A fast sampler's aperture behaves as a rectangular window of width and height

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electronics, Instrumentation & Communication · Signals, systems, control and digital signal processing

Electronics for Measurement, Ch Linear systems, the impulse response and convolution

Problem

A fast sampler's aperture behaves as a rectangular window of width tw=0.4 mst_w = 0.4\,\mathrm{ms} and height 1/tw1/t_w, centred on the sampling instant t0=5 mst_0 = 5\,\mathrm{ms}, rather than an ideal δ(t−t0)\delta(t-t_0). It samples a signal f(t)=t2f(t) = t^2 (with tt in milliseconds, so ff is in ms2\mathrm{ms}^2). ((a)) Calculate the exact value of the sampled average 1tw∫t0−tw/2t0+tw/2f(t) dt\dfrac{1}{t_w}\displaystyle\int_{t_0 - t_w/2}^{t_0+t_w/2} f(t)\,\mathrm{d}t. ((b)) Compare this with the value the sifting property would give for an ideal delta function, f(t0)f(t_0), and state the error.

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