superposition

Definition

The superposition principle states that for a linear system, the response to a sum of inputs equals the sum of the responses to each individual input.

If $\mathcal{L}$ is a linear operator, then:

$$ \mathcal{L}\{a_1 f_1(x) + a_2 f_2(x)\} = a_1 \mathcal{L}\{f_1(x)\} + a_2 \mathcal{L}\{f_2(x)\} $$

for all scalars $a_1, a_2$ and all inputs $f_1, f_2$ in the domain of $\mathcal{L}$.

Remarks

• Superposition is the defining property of linearity in systems theory.
• It allows complex inputs to be decomposed into simpler components, each analyzed separately, then recombined.
• In optics and signal processing, this principle enables techniques like Fourier analysis, where inputs are expressed as sums of eigenfunctions (complex exponentials).

Superposition Integral

For a linear system with impulse response $h(x; x_0)$, the output is given by the superposition integral:

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

This represents the response as a weighted sum (integral) of impulse responses.

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