Impulse Response

#Engineering

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

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