A Gaussian pulse with is fed into a system whose response function is with

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 Delta functions, distributions and generalised functions

Problem

A Gaussian pulse f(t)=e−t2/2σ2f(t) = e^{-t^2/2\sigma^2} with σ=0.50 s\sigma = 0.50\,\mathrm{s} is fed into a system whose response function is h(t)=δ(t2−a2)h(t) = \delta(t^2 - a^2) with a=1.0 sa = 1.0\,\mathrm{s}. This is a symmetric smoothing filter rather than a physical echo, since part of its response comes before its input. The output is the convolution (f∗h)(t)=∫−∞∞f(t−τ) h(τ) dτ.(f*h)(t) = \int_{-\infty}^{\infty} f(t-\tau)\,h(\tau)\,\mathrm{d}\tau. ((a)) Reduce h(τ)=δ(τ2−a2)h(\tau) = \delta(\tau^2-a^2) to a sum of ordinary delta functions. ((b)) Hence evaluate (f∗h)(0)(f*h)(0), the output at t=0t=0.

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