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

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} (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{align}}

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)}