Digital Signal Processing - Linear Systems

A pnear system follows the laws of superposition. This law is necessary and sufficient condition to prove the pnearity of the system. Apart from this, the system is a combination of two types of laws −

Law of additivity

Law of homogeneity

Both, the law of homogeneity and the law of additivity are shown in the above figures. However, there are some other conditions to check whether the system is pnear or not.

The conditions are −

The output should be zero for zero input.

There should not be any non-pnear operator present in the system.

Examples of non-pnear operators −

(a) Trigonometric operators- Sin, Cos, Tan, Cot, Sec, Cosec etc.

(b) Exponential, logarithmic, modulus, square, Cube etc.

(c) sa(i/p) , Sinc (i/p) , Sqn (i/p) etc.

Either input x or output y should not have these non-pnear operators.

Examples

Let us find out whether the following systems are pnear.

a) $y(t) = x(t)+3$

This system is not a pnear system because it violates the first condition. If we put input as zero, making x(t) = 0, then the output is not zero.

b) $y(t) = sin tx(t)$

In this system, if we give input as zero, the output will become zero. Hence, the first condition is clearly satisfied. Again, there is no non-pnear operator that has been appped on x(t). Hence, second condition is also satisfied. Therefore, the system is a pnear system.

c) $y(t) = sin (x(t))$

In the above system, first condition is satisfied because if we put x(t) = 0, the output will also be sin(0) = 0. However, the second condition is not satisfied, as there is a non-pnear operator which operates x(t). Hence, the system is not pnear.