Difference between revisions of "HW2.C Nicholas Browdues ECE301Fall2008mboutin" - Rhea

Revision as of 06:39, 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 $ \mathbf{y}[n]=\mathbf{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_1=1 $ $ k_2=2 $

If the system is linear these properties hold:

$ y[a+b]=y[a]+y[b] \, $

$ y[kb]=ky[b] \, $

Here is the proof that the first prop holds.

$ \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} $