Previous Year Questions- Fourier Transform

Q1: Let X(ω) be the Fourier transform of the signal,
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: 
Q2: The discrete-time Fourier transform of a signal x[n] is X(Ω) = 1(1+cosΩ)e^-jΩ. Consider that x_p[n] is a periodic signal of period N = 5 such that
Note that 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 : x_p(n) is a period signal of period N = 5.
We have,

Q3: The Fourier transform X(ω) of the signal x(t) is given by  
Which one of the following statements is true?       (2023)
(a) $𝑥\left(𝑡\right)$x(t) tends to be an impulse as W₀→∞
(b) x(0) decreases as W₀ increases
(c) (d) Ans: (a)
Sol: Given,
We know,
By duality,
Given,
Thus,
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,
The output of the system is      (2022)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: Fourier transform of signal  is given by
Now, impulse response
Using property,
Therefore, Fourier transform of impulse response
Cut-off frequencies,
ω_L= 30π rad/sec
ω_H = 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 The maximum magnitude of X(ω) is _____.        (2021)
(a) 1
(b) 2
(c) 3
(d) 4
Ans: (a)
Sol: Fourier transform,
As A = 1, τ = 1
∵ Peak value of sampling function occurs at ω = 0  
Peak value = 1

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   Suppose  for all ω, and F(0) = 1. Then      (2021)
(a) f(0) < 1
(b) f(0) > 1
(c) $𝑓\left(0\right)=1$f(0) = 1
(d) f(0) = 0
Ans: (a)
Sol: The following informations are given about By solving the above linear differential equations, (by mathematics)
⇒
Put ω = 0, F(0) = K
⇒ 1 = K (From info.
From (iv),
As we know,
Thus,
At t = 0,

Q7: The Fourier transform of a continuous-time signal x(t) is given by
where  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: By taking inverse Fourier transform,

Q8: The value of the integral  is equal to      (SET-2 (2016))
(a) 0
(b) 0.5
(c) 1
(d) 2
Ans: (d)
Sol: The Fourier transform of

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) = f₁ + f₂ = 5 + 1 = 6kHz

Q10: Suppose x₁(t) and x₂(t) have the Fourier transforms as shown below.
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) $𝑥1\left(𝑡\right)𝑎𝑛𝑑𝑥2\left(𝑡\right)$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   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,
Now,  X₁(jω) is real. Therefore,  x₁(t) will be conjugate symmetric.  

Q11: Consider a signal defined by
Its Fourier Transform is       (SET-2 (2015))
(a)
(b)
(c)
(d)
Ans: (a)
Sol: Since,
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))
(a) $\mid 𝐻\left(0\right)\mid >\mid 𝐻\left(𝜔\right)\mid ;\mid 𝜔\mid >0$∣H(0)∣ > ∣H(ω)∣;∣ω∣ > 0
(b) $\mid 𝐻\left(𝜔\right)\mid$∣H(ω)∣ has multiple maxima, at ω₁ and ω₂ 
(c) $\mid 𝐻\left(0\right)\mid <\mid 𝐻\left(𝜔\right)\mid ;\mid 𝜔\mid >0$∣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
The Fourier transform of the convolved signal y(t) = x(2t) ∗ x(t/2) is       (SET-3 (2014))
(a)
(b)
(c)
(d)
Ans: (a)
Sol: Given signal can be drawn as
Therefore,
then by time scaling,
Convolution in time domain multiplication in frequency domain

Q15: A function f(t) is shown in the figure.
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
Define g(t) by
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,

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: 
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  {where X(ω) is the Fourier transform of x(t)} is.      (2010)
(a) 2
(b) 2π
(c) 4
(d) 4π
Ans: (d)
Sol: By using Parseval's theorem,

Q19: Let  and zero otherwise. Then if   the Fourier Transform of x(t)+x(-t) will be given by       (2008)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: 
Q20: A signal x(t) = sinc(αt) where α is a real constant 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: So, output is of the form k sin c(γt)
where, γ = min(α, β)