7.4: Parseval's Theorem

Learning Objectives

Properties of the Fourier transform and some useful transform pairs are provided in this table. Especially important among these properties is Parseval's Theorem, which states that power computed in either domain equals the power in the other.

$\int_{-\infty }^{\infty }s^{2}(t)dt=\int_{-\infty }^{\infty }\left ( \left | S(f) \right | \right )^{2}df$

Of practical importance is the conjugate symmetry property: When s(t) is real-valued, the spectrum at negative frequencies equals the complex conjugate of the spectrum at the corresponding positive frequencies. Consequently, we need only plot the positive frequency portion of the spectrum (we can easily determine the remainder of the spectrum).

Exercise $$\PageIndex{1}$$

How many Fourier transform operations need to be applied to get the original signal back:

$\mathfrak{F}(...(\mathfrak{F}(s)))=s(t)$

Solution

$\mathfrak{F}(\mathfrak{F}(\mathfrak{F}(\mathfrak{F}(s(t)))))=s(t)$

We know that

$\mathfrak{F}(S(f))=\int_{-\infty }^{\infty }S(f)e^{-(i2\pi ft)}df=\int_{-\infty }^{\infty }S(f)\overline{e^{i2\pi f(-t)}}df=s(-t)$

Therefore, two Fourier transforms applied to s(t) yields s(-t). We need two more to get us back where we started.

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