Linearity
A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems:
- Suppose there are two inputs
- $ \,x1(t) $
- $ \,x2(t) $
- with outputs
- $ \,y1(t) = C\left\{x1(t)\right\} $
- $ \,y2(t) = C\left\{x2(t)\right\} $
- A linear system must satisfy the condition
- $ \,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\} $
Example of a Linear System
- $ \,x1(t) = sin(t) $
- $ \,x2(t) = cos(t) $
- $ \,y1(t) = C\left\{x1(t)\right\} = C(sin(t)) $
- $ \,y2(t) = C\left\{x2(t)\right\} = C(cos(t)) $
- $ \,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\} $
Thus, $ \,y(t) = Cx(t) $ is a linear system.
