GATE 2021 EC – Question 49
The exponential Fourier series representation of a continuous-time periodic signal $x(t)$ is defined as
$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t}$$
where $\omega_0$ is the fundamental angular frequency of $x(t)$ and the coefficients of the series are $a_k$. The following information is given about $x(t)$ and $a_k$.
I. $x(t)$ is real and even, having a fundamental period of 6
II. The average value of $x(t)$ is 2
III. $a_k=\begin{cases}k,&1\le k\le3\\0,&k>3\end{cases}$
The average power of the signal $x(t)$ (rounded off to one decimal place) is ________.
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Correct answer: 32
Explanation
The average value gives $a_0=2$. Since $x(t)$ is real and even, $a_{-k}=a_k$, so the non-zero coefficients are $a_0=2$, $a_{\pm1}=1$, $a_{\pm2}=2$ and $a_{\pm3}=3$. By Parseval, $P=\sum|a_k|^2=2^2+2(1^2+2^2+3^2)=4+28=32$.