Q31P Verify Parseval鈥檚 theorem (12.... [FREE SOLUTION]

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Chapter 7: Q31P (page 386)

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.
31. $f (x)$ as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate ${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪$ .

Short Answer

Expert verified

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

Step by step solution

Given Information.

The given function is ${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪$ .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 the function and its Fourier transform are

$\begin{array}{l} f (x) = \frac{2}{蟺} {鈭�}_{0}^{鈭�} \frac{s i n 伪 c o s (伪 x)}{伪} d 伪 \\ g (伪) = \frac{s i n 伪}{伪蟺} \end{array}$

Thus, $\begin{matrix} \frac{1}{2 蟺} {鈭�}_{鈭� 鈭�}^{鈭�} {| f (x) |}^{2} d x = \frac{1}{2 蟺} {鈭�}_{- 1}^{1} d x \\ = \frac{1}{蟺} \end{matrix}$

And

$\begin{matrix} {鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{1}{蟺^{2}} {鈭�}_{- 鈭�}^{鈭�} \frac{s i n^{2} 伪}{伪^{2}} d 伪 \\ = \frac{1}{蟺^{2}} {鈭�}_{- 鈭�}^{鈭�} \frac{d}{d 伪} (- \frac{s i n^{2} 伪}{伪}) d 伪 + \frac{1}{蟺^{2}} {鈭�}_{- 鈭�}^{鈭�} \frac{d}{d 伪} (\frac{2 s i n 伪 c o s 伪}{伪}) d 伪 \\ = 0 + \frac{4}{蟺^{2}} {鈭�}_{0}^{鈭�} \frac{s i n 伪 c o s 伪}{伪} d 伪 \end{matrix}$

It is known that ${鈭�}_{0}^{鈭�} \frac{s i n 伪 c o s 伪}{伪} d 伪 = \frac{蟺}{4}$

Therefore

$\begin{matrix} {鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{4}{蟺^{2}} {鈭�}_{0}^{鈭�} \frac{s i n 伪 c o s 伪}{伪} d 伪 \\ = \frac{4}{蟺^{2}} \frac{蟺}{4} \\ = \frac{1}{蟺} \end{matrix}$

${鈭�}_{鈭� 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪 = \frac{1}{蟺}$ .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.
31. $f (x)$ as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate ${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪$ .

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

Fundamentals of Physics

Magnetism and Electromagnetic Induction

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91影视

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.31. f(x)as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate鈭�-鈭�鈭�|g(伪)|2d伪.

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

Fundamentals of Physics

Magnetism and Electromagnetic Induction

Electromagnetism

Quantum Physics

Electrostatics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.
31. $f (x)$ as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate ${鈭�}_{- 鈭�}^{鈭�} {| g (伪) |}^{2} d 伪$ .