아날로그신호와 디지털신호의 정의

Definition¹

The elements of the $L^{p}$ space are referred to as analog signals or continuous-time signals.

$$f \in L^{p}(\mathbb{R})$$

The elements of the $\ell^{p}$ space are referred to as digital signals or discrete-time signals.

$$x_{n} = \left\{ x_{n} \right\} \in \ell^{p}(\mathbb{N})$$

Explanation

By definition, analog signals and digital signals possess almost identical characteristics, with the primary difference being whether they are functions or sequences. It is important to note that although another name for analog signals is continuous-time signals, this does not imply that they are continuous functions. It signifies that the domain, time $t \in \mathbb{R}$, is continuous. Similarly, in discrete-time signals, “discrete” describes the domain of the signal, which in this case, also implies that the values are discrete.

Digital signals are commonly represented as $x[n] = x_{n}$, and their Fourier transform is denoted in uppercase as $X[n] = \hat{x}$. Furthermore, if a variable is indicated with parentheses, it signifies an analog signal, whereas brackets indicate a digital signal.

$$아날로그신호\ f(t) \qquad 디지털신호\ f[n]$$

Since the sets of signals are $L^{p}$ and $\ell^{p}$ respectively, the norm of a signal is well-defined. In particular, in the field of signal processing, the $2$-norm of a signal is referred to as energy.