MATH 414 Lecture 16

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Convolution Theorem

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{F} \left[ f * g \right] &= \sqrt{2\pi} \hat{f}(\lambda) \, \hat{g}(\lambda) \\ (f * g)(t) &= \frac{1}{\sqrt{2\pi}} \, \mathcal{F}^{-1} \left[ \hat{f}(\lambda) \, \hat{g}(\lambda) \right] \end{align}}

Example

Let $f(t)={\begin{cases}1&x\in \left[-\pi ,\pi \right]\\0&{\mbox{otherwise}}\end{cases}}$

${\begin{aligned}(f*f)(t)&=\int _{-\infty }^{\infty }f(\tau )\,f(t-\tau )\,\mathrm {d} \tau \\&=\int _{t-\pi }^{t+\pi }f(\tau )\,\mathrm {d} \tau \\\end{aligned}}$

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(\tau) = \tau \, \theta(\tau + \pi) - \tau \, \theta(\tau - \pi)}

Observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G'(\tau) = f(\tau)} , so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f*f)(t) = \int_{t-\pi}^{t+\pi} G'(\tau) \,\mathrm{d}\tau = G(t + \pi) - G(t - \pi)} by the fundamental theorem of calculus

In this case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f*f)(t) = (t+\pi) \, \theta(t+2\pi) - (t + \pi) \, \theta(t) - ((t-\pi) \, \theta(t) - (t-\pi) \, \theta(t - 2\pi)) = (t+\pi) \, \theta(t + 2\pi) + (t-\pi) \, \theta(t - 2\pi)}