Definition of Analog Signals and Digital Signals

Definition¹

An element of the $L^{p}$ space is called an analog signal or a continuous-time signal.

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

An element of the $\ell^{p}$ space is called a digital signal or a discrete-time signal.

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

Explanation

By definition, analog signals and digital signals have almost the same characteristics, and the only difference between the two is whether it is a function or a sequence. Note that although another name for an analog signal is continuous-time signal, this does not mean that it is a continuous function. It means that the domain, the time $t \in \mathbb{R}$, is continuous. Likewise, “discrete” in discrete-time signal is also a word that modifies the time, which is the domain of the signal (in this case, of course, the values are discrete as well).

A digital signal is commonly denoted as $x[n] = x_{n}$, and its Fourier transform is denoted with a capital letter as $X[n] = \hat{x}$. Also, if the variable is enclosed in round parentheses it means an analog signal, and if enclosed in square brackets it means a digital signal.

$$ \text{analog signal}\ f(t) \qquad \text{digital signal}\ f[n] $$

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 called the energy.