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Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) PDF Download

Q1: The discrete time Fourier series representation of a signal x[n] with period N is written as Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) A discrete time periodic signal with period N = 3, has the non-zero Fourier series coefficients: a−3 = 2 and a4 = 1. The signal is     (2022)
(a) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Ans: (b)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)We have,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Put n = 0 in eq. (1)
x(0) = a0+a= 2+1 = 3
Put n = 1 in eq. (1)
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)These two conditions satisfy by the option (B). Hence, option (B) will be correct.

Q2: A periodic function f(t), with a period of 2π, is represented as its Fourier series,  
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)the Fourier series coefficients a1 and b1 of f(t) are      (2019)
(a) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Ans: (d)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q3: Let the signal Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) be passed through an LTI system with frequency response H(ω) , as given in the figure below
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)The Fourier series representation of the output is given as      (SET-1 (2017))
(a) 4000+4000𝑐𝑜𝑠(2000𝜋𝑡)4000 + 4000cos(2000πt) + 4000cos(4000πt)
(b) 2000+2000cos(2000πt) + 2000cos(4000πt)
(c) 4000𝑐𝑜𝑠(2000𝜋𝑡)4000cos (2000πt)
(d) 2000𝑐𝑜𝑠(2000𝜋𝑡)2000cos (2000πt)
Ans:
(c)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Time period:
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Since, x(t) is even-halt wave symmetric. So, its expansion will contain only odd harmonics of cos. Therefore, coefficient of fundamental harmonic is
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Now, frequency components available in expansion are
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)As, LTI system given in the question will pass upto 5000 π rad/sec frequency component of input.So, output will have only one component of sfrequency 2000π rad/sec
Thus, y(t) = expansion of output = a1cosω0t = 4000cos 2000πt  

Q4: Consider Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) 
Here, ⌊t⌋ represents the largest integer less than or equal to t and ⌈t⌉ denotes the smallest integer greater than or equal to t. The coefficient of the second harmonic component of the Fourier series representing g(t) is _________.       (SET-1   (2017))
(a) 0
(b) 1
(c) 2
(d) 3
Ans:
(a)
Sol: Given that,  Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
where,
⌊t⌋= greatest integer less than or equal to 't'.
⌈t⌉= smallest integer greater than or equal to 't'.
Now,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Since, g(t) is nonperiodic. So, there is no Fourier series expansion of this signal and hence no need to calculate harmonic here.

Q5: Let f(x) be a real, periodic function satisfying f(-x) = -f(x). The general form of its Fourier series representation would be      (SET-2  (2016))
(a) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Ans: (b)
Sol: Given that,
 f(−x) = −f(x)
So, function is an odd function.
So, the fourier series will have sine term only. So,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q6: The signum function is given by
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)The Fourier series expansion of sgn(cos(t)) has      (SET-1(2015))
(a) only sine terms with all harmonics
(b) only cosine terms with all harmonics.
(c) only sine terms with even numbered harmonics.
(d) only cosine terms with odd numbered harmonics.
Ans: 
(d)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)So, cos(t) is
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)So, sgn(cos t) is a rectangular signal which is even and has half wave symmetry.
So, Fourier series will have only cosine terms with add harmonics only.

Q7: Let g : [0, ∞) → [0, ∞) be a function defined by g(x) = x-[x], where [x] represents the integer part of x . (That is, it is the largest integer which is less than or equal to x ). The value of the constant term in the Fourier series expansion of g(x) is____.       (SET-1 (2014))
(a) 0
(b) 0.5
(c) 0.75
(d) 1
Ans:
(b)
Sol: Given function g(t) = x−[x]
where, [x] is a integer part of x
Then function g(x) will be
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)The value of the constant term (or) dc term in the Fourier series expansion of g(x) is
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q8: For a periodic square wave, which one of the following statements is TRUE ?     (SET-1 (2014))
(a) The Fourier series coefficients do not exist
(b) The Fourier series coefficients exist but the reconstruction converges at no point
(c) The Fourier series coefficients exist but the reconstruction converges at most point
(d) The Fourier series coefficients exist and the reconstruction converges at every point.
Ans: 
(c)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

Reconstruction of signal by its Fourier series coefficient is not possible at those points where signal is discontinuous.
In the above figure, at integer multiples of 'T/2', signal recovery is not possible by using its coefficient.
Therefore, reconstruction of x(t) by using its coefficient is possible at most of the points except those instants where x(t) is discontinous.

Q9: For a periodic signal
v(t) = 30sin100t + 10cos300t + 6sin(500t + π/4),
the fundamental frequency in rad/s       (2013)
(a) 100
(b) 300
(c) 500
(d) 1500
Ans: 
(a)

Q10: The fourier series expansion Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) of the periodic signal shown below will contain the following nonzero terms       (2011)
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)(a) a0 and bn, n = 1, 3, 5,... ∞
(b) 𝑎0𝑎𝑛𝑑𝑎𝑛,𝑛=1,2,3,...a0 and an, n = 1, 2, 3 ,... ∞
(c) 𝑎0,𝑎𝑛𝑎𝑛𝑑𝑏𝑛,𝑛=1,3,5,...a0, an and bn, n = 1, 3, 5,... ∞
(d) a0 and an, n = 1, 3, 5 ,... ∞
Ans:
(d)
Sol: Let, x(t) = Even and Hws
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Fourier series expansion of x(t) contains cos terms with odd harmonics.
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Now, f(t) = 1 + x(t)  
Fourier series of f(t) contains dc and cos terms with odd harmonics.

