Chapter 7: Q31P (page 386)

Verify Parseval’s 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 $\int_{- \infty}^{\infty} {| g (α) |}^{2} d α$ .

Short Answer

Expert verified

$\int_{- \infty}^{\infty} {| g (α) |}^{2} d α = \frac{1}{π}$ .Thus the Parseval theorem is confirmed.

Step by step solution

Given Information.

The given function is $\int_{- \infty}^{\infty} {| g (α) |}^{2} d α$ .Parseval theorem is to be verified for this special case.

Definition of Parseval’s theorem

Parseval’s theorem is a theorem stating that the energy of a signal can be expressed as its frequency components’ average energy.

Verify Parseval Theorem

It is known that the function and its Fourier transform are

$\begin{array}{l} f (x) = \frac{2}{π} \int_{0}^{\infty} \frac{s i n α c o s (α x)}{α} d α \\ g (α) = \frac{s i n α}{α π} \end{array}$

Thus, $\begin{matrix} \frac{1}{2 π} \int_{- \infty}^{\infty} {| f (x) |}^{2} d x = \frac{1}{2 π} \int_{- 1}^{1} d x \\ = \frac{1}{π} \end{matrix}$

And

$\begin{matrix} \int_{- \infty}^{\infty} {| g (α) |}^{2} d α = \frac{1}{π^{2}} \int_{- \infty}^{\infty} \frac{s i n^{2} α}{α^{2}} d α \\ = \frac{1}{π^{2}} \int_{- \infty}^{\infty} \frac{d}{d α} (- \frac{s i n^{2} α}{α}) d α + \frac{1}{π^{2}} \int_{- \infty}^{\infty} \frac{d}{d α} (\frac{2 s i n α c o s α}{α}) d α \\ = 0 + \frac{4}{π^{2}} \int_{0}^{\infty} \frac{s i n α c o s α}{α} d α \end{matrix}$

It is known that $\int_{0}^{\infty} \frac{s i n α c o s α}{α} d α = \frac{π}{4}$

Therefore

$\begin{matrix} \int_{- \infty}^{\infty} {| g (α) |}^{2} d α = \frac{4}{π^{2}} \int_{0}^{\infty} \frac{s i n α c o s α}{α} d α \\ = \frac{4}{π^{2}} \frac{π}{4} \\ = \frac{1}{π} \end{matrix}$

$\int_{- \infty}^{\infty} {| g (α) |}^{2} d α = \frac{1}{π}$ .Thus the Parseval theorem is confirmed.

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Verify Parseval’s 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 $\int_{- \infty}^{\infty} {| g (α) |}^{2} d α$ .

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Verify Parseval Theorem

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Verify Parseval’s 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∫-∞∞|g(α)|2dα.

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

Magnetism

Mechanics and Materials

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Study anywhere. Anytime. Across all devices.

Verify Parseval’s 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 $\int_{- \infty}^{\infty} {| g (α) |}^{2} d α$ .