1.2.1 Linear Systems and Shift-Invariant Systems

 

    This definition of a system is very broad.  Only in very special circumstances do our systems analysis problems become mathematically tractable. To make any progress at all, we must restrict the class of systems which we will consider. Without any restrictions, char­acterizing a system requires a complete input-output relationship. Knowing the output of a system to one set of inputs does not generally allow us to determine the output of the system to any other set of inputs. Two types of restriction which greatly simplify the characterization and analysis of a system are linearity and shift invariance. In practice, fortunately, many systems can be approximated to be linear and shift invariant.

 

 

The linearity of a system T is defined as  ;

 

 

 

Linearity Û T[ax1(n1, n2) + bx2(n1, n2] = ay1(n1, n2) + by2(n1, n2)        (1.12)

 

 

 

where

 

 

               T[x1(n1, n2] = y1(n1, n2),    T[x2(n1, n2)] = y2(n1, n2) 

 

 

 

a and b are any scalar constants, and AÛB means that A implies B and B implies A. The condition in (1.12) is called the principle of superposition. To illustrate this concept, a linear system and a nonlinear system are shown in Figure 1.13(a).

    Suppose we first apply  x1(n1, n2) as the input and  find that  y1(n1, n2)  as the response. Now what would happen if we had used ax1(n1, n2) + bx2(n1, n2)         as the input where a and b are constants? In general, if the system is not linear, the response to this combination or superposition of inputs could be quite arbitrary. But a linear system has the defining property that the superposition of inputs ax1(n1, n2) + bx2(n1, n2) produces an output which is a corresponding superposition ay1(n1, n2) + by2(n1, n2). By definition this property must hold for any x1(n1, n2), x2(n1, n2), a, and b.

 

 

 

The shift invariance (SI) or space invariance of a system is defined as

 

 

 

                 Shift-invarianceÛ  T[x(n1-m1, n2 - m2)] = y(n1 - m1, n2 - m2 )                        (1.13)

 

 

 

where

 

 

               y(n1, n2) = T[x(n1, n2)] and m1 and m2 are any integers.

 

 

 

A system is shift-invariant if it has the property that system behaviour does not depend on where the input is applied. More precisely, if we know that   y(n1, n2)  is the response to  x(n1, n2), then the defining property of the space-invariant system is that  y(n1 - m1, n2 - m2 ) is the response to x(n1 - m1, n2 - m2 ) where m1 and m2  are any constants. We insist that this property be true for any choice of x(n1, n2)   and   m1 , m2  .

 

For convenience refer to Figure 1.13(b).

 

 

 

 

 

 

In the  example above in Figure 1.13 (a) there is a system T .  What can we say about the systems ;

·          linearity?

·          shift-invariance ?

 

 

Is the system linear ?

 

The system T is given as  ;

 

y(n1, n2)=T[x(n1, n2)]= x(n1, n2) . g(n1, n2)

 

The principle of superposition (linearity rule) is  given as  ;

 

T[ax1(n1, n2) + bx2(n1, n2] = ay1(n1, n2) + by2(n1, n2)

 

Then  ;

 

T[x(n1, n2)]= x(n1, n2) . g(n1, n2)    

T[ax1(n1, n2) + bx2(n1, n2] = [ax1(n1, n2) + bx2(n1, n2]. g(n1, n2)    (Expression-a)

 

and

 

y(n1, n2)= x(n1, n2) . g(n1, n2)

ay1(n1, n2) + by2(n1, n2)= [ax1(n1, n2). g(n1, n2)] + [bx2(n1, n2). g(n1, n2)]

                                   = [ax1(n1, n2) + bx2(n1, n2)]. g(n1, n2)         (Expression-b)

 

 

Expression-a  ® T[ax1(n1, n2) + bx2(n1, n2] = [ax1(n1, n2) + bx2(n1, n2]. g(n1, n2) 

Expression-b  ® ay1(n1, n2) + by2(n1, n2)= [ax1(n1, n2) + bx2(n1, n2)]. g(n1, n2)        

 

 As Expression-a=Expression-b    the rule of linearity is satisfied and the system is said to be linear.

 

Now let’s look if the system is shift-invariant ?

 

The system T is given as  ;    y(n1, n2)=T[x(n1, n2)]= x(n1, n2) . g(n1, n2)

Shift-invariance rule    :        T[x(n1-m1, n2 - m2)] = y(n1 - m1, n2 - m2 )

 

T[x(n1, n2)]= x(n1, n2) . g(n1, n2)            

T[x(n1-m1, n2 - m2)] = x(n1-m1, n2 - m2) . g(n1, n2)          (Expression-c)

 

and

 

y(n1, n2)= x(n1, n2) . g(n1, n2)

y(n1 - m1, n2 - m2 )= x(n1-m1, n2 - m2) . g(n1-m1, n2 - m2)     (Expression-d)    

 

 As Expression-c ¹ Expression-d   the rule of  shift-invariance is not satisfied and the system is said to be not shift-invariant  or other words the system is shift-variant.

