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== linear system == | == linear system == | ||
| + | <math>y(t) = 4x(t)</math> | ||
| + | <math>x_1(t)\rightarrow y_1(t) = 4x_1(t)</math> | ||
| + | <math>x_2(t)\rightarrow y_2(t) = 4x_2(t)</math> | ||
| + | <math>x_3 = ax_1(t) + bx_2(t)</math> | ||
| + | <math>w(t) = 4t_3(t)</math> | ||
| + | |||
| + | <math>=4(ax_1(t) + bx_2(t))</math> | ||
| + | |||
| + | <math>=4ax_1(t) + 4bx_2(t)</math> | ||
| + | |||
| + | <math>=ay_1(t) + by_2(t)</math> | ||
| + | |||
| + | so it is linear | ||
== non linear system == | == non linear system == | ||
| + | |||
| + | <math>y[n] = 4x[n] + 5</math> | ||
| + | |||
| + | <math>x_1 \rightarrow y_1[n] = 4x_1[n] +5 = 21 </math> | ||
| + | |||
| + | <math>x_2 \rightarrow y_2[n] = 4x_2[n] + 5 = 25</math> | ||
| + | |||
| + | <math>w[n] = 4[x_1[n] + x_2[n]] + 5 = 41</math> | ||
| + | |||
| + | <math>w[n]\neq y_1[n] + y_2[n] </math> | ||
| + | |||
| + | so it is non linear | ||
Latest revision as of 10:53, 11 September 2008
Def of linear system
Linear system is a system that possesses the important property of superposition.(Text book P.53)
linear system
$ y(t) = 4x(t) $
$ x_1(t)\rightarrow y_1(t) = 4x_1(t) $
$ x_2(t)\rightarrow y_2(t) = 4x_2(t) $
$ x_3 = ax_1(t) + bx_2(t) $
$ w(t) = 4t_3(t) $
$ =4(ax_1(t) + bx_2(t)) $
$ =4ax_1(t) + 4bx_2(t) $
$ =ay_1(t) + by_2(t) $
so it is linear
non linear system
$ y[n] = 4x[n] + 5 $
$ x_1 \rightarrow y_1[n] = 4x_1[n] +5 = 21 $
$ x_2 \rightarrow y_2[n] = 4x_2[n] + 5 = 25 $
$ w[n] = 4[x_1[n] + x_2[n]] + 5 = 41 $
$ w[n]\neq y_1[n] + y_2[n] $
so it is non linear
