Part C. Linearity
If, for any two inputs, x1(t) and x2(t), you can apply each to a system to produce y1(t) and y2(t) respectively, then multiply each y(t) by a constant complex number, called a and b respectively, then add ay1(t)+by2(t) to produce a final output z(t). Then, if you again take x1(t) and x2(t), but this time first multiply each by a and b respectively, where a and b are again constant complex numbers, then add ax1(t)+bx2(t) and apply this input to the system to get an output w(t). If z(t)=w(t), then this system can be called a linear system.
Example of a Linear System
Suppose a system produces the output y(t)=2x(t-1) and let a=5 and b=2, then
x1(t) -> system -> 2x1(t-1) -> *5 -> 10x1(t-1) }
} -> + -> 10x1(t-1) + 4x2(t-1)
x2(t) -> system -> 2x2(t-1) -> *2 -> 4x2(t-1) }
x1(t) -> *5 -> 5x1(t) }
} -> + -> 5x1(t) + 2x2(t) -> system -> 10x1(t-1) + 4x2(t-1)
x2(t) -> *2 -> 2x2(t) }
Since the outputs of both systems above are the same and it does not matter whether you multiply and add first or if you apply the system first, as proved above, this system can be called linear.
Example of a Non-Linear System
Suppose a system produces the output y(t)=exp(t) and let a=5 and b=2, then
x1(t) -> system -> exp(x1(t)) -> *5 -> 5exp(x1(t)) }
} -> + -> 5exp(x1(t)) + 2exp(x2(t))
x2(t) -> system -> exp(x2(t)) -> *2 -> 2exp(x2(t)) }
x1(t) -> *5 -> 5x1(t) }
} -> + -> 5x1(t) + 2x2(t) -> system -> exp(5x1(t) + 2x2(t))
x2(t) -> *2 -> 2x2(t) }
Since the outputs of both systems are not the same and it DOES matter whether you multiply and ass first or if you apply the system first, as proved above, this system is called NON-linear.
