Inspection Method
Fourier Series CT Fourier Transform
Introduction Direct Method Method of Inspection
Table of Contents

Example

$x(t)=3cos(\frac{\pi t}{2}+\frac{\pi}{4})=\sum_{k=-\infty}^{\infty}X[k]e^{jkw_0t}$

$3cos(\frac{\pi t}{2}+\frac{\pi}{4})=.......+x[-2]e^{-j2w_0t}+x[-1]e^{-jw_0t}+x[0]+x[1]e^{jw_0t}+x[2]e^{j2w_0t}+.......$

Time period $T=4$,$w_0=\frac{2\pi}{4}=\frac{\pi}{2}$

$\frac{3}{2}\big[e^{j(\frac{\pi t}{2}+\frac{\pi}{4})}+e^{-j(\frac{\pi t}{2}+\frac{\pi}{4})}\big]=\frac{3}{2}e^{-j\frac{\pi}{4}}e^{-j\frac{\pi t}{2}}+ \frac{3}{2}e^{j\frac{\pi}{4}}e^{j\frac{\pi t}{2}}= .......+x[-1]e^{-j\frac{\pi t}{2}}+x[0]+x[1]e^{j\frac{\pi t}{2}}+x[2]e^{j\pi t}+.......$

So, Fourier Series Coefficients are given by:

(1)
\begin{align} X[k]= \begin{cases} \frac{3}{2}e^{-j\frac{\pi}{4}} & \text{if}\ k=-1 \\ \nonumber \frac{3}{2}e^{j\frac{\pi}{4}} & \text{if}\ k=1\\ \nonumber 0 & \text{if}\ k\neq -1,1 \end{cases} \end{align}

Therefore,

(2)
\begin{align} X[k]=\frac{3}{2}e^{-j\frac{\pi}{4}}\delta[k+1] + \frac{3}{2}e^{j\frac{\pi}{4}}\delta[k-1] \end{align}

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