambda \,\!$ $= -\int_{\infty}^{-\infty} e^{j \omega_n (t-u)} h(u)\, du$ The order of integration switched due to changing from $-\lambda = u\,\!$ $=\underbrace{\left (\int_{-\infty}^{\infty} e^{-j \omega_nu} h(u)\, du \right )}_{eigenvalue} \underbrace{e^{j2\pi \omega_nt}}_{eigenfunction}$ $=\left \langle h(u) \mid e^{j \omega_n u} \right \rangle e^{j \omega_n t}$ Different notation $=H(\omega_n)e^{j \omega_n t}$ Different notation

## Example 2

Let $x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} = \sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t}$

 $\sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t}$ $=\sum_{n=-\infty}^{\infty} \alpha_n H(\omega_n)e^{j \omega_n t}$ From Example 1 $=\sum_{n=-\infty}^{\infty} \frac {1}{T} \left \langle x(t) \mid e^{j\omega_n t}\right \rangle \left \langle h(u) \mid e^{j \omega_n u}\right \rangle e^{j \omega_n t}$ 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.