Question

Parseval's Theorem for Fourier Integral question:

If x(t) = sin(2t)/(pi*t), compute the energy of d²x/dt² using Parseval's Theorem for Fourier Integral

Answer 1

Parseval's Theorem for Fourier Integral states that the integral of the square of a function in the time domain is equal to the integral of the square of its Fourier transform in the frequency domain.

Step-by-step explanation:

Parseval's Theorem for Fourier Integral states that the integral of the square of a function in the time domain is equal to the integral of the square of its Fourier transform in the frequency domain. In this case, we need to compute the energy of the second derivative of x(t).

The Fourier transform of x(t) can be obtained as X(w) = F{x(t)} = π * [δ(w-2) - δ(w+2)] / 2, where δ(w) is the Dirac delta function.

Using Parseval's Theorem, the energy of the second derivative of x(t) can be calculated as E = (1/2π) * ∫|d²x/dt²|² dt = (1/2π) * ∫|(-4)w²X(w)|² dw = (1/2π) * ∫(16w⁴)|X(w)|² dw = 16 * ∫(w⁴)|X(w)|² dw.