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Convolution

Source lecture(s): PHY621 Lec3-4

Intuition

Convolution in one domain is simple multiplication in the other—why Fourier/Laplace methods simplify linear systems.

Formal Definition

\[(f*g)(t)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau\]

Mathematical Formulation

\[\mathcal{F}\{f*g\}=\hat{f}\cdot\hat{g}$$ $$\mathcal{F}\{f\cdot g\}=\hat{f}*\hat{g}\]

Derivation

Insert the Fourier representation of \(g\) into the convolution integral and exchange order of integration.

Worked Example

Convolving two Gaussians yields another Gaussian with variance equal to the sum of variances.

Common Mistakes

  • Dropping integration limits when switching domains.
  • Confusing correlation with convolution.

Quiz

Q1. What operation in frequency space corresponds to time-domain convolution?

Answer

Multiplication.