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| − | <math>y[a+b]=y[a]+y[b] \,</math> | + | <math>y[\mathbf{a}+\mathbf{b}]=y[\mathbf{a}]+y[\mathbf{b}] \,</math> |
| − | <math>y[ | + | <math>y[k\mathbf{a}]=ky[\mathbf{a}] \,</math> |
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And the second: | And the second: | ||
| − | <math>ky[a] = \,</math> | + | <math>ky[\mathbf{a}] =\begin{bmatrix} 24 &36 \end{bmatrix} \,</math> |
| + | |||
| + | <math>y[k\mathbf{a}] =\begin{bmatrix} 6 &6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \,</math> | ||
| + | |||
| + | =\begin{bmatrix} 24 &36 \end{bmatrix} | ||
Revision as of 06:49, 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}] \, $
$ 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}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}
