Linearity:
Definition
A discrete-time system T[.] is linear if the response to the weighted sum of the inputs x1(n) and x2(n) is a weighted sum (with the same weights) of the responses of the inputs individually for all weights and all acceptable inputs. Therefore the system y(n) = T[x(n)] is linear if for all a1, a2, x1(n) and x2(n) and we have the following equation
T[a1x1(n)+a2x2(n)] = a1T[x1(n)] + a2T[x2(n)]
Another way of represent this is that if the inputs x1(n) and x2(n) produce the outputs y1(n) and y2(n), respectively, then input a1x1(n) + a2x2(n) produces the output a1 y1(n) + a2 y2(n). This is known as superposition principle. The a1, a2, x1(n) and x2(n) are complex-valued. The above definition combines 2 properties, which are stated as follows,
1. Additivity, that is, T[x1(n)+x2(n)] = T[x1(n)] + T[x2(n)], and
2. Scaling (or homogeneity), that is, T[c x(n)] = c T[x(n)]
The process of checking the linearity is:
1. Find outputs y1(n) and y2(n) for inputs x1(n) and x2(n) respectively.
2. Form sum a1 y1(n) + a2 y2(n)
3. Find output y3(n) with respect to input a1x1(n) + a2x2(n)
4. Compare results of the steps 2 and 3
Examples of linear systems are stated as follows:
1. y(n) = x(n) + x(n-1) + x(n-2)
2. y(n) = y(n-1) + x(n)
3. y(n) = 0
4. y(n) = n x(n) (But time-varying)
Examples of nonlinear systems are stated as follows:
1. y(n) = x2(n)
2. y(n) = 2 x(n)+3.
This is a linear equation however, this system is made from linear part, 2 x(n), and a zero-input response,
3. This is known as incrementally linear system, because it responds linearly to make changes in the input.
Example 1 Show that whether the system y(n) = T[x(n)] = x(-n) is linear or nonlinear.
Answer To determine the outputs y1(.) and y2(.) corresponding to the 2 input sequences x1(n) and x2(n) and form the weighted sum of outputs:
y1(n) = T[x1(n)] = x1(-n)
y2(n) = T[x2(n)] = x2(-n)
The weighted sum of outputs = a1 x1(-n) + a2 x2(-n) → (A).
Now determine the output y3 because of a weighted sum of inputs:
y3(n) = T[a1 x1(n) + a2 x2(n)] = a1 x1(-n) + a2 x2(-n) → (B)
Check that weather (A) and (B) are equal. In this case (A) and (B) are equal; Therefore the system is linear.
Example 2 Examine y(n) = T[x(n)] = x(n) + n x(n+1) for linearity.
Answer The outputs of x1(n) and x2(n) are:
y1(n) = T[x1(n)] = x1(n) + n x1(n+1)
y2(n) = T[x2(n)] = x2(n) + n x2(n+1)
The weighted sum of outputs = a1 x1(n) + a1 n x1(n+1) + a2 x2(n) + a2 n x2(n+1) → (A)
The output of the a weighted sum of inputs is
y3(n) = T[a1 x1(n) + a2 x2(n)]
= a1 x1(n) + a2 x2(n) + n (a1 x1(n+1) + a2 x2(n+1))
= a1 x1(n) + a2 x2(n) + n a1 x1(n+1) + n a2 x2(n+1) → (B)
As (A) and (B) are equal the system is linear.
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