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Symmetry in Trignometric Fourier Series Coefficients

The Fourier S...

Question

The Fourier Series coefficients, of a periodic signal x (t) expressed as,

$x (t) = \sum_{k = - \infty}^{\infty} a_{k} e^{j 2 π k t / T}$ are given by
$a_{- 2} = 2 - j 1;$
$a_{- 1} = 0.5 + j 0.2;$
$a_{0} = j 2;$
$a_{1} = 0.5 - j 0.2;$
$a_{2} = 2 + j 1;$
and $a_{k} = 0; f o r | k | > 2$
Which of the following is true?

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Solution

$x (t) = a_{- 2} e^{- j 2 ω_{0} t} + a_{- 1} e^{- j ω_{0} t} + a_{0}$
$+ a_{1} e^{j ω_{0} t} + a_{2} e^{j 2 ω_{0} t}$
$= (2 - j) e^{- j 2 ω_{0} t} + (2 + j) e^{j 2 ω t}$
$+ (0.5 + j 0.2) e^{- j ω_{0} t} + (0.5 - j 0.2) e^{j ω_{0} t} + 2 j$
$= 2 (e^{- 2 ω_{0} t} + e^{j 2 ω_{0} t}) + j [e^{j 2 ω_{0} t} - e^{- j 2 ω_{0} t}]$
$+ 0.5 (e^{j ω_{0} t} + e^{- j ω_{0} t}) - j 0.2 (e^{j ω_{0} t} - e^{- j ω_{0} t}) + 2 j$
$= 4 cos 2 ω_{0} t - 2 sin 2 ω_{0} t + cos ω_{0} t$
$+ 0.4 sin ω_{0} t + 2 j$
$= R e a l [x (t)] + j I m g [x (t)]$
Where,
Real [x(t)] $= 4 cos 2 ω_{0} t - 2 sin 2 ω_{0} t + cos ω_{0} t + 0.4 sin ω_{0} t$
= neither even nor odd signal
lmg [x(t)] = 2 = constant

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