A radar receiver records a rectangular pulse of unit (dimensionless) amplitude

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 radar receiver records a rectangular pulse of unit (dimensionless) amplitude, present for 9 μs<t<11 μs9\,\mu\text{s} < t < 11\,\mu\text{s} and zero outside that interval. ((a)) Using the rectangle transform pair together with the shift theorem, write down f~(ω)=∫−∞∞f(t)eiωt dt\tilde f(\omega) = \int_{-\infty}^{\infty} f(t)e^{i\omega t}\,\mathrm{d}t, the time convention of the text, as an explicit expression in ω\omega. ((b)) Evaluate ∣f~(ω)∣|\tilde f(\omega)| at ω=π×106 rad s−1\omega = \pi\times10^{6}\,\mathrm{rad\,s^{-1}}.

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