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Consider the system | Consider the system | ||
| − | <math> \mathbf{ | + | <math> \mathbf{y}[n]=\mathbf{x}[n]\cdot\mathbf{M} </math> |
where | where | ||
<math> \mathbf{a} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, | <math> \mathbf{a} = \begin{bmatrix}4 & 1 \end{bmatrix} </math>, | ||
<math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math>, | <math> \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} </math>, | ||
and <math> \mathbf{b} = \begin{bmatrix}7 & 12 \end{bmatrix} </math> | and <math> \mathbf{b} = \begin{bmatrix}7 & 12 \end{bmatrix} </math> | ||
Revision as of 09:08, 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}[n]=\mathbf{x}[n]\cdot\mathbf{M} $ where $ \mathbf{a} = \begin{bmatrix}4 & 1 \end{bmatrix} $, $ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $, and $ \mathbf{b} = \begin{bmatrix}7 & 12 \end{bmatrix} $