Q11: The second harmonic component of the periodic waveform given in the figure has an amplitude of
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)(a) 0
(b) 1
(c) 2/π
(d) √5
Ans: 
(a)
Sol: The given signal is odd as wel as having half wave symmetry.
So, it has only sine terms with odd harmonics. So, for second harmonic term amplitude = 0.

Q12: The Fourier Series coefficients of a periodic signal x(t), expressed as 𝑥(𝑡) = Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) are given by a−2 = 2−j1, 𝑎−1 = 0.5 + 𝑗0.2, a0 = j2, a= 0.5− j0.2,  𝑎= 2+𝑗1 and  ak = 0 for ∣k∣ > 2 Which of the following is true ?      (2009)
(a) x(t) has finite energy because only finitely many coefficients are non-zero
(b) x(t) has zero average value because it is periodic
(c) The imaginary part of x(t) is constant
(d) The real part of x(t) is even
Ans:
(c)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q13: Let x(t) be a periodic signal with time period T, Let 𝑦(𝑡) = x(t − t0) + x(t + t0) for some t0. The Fourier Series coefficients of y(t) are denoted by bk. If b= 0 for all odd k , then t0 can be equal to      (2008 )
(a) T/8
(b) T/4
(c) T/2
(d) 2T
Ans: 
(b)
Sol: y(t) = x(t − t0) + x(t + x0)
Since, x(t) is periodic with period T.
Therefore, x(t − t0) and x(t + t0) will also be periodic with period T.  
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)ak is Fourier series coefficient of signal x(t)
therefore,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q14: A signal x(t) is given by  
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Which among the following gives the fundamental fourier term of x(t)?     (2007)
(a) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Ans: (a)
Sol: According to defination of signal given in question the x(t) will be as
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)So it is periodic with period,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Therefore, fundamental angular frequency
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Now,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Comparing it with
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q15: x(t) is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency ω = 2π(2k)/T; k = 1, 2... Also, no sine terms are present. Then x(t) satisfies the equation        (2006)
(a) 𝑥(𝑡)=𝑥(𝑡𝑇)x(t) = −x(t − T)
(b) x(t) = x(T − t) = −x(−t)
(c) 𝑥(𝑡)=𝑥(𝑇𝑡)=𝑥(𝑡𝑇/2)x(t) = x(T − t) = −x(t − T/2)
(d) x(t) = x(t − T) = x(t − T/2)
Ans: 
(c)
Sol: Since trigonometric fourier series has no sine terms and has only cosine terms therefore this will be an even signal i.e. it will satisfy.
x(t) = x(−t)
or, we can write,
 x(t − T) = x(−t + T)
but signal is periodic with period T.  
therefore x(t − T) = x(t)
therefore, x(t) = x(T − t)...(i)
Now, since signal contains only odd harmonics i.e. no terms of frequency
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)i.e. no even harmonics.
This means signal contains half wave symmetry
this implies that,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q16: The Fourier series for the function f(x) = sin2x is    (2005)
(a) sin𝑥+sin2𝑥sinx + sin2x
(b) 1cos2𝑥1 − cos2x  
(c) sin2x + cos2x
(d) 0.50.5cos2𝑥0.5 − 0.5cos2x  
Ans:
(d)
Sol: f(n) = sin2x
for finding the fourier series expansion
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q17: For the triangular wave from shown in the figure, the RMS value of the voltage is equal to    (2005)
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)(a)  Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) 1/3
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

Ans: (a)
Sol: From the wave symmetry,
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q18: The rms value of the periodic waveform given in figure is
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)(a) 2√6 A
(b) 6√2 A
(c) √4/3 A
(d) 1.5 A
Ans:
(a)
Sol: Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Q19: Fourier Series for the waveform, f(t) shown in figure is    (2002)
Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)(a) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE)
Ans: (c)
Sol: ∵ f(t) is an even function with half waves symmetry,
∴ dc term as well as sine terms = 0
Only the cosine terms with odd harmonics will be present.

The document Previous Year Questions- Fourier Series | Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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FAQs on Previous Year Questions- Fourier Series - Signals and Systems - Electrical Engineering (EE)

1. What is a Fourier series in electrical engineering?
Ans. In electrical engineering, a Fourier series is a mathematical tool used to represent a periodic function as a sum of sine and cosine functions. This representation helps in analyzing and understanding the behavior of signals in electrical systems.
2. How is a Fourier series used in signal processing?
Ans. Fourier series are commonly used in signal processing to analyze and manipulate signals. By decomposing a signal into its frequency components using Fourier series, engineers can filter out unwanted noise, compress data, and extract useful information from the signal.
3. What is the difference between Fourier series and Fourier transform?
Ans. Fourier series is used for periodic functions, while Fourier transform is used for non-periodic functions. Fourier series breaks down a periodic function into a sum of sine and cosine functions, while Fourier transform breaks down a non-periodic function into its frequency components.
4. How do engineers apply Fourier series in circuit analysis?
Ans. Engineers use Fourier series in circuit analysis to analyze the behavior of signals in circuits. By applying Fourier series, engineers can determine the harmonic components present in a signal, calculate power consumption, and design filters to control the signal's frequency content.
5. Can Fourier series be used to analyze real-world electrical signals?
Ans. Yes, Fourier series can be used to analyze real-world electrical signals. By decomposing a signal into its frequency components, engineers can identify noise, distortions, and other characteristics of the signal that are crucial for designing and optimizing electrical systems.
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