Parsevals theorem for fourier transforms

Question

Assignment:

Evaluate the following integrals using Rayleigh’s energy theorem (This is Parseval’s theorem for Fourier transforms).  All integrals spans between –∞ and ∞.

I1 = ∫ df / [α² + (2πf)²]

I2 = ∫ sinc2(τf) df

I3 = ∫ df / [α² + (2πf)²]²

I4 = ∫ sinc⁴(τf) df

Using Rayleigh’s Theorem

Rayleigh’s energy theorem (Parseval’s theorem for Fourier transforms) is convenient for finding the energy in a signal whose square is not easily integrated in the time domain. or vie versa.

For example, the signal x(t) = 40 sinc (20t) has energy density

G( f ) =  | X( f ) |2 =  [2Π ( f/20 )]2 = 4Π ( f/20 ) where Π(f/20) need not be squared because it has unity amplitude.  Using Rayleigh’s energy theorem, you can find that energy in x(t) is

E = ∫G(f) df = ∫ 4 df = 80 J  (1st integral span between -∞ and ∞ and 2nd integral span between -10 and 10).

This would check with the result that is obtained by integrating x2(t) over all t using the definite integral ∫ sinc2 (u) du = 1.

The energy contained in the frequency interval (0, W) can be found from the integral

Ew = ∫ G(f) df = 2 ∫ [2Π ( f/20 )]2 df (1st integral span -W and W, 2nd integral span 0 and W).

= 8W when W ≤ 10
= 80 when W > 10

Provide complete and step by step solution for the question and show calculations and use formulas.