Time Differentiation MCQ Quiz - Objective Question with Answer for Time Differentiation - Download Free PDF

Concept:

Fourier Transform:

• The Fourier transform of a function converts it from time domain to frequency domain.
• If f(t) is a time-domain function, its Fourier Transform is defined as:
• For Gaussian functions like  , their Fourier transform is also Gaussian.
• The Fourier transform of    is  .
• Multiplying by t in time domain corresponds to taking derivative with respect to p in frequency domain:

Calculation:

Given,

Let f(t) = t e^−t²⁄2

Let F(p) = Fourier transform of e^−t²⁄2 = e^−p²⁄2

⇒ Fourier transform of t f(t) = i × d/dp (e^−p²⁄2)

⇒ = i × (−p e^−p²⁄2) = −i p e^−p²⁄2

⇒ Now t e^−t²⁄2 = f(t),

so full FT is i × d/dp (F(p))

⇒ Add F(p) itself:

Final result = (1 + i p) e^{−(p² − 1)/2}

∴ The correct Fourier transform is:   

Let

The

Now,

Let

Taking Fourier transform

Thus,

Concept:

Fourier Transform:

• The Fourier transform of a function converts it from time domain to frequency domain.
• If f(t) is a time-domain function, its Fourier Transform is defined as:
• For Gaussian functions like  , their Fourier transform is also Gaussian.
• The Fourier transform of    is  .
• Multiplying by t in time domain corresponds to taking derivative with respect to p in frequency domain:

Calculation:

Given,

Let f(t) = t e^−t²⁄2

Let F(p) = Fourier transform of e^−t²⁄2 = e^−p²⁄2

⇒ Fourier transform of t f(t) = i × d/dp (e^−p²⁄2)

⇒ = i × (−p e^−p²⁄2) = −i p e^−p²⁄2

⇒ Now t e^−t²⁄2 = f(t),

so full FT is i × d/dp (F(p))

⇒ Add F(p) itself:

Final result = (1 + i p) e^{−(p² − 1)/2}

∴ The correct Fourier transform is:   

(differentiation property in frequency domain)

( differentiation property in time domain)

at  t = 1/3, δ(t) = 0

We know that,

From the property of delta function

F (t) δ (t – t₀) = f (t₀) δ (t – t₀)

⇒ t e^-4t δ (t) = 0 e^-4(0) δ (t) = 0

⇒ x (t) = -4te^-4t u(t) + e^-4t u(t)

Differentiating given signal w.r.t time

Again differentiating w.r.t time we get

The Fourier transform of above signal is

≃ 0.26

Let 

The 

Now,

Let 

Taking Fourier transform

Thus,

We have 

Taking Fourier transform we have,