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Matt L.
Matt L.
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Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{st}=e^{\sigma t}e^{j\omega t}$$$$e^{s_{\infty}t}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma<0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.

Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{st}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma<0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.

Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{s_{\infty}t}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma<0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.

Source Link
Matt L.
Matt L.
  • 92.4k
  • 10
  • 81
  • 184

Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{st}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma<0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.