From a general point of view, signals are functions of one or several independent variables. There two types of signals – descrete-time and continuous-time signals. Descrete-time signals are defined at the discrete moment of time and the mathemetical function takes the descrete 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 *∈ [*t*_{1}, *t*_{2}] is:

Where |*x(t)| *is the magnitude of the function *x(t)*.

Here the brackets are describing the time-continuous interval *t*_{1}≤*t≤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 ∈ [n _{1},n_{2}] *is the sum:

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:

${E}_{\infty}={\int}_{{\u2013}_{\infty}}^{{+}_{\infty}}\left|x\right(t){|}^{2}dt$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_{∞ }→∞. For converging integrals and sums, signals have a finite energy E_{∞ }˂ ∞.

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_{∞ }˂ ∞ are characterised with zero average power P_{∞ }= 0. The signals with infinite total energy E_{∞ }= ∞ are characterised by P_{∞ }˃ 0.

**Transformation of the independent variable**

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) → 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*|*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]**→ x[an*+*b]*.

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

**Types of 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 *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(t) *with period* T, *are characterised by the feature *x(t) = x(t + T)*. Also we can deduce that *x(t) = 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[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.*

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:

$Ev\left\{x\left(t\right)\right\}=\frac{{}_{1}}{{}_{2}}\left[x\right(t)+x(\u2013t\left)\right]\mathrm{and}Od\mathit{\left\{}x\mathit{\right(}t\mathit{\left)}\mathit{\right\}}\mathit{}\mathit{=}\frac{{}_{\mathit{1}}}{{}_{\mathit{2}}}\mathit{}\mathit{\left[}x\mathit{\right(}t\mathit{)}\mathit{}\mathit{\u2013}\mathit{}x\mathit{(}\mathit{\u2013}t\mathit{\left)}\mathit{\right]}\mathit{,}\mathit{}\mathrm{for}\mathrm{discrete}\u2013\mathrm{time}\mathrm{signal}Ev\mathit{\left\{}x\mathit{\right[}n\mathit{\left]}\mathit{\right\}}\mathit{}\mathit{=}\mathit{}\frac{\mathit{1}}{{}_{\mathit{2}}}\mathit{}\mathit{\left(}x\mathit{\right[}n\mathit{]}\mathit{}\mathit{+}\mathit{}x\mathit{[}\mathit{\u2013}n\mathit{\left]}\mathit{\right)}\mathit{}\mathrm{and}\mathit{}Od\mathit{\left\{}x\mathit{\right[}n\mathit{\left]}\mathit{\right\}}\mathit{}\mathit{=}\mathit{}\frac{\mathit{1}}{{}_{\mathit{2}}}\mathit{}\mathit{\left(}x\mathit{\right[}n\mathit{]}\mathit{}\mathit{+}\mathit{}x\mathit{[}\mathit{\u2013}n\mathit{\left]}\mathit{\right)}\mathit{.}$