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

Revision as of 06:03, 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\, $

If the system is linear these properties hold:

$ y(t_0+t_1)=y(t_0)+y(t_1) \, $

$ 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} $