Devise a Procedure to Construct Xsn From Xdn Without Constructing Any Continuoustime 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 $t\in \left[t_1,t_2\right]$ 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_{$t_2–t_1$} .

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$n_2–n_1+1$.

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

For continuous-time, and $E_{\infty }=\sum _{–\infty }^{+\infty }\left|x\right[n]{|}^{{}_2}\mathrm{for}\mathrm{discrete}\mathrm{time}.$

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:

$P_{\infty }=\underset{N\to \infty }{\lim }\frac{{}_1}{{}_{2N+1}}\sum _{–N}^{+N}\left|x\right[n]{|}^2\mathrm{and}{\mathrm{P}}_{\infty }=\underset{\mathrm{T}\to \infty }{\lim }\frac{1}{2\mathrm{T}}{\int }_{–\mathrm{T}}^{+\mathrm{T}}|x\mathit{\left(}t\mathit{\right)}{\mathit{|}}^{\mathit{2}}dt\mathit{.}$

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.

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 $x(t–t_0)$.
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 $x(–t)$ is a $x\left(t\right)$ reflection over $t=0$.
3. 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.

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$x\left[n\right]$ 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.

Figure 2. Perisodic discrete-time 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.

Figure 3. Periodic continuous-time 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$x\left(t\right)=x(–t)$ . And the signals are odd, if$x\left[n\right]=–x\left[n\right]$ and $x\left(t\right)=–x(–t)$ . Odd signals are always 0 when $n=0$ , or $t=0$.

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

Figure 4. Even and odd discrete-time signals.

Figure 5. Even and odd continuous-time signals.

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

$$\begin{gathered} Ev\left\{x\left(t\right)\right\}=\frac{{}_1}{{}_2}\left[x\right(t)+x(–t\left)\right] \\ Od\mathit{\left\{}x\mathit{\right(}t\mathit{\left)}\mathit{\right\}}\mathit{}\mathit{=}\frac{{}_{\mathit{1}}}{{}_{\mathit{2}}}\mathit{}\mathit{\left[}x\mathit{\right(}t\mathit{)}\mathit{}\mathit{–}\mathit{}x\mathit{(}\mathit{–}t\mathit{\left)}\mathit{\right]} \end{gathered}$$

for discrete-time signals:

$$\begin{gathered} Ev\left\{x\left[n\right]\right\}=\frac{1}{2}(x\left[n\right]+x\left[–n\right]) \\ Od\left\{x\left[n\right]\right\}=\frac{1}{2}(x\left[n\right]+x\left[–n\right]) \end{gathered}$$

More educational tutorials can be accesses as well via Reddit community r/ElectronicsEasy.

Types of signals

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Source: https://www.student-circuit.com/learning/year2/signals-and-systems-intermediate/discrete-and-continuous-signals/

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