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 ( t ) over the interval t t 1 , t 2 is t 1 t 2 | x ( t ) | 2 d t .

Here 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 n 1 n 2 | x [ 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 = + | x ( t ) | 2 d t

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 = lim N 1 2 N + 1 N + N | x [ n ] | 2 and P = lim T 1 2 T T + T | x ( t ) | 2 d t .

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.

  1. 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 ) .
  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 ( a t + 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 a n + b  .

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

What is the difference between continuous and discrete signal
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 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 2 N , … k N 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.

What is the difference between continuous and discrete signal
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 + m T ) , 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.

What is the difference between continuous and discrete signal
Figure 3. Periodic continuous-time 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.

SSFig4
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:

E v { x ( t ) } = 1 2 [ x ( t ) + x ( t ) ] O d { x ( t ) } = 1 2 [ x ( t ) x ( t ) ]

for discrete-time signals:

E v x n = 1 2 ( x n + x n ) O d x n = 1 2 ( x n + x n )

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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