Q32P Verify Parseval鈥檚 theorem (12.... [FREE SOLUTION]

Chapter 7: Q32P (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 21.

Short Answer

Expert verified

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

Step by step solution

Given Information.

The given functions are $f (x) = e^{- \frac{x^{2}}{2 蟽^{2}}}$ and $g (伪) = \frac{蟽}{\sqrt{2 蟺}} e^{- \frac{蟽^{2} 伪^{2}}{2}}$ .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

It is known that Gaussian integral is ${鈭�}_{- 鈭�}^{鈭�} e^{- x^{2}} d x = \sqrt{蟺}$

Use the Gaussian integral to verify Parseval theorem

$\begin{matrix} \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} e^{\frac{- x^{2}}{蟽^{2}}} d (x 蟽) 蟽 = \frac{蟽}{2 蟺} \sqrt{蟺} \\ = \frac{蟽}{2 \sqrt{蟺}} \end{matrix}$

And

$\begin{matrix} \frac{蟽^{2}}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} e^{- 蟽^{2} 伪^{2}} d (伪蟽) \frac{1}{蟽} = \frac{蟽}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} e^{- x^{2}} d x \\ = \frac{蟽}{2 蟺} \sqrt{蟺} \\ = \frac{蟽}{2 \sqrt{蟺}} \end{matrix}$

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

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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 21.

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Verify Parseval Theorem

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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 21.

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

Nuclear Physics

Quantum Physics

Astrophysics

Physics of Motion

Turning Points in Physics

Force

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 21.