You are using a video microscope to count infected cells. When a cell passes through the observation window, the time course of intensity at a particular pixel is well approximated by x(t) = sinc^2(t) * cos(t).
a) Derive the Fourier transform X(w) of x(t).
b) Sketch X(w). Make sure that your answer is maximally simplified.
c) Determine the Nyquist frequency.
d) If x(t) is sampled at rates of 5 rad/sec and 10 rad/sec, sketch the corresponding Fourier transforms.
e) Determine the appropriate filter that could be applied to minimize sampling artifacts if you have to record below the Nyquist frequency. When would this filter be applied?
