SIGNALS AND SYSTEMS-I PROPERTIES OF SYSTEMS

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

A system may be stated as an entity made up of a combination of several different elements connected to each other in an ordered fashion, or according to some finite rule. A system can have one or more inputs to feed signals into it, and one or more outputs to take data out from it. The system input(s) and output(s) can be seen to be related to each other by some finite rules or algorithms. Computers, electrical machines, radio transmitters, and receivers are a few among an infinite number of systems that exist in this world.

            The behavior of a system can be described by one or more of the following properties:

·               Linearity

·               Causality

·               Time-variance

·               Convolution

·               Stability

·               Memory

2. LINEARITY

A system is said to be linear, if and only if it obeys the principle of superposition. The principle of superposition may be explained with the help of the following.

Consider a system, whose output y(t) is dependent on a certain input x(t). Let us define the relations existing between x(t) and y(t) as

                                                              y(t) = f [x(t)]   (1)                                                      

where f represents a function. For example, let us consider the relations

                                    x₃(t) = ax₁(t) + bx₂(t)  (2)                                          

                                    y₃(n) = ay₁(n) + by₂(n) (3)                                                    

where x₁(t), x₂(t), and x₃(t) are the values of x(t) at three instants of time and y₁(t), y₂(t), and y₃(t) are the corresponding values of y(t) at the same intervals of time, and a and b are constants. Since

                                    y(t) = f [x(t)]                                                               

we can express the relations in the form

           y₃(t) = ay₁(t) + by₂(t) = f [x₃(t)] =  f [ax₁(t) + bx₂(t)] (4)                    

Equation (4) may be written in the form:

            y₃(t) = ay₁(t) + by₂(t) = a f [x₁(t)] + b f [x₂(t)] (5)                           

            Now, the principle of superposition says that if the system is to be linear, then we must have

                       

                                    f [ax₁(t)+ bx₂(t)] = a f [x₁(t)] + b f [x₂(t)]  (6)                                    

Notice that the relations

  y₃(t) = f [ax₁(t) + bx₂(t)]

and

  y₃(t) = a f [x₁(t)] + b f [x₂(t)]

were obtained through two independent methods. We may now state:

            If a given system is to be linear, then its response (or output) to a weighted sum of inputs must be equal to the corresponding weighted sum of its responses to each of the individual inputs.

ILLUSTRATIVE EXAMPLE 1: Determine whether the system governed by the equation  y(t) = 3 x(t) is linear or not. .

Solution: To determine whether a given system is linear or not, we adopt the following procedure, which is based on the principle of superposition given above. Based on that, we may write

                                                            y₁(t) = 3x₁(t)  (1)

                                                            y₂(t) = 3x₂(t)  (2)

                                                            y₃(t) =  x₃(t)  (3)

Let                                                                                                                                   

                                                   x₃(t) = ax₁(t) + bx₂(t) (4)

and                                            

                                                    y₃(t) = ay₁(t) + by₂(t) (5)

where a and b are constants. Now, substituting (4) into (3) yields

                                          y₃(t) = 3x₃(t) = 3[ax₁(t) + bx₂(t)]

                                                      = 3ax₁(t) + 3bx₂(t)

                               = a[3x₁(t)] + b[3x₂(t)]

                                 

                                 = ay₁(t) +by₂(t) (6)                                

Comparison of (6) and (5) shows that they are the same. Hence, we conclude that the relation y₃(t) = 3x₃(t) represents a linear system.

            In the above method, we arrived at the final conclusion through two different paths using the same equation that govern the given system. If the two paths help us to arrive at the same final solution, then we say that the system obeys the principle of superposition, and hence is linear.

However, it must be carefully noted that all equations representing straight lines need not necessarily represent linear systems. The following example will illustrate this idea.

ILLUSTRATIVE EXAMPLE 2: Test whether the system governed by the equation y(t) = Ax(t)+B is linear or not.

                       

Solution: We know that the equation y(t) = Ax(t)+B, where A and B are constants, represents a straight line. We now show that the system represented by this equation is not linear. To prove this statement, we proceed as in Illustrative Example 1. Following the same procedure, let

                                                            y₁(t) = Ax₁(t)+B  (1)

                                                            y₂(t) = Ax₂(t)+B  (2)

                                                            y₃(t) = Ax₃(t)+B  (3)

Let                                                                                                                                   

                                                   x₃(t) = ax₁(t) + bx₂(t) (4)

and                                            

                                                    y₃(t) = ay₁(t) + by₂(t) (5)

where a and b are also constants. Now, substituting (4) into (3) yields

                                          y₃(t) = Ax₃(t)+B = A[ ax₁(t) + bx₂(t)] + B

                                                      = Aax₁(t) + Abx₂(t) + B

                                = a[Ax₁(t)] + b[Ax₂(t)] + B                                

                                ≠ ay₁(t) + by₂(t)              (6)                  

                                                                                                                                  

 Equation (6) shows that the two paths for arriving at the final result do not agree with each other; We therefore conclude that the system governed by the equation y(t) = Ax(t)+B is not linear.

We now conclude that straight-line equations passing through the origin and extending from ‒∞ to +∞ will represent linear systems. Figure 1 represents a linear system and Fig. 2 represents a nonlinear system.

Both linearity and nonlinearity are desirable properties of practical systems. For example, amplifiers are linear systems. In an amplifier, the output is directly proportional to the input. Any nonlinearity in the amplifier system, as stated above, will produce distortion and noise in its output. However, when the same amplifier is used as a switch, we operate it in the nonlinear regions (for example, in the saturation and cut-off regions) of the system.

3. CAUSALITY

The term causal represents the idea “that which causes”. A system is said to be causal, if the value of its present output(s) depend(s) only on the present and past values of its inputs [which may include inputs derived from the output(s) through feedback connections], and does not in any way depend on the future values of the inputs.

            It is easy to see that all physically realizable systems are causal. Let us consider the example of a student writing examination on a given subject. Before writing the examination, he must have studied the subject thoroughly, and only these studies will help him in writing the examination. Any studies that he may make on that subject after he has written the examination will not help him in any way in writing an examination that has already been over!

In causal systems, inputs applied cause them to produce outputs. So, a method to check for the causality of a system is to check the time period(s) of its input(s) and output(s), and see whether they contain a term or terms with future value(s) in them. The following example will illustrate our procedure for testing causality.

ILLUSTRATIVE EXAMPLE 3: Test the causality of the system whether the system governed by the expression:

                                 y(t) = ay(t‒1) + by(t‒2) + cy(t‒3) + dx(t) (1)

                                                       

Solution: We know that terms containing t in them represent present values, and (t-1), (t-2), etc. represent present values delayed one unit of time, two units of time and so on. It can be seen that the terms in (1) contain only present and past values of input and output among them. They do not contain any value representing a future input. Hence we state that the system governed by (1) is causal and is physically realizable.

ILLUSTRATIVE EXAMPLE 4: Test the causality of the system governed by the expression

                                         y(t) = ay(t‒1) + by(t‒2) + cy(t‒3) + dx(t + 1) (1)

                         

Solution: inspection of (1) reveals that it contains the future term x(t+1), and therefore it is non-causal.