A dimensionless signal is sampled times at

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Computational Physics, Ch Spectra of sampled signals

Problem

A dimensionless signal is sampled N=4N=4 times at fs=4 Hzf_s = 4\,\mathrm{Hz}, giving yn=cos⁡(πn/2)y_n = \cos(\pi n/2) for n=0,1,2,3n = 0,1,2,3, so y0=1y_0=1, y1=0y_1=0, y2=−1y_2=-1, y3=0y_3=0. (a) Using y~k=∑n=03yne−2πikn/4\tilde y_k = \sum_{n=0}^{3} y_n e^{-2\pi i k n/4}, compute y~0,y~1,y~2,y~3\tilde y_0, \tilde y_1, \tilde y_2, \tilde y_3 by hand. (b) Using the inverse transform yn=14∑k=03y~ke2πikn/4y_n = \frac{1}{4}\sum_{k=0}^{3}\tilde y_k e^{2\pi i kn/4}, recover y0y_0 from the coefficients found in part (a) and confirm it matches the given value.

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