Question

The exponential Fourier series representation of a continuous - time periodic signal x(t) is defined as $x\left(t\right)={\sum }_{k=-\infty }^{\infty }{a}_k{e}^{jk{\omega }_o}t$ where ${\omega }_o$ is the fundamental angular frequency of x(t) and the coefficient 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. $a_k=\{\begin{array}{cc}2,& 1\le k\le 3\\ 0,& k>3\end{array}$
The average power of the signal x(t) (rounded off to one decimal place) is

• A. 32

Answer 1

The correct option is A 32
1. x(t) is real and even so $a_k$ is also real and even $a_k$ = $a_{-k}$

2. Average of x(t) is 2 i.e., $a_0$ = 2.

3. x(t) $\to$ $a_k$ = k $1\le k\le 3$ $a_1=1$ $a_{-1}=1$

$a_2=2$ $a_{-2}=2$
$a_3=3$ $a_{-3}=3$

4. $T_0=6$

Parseval's Power Theorem

$\frac{1}{T}{\int }_0^T|x(t{)}^2|dt={\sum }_{n=-\infty }^{+\infty }|a_k{|}^2$

$P={\sum }_{n=-\infty }^{+\infty }|a_k{|}^2$
$=|a_{-3}{|}^2|a_{-2}{|}^2+|a_{-1}{|}^2+|a_0{|}^2+|a_1{|}^2+|a_2{|}^2+|a_3{|}^2$

$=2|a_1{|}^2+2|a_2{|}^2+2|a_3{|}^2+|a_0{|}^2$

$=2\times 1^2+2\times (2{)}^2+2(3{)}^2+(2{)}^2$

$=2+8+18+4$

$P_{x\left(t\right)}=32$