Definition

If

 $ x_{1}(t) \to sys \to *a \to $
                      $ + \to a x_{1}(t) + b x_{2}(t) $
 $ x_{2}(t) \to sys \to *b \to $

And

 $ x_{1}(t) \to *a \to $
               $ + \to sys \to a x_{1}(t) + b x_{2}(t) $
 $ x_{2}(t) \to *b \to $

And $ a $ and $ b $ are any complex number,

Then the system is linear.


Example of a Linear System

Equation: $ y[n] = 2 x[n] $

 $ x_{1}[n] \to sys \to *a \to $
                      $ + \to 2a x_{1}[n] + 2b x_{2}[n] $
 $ x_{2}[n] \to sys \to *b \to $
 $ x_{1}[n] \to *a \to $
               $ + \to sys \to 2a x_{1}[n] + 2b x_{2}[n] $
 $ x_{2}[n] \to *b \to $

Since $ 2a x_{1}[n] + 2b x_{2}[n] $ and $ 2a x_{1}[n] + 2b x_{2}[n] $ are equal, the system is linear.

Example of a Non-Linear System

Equation: $ y[n] = x[n]^2 $

 $ x_{1}[n] \to sys \to *a \to $
                      $ + \to a x_{1}[n]^2 + b x_{2}[n]^2 $
 $ x_{2}[n] \to sys \to *b \to $
 $ x_{1}[n] \to *a \to $
               $ + \to sys \to (a x_{1}[n] + b x_{2}[n])^2 $
 $ x_{2}[n] \to *b \to $

Since $ a x_{1}[n]^2 + b x_{2}[n]^2 $ and $ (a x_{1}[n] + b x_{2}[n])^2 $ are not equal, the system is not linear.

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang