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Consider the system | Consider the system | ||
| − | <math> y(t)=x( | + | <math> y(t)=x(t)^2 \,</math> |
let | let | ||
| − | <math> | + | <math> x(t_0)=2\,</math> |
| − | + | ||
| + | <math> x(t_1)=-2\,</math> | ||
<math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math> | <math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math> | ||
Revision as of 06:01, 11 September 2008
Linear System Definition
A system takes a given input and produces an output. For the system to be linear it must preserve addition and multiplication. In mathematical terms:
$ x(t+t_0)=x(t) + x(t_0)\, $
and
$ x(kt)=kx(t)\, $
Linear System Example
Consider the system $ y[n]=x[n]\cdot\mathbf{M} $
let
$ \mathbf{a} = \begin{bmatrix}2 & 2 \end{bmatrix} $
$ \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} $
$ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $
$ k=3\, $
If the system is linear these properties hold:
$ y[\mathbf{a}+\mathbf{b}]=y[\mathbf{a}]+y[\mathbf{b}] \, $
thus the system is linear.
$ y[k\mathbf{a}]=ky[\mathbf{a}] \, $
Here is the proof that the first prop holds:
$ y[a] = \begin{bmatrix}8 & 12 \end{bmatrix} $
$ y[a+b] = \begin{bmatrix} 6 &3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} =\begin{bmatrix}24 & 18 \end{bmatrix} $
$ y[a]+y[b]= \begin{bmatrix}8 & 12 \end{bmatrix} +\begin{bmatrix}16 & 6 \end{bmatrix} = \begin{bmatrix}24 & 18 \end{bmatrix} $
And the second:
$ ky[\mathbf{a}] =\begin{bmatrix} 24 &36 \end{bmatrix} \, $
$ y[k\mathbf{a}] =\begin{bmatrix} 6 &6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \, $ $ =\begin{bmatrix} 24 &36 \end{bmatrix} $
Non-Linear System Example
Consider the system $ y(t)=x(t)^2 \, $
let
$ x(t_0)=2\, $
$ x(t_1)=-2\, $ $ \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} $
$ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $
$ k=3\, $
If the system is linear these properties hold:
$ y[\mathbf{a}+\mathbf{b}]=y[\mathbf{a}]+y[\mathbf{b}] \, $
thus the system is linear.
$ y[k\mathbf{a}]=ky[\mathbf{a}] \, $
Here is the proof that the first prop holds:
$ y[a] = \begin{bmatrix}8 & 12 \end{bmatrix} $
$ y[a+b] = \begin{bmatrix} 6 &3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} =\begin{bmatrix}24 & 18 \end{bmatrix} $
$ y[a]+y[b]= \begin{bmatrix}8 & 12 \end{bmatrix} +\begin{bmatrix}16 & 6 \end{bmatrix} = \begin{bmatrix}24 & 18 \end{bmatrix} $
And the second:
$ ky[\mathbf{a}] =\begin{bmatrix} 24 &36 \end{bmatrix} \, $
$ y[k\mathbf{a}] =\begin{bmatrix} 6 &6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \, $ $ =\begin{bmatrix} 24 &36 \end{bmatrix} $