 

 

 

 

 

 

In the example above in Figure 1.13 (b) there is a system T.  What can we say about the systems ;

·          linearity?

·          shift-invariance ?

 

 

Is the system linear ?

 

The system T is given as  ;

 

                        y(n1, n2)= T[x(n1, n2)]= x2(n1, n2)

 

The principle of superposition (linearity rule) is  given as  ;

 

                        T[ax1(n1, n2) + bx2(n1, n2)] = ay1(n1, n2) + by2(n1, n2)

 

Then  ;

If  input x(n1, n2)  can be written as the sum of two other functions such as  x1(n1, n2) + x2(n1, n2)

    x(n1, n2)  = x1(n1, n2) + x2(n1, n2)

Then  we have to say that the square of the input  is equal to  square of the sum such as ;

 

    x2(n1, n2)=[ x1(n1, n2) + x2(n1, n2)]2

 

 If    T[x(n1, n2)]= x2(n1, n2)   then

       T[ax1(n1, n2) + bx2(n1, n2)]= )=[ x1(n1, n2) + x2(n1, n2)]2                    (Expression-a)    

 

 

On the other hand   ;

 

y(n1, n2)= x2(n1, n2)

 

From property the definition we know that

 

y1(n1, n2) = T[x1(n1, n2)]   which makes  y1(n1, n2) = x12(n1, n2)

 

Same as y1(n1, n2) we can write

 

Y2(n1, n2) = T[x2(n1, n2)]= x22(n1, n2)

 

Therefore

 

ay1(n1, n2) + by2(n1, n2)=ax12(n1, n2)+bx22(n1, n2)                            (Expression-b)    

 

 

Expression-a  ® T[ax1(n1, n2) + bx2(n1, n2] = [ax1(n1, n2) + bx2(n1, n2]2

Expression-b  ® ay1(n1, n2) + by2(n1, n2)= ax12(n1, n2) + bx22(n1, n2)        

 

We see that     Expression-a ¹ Expression-b   and  the system is not linear and said to be non-linear.

 

 

 

Now let’s look if the system is shift-invariant ?

 

The system T is given as  ;    y(n1, n2)=T[x(n1, n2)]= x2(n1, n2)

Shift-invariance rule    :        T[x(n1-m1, n2 - m2)] = y(n1 - m1, n2 - m2 )

T[x(n1, n2)]= x2(n1, n2)           

T[x(n1-m1, n2 - m2)] = x2(n1-m1, n2 - m2)                         (Expression-c)

 

and

 

y(n1, n2)= x2(n1, n2)

y(n1 - m1, n2 - m2 )= x2(n1-m1, n2 - m2)                                 (Expression-d)    

 

 As Expression-c = Expression-d   the rule of  shift-invariance is satisfied and the system is said to be shift-invariant.

 

 

 

    Consider a linear system T. Using (1.2) and (1.12). we can express the output y(n1, n2) for an input x(n1, n2) as

 

                            x(n1, n2) =

 

and (1.12),

 

                                   T[ax1(n1, n2) + bx2(n1, n2] = ay1(n1, n2) + by2(n1, n2)

 

we can express the output y(n1, n2) for an input x(n1, n2) as

 

 

                        (1.14)

 

From (1.14), a linear system can be completely characterized by the response of the system to the impulse d(n1, n2) and its shifts d(n1 - k1, n2 - k2). If we know T[d(n1 — k1, n2 — k,)] for all integer values of k1 and k2  the output of the linear system to any input x(n1, n2)  can be obtained from (1.14). For a nonlinear system. knowledge of T[d(n1-k1,n2-k2)] for all integer values of k1 and k2 does not tell us the output of the system when the input x(n1,n2) is 2d(n1, n2), d(n1, n2) +d(n1-1, n2), or many other sequences.

    System characterization is further simplified if we impose the additional re­striction of shift-invariance. Suppose we denote the response of a system T to an input d(n1, n2) by h(n1, n2);

 

                                    h(n1, n2) = T[d(n1, n2)].                                                  (1.15)

 

From (1.13)

 

                                   T[x(n1-m1, n2 - m2)] = y(n1 - m1, n2 - m2 )

 

and (1.15), we can write

 

h(n1 — k1, n2 — k2) = T[d(n1 — k1,n2 — k2)]                                                  (1.16)

 

for a shift-invariant system T. For a linear and shift-invariant (LSI) system, then from (1.14) and (1.16), the input-output relation is given by

 

             (1.17)

 

Equation (1.17) states that an LSI system is completely characterized by the impulse response h(n1, n2) Specifically, for an LSI system, knowledge of h(n1, n2) alone allows us to determine the output of the system to any input from (1.17). Equation (1.17) is referred to as convolution, and is denoted by the convolution operator “*“ as follows:

 

For an LSI system.

 

             (1.18)

 

 

Note that the impulse response h(n1, n2) which plays such an important role for an LSI system, loses its significance for a nonlinear or shift-variant system. Note also that an LSI system can be completely characterized by the system response to one of many other input sequences. The choice of d(n1, n2) as the input in characterizing an LSI system is the simplest both conceptually and in practice.