신호의 에너지와 평균 전력

Definition¹

Analog Signal

The energy of an analog signal $f \in L^{p}$ $E_{f}$ is defined as follows.

$$ E_{f} := \int_{-\infty}^{\infty} \left| f(t) \right|^{2} dt = \left\| f \right\|_{2}^{2} $$

If $E_{f} \lt \infty$, then $f$ is called an energy signal. For $f$, which is not an energy signal, the mean power $P_{f}$ is defined as follows.

$$ P_{f} := \lim\limits_{T \to \infty} \dfrac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} \left| f(t) \right|^{2}dt $$

If $P_{f} \lt \infty$, then $f$ is called a power signal.

Digital Signal

The energy $E_{x_{n}}$ of a digital signal $x_{n} = \left\{ x_{n} \right\} \in \ell^{p}$ is defined as follows. $$ E_{x_{n}} := \sum\limits_{n \in \mathbb{N}} \left| x_{n} \right|^{2} = \left\| x_{n} \right\|_{2}^{2} $$

If $E_{x_{n}} \lt \infty$, then $x_{n}$ is called an energy signal. For $x_{n}$, which is not an energy signal, the mean power $P_{x_{n}}$ is defined as follows.

$$ P_{x_{n}} := \lim\limits_{T \to \infty} \dfrac{1}{T} \sum\limits_{n=1}^{T} \left| x_{n} \right|^{2} $$

If $P_{x_{n}} \lt \infty$, then $x_{n}$ is called a power signal.

Description

In other words, an element of the $L^{2}$ / $\ell^{2}$ space is called an energy signal, and the $2$-norm $\left\| \cdot \right\|_{2}$ of the energy signal is called the energy. Since energy is the $2$-norm, we obtain the following from the Plancherel’s Theorem.

Plancherel’s Theorem

$$\| \hat{f} \|_{2}^{2} = 2\pi \| f \|_{2}^{2}$$

If $\hat{f}$ is the Fourier transform of $f$,

$$ E_{f} = \int \left| f(t) \right|^{2} dt = \dfrac{1}{2\pi} \int | \hat{f}(\omega) |^{2} d\omega $$

then $| \hat{f}(\omega) |^{2}$ is called the energy spectrum (density) of $f$ and is denoted as follows.

$$ S_{f}(\omega) = S_{ff}(\omega) := | \hat{f}(\omega) |^{2} $$