# 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 $=\left(2 \pi a_{5}\right)^{2}$
$Ratio =\left(\frac{a_{7}}{a_{5}}\right)^{2}=\left(\frac{5}{7}\right)^{2}=\frac{25}{49}=0.51$
 Question 6
The trigonometric Fourier series of an even function does not have the
 A dc term B cosine terms C sine terms D odd harmonic terms
GATE EC 2011   Signals and Systems
Question 6 Explanation:
Trigonometric Fourier series of an even function has de and cosine terms only.
 Question 7
The trigonometric Fourier series for the waveform $f(t)$ shown below contains
 A only cosine terms and zero values for the dc components B only cosine terms and a positive value for the dc components C only cosine terms and a negative value for the dc components D only sine terms and a negative value for the dc components
GATE EC 2010   Signals and Systems
Question 7 Explanation:
since f(t) is an even function, its trigonometric Fourier series contains only cosine terms.
D.C. component,
\begin{aligned} A_{0} &=\frac{1}{T} \int_{-T / 2}^{T / 2} f(t) d t=\frac{2}{T} \int_{0}^{T / 2} f(t) d t \\ &=\frac{2}{T}\left[\int_{0}^{T / 4} A d t+\int_{T / 4}^{T / 2}(-2 A) d t\right] \\ &=\frac{2}{T}\left[\frac{A T}{4}-2 A\left(\frac{T}{2}-\frac{T}{4}\right)\right] \\ &=\frac{2}{T}\left[-\frac{A T}{4}\right]=-\frac{A}{2} \end{aligned}
Therefore, the trigonometric Fourier series for the waveform f(t) contains only cosine terms and a negative value for the dc component.
 Question 8
The Fourier series of a real periodic function has only
(P) cosine terms if it is even
(Q) sine terms if it is even
(R) cosine terms if it is odd
(S) sine terms if it is odd
Which of the above statements are correct ?
 A P and S B P and R C Q and S D Q and R
GATE EC 2009   Signals and Systems
Question 8 Explanation:
The Fourier series of a real periodic function has only cosine terms if it is even and only sine terms if it is odd.
 Question 9
Choose the function $f (t); -\infty \lt t \lt \infty$ for which a Fourier series cannot be defined.
 A 3sin(25t) B 4cos(20t+3)+2sin(10t) C exp(-|t|)sin(25t) D 1
GATE EC 2005   Signals and Systems
Question 9 Explanation:
All other functions are either periodic or constant function.
 Question 10
The Fourier series expansion of a real periodic signal with fundamental frequency $f_{0}$ is given by $g_{p}(t)=\sum_{n=-\infty }^{\infty}c_{n}e^{j2\pi nf_{0}t}$. It is given that $c_{3}=3+j5$. Then $c_{-3}$ is
 A $5+j3$ B $-3-j5$ C $-5+j3$ D $3-j5$
GATE EC 2003   Signals and Systems
Question 10 Explanation:
$\begin{array}{l} c_{-k}=c_{k}^{*} \\ c_{3}=3+j 5 \\ c_{-3}=c_{k}^{*}=3-j 5 \end{array}$
There are 10 questions to complete.