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<math> x(t) = a*x1(t) + b*x2(t) = a*y1(t) + b*y2(t) </math> | <math> x(t) = a*x1(t) + b*x2(t) = a*y1(t) + b*y2(t) </math> | ||
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| + | |||
| + | |||
| + | Take for a simple example: | ||
| + | |||
| + | Ex) What is the output of: | ||
| + | |||
| + | <math> x[n] = e^{j*\pi*n} -> n*e^{-j*\pi*n} </math> | ||
| + | |||
| + | <math> x[n] \to Sys 1 \to n*x[-n] </math> | ||
| + | |||
| + | |||
| + | from: | ||
| + | <math> e^{j*n*y} = cos(n*y) + j*sin(n*y) </math> | ||
| + | |||
| + | we determine: | ||
| + | <math> x[n] = cos(\pi*n) = \frac{e^{j*\pi*n} + e^{-j*\pi*n}}{2} </math> | ||
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| + | |||
| + | |||
| + | yields to <math> x[t*n] = \frac{te^{j*\pi*n} + t*e^{-j*\pi*n}}{2} </math> | ||
| + | |||
| + | <math> == cos(\pi*n) \to -1 </math> on every even integer interval. | ||
Latest revision as of 14:35, 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(\pi*n) = \frac{e^{j*\pi*n} + e^{-j*\pi*n}}{2} $
yields to $ x[t*n] = \frac{te^{j*\pi*n} + t*e^{-j*\pi*n}}{2} $
$ == cos(\pi*n) \to -1 $ on every even integer interval.
