Q33P Verify Parseval鈥檚 theorem (12.... [FREE SOLUTION]

Chapter 7: Q33P (page 386)

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.
32. $f (x)$ and $g (伪)$ as in problem 24a.

Short Answer

Expert verified

${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} {| f (x) |}^{2} d x = \frac{1}{2 蟺}$ .Thus the Parseval theorem is confirmed.

Step by step solution

Given Information.

The given functions are $f (x) = e^{- | x |}$ and $g (伪) = \frac{1}{蟺 (1 + 伪^{2})}$ .Parseval theorem is to be verified for this special case.

Definition of Parseval’s theorem

Parseval鈥檚 theorem is a theorem stating that the energy of a signal can be expressed as its frequency components鈥� average energy.

Verify Parseval Theorem

$\begin{matrix} \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} {| f (x) |}^{2} d x = \frac{1}{蟺} {鈭�}_{0}^{鈭�} e^{- 2 x} d (- 2 x) \frac{- 1}{2} \\ = - \frac{1}{2 蟺} {[e^{- 2 x}]}_{0}^{鈭�} \\ = \frac{1}{2 蟺} \end{matrix}$

And

$\begin{matrix} {鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{1}{蟺^{2}} {鈭�}_{- 鈭�}^{鈭�} \frac{d 伪}{{(1 + 伪^{2})}^{2}} \\ = \frac{1}{蟺^{2}} {[\frac{伪}{2 (1 + 伪^{2})} + \frac{1}{2} a r c t a n 伪]}_{- 鈭�}^{鈭�} \\ = \frac{1}{2 蟺^{2}} (\frac{蟺}{2} - (- \frac{蟺}{2})) \\ = \frac{1}{2 蟺} \end{matrix}$

${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} {| f (x) |}^{2} d x = \frac{1}{2 蟺}$ .Thus the Parseval theorem is confirmed.

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Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.
32. $f (x)$ and $g (伪)$ as in problem 24a.

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Verify Parseval Theorem

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.32. f(x)and g(伪)as in problem 24a.

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Verify Parseval Theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Rotational Dynamics

Particle Model of Matter

Magnetism and Electromagnetic Induction

Electricity

Engineering Physics

Study anywhere. Anytime. Across all devices.

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.
32. $f (x)$ and $g (伪)$ as in problem 24a.