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 ∈ [t1, t2] is:
Where |x(t)| is the magnitude of the function x(t).
Here the brackets are describing the time-continuous interval t1≤t≤t2. The parentheses (t1, t2) can be used for describing the time-continuous interval t1 ˂ t ˂ t2. The continuous-time power can be obtained by deriving the energy by the time interval t2 – t1.
The total energy of the discrete-time signal x[n] over the interval n ∈ [n1,n2] is the sum:
Where the average power over the indicated interval can be obtained with energy derived by the n2 – n1 + 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∞ →∞. 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:
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.
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 – n0] are the same but are displaced relatively to each other. The same for time-continuous signals x(t) and x(t – t0).
- 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.
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 N0 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 T0 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 signalx(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: