Convolution in Time Domain MCQ Quiz - Objective Question with Answer for Convolution in Time Domain - Download Free PDF

Last updated on Apr 3, 2025

Latest Convolution in Time Domain MCQ Objective Questions

Convolution in Time Domain Question 1:

Let x1(t) = u(t + 1.5) − u(t − 1.5) and x2(t) is shown in the figure below. For y(t) = x1(t) ∗ x2(t), the y(t)dt is ________ (rounded off to the nearest integer). 

F1 Engineering Arbaz 27-12-23 D8

Answer (Detailed Solution Below) 15

Convolution in Time Domain Question 1 Detailed Solution

F1 Engineering Arbaz 27-12-23 D9

x1(t) = u(t + 1.5) - u(t - 1.5)

x1(t)=rect(t3)

x1(t)=rect(t3)FT3Sa(1.5ω)

Now, x2(t)=δ(t+3)+rect(t2)+2δ(t2)

Taking Fourier transform

X2(ω)=e3jω+2Sa(ω)+2e2kω

∵  y(t)=x1(t)x2(t)

Y(ω)=X1(ω)X2(ω)

We know, Y(ω)=y(t)ejωtdt

∴ y(t)=Y(0)

∴ Y(0) = X1(0).X2(0)

= 3[1 + 2 + 2] = 15

Convolution in Time Domain Question 2:

Let x1(t) = e-tu(t) and x2(t) = u(t) - u(t - 2), where u(⋅) denotes the unit step function.

If y(t) denotes the convolution of x1(t) and x2(t), then limty(t)= __________ (rounded off to one decimal place).

Answer (Detailed Solution Below) 0

Convolution in Time Domain Question 2 Detailed Solution

Correct answer of the question is 0.

Solution:

Given that,       

 x1(t)=etu(t)x2(t)=u(t)u(t2)y(t)=x1(t)x2(t)                         

By applying Laplace Transform

x1(t)=etu(t)X1(s)=1s+1x2(t)=u(t)u(t2)X2(s)=1se2ss=1s(1e2s)y(t)=x1(t)x2(t)Y(s)=X1(s)×X2(s)Y(s)=1e2ss(s+1)                          

By applying Final Value Theorem:

limtf(t)=lims0s.F(s)=lims0s(1e2s)s(s+1)=(11)1=0                                                                  

Hence, the answer to the question is 0.

Convolution in Time Domain Question 3:

Let z(t) be the output of the first system and the input to the second system in the cascade. Find the output y(t).

quesOptionImage905

  1. y(t) = x(t) ⊗ b2(t)
  2. y(t) = z(t) ⊗ b2(t)
  3. y(t) = z(t) ⊗ x(t)
  4. y(t) = z(t) + b2(t)
  5. y(t) = b1(t) ⊗ b2(t)

Answer (Detailed Solution Below)

Option 2 : y(t) = z(t) ⊗ b2(t)

Convolution in Time Domain Question 3 Detailed Solution

Analysis:

When two blocks are cascaded their impulse responses are to be convolved.

quesImage3891

Here two blocks are cascaded, then equivalent impulse response of the whole system will be:

b1(t) ⊗ b2(t)

In the above figure:

z(t) = x(t) ⊗ b1(t)

y(t) = z(t) ⊗ b2(t)

The expression for y(t) in terms of x(t) will be:

y(t) =x(t) ⊗ (b1(t)⊗b2(t))

Important Points

Properties of convolution:

Associativity: f* (f* f3) = (f* f2) * f3

Commutativity: f* f2 = f* f

Distribitivity: f* (f2 + f3) = f* f2 + f* f3

Multilinearity: a(f* f2) = (af1) * f2 = f1(af2

Convolution in Time Domain Question 4:

Let z(t) be the output of the first system and the input to the second system in the cascade. Find the output y(t).

quesOptionImage905

  1. y(t) = x(t) ⊗ b2(t)
  2. y(t) = z(t) ⊗ b2(t)
  3. y(t) = z(t) ⊗ x(t)
  4. y(t) = z(t) + b2(t)

