# Fourier Series

 Question 1
Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $a_{k}$, that is

$x(t)=\sum_{k=-\infty }^{\infty }a_{k}e^{jk\frac{2\pi }{T}t}$

The same function x(t) can also be considered function with period ${T=40}'$. Let $b_{k}$ be the fourier series cofficients when period is taken as ${T}'$. If $\sum_{k=-\infty}^{\infty}|a_{k}|=16$ ,then $\sum_{k=-\infty}^{\infty}|a_{k}|$ is equal to
 A 256 B 64 C 16 D 4
GATE EC 2018   Signals and Systems
Question 1 Explanation:
Change in only time period or frequency does not change in the value of Fourier series coefficients
So,
\begin{aligned} b_{k} &=a_{k} \\ \sum_{k=-\infty}^{\infty}\left|b_{k}\right| &=\sum_{k=-\infty}^{\infty}\left|a_{k}\right|=16 \end{aligned}
 Question 2
Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let $\{a_{k}\}$ be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):

I. The complex Fourier series coefficients of x(3t) are $\{a_{k}\}$ where k is integer valued
II. The complex Fourier series coefficients of x(3t) are $\{3{a_{k}}\}$ where k is integer valued
III. The fundamental angular frequency of x(3t) is 6$\pi$ rad/s

For the three statements above, which one of the following is correct?
 A only II and III are true B only I and III are true C only III is true D only I is true
GATE EC 2017-SET-1   Signals and Systems
Question 2 Explanation:
Initially $T = 1\text{sec}$, so $\omega_{0} = 2\pi \text{rad/sec}.$ when is compressed by 3, frequency will expand by same factor but there is no change in values of $a_{k}.$
So, both statement I and Ill are correct

 Question 3
A periodic signal x(t) has a trigonometric Fourier series expansion
$x(t)=a_{0}+\sum_{n=1}^{\infty}(a_{n}cosn\omega_{0}t)$

If $x(t)=-x-(t)=-x(t-\pi/\omega_{0})$, we can conclude that
 A $a_{n}$ are zero for all n and $b_{n}$ are zero for n even B $a_{n}$ are zero for all n and $b_{n}$ are zero for n odd C $a_{n}$ are zero for n even and $b_{n}$ are zero for n odd D $a_{n}$ are zero for n odd and $b_{n}$ are zero for n even
GATE EC 2017-SET-1   Signals and Systems
Question 3 Explanation:
Signal has odd and half wave symmetries.So at $a_{n}$ are zero and $b_{n}$ are zero to n even.
 Question 4
The magnitude and phase of the complex Fourier series coefficients $a_{k}$ of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation: C is the set of complex numbers, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

 A $x(t)\in R$ B $x(t)\in P$ C $x(t)\in (C-R)$ D the information given is not sufficient to draw any conclusion about x(t)
GATE EC 2015-SET-2   Signals and Systems
Question 4 Explanation:
\begin{aligned} \left|a_{k}\right|&-\text{ even symmetry} \\ \angle a_{k}&-\text{ Odd symmetry} &(\pi \text{ can be } -\pi) \end{aligned}
$\Rightarrow x(t)$is real.
 Question 5
Consider the periodic square wave in the figure shown.

The ratio of the power in the $7^{th}$ harmonic to the power in the $5^{th}$ harmonic for this waveform is closest in value to ______.
 A 0.51 B 0.25 C 0.75 D 1
GATE EC 2014-SET-2   Signals and Systems
Question 5 Explanation:

\begin{aligned} X(\omega) &=\sum_{K=-\infty}^{\infty} 2 \pi a_{k} \delta\left(0-K \omega\right) \\ a_{K} &=\frac{1}{2 \pi} \int_{0}^{2} x(t) e^{-j K \omega_{0} t} d t \end{aligned}
\begin{aligned} &=\frac{1}{2 \pi}\left\{\int_{0}^{1} 1 \cdot e^{-j K \omega_{o} t} d \omega-\int_{1}^{2} 1 \cdot e^{-j K \omega_{o} t} d \omega\right\}\\ &=\frac{1}{2\pi}\left[\frac{1}{-j K \omega_{o}}[e^{-j K \omega_{o}}-1]-\frac{1}{-j K \omega_{o}}[e^{(-j 2K \omega_{o})}-e^{(-j 2K \omega_{o})}] \right ]\\ &=\frac{1}{2\pi}\left[\frac{e^{-j K \omega_{o}l2}}{j K \omega_{o}}.2j\sin\frac{K\omega_{o}}{2}-\frac{1}{j K \omega_{o}}e^{-\frac{j K \omega_{o}3}{2}}\left(2j\sin\frac{K\omega_{0}}{2} \right ) \right ]\\ &=\frac{1}{\pi}\left[\frac{\sin \frac{K \omega_{o}}{2}}{K \omega_{o}}\left(e^{-j K \omega_{0}}\right)\left(e^{\frac{j K \omega_{0}}{2}}-e^{-\frac{K_{m_{2}}}{2}}\right)\right] \\ &=\frac{1}{\pi}\left(\frac{\sin K \omega_{0} / 2}{K \omega_{o}}\right) e^{-j K \omega_{0}} \times 2 j \sin \left(\frac{\left.K_{\left(0_{0}\right.}\right)}{2}\right) \\ &=\frac{1}{\pi^{2}} \frac{\left(\sin K \frac{\pi}{2}\right)^{2}}{K} 2 j \times \cos K \pi \end{aligned}\\
$7^{\text {th }}$ harmonic power $=\left(2 \pi a_{7}\right)^{2}$
$5^{\text {th }}$ harmonic power $=\left(2 \pi a_{5}\right)^{2}$
$Ratio =\left(\frac{a_{7}}{a_{5}}\right)^{2}=\left(\frac{5}{7}\right)^{2}=\frac{25}{49}=0.51$

There are 5 questions to complete.