Impulse Response

$h(t,\tau )=H(\delta (t-\tau ))$

• Impulse response
• $H(\cdot )$ is the system
• $\delta (t-\tau )$ is an impulse function where you replace every instance of $x(t-\tau )$ with this
• $t$ in $h(t,\tau )$ is a specific value of time
• $t$ in $\delta (t-\tau )$ is an independent dummy variable and varies across all time
  • Can be more rigorously defined as $h(t,\tau ):=H[\delta (s-\tau )]{|}_{s=t}$
• $\tau$ is the time of the impulse function