10/3,6 - The Game

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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.