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=8N=8 times at fs=8 Hzf_s = 8\,\mathrm{Hz}, giving yn=sin⁡(πn/4)y_n = \sin(\pi n/4) for n=0,…,7n=0,\dots,7: y0=0y_0=0, y1=0.70711y_1=0.70711, y2=1y_2=1, y3=0.70711y_3=0.70711, y4=0y_4=0, y5=−0.70711y_5=-0.70711, y6=−1y_6=-1, y7=−0.70711y_7=-0.70711. (a) Write yny_n as 12i(ei2πn/8−e−i2πn/8)\frac{1}{2i}\left(e^{i2\pi n/8}-e^{-i2\pi n/8}\right) and use the orthogonality of the eight discrete exponentials to find y~k\tilde y_k for every kk, without summing term by term. (b) Use the inverse transform to recover y2y_2 from your coefficients and check it against the value given.

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