**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 $x\left(t\right)$ *over the interval* $t\in \left[{t}_{1},{t}_{2}\right]$* is ${\int}_{{t}_{1}}^{{t}_{2}}\left|x\right(t){|}^{2}dt$.

Here $\left|x\left(t\right)\right|$ is the magnitude of the function *$x\left(t\right)$*.

Here the brackets are describing the time-continuous interval ${t}_{1}\le t\le {t}_{2}$*. *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 _{ ${t}_{2}\u2013{t}_{1}$}.

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

Where the average power over the indicated interval can be obtained with energy derived by the ${n}_{2}\u2013{n}_{1}+1$.

Many systems exist over the infinite interval of the independent variable. For these systems ${E}_{\infty}={\int}_{{\u2013}_{\infty}}^{{+}_{\infty}}\left|x\right(t){|}^{2}dt$

For continuous-time, and ${E}_{\infty}=\sum _{\u2013\infty}^{+\infty}\left|x\right[n]{|}^{{}_{2}}\mathrm{for}\mathrm{discrete}\mathrm{time}.$

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:

${P}_{\infty}=\underset{N\to \infty}{\mathrm{lim}}\frac{{}_{1}}{{}_{2N+1}}\sum _{\u2013N}^{+N}\left|x\right[n]{|}^{2}\mathrm{and}{\mathrm{P}}_{\infty}=\underset{\mathrm{T}\to \infty}{\mathrm{lim}}\frac{1}{2\mathrm{T}}{\int}_{\u2013\mathrm{T}}^{+\mathrm{T}}|x\mathit{\left(}t\mathit{\right)}{\mathit{|}}^{\mathit{2}}dt\mathit{.}$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 $x\left[n\right]$ and $x\left[n\u2013{n}_{0}\right]$ are the same but are displaced relatively to each other. The same for time-continuous signals $x\left(t\right)$ and $x(t\u2013{t}_{0})$.*Time reversal is*when the signal $x\left[\u2013n\right]$ is obtained from $x\left[n\right]$ by reflecting the signal relatively $\u2013n=0$. For continuous-time signals $x(\u2013t)$ is a $x\left(t\right)$ reflection over $t=0$.- Transformation $x\left(t\right)\to x(at+b)$
*,*is where $a$ and $b$ are given numbers. Here the transformation depends on the value and sign of numbers, so if $a>0$ and*$\left|a\right|>1$*the signal is extended, if $a>0$ and $\left|a\right|<1$ 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 $x\left[n\right]\to x\left[an+b\right]$ .

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

### Periodic signals.

*The periodic* *discrete-time signals $x\left[n\right]$ *with the period* $N$, *where $N$* *is the positive integer number, are characterised by the feature *$x\left[n\right]=x\left[n+N\right]$ *for all* $n$ *values. This equation also works for $2N$*, … $kN$* 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 $x\left(t\right)$ *with period* $T$, *are characterised by the feature $x\left(t\right)=x(t+T)$. Also we can deduce that *$x\left(t\right)=x(t+mT)$*, 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 $x\left[n\right]$* *and continuous-time signal *$x\left(t\right)$ *are even if they are equal to their time-reversed counterparts, $x\left[n\right]=x\left[\u2013n\right]$ and $x\left(t\right)=x(\u2013t)$*.* And the signals are odd, if $x\left[n\right]=\u2013x\left[n\right]$ and $x\left(t\right)=\u2013x(\u2013t)$*. *Odd signals are always 0 when $n=0$*, *or* $t=0$.*

Figures 4 and 5 depict even and odd discrete- and continuous-time signals.

Any continuous-or discrete-time signals can be presented as a sum of odd and even signals. For continuous-time signals:

$Ev\left\{x\left(t\right)\right\}=\frac{{}_{1}}{{}_{2}}\left[x\right(t)+x(\u2013t\left)\right]\phantom{\rule{0ex}{0ex}}Od\mathit{\left\{}x\mathit{\right(}t\mathit{\left)}\mathit{\right\}}\mathit{}\mathit{=}\frac{{}_{\mathit{1}}}{{}_{\mathit{2}}}\mathit{}\mathit{\left[}x\mathit{\right(}t\mathit{)}\mathit{}\mathit{\u2013}\mathit{}x\mathit{(}\mathit{\u2013}t\mathit{\left)}\mathit{\right]}\phantom{\rule{0ex}{0ex}}$for discrete-time signals:

$Ev\left\{x\left[n\right]\right\}=\frac{1}{2}(x\left[n\right]+x\left[\u2013n\right])\phantom{\rule{0ex}{0ex}}Od\left\{x\left[n\right]\right\}=\frac{1}{2}(x\left[n\right]+x\left[\u2013n\right])$More educational tutorials can be accesses as well via Reddit community **r/ElectronicsEasy**.