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

Q1: Let X(ω) be the Fourier transform of the signal,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The value of the derivative of X(ω) at ω = 0 at is _____ (rounded off to 1 decimal place)     (2024)
(a) 0
(b) 0.2
(c) 0.4
(d) 0.8
Ans:
(a)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q2: The discrete-time Fourier transform of a signal x[n] is X(Ω) = 1(1+cosΩ)e−jΩ. Consider that xp[n] is a periodic signal of period N = 5 such that
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Note that Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The magnitude of the Fourier series coefficient a3 is ____ (Round off to 3 decimal places).       (2023)
(a) 0.038
(b) 0.025
(c) 0.068
(d) 0.012
Ans: 
(a)
Sol: Given : xp(n) is a period signal of period N = 5.
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)We have,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q3: The Fourier transform X(ω) of the signal x(t) is given by  
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Which one of the following statements is true?       (2023)
(a) 𝑥(𝑡)x(t) tends to be an impulse as W0→∞
(b) x(0) decreases as W0 increases
(c) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)(d) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Ans:
(a)
Sol: Given,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)We know,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)By duality,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Given,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Thus,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)From rectangular function, At W ⇒ ∞, X(w) = 1
Taking inverse fourier transform x(t) = δ(t)
Option (A) will be correct.

Q4: Let an input x(t) = 2sin(10πt) + 5cos(15πt) + 7sin(42πt) + 4cos(45πt) is passed through an LTI system having an impulse response,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The output of the system is      (2022)
(a) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (c)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Fourier transform of signal Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) is given by
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Now, impulse response
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Using property, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Therefore, Fourier transform of impulse response
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Cut-off frequencies,
ωL= 30π rad/sec
ωH = 50π rad/sec
Thus, output of the system = 7sin 42πt + 4cos 45πt

Q5: Consider a continuous-time signal x(t) defined by x(t) = 0 for ∣t∣ >1, and x(t) = 1 − ∣t∣ for ∣t∣ ≤ 1. Let the Fourier transform of x(t) be defined as Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The maximum magnitude of X(ω) is _____.        (2021)
(a) 1
(b) 2
(c) 3
(d) 4
Ans:
(a)
Sol: Fourier transform, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
As A = 1, τ = 1
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)∵ Peak value of sampling function occurs at ω = 0  
Peak value = 1
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q6: Let f(t) be an even function, i.e. f(−t) = f(t) for all t. Let the Fourier transform of f(t) be defined as  Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) Suppose Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) for all ω, and F(0) = 1. Then      (2021)
(a) f(0) < 1
(b) f(0) > 1
(c) 𝑓(0)=1f(0) = 1
(d) f(0) = 0
Ans:
(a)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The following informations are given about Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)By solving the above linear differential equations, (by mathematics)
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Put ω = 0, F(0) = K
⇒ 1 = K (From info.
From (iv), Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
As we know, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Thus, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
At t = 0, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)

Q7: The Fourier transform of a continuous-time signal x(t) is given by Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
where Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) and ω denotes frequency. Then the value of |lnx(t)| at t = 1 is ___________ (up to 1 decimal place). ( ln denotes the logarithm to base e)        (2018)
(a) 10.0
(b) 7.5
(c) 11.8
(d) 2.8
Ans:
(a)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)By taking inverse Fourier transform,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q8: The value of the integral Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) is equal to      (SET-2 (2016))
(a) 0
(b) 0.5
(c) 1
(d) 2
Ans: 
(d)
Sol: The Fourier transform of
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q9: Suppose the maximum frequency in a band-limited signal x(t) is 5 kHz. Then, the maximum frequency in x(t) cos(2000πt), in kHz, is ________.       (SET-2 (2016))
(a) 5
(b) 6
(c) 7
(d) 8
Ans
: (b)
Sol: Maximum possible frequency of x(t)(2000πt) = f+ f2 = 5 + 1 = 6kHz

Q10: Suppose x1(t) and x2(t) have the Fourier transforms as shown below.
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Which one of the following statements is TRUE?      (SET-1 (2016))
(a) x1(t) and x2(t) are complex and x1(t) x2(t) is also complex with nonzero imaginary part
(b) x1(t) and x2(t) are real and x1(t) x2(t) is also real
(c) x1(t) and x2(t) are complex but x1(t) x2(t) is real
(d) 𝑥1(𝑡)𝑎𝑛𝑑𝑥2(𝑡)x1(t) and x2(t) are imaginary but x1(t) x2(t) is real
Ans:
(c)
Sol: By observing X1(jω) and X2(jω), we can say that they are not conjugate symmetric. Since, the fourier transform is not conjugate symmetric the signal will not be real. So, x1(t), x2(t) are not real. Now the fourier transform of  x1(t)⋅x2(t) will be  Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) and by looking at X1(jω) and X2(jω), we can say that X1(jω)∗X2(jω) will be conjugate symmetric and thus x1(t)⋅x2(t) will be real.
By observing X1(jω) and  X2(jω), we can say,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Now,  X1(jω) is real. Therefore,  x1(t) will be conjugate symmetric.  
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q11: Consider a signal defined by
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Its Fourier Transform is       (SET-2 (2015))
(a) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (a)
Sol: Since, Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q12: A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X(ω) and Y(ω). Which of the following statments is TRUE ?       (SET-3 ( 2014))
(a) X(ω) and Y(ω) are both real
(b) X(ω) is real and Y(ω) is imaginary
(c) X(ω) and Y(ω) are both imaginary
(d) X(ω) is imaginary and Y(ω) is real
Ans:
(b)
Sol: For real even function x(t), the Fourier transform X(ω) is always real even. y(t) is a derivative of x(t) which is a real odd function because derivative of even function is an odd function and hence, Fourier transform Y(ω) is imaginary odd.

