Signals and SystemsYear 2

Signals and systems: The definition and characterising of signals

Signalsandsystems1

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∈ [t1, t2] is:

t1t2|x(t)|2dt

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

Here the brackets are describing the time-continuous interval t1t≤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 t2t1.

The total energy of the discrete-time signal x[n] over the interval n ∈ [n1,n2is the sum:

n1n2|x[n]|2

Where the average power over the indicated interval can be obtained with energy derived by the n2n1 + 1.

Many systems exist over the infinite interval of the independent variable. For these systems:

E = + |x(t)|2dt

For continuous-time, and:

E =+|x[n]|2 for discrete 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 =limN 12N+1 N +N|x[n]|2 and P = limT 12TT+T|x(t)|2dt.

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.

  1. Time shift is the transformation when two signals x[n] and x[nn0] are the same but are displaced relatively to each other. The same for time-continuous signals x(t) and x(tt0).
  2. 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.
  3. 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.

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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.
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 Nis the smallest period value where this equation works. Figure 2 depicts an example of discrete-time periodic signal.

SSFig2
Figure 2. Perisodic discrete-time 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 Tis the smallest period value where this equation works. Figure 3 depicts an example of discrete-time periodic signal.

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Figure 3. Periodic continuous-time 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.

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Figure 4. Even and odd discrete-time signals.
SSFig5
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:

Ev{x(t)} =12 [x(t)+x(t)] and Od{x(t)} =12 [x(t)  x(t)], for discretetime signal Ev{x[n]} = 12 (x[n] + x[n]) and Od{x[n]} = 12 (x[n] + x[n]).

#2 Types of signals

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