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 xp[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 : xp(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(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,
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 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(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(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) ∣H(0)∣ > ∣H(ω)∣;∣ω∣ > 0
(b) ∣H(ω)∣ has multiple maxima, at ω1 and ω2
(c) ∣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)