What is the difference between continuous and discrete signal

This post answers the question “What is the difference between continuous and discrete signal?” From a general point of view, signals are functions of one or several independent variables. There are two types of signals – discrete-time and continuous-time signals. Discrete-time signals are defined at the discrete moment of time and the mathematical function takes the discrete set of values.

Continuous-time signals are characterised by independent variables that are continuous and define a continuous set of values. Usually the variable indicates the continuous time signals, and the variable n indicates the discrete-time system. Also the independent variable is enclosed at parentheses for continuous-time signals and to the brackets for discrete-time systems. The feature of the discrete-time signals is that they are sampling continuous-time signals.

The signals we are describing are obviously related to the features of the system as power and energy. The total energy of the continuous-time signal $x (t)$ over the interval $t \in [t_{1}, t_{2}]$ is $\int_{t_{1}}^{t_{2}} | x (t) |^{2} d t$ .

Here $|x (t)|$ is the magnitude of the function $x (t)$ .

Here the brackets are describing the time-continuous interval $t_{1} \leq t \leq t_{2}$ . The parentheses $(t_{1}, t_{2})$ can be used for describing the time-continuous interval $t_{1} < t < t_{2}$ . The continuous-time power can be obtained by deriving the energy by the time interval _{$t_{2} - t_{1}$}.

The total energy of the discrete-time signal $x [n]$ over the interval $n \in [n_{1}, n_{2}]$ is the sum $\sum_{n_{1}}^{n_{2}} | x [n] |^{2}$ .

Where the average power over the indicated interval can be obtained with energy derived by the $n_{2} - n_{1} + 1$ .

Many systems exist over the infinite interval of the independent variable. For these systems $E_{\infty} = \int_{-_{\infty}}^{+_{\infty}} | x (t) |^{2} d t$

For continuous-time, and $E_{\infty} = \sum_{- \infty}^{+ \infty} | x [n] |^{_{2}} for discrete time .$

Some integrals and sums may not converge. These systems are characterised by the infinite energy $E_{\infty} \to \infty$ . For converging integrals and sums, signals have a finite energy $E_{\infty} < \infty$ .

The average power for discrete-time and continuous-time signals for an infinite period of time are:

P_{\infty} = \lim_{N \to \infty} \frac{_{1}}{_{2 N + 1}} \sum_{- N}^{+ N} | x [n] |^{2} and P_{\infty} = \lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{+ T} | x (t) |^{2} d t .

The signals with a finite total energy $E_{\infty} < \infty$ are characterised with zero average power $P_{\infty} = 0$ . The signals with infinite total energy $E_{\infty} = \infty$ are characterised by $P_{\infty} > 0$ .

We are considering here the most simple and frequent variable transformations that can be combined, resulting in complex transformations.

Time shift is the transformation when two signals $x [n]$ and $x [n - n_{0}]$ are the same but are displaced relatively to each other. The same for time-continuous signals $x (t)$ and $x (t - t_{0})$ .
Time reversal is when the signal $x [- n]$ is obtained from $x [n]$ by reflecting the signal relatively $- n = 0$ . For continuous-time signals $x (- t)$ is a $x (t)$ reflection over $t = 0$ .
Transformation $x (t) \to x (a t + b)$ , is where $a$ and $b$ are given numbers. Here the transformation depends on the value and sign of numbers, so if $a > 0$ and $|a| > 1$ the signal is extended, if $a > 0$ and $|a| < 1$ the signal is compressed, if $a < 0$ , the signal is reversed and can be extended or compressed, depending on the $b$ magnitude and sign of the signal is shifted right or left. For discrete-time variables the transformations are the same $x [n] \to x [a n + b]$ .

Figure 1 depicts different kinds of signal transformations for continuous-time and discrete-time variables.

What is the difference between continuous and discrete signal — Figure 1. a, b – the time shift transformation for continuous-time and discrete-time signals; c, d – reverse transformation for continuous-time and discrete-time signals; e, f – scale transformation for continuous-time and discrete-time signals.

Periodic signals.

The periodic discrete-time signals $x [n]$ with the period $N$ , where $N$ is the positive integer number, are characterised by the feature $x [n] = x [n + N]$ for all $n$ values. This equation also works for $2 N$ , … $k N$ period. The fundamental period $N_{0}$ is the smallest period value where this equation works. Figure 2 depicts an example of discrete-time periodic signal.

The continuous-time periodic signals $x (t)$ with period $T$ , are characterised by the feature $x (t) = x (t + T)$ . Also we can deduce that $x (t) = x (t + m T)$ , where $m$ is an integer number. The fundamental period $T_{0}$ is the smallest period value where this equation works. Figure 3 depicts an example of discrete-time periodic signal.

Even and odd signals.

The discrete-time signal $x [n]$ and continuous-time signal $x (t)$ are even if they are equal to their time-reversed counterparts, $x [n] = x [- n]$ and $x (t) = x (- t)$ . And the signals are odd, if $x [n] = - x [n]$ and $x (t) = - x (- t)$ . Odd signals are always 0 when $n = 0$ , or $t = 0$ .