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Determine whether the functions given can be represented by a Fourier series, a) $x(t)=\cos 6 t+\sin 8 t+e^{i 2 t}$ b) $x(t)=\cos (t)+\sin \Pi t$
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Solution:

(a) $x(t)=\cos 6 t+\sin 8 t+e^{i 2 t}$

$ \begin{aligned}\\ & T_1=\frac{2 \Pi}{6}=\frac{\Pi}{3} \quad T_2=\frac{2 \Pi}{8}=\frac{\Pi}{4} \quad T_3=2 \Pi / 2=\Pi \\\\ & \frac{T_1}{T_2}=\frac{\pi / 3}{\Pi / 4}=\frac{4}{3} \quad \frac{T_1}{T_3}=\frac{\Pi / 3}{\Pi}=\frac{1}{3}\\ \end{aligned} $

Rational numbers, so periodic, so it can be represented by Fourier series.

$ \begin{gathered} \text { b) } x(t)=\cos t+\sin \pi t \\\\ T_1=\frac{2 \pi}{1}=2 \Pi \quad T_2=\frac{2 \Pi}{\Pi}=2 \\\\ \frac{T_1}{T_2}=\Pi\\ \end{gathered} $

Irrational number, so aperiodic, no Fourier series exist

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