A photodetector pulse

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Fourier transforms, spectra and convolution

Problem

A photodetector pulse f(t)=e−∣t∣/τf(t) = e^{-|t|/\tau}, with τ=5.0 ns\tau = 5.0\,\mathrm{ns}, has transform f~(ω)=2τ1+ω2τ2\tilde f(\omega) = \dfrac{2\tau}{1 + \omega^2\tau^2} under the time convention f~(ω)=∫f(t)eiωt dt\tilde f(\omega) = \int f(t)e^{i\omega t}\,\mathrm{d}t. You may use ∫−∞∞du(1+u2)2=π2\int_{-\infty}^{\infty}\dfrac{\mathrm{d}u}{(1+u^2)^2} = \dfrac{\pi}{2}. ((a)) Compute ∫−∞∞∣f(t)∣2 dt\int_{-\infty}^{\infty}|f(t)|^2\,\mathrm{d}t directly. ((b)) Compute 12π∫−∞∞∣f~(ω)∣2 dω\dfrac{1}{2\pi}\int_{-\infty}^{\infty}|\tilde f(\omega)|^2\,\mathrm{d}\omega, and compare it with part (a). ((c)) Evaluate the right side of part (b) with the factor 1/2π1/2\pi left out, and say whether checking the units of the two sides would have caught the omission.

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