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
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
256  
64  
16  
4 
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}\leftb_{k}\right &=\sum_{k=\infty}^{\infty}\lefta_{k}\right=16 \end{aligned}
So,
\begin{aligned} b_{k} &=a_{k} \\ \sum_{k=\infty}^{\infty}\leftb_{k}\right &=\sum_{k=\infty}^{\infty}\lefta_{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?
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?
only II and III are true  
only I and III are true  
only III is true  
only I is true 
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
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
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_{n} are zero for all n and b_{n} are zero for n even  
a_{n} are zero for all n and b_{n} are zero for n odd  
a_{n} are zero for n even and b_{n} are zero for n odd  
a_{n} are zero for n odd and b_{n} are zero for n even 
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.
x(t)\in R  
x(t)\in P  
x(t)\in (CR)  
the information given is not sufficient to draw any conclusion about x(t) 
Question 4 Explanation:
\begin{aligned} \lefta_{k}\right&\text{ even symmetry} \\ \angle a_{k}&\text{ Odd symmetry} &(\pi \text{ can be } \pi) \end{aligned}
\Rightarrow x(t) is real.
\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 ______.
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 ______.
0.51  
0.25  
0.75  
1 
Question 5 Explanation:
\begin{aligned} X(\omega) &=\sum_{K=\infty}^{\infty} 2 \pi a_{k} \delta\left(0K \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
dc term  
cosine terms  
sine terms  
odd harmonic terms 
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
only cosine terms and zero values for the dc components  
only cosine terms and a positive value for the dc components  
only cosine terms and a negative value for the dc components  
only sine terms and a negative value for the dc components 
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.
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 ?
(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 ?
P and S  
P and R  
Q and S  
Q and R 
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.
3sin(25t)  
4cos(20t+3)+2sin(10t)  
exp(t)sin(25t)  
1 
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
5+j3
 
3j5
 
5+j3
 
3j5

Question 10 Explanation:
\begin{array}{l} c_{k}=c_{k}^{*} \\ c_{3}=3+j 5 \\ c_{3}=c_{k}^{*}=3j 5 \end{array}
There are 10 questions to complete.