| Line 12: | Line 12: | ||
Ex) What is the output of: | Ex) What is the output of: | ||
| − | <math> x[n] = e^{j*pi*n} -> n*e^{-j*pi*n} </math> | + | <math> x[n] = e^{j*\pi*n} -> n*e^{-j*\pi*n} </math> |
<math> x[n] \to Sys 1 \to n*x[-n] </math> | <math> x[n] \to Sys 1 \to n*x[-n] </math> | ||
| Line 18: | Line 18: | ||
from: | from: | ||
| − | <math> e^{j*n*y} = cos(n*y) + j*sin(n*y) | + | <math> e^{j*n*y} = cos(n*y) + j*sin(n*y) </math> |
we determine: | we determine: | ||
x[n] = cos( | x[n] = cos( | ||
Revision as of 15:26, 18 September 2008
The Basics of Linearity
A system is linear if its inputs are sequentially equal to the outputs for a certain function:
$ x(t) = a*x1(t) + b*x2(t) = a*y1(t) + b*y2(t) $
Take for a simple example:
Ex) What is the output of:
$ x[n] = e^{j*\pi*n} -> n*e^{-j*\pi*n} $
$ x[n] \to Sys 1 \to n*x[-n] $
from:
$ e^{j*n*y} = cos(n*y) + j*sin(n*y) $
we determine: x[n] = cos(
