Linear Shift-Invariant System

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A linear shift-invariant (LSI) system is an operator $\mathcal{H}$ that satisfies two properties:

Definition

  1. Linearity: For any inputs $g_1(x)$, $g_2(x)$ and scalars $a$, $b$:
$$ \mathcal{H}\{a g_1(x) + b g_2(x)\} = a\mathcal{H}\{g_1(x)\} + b\mathcal{H}\{g_2(x)\} $$
  1. Shift-Invariance: If $\mathcal{H}\{g(x)\} = h(x)$, then:
$$ \mathcal{H}\{g(x - x_0)\} = h(x - x_0) $$

Physically, shifting the input shifts the output by the same amount without changing its form.

Characterization

An LSI system is completely characterized by its impulse response $h(x)$. The output for any input is given by convolution:

$$ g_{\text{out}}(x) = g_{\text{in}}(x) * h(x) = \int_{-\infty}^{\infty} g_{\text{in}}(\xi) \, h(x - \xi) \, d\xi $$

Eigenfunctions

Complex exponentials $e^{j2\pi f x}$ are eigenfunctions of all LSI systems:

$$ \mathcal{H}\{e^{j2\pi f x}\} = H(f) \cdot e^{j2\pi f x} $$

where $H(f) = \mathcal{F}\{h(x)\}$ is the transfer function (eigenvalue).

Examples in Optics

Shift-invariant:

Not shift-invariant:

  • Coma
  • Astigmatism
  • Distortion