Q13: A continuous-time LTI system with system function H(ω) has the following polezero plot. For this system, which of the alternatives is TRUE ?       (SET-3  (2014))
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)(a) 𝐻(0)>𝐻(𝜔);𝜔>0∣H(0)∣ > ∣H(ω)∣;∣ω∣ > 0
(b) 𝐻(𝜔)∣H(ω)∣ has multiple maxima, at ωand ω2 
(c) 𝐻(0)<𝐻(𝜔);𝜔>0∣H(0)∣ < ∣H(ω)∣ ; ∣ω∣ > 0
(d) ∣H(ω)∣ = constant;  −∞ < ω < ∞
Ans:
(d)
Sol: ⇒ Symmetrically located pole and zero.
⇒ All pass filter.
⇒ Constant magnitude
(−∞ ≤ ω ≤ ∞)

Q14: A signal is represented by
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The Fourier transform of the convolved signal y(t) = x(2t) ∗ x(t/2) is       (SET-3 (2014))
(a) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (a)
Sol: Given signal can be drawn as
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Therefore,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)then by time scaling,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Convolution in time domain multiplication in frequency domain
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q15: A function f(t) is shown in the figure.
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)The Fourier transform F(ω) of f(t) is      (SET-3 (2014))
(a) real and even function of w
(b) real and odd function of w
(c) imaginary and odd function of w
(d) imaginary and even function of w
Ans:
(c)
Sol: Fiven signal f(t) is an odd signal. Hence, F(ω) is imaginary and odd function of ω.

Q16: Let f(t) be a continuous time signal and let F(ω) be its Fourier Transform defined by Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Define g(t) by Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
What is the relationship between f(t) and g(t) ?        (SET-1 (2014))
(a) g(t) would always be proportional to f (t)
(b) g(t) would be proportional to f(t) if f(t) is an even function
(c) g(t) would be proportional to f(t) only if f(t) is a sinusoidal function
(d) g(t) would never be proportional to f(t)
Ans
: (b)
Sol: Given that,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q17: The Fourier transform of a signal h(t) is H(jω) = (2cosω)(sin2ω)/ω. The value of h(0) is     (2012)
(a) 1/4
(b) 1/2
(c) 1
(d) 2
Ans:
(c)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q18: x(t) is a positive rectangular pulse from t = -1 to t = +1 with unit height as shown in the figure. The value of Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) {where X(ω) is the Fourier transform of x(t)} is.      (2010)
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)(a) 2
(b) 2π
(c) 4
(d) 4π
Ans:
(d)
Sol: By using Parseval's theorem,
Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q19: Let Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE) and zero otherwise. Then if Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)  the Fourier Transform of x(t)+x(−t) will be given by       (2008)
(a) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (c)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)
Q20: A signal x(t) = sinc(αt) where α is a real constant Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)is the input to a Linear Time Invariant system whose impulse response h(t) = sinc(βt), where β is a real constant. If min (α, β) denotes the minimum of α and β and similarly, max (α, β) denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the output of the system ?        (2008)
(a) It will be of the form Ksinc(γt) where γ = min(α, β)
(b) It will be of the form Ksinc(γt) where γ = max (α, β)
(c) It will be of the form Ksinc(αt)
(d) It can not be a sinc type of signal
Ans:
(a)
Sol: Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Fourier Transform | Signals and Systems - Electrical Engineering (EE)So, output is of the form k sin c(γt)
where, γ = min(α, β)  

The document Previous Year Questions- Fourier Transform | 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 Transform - Signals and Systems - Electrical Engineering (EE)

1. What is the Fourier Transform in electrical engineering?
Ans. The Fourier Transform in electrical engineering is a mathematical tool used to analyze and transform signals from the time domain to the frequency domain. It decomposes a signal into its constituent frequencies, allowing for a better understanding of the signal's behavior.
2. How is the Fourier Transform calculated for a continuous signal?
Ans. The Fourier Transform for a continuous signal is calculated using the integral formula: F(ω) = ∫ f(t) e^(-jωt) dt, where F(ω) is the frequency domain representation of the signal, f(t) is the time domain signal, ω is the frequency variable, and j is the imaginary unit.
3. What is the difference between the Fourier Transform and the Laplace Transform in electrical engineering?
Ans. The Fourier Transform is used for signals that are periodic and have a finite energy, while the Laplace Transform is used for signals that are transient and have infinite energy. Additionally, the Laplace Transform includes information about the initial conditions of a system, which the Fourier Transform does not.
4. What are some common applications of the Fourier Transform in electrical engineering?
Ans. The Fourier Transform is used in various applications such as signal processing, communication systems, image processing, and control systems. It is particularly useful for analyzing the frequency content of signals and filtering out unwanted noise.
5. How can the Fourier Transform be used to analyze the frequency response of a system in electrical engineering?
Ans. By taking the Fourier Transform of the system's impulse response, the frequency response of the system can be obtained. This allows engineers to analyze how the system responds to different frequencies and design filters or control systems accordingly.
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