Signals and Systems

# Discrete-time and continuous-time signals

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 is ${\int }_{{t}_{1}}^{{t}_{2}}|x\left(t\right){|}^{2}dt$.

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  can be used for describing the time-continuous interval ${t}_{1}. 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 $x\left[n\right]$ over the interval is the sum $\sum _{{n}_{1}}^{{n}_{2}}|x\left[n\right]{|}^{2}$.

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

For continuous-time, and

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:

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.

1. Time shift is the transformation when two signals $x\left[n\right]$ and $x\left[n–{n}_{0}\right]$ are the same but are displaced relatively to each other. The same for time-continuous signals $x\left(t\right)$ and $x\left(t–{t}_{0}\right)$.
2. Time reversal is when the signal $x\left[–n\right]$ is obtained from $x\left[n\right]$ by reflecting the signal relatively  $–n=0$. For continuous-time signals $x\left(–t\right)$  is a  $x\left(t\right)$ reflection over $t=0$.
3. Transformation $x\left(t\right)\to x\left(at+b\right)$, 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 $x\left(t\right)=x\left(t+T\right)$. Also we can deduce that $x\left(t\right)=x\left(t+mT\right)$, 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, $x\left[n\right]=x\left[–n\right]$ and $x\left(t\right)=x\left(–t\right)$. And the signals are odd, if $x\left[n\right]=–x\left[n\right]$ and $x\left(t\right)=–x\left(–t\right)$. 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:

for discrete-time signals:

$Ev\left\{x\left[n\right]\right\}=\frac{1}{2}\left(x\left[n\right]+x\left[–n\right]\right)\phantom{\rule{0ex}{0ex}}Od\left\{x\left[n\right]\right\}=\frac{1}{2}\left(x\left[n\right]+x\left[–n\right]\right)$

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