impulse_response

Definition

The impulse response of a linear system is the output of the system when the input is a delta function (impulse). For a linear operator $\mathcal{L}$, the impulse response $h(x; x_0)$ is defined by:

$$ \mathcal{L}\{\delta(x - x_0)\} = h(x; x_0) $$

where $h(x; x_0)$ is the response at position $x$ due to an impulse at position $x_0$.

Remarks

• The impulse response completely characterizes a linear system.
• For shift-invariant systems, $h(x; x_0) = h(x - x_0)$, and the system depends only on the relative displacement.
• In optics, the impulse response is called the point_spread_function.
• In PDEs, the impulse response is called the Green’s function.

Superposition Integral

Any input can be expressed as a weighted sum of impulses. By superposition, the output is:

$$ g(x) = \int f(x_0) \, h(x; x_0) \, dx_0 $$

For shift-invariant systems, this becomes the convolution integral:

$$ g(x) = \int f(x_0) \, h(x - x_0) \, dx_0 = f(x) * h(x) $$

Related Articles

• point_spread_function
• superposition
• convolution
• Dirac_delta_function