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<math> \mathbf{y}[b]=\mathbf{b}\cdot\mathbf{M} </math> | <math> \mathbf{y}[b]=\mathbf{b}\cdot\mathbf{M} </math> | ||
where | where | ||
| − | <math> \mathbf{y}[b] = \begin{bmatrix} | + | <math> \mathbf{y}[b] = \begin{bmatrix}16 & 6 \end{bmatrix} </math>, |
<math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, and | <math> \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, and | ||
<math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math> | <math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math> | ||
| Line 27: | Line 27: | ||
| − | <math>y[a+b]= | + | <math>y[a+b]=y[a]+y[b] \,</math> |
<math>y[kb]=ky[b] \,</math> | <math>y[kb]=ky[b] \,</math> | ||
| + | |||
| + | by performing the math | ||
| + | |||
| + | <math> \mathbf{y}[a] = \begin{bmatrix}8 & 12 \end{bmatrix} </math> | ||
| + | |||
| + | |||
| + | <math> \mathbf{y}[a+b] = \begin{bmatrix}24 & 18 \end{bmatrix} </math> | ||
| + | |||
| + | |||
| + | <math>y[a]+y[b]= \begin{bmatrix}8 & 12 \end{bmatrix} +\begin{bmatrix}16 & 6 \end{bmatrix} = \begin{bmatrix}24 & 18 \end{bmatrix}</math> | ||
Revision as of 10:28, 10 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+t0)=x(t) + x(t0) $
and
$ x(k*t)=k*x(t) $
Linear System Example
Consider the system $ \mathbf{y}[b]=\mathbf{b}\cdot\mathbf{M} $ where $ \mathbf{y}[b] = \begin{bmatrix}16 & 6 \end{bmatrix} $, $ \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} $, and $ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $
let $ \mathbf{a} = \begin{bmatrix}2 & 2 \end{bmatrix} $
If the system is linear these properties hold:
$ y[a+b]=y[a]+y[b] \, $
$ y[kb]=ky[b] \, $
by performing the math
$ \mathbf{y}[a] = \begin{bmatrix}8 & 12 \end{bmatrix} $
$ \mathbf{y}[a+b] = \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} $