Answer (Detailed Solution Below)

Option 2 : y(t) = z(t) ⊗ b2(t)

Convolution in Time Domain Question 4 Detailed Solution

Analysis:

When two blocks are cascaded their impulse responses are to be convolved.

quesImage3891

Here two blocks are cascaded, then equivalent impulse response of the whole system will be:

b1(t) ⊗ b2(t)

In the above figure:

z(t) = x(t) ⊗ b1(t)

y(t) = z(t) ⊗ b2(t)

The expression for y(t) in terms of x(t) will be:

y(t) =x(t) ⊗ (b1(t)⊗b2(t))

Important Points

Properties of convolution:

Associativity: f* (f* f3) = (f* f2) * f3

Commutativity: f* f2 = f* f

Distribitivity: f* (f2 + f3) = f* f2 + f* f3

Multilinearity: a(f* f2) = (af1) * f2 = f1(af2

Convolution in Time Domain Question 5:

Consider the system with x(t) as input and y(t) as output. The frequency-domain characteristics are shown in the figure. Which combination of A and B will give ‘y’ as a result?

F1 S.B Madhu 7.11.19 D 5

  1. F1 S.B Madhu 7.11.19 D 6
  2. F1 S.B Madhu 7.11.19 D 7
  3. F1 S.B Madhu 7.11.19 D 8
  4. F1 S.B Madhu 7.11.19 D 9

Answer (Detailed Solution Below)

Option 1 : F1 S.B Madhu 7.11.19 D 6

Convolution in Time Domain Question 5 Detailed Solution

Calculation:

Analyzing the circuit by taking each Option into consideration we proceed as follows:

First considering option (1)

i.e. if the Frequency Spectrum of A is as shown:

F1 S.B. Nita 14.11.2019 D 3

The output when the signal x(t) passes through A will simply be the multiplication of j in the frequency spectrum of x(t). The output signal spectrum from Block A will be, therefore:

F1 S.B. Nita 14.11.2019 D 4

When the above spectrum is multiplied with j (as seen in the circuit given), the output becomes;

F1 S.B. Nita 14.11.2019 D 5

When the above signal is then passed through a summer, the resultant spectrum will be as shown:

F1 S.B. Nita 14.11.2019 D 6

But given y(t) is a shifted spectrum of the above, and we know that;

If x(t) « X(f)

x(t). ejωot    X(f+fo)

i.e. when multiplied with a single negative frequency exponent, the spectrum is shifted to the left by f0.

If B = F1 S.B. Nita 14.11.2019 D 7

The output spectrum will be:

F1 S.B. Nita 14.11.2019 D 8

So Option (1) shows the Correct representation of A and B.

Top Convolution in Time Domain MCQ Objective Questions

If the signal x(t)=sin(t)πtsin(t)πt  with ∗ denoting the convolution operation, then x(t) is equal to

  1. sin(t)πt
  2. sin(2t)2πt
  3. 2sin(t)πt
  4. (sin(t)πt)2

Answer (Detailed Solution Below)

Option 1 : sin(t)πt

Convolution in Time Domain Question 6 Detailed Solution

Download Solution PDF

Concept:

x(t)={A;|t|τ20;|t|>τ2

x(t)={A;τ2tτ20;otherwise

This signal can be represented as:

F1 U.B Deepak 28.03.2020 D2

x(t)=A[u(tτ2)u(τ+12)]

By applying the Fourier transform,

X(ω)=τ2τ2A.ejωtdt

=Ajω[ejω2ejω2]

=2Aejωτ2ejωτ22jω

=2Aωsinωτ2

=2Aτωτsinωτ2

=Aτsinωτ2ωτ2

By putting ω = 2πF

=AτsinπFτπFτ

Calculation:

x(t)=sintπtsintπt

The Fourier transform of the given since function will be a rectangular wave as shown:

F2 S.B 25.6.20 Pallavi D2

Since the convolution time domain is the multiplication in the frequency domain, the Fourier transform (Spectrum) of x(t) will be:

F2 S.B 25.6.20 Pallavi D3

Now inverse Fourier, we get the time-domain expression of x(t) as:

Gate EC 2016 paper 3 Images-Q7.1

Thus, sintπtsintπt=sintπt

Let z(t) be the output of the first system and the input to the second system in the cascade. Find the output y(t).

quesOptionImage905

  1. y(t) = x(t) ⊗ b2(t)
  2. y(t) = z(t) ⊗ b2(t)
  3. y(t) = z(t) ⊗ x(t)
  4. y(t) = z(t) + b2(t)

Answer (Detailed Solution Below)

Option 2 : y(t) = z(t) ⊗ b2(t)

Convolution in Time Domain Question 7 Detailed Solution

Download Solution PDF

Analysis:

When two blocks are cascaded their impulse responses are to be convolved.

quesImage3891

Here two blocks are cascaded, then equivalent impulse response of the whole system will be:

b1(t) ⊗ b2(t)

In the above figure:

z(t) = x(t) ⊗ b1(t)

y(t) = z(t) ⊗ b2(t)

The expression for y(t) in terms of x(t) will be:

y(t) =x(t) ⊗ (b1(t)⊗b2(t))

Important Points

Properties of convolution:

Associativity: f* (f* f3) = (f* f2) * f3

Commutativity: f* f2 = f* f

Distribitivity: f* (f2 + f3) = f* f2 + f* f3

Multilinearity: a(f* f2) = (af1) * f2 = f1(af2

Consider the system with x(t) as input and y(t) as output. The frequency-domain characteristics are shown in the figure. Which combination of A and B will give ‘y’ as a result?

F1 S.B Madhu 7.11.19 D 5

  1. F1 S.B Madhu 7.11.19 D 6
  2. F1 S.B Madhu 7.11.19 D 7
  3. F1 S.B Madhu 7.11.19 D 8
  4. F1 S.B Madhu 7.11.19 D 9

Answer (Detailed Solution Below)

Option 1 : F1 S.B Madhu 7.11.19 D 6

Convolution in Time Domain Question 8 Detailed Solution

Download Solution PDF

Calculation:

Analyzing the circuit by taking each Option into consideration we proceed as follows:

First considering option (1)

i.e. if the Frequency Spectrum of A is as shown:

F1 S.B. Nita 14.11.2019 D 3

The output when the signal x(t) passes through A will simply be the multiplication of j in the frequency spectrum of x(t). The output signal spectrum from Block A will be, therefore:

F1 S.B. Nita 14.11.2019 D 4

When the above spectrum is multiplied with j (as seen in the circuit given), the output becomes;

F1 S.B. Nita 14.11.2019 D 5

When the above signal is then passed through a summer, the resultant spectrum will be as shown:

F1 S.B. Nita 14.11.2019 D 6

But given y(t) is a shifted spectrum of the above, and we know that;

If x(t) « X(f)

x(t). ejωot    X(f+fo)

i.e. when multiplied with a single negative frequency exponent, the spectrum is shifted to the left by f0.

If B = F1 S.B. Nita 14.11.2019 D 7

The output spectrum will be:

F1 S.B. Nita 14.11.2019 D 8

So Option (1) shows the Correct representation of A and B.

Let x1(t) = e-tu(t) and x2(t) = u(t) - u(t - 2), where u(⋅) denotes the unit step function.

If y(t) denotes the convolution of x1(t) and x2(t), then limty(t)= __________ (rounded off to one decimal place).

Answer (Detailed Solution Below) 0

Convolution in Time Domain Question 9 Detailed Solution

Download Solution PDF

Correct answer of the question is 0.

Solution:

Given that,       

 x1(t)=etu(t)x2(t)=u(t)u(t2)y(t)=x1(t)x2(t)                         

By applying Laplace Transform

x1(t)=etu(t)X1(s)=1s+1x2(t)=u(t)u(t2)X2(s)=1se2ss=1s(1e2s)y(t)=x1(t)x2(t)Y(s)=X1(s)×X2(s)Y(s)=1e2ss(s+1)                          

By applying Final Value Theorem:

limtf(t)=lims0s.F(s)=lims0s(1e2s)s(s+1)=(11)1=0                                                                  

Hence, the answer to the question is 0.

Convolution in Time Domain Question 10:

If x(t)=2eπt2 and h(t)=eπt24. If y(t) is the convolution of these two signals, then the value of y(t) at t = 0 is  _____

Answer (Detailed Solution Below) 1.7 - 1.9

Convolution in Time Domain Question 10 Detailed Solution

y(t) = x(t) * h(t)

Y(f) = X(f) × H(f)

x(t)=2eπt22eπf2

h(t)=eπt242eπ4f2

Y(f)=2eπf22e4πf2=4e5πf2

y(t)=Y(f)ej2πftdf

 y(0)=Y(f)df=4e5πf2df

5f2 = u2

5f=u

⇒ du = √5 df

y(0)=4eπu2du5=45eπu2du

y(0)=45×1=45

Convolution in Time Domain Question 11:

If the signal x(t)=sin(t)πtsin(t)πt  with ∗ denoting the convolution operation, then x(t) is equal to

  1. sin(t)πt
  2. sin(2t)2πt
  3. 2sin(t)πt
  4. (sin(t)πt)2

Answer (Detailed Solution Below)

Option 1 : sin(t)πt

Convolution in Time Domain Question 11 Detailed Solution

Concept:

x(t)={A;|t|τ20;|t|>τ2

x(t)={A;τ2tτ20;otherwise

This signal can be represented as:

F1 U.B Deepak 28.03.2020 D2

x(t)=A[u(tτ2)u(τ+12)]

By applying the Fourier transform,

X(ω)=τ2τ2A.ejωtdt

=Ajω[ejω2ejω2]

=2Aejωτ2ejωτ22jω

=2Aωsinωτ2

=2Aτωτsinωτ2

=Aτsinωτ2ωτ2

By putting ω = 2πF

=AτsinπFτπFτ

Calculation:

x(t)=sintπtsintπt

The Fourier transform of the given since function will be a rectangular wave as shown:

F2 S.B 25.6.20 Pallavi D2

Since the convolution time domain is the multiplication in the frequency domain, the Fourier transform (Spectrum) of x(t) will be:

F2 S.B 25.6.20 Pallavi D3

Now inverse Fourier, we get the time-domain expression of x(t) as:

Gate EC 2016 paper 3 Images-Q7.1

Thus, sintπtsintπt=sintπt

Convolution in Time Domain Question 12:

Let z(t) be the output of the first system and the input to the second system in the cascade. Find the output y(t).

quesOptionImage905

  1. y(t) = x(t) ⊗ b2(t)
  2. y(t) = z(t) ⊗ b2(t)
  3. y(t) = z(t) ⊗ x(t)
  4. y(t) = z(t) + b2(t)

Answer (Detailed Solution Below)

Option 2 : y(t) = z(t) ⊗ b2(t)

Convolution in Time Domain Question 12 Detailed Solution

Analysis:

When two blocks are cascaded their impulse responses are to be convolved.

quesImage3891

Here two blocks are cascaded, then equivalent impulse response of the whole system will be:

b1(t) ⊗ b2(t)

In the above figure:

z(t) = x(t) ⊗ b1(t)

y(t) = z(t) ⊗ b2(t)

The expression for y(t) in terms of x(t) will be:

y(t) =x(t) ⊗ (b1(t)⊗b2(t))

Important Points

Properties of convolution:

Associativity: f* (f* f3) = (f* f2) * f3

Commutativity: f* f2 = f* f

Distribitivity: f* (f2 + f3) = f* f2 + f* f3

Multilinearity: a(f* f2) = (af1) * f2 = f1(af2

Convolution in Time Domain Question 13:

Let z(t) be the output of the first system and the input to the second system in the cascade. Find the output y(t).

quesOptionImage905

  1. y(t) = x(t) ⊗ b2(t)
  2. y(t) = z(t) ⊗ b2(t)
  3. y(t) = z(t) ⊗ x(t)
  4. y(t) = z(t) + b2(t)
  5. y(t) = b1(t) ⊗ b2(t)

Answer (Detailed Solution Below)

Option 2 : y(t) = z(t) ⊗ b2(t)

Convolution in Time Domain Question 13 Detailed Solution

Analysis:

When two blocks are cascaded their impulse responses are to be convolved.

quesImage3891

Here two blocks are cascaded, then equivalent impulse response of the whole system will be:

b1(t) ⊗ b2(t)

In the above figure:

z(t) = x(t) ⊗ b1(t)

y(t) = z(t) ⊗ b2(t)

The expression for y(t) in terms of x(t) will be:

y(t) =x(t) ⊗ (b1(t)⊗b2(t))

Important Points

Properties of convolution:

Associativity: f* (f* f3) = (f* f2) * f3

Commutativity: f* f2 = f* f

Distribitivity: f* (f2 + f3) = f* f2 + f* f3

Multilinearity: a(f* f2) = (af1) * f2 = f1(af2

Convolution in Time Domain Question 14:

Let x1(t) = u(t + 1.5) − u(t − 1.5) and x2(t) is shown in the figure below. For y(t) = x1(t) ∗ x2(t), the y(t)dt is ________ (rounded off to the nearest integer). 

F1 Engineering Arbaz 27-12-23 D8

Answer (Detailed Solution Below) 15

Convolution in Time Domain Question 14 Detailed Solution

F1 Engineering Arbaz 27-12-23 D9

x1(t) = u(t + 1.5) - u(t - 1.5)

x1(t)=rect(t3)

x1(t)=rect(t3)FT3Sa(1.5ω)

Now, x2(t)=δ(t+3)+rect(t2)+2δ(t2)

Taking Fourier transform

X2(ω)=e3jω+2Sa(ω)+2e2kω

∵  y(t)=x1(t)x2(t)

Y(ω)=X1(ω)X2(ω)

We know, Y(ω)=y(t)ejωtdt

∴ y(t)=Y(0)

∴ Y(0) = X1(0).X2(0)

= 3[1 + 2 + 2] = 15

Convolution in Time Domain Question 15:

Consider the system with x(t) as input and y(t) as output. The frequency-domain characteristics are shown in the figure. Which combination of A and B will give ‘y’ as a result?

F1 S.B Madhu 7.11.19 D 5

  1. F1 S.B Madhu 7.11.19 D 6
  2. F1 S.B Madhu 7.11.19 D 7
  3. F1 S.B Madhu 7.11.19 D 8
  4. F1 S.B Madhu 7.11.19 D 9

Answer (Detailed Solution Below)

Option 1 : F1 S.B Madhu 7.11.19 D 6

Convolution in Time Domain Question 15 Detailed Solution

Calculation:

Analyzing the circuit by taking each Option into consideration we proceed as follows:

First considering option (1)

i.e. if the Frequency Spectrum of A is as shown:

F1 S.B. Nita 14.11.2019 D 3

The output when the signal x(t) passes through A will simply be the multiplication of j in the frequency spectrum of x(t). The output signal spectrum from Block A will be, therefore:

F1 S.B. Nita 14.11.2019 D 4

When the above spectrum is multiplied with j (as seen in the circuit given), the output becomes;

F1 S.B. Nita 14.11.2019 D 5

When the above signal is then passed through a summer, the resultant spectrum will be as shown:

F1 S.B. Nita 14.11.2019 D 6

But given y(t) is a shifted spectrum of the above, and we know that;

If x(t) « X(f)

x(t). ejωot    X(f+fo)

i.e. when multiplied with a single negative frequency exponent, the spectrum is shifted to the left by f0.

If B = F1 S.B. Nita 14.11.2019 D 7

The output spectrum will be:

F1 S.B. Nita 14.11.2019 D 8

So Option (1) shows the Correct representation of A and B.

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