Consider the following continuous time signal x(t) :

If $C_{k}$ is the complex Fourier series coefficient of x(t), then the value of | $C_{2}$ | will be _______.
0.276

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Solution

The correct option is A 0.276
$C_{k} = \frac{1}{T} \int_{< T >} x (t) e^{- j k ω_{0} t} d t$

$C_{2} = \frac{1}{3} \int_{- 1}^{1} 2 e^{- j 2 \times \frac{2 π}{3} t} d t = \frac{2}{3} \int_{- 1}^{1} e^{- j \frac{4 π}{3} t} d t = \frac{2}{3} {⎡ ⎣ \frac{e^{- j \frac{4 π}{3} t}}{- j \frac{4 π}{3}} ⎤ ⎦}_{- 1}^{1}$
$= \frac{2}{3} \times \frac{3}{2 π} ⎡ ⎣ \frac{e^{j \frac{4 π}{3}} - e^{- j \frac{4 π}{3}}}{2 j} ⎤ ⎦$
= $\frac{1}{π} s i n (\frac{4 π}{3}) = - 0.276$

$| C_{2} | = 0.276$

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Consider the following continuous time signal x(t) : If Ck is the complex Fourier series coefficient of x(t), then the value of | C2| will be _______. 0.276

The correct option is A 0.276Ck=1T∫<T>x(t)e−jkω0tdt C2=13∫1−12e−j2×2π3tdt=23∫1−1e−j4π3tdt=23⎡⎣e−j4π3t−j4π3⎤⎦1−1 =23×32π⎡⎣ej4π3−e−j4π32j⎤⎦ = 1πsin(4π3)=−0.276 |C2|=0.276

Consider the following continuous time signal x(t) :

If $C_{k}$ is the complex Fourier series coefficient of x(t), then the value of | $C_{2}$ | will be _______.
0.276