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A*x1(t) + B*x2(t) => A*y1(t) + B*y2(t) .... extendable for any amount of complex numbers (A, B, C...) and functions (x1, x2, x3...) | A*x1(t) + B*x2(t) => A*y1(t) + B*y2(t) .... extendable for any amount of complex numbers (A, B, C...) and functions (x1, x2, x3...) | ||
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Linear System: y[n] = 4 * x[n] | Linear System: y[n] = 4 * x[n] | ||
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Let x1[n] = <math>2n</math> | Let x1[n] = <math>2n</math> | ||
x2[n] = <math>n^2</math> | x2[n] = <math>n^2</math> | ||
| − | + | ||
x1[n] + x2[n] = xtot[n] = <math>2n + n^2</math> ====> ytot[n] = 4* xtot[n] = <math>8n + 4n^2</math> | x1[n] + x2[n] = xtot[n] = <math>2n + n^2</math> ====> ytot[n] = 4* xtot[n] = <math>8n + 4n^2</math> | ||
| − | + | ||
x1[n] => y1[n] = 8n x2[n] => y2[n] = <math>4n^2</math> ====> ytot = y1[n] + y2[n] = <math>8n + 4n^2</math> | x1[n] => y1[n] = 8n x2[n] => y2[n] = <math>4n^2</math> ====> ytot = y1[n] + y2[n] = <math>8n + 4n^2</math> | ||
| − | + | ||
<math>8n + 4n^2</math> = <math>8n + 4n^2</math> | <math>8n + 4n^2</math> = <math>8n + 4n^2</math> | ||
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| + | Since the output of the two is the same, the system is linear. | ||
| − | Since the output of the two | + | |
| + | |||
| + | Nonlinear System: y[n] = x[n]<math>^2</math> | ||
| + | |||
| + | Let x1[n] = <math>2n</math> | ||
| + | x2[n] = <math>n^2</math> | ||
| + | |||
| + | x1[n] + x2[n] = xtot[n] = <math>2n + n^2</math> ====> ytot[n] = xtot[n]<math>^2</math> | ||
| + | = <math>4n^2 + 4n^3 + n^4</math> | ||
| + | |||
| + | x1[n] => y1[n] = 4n^2 x2[n] => y2[n] = <math>n^4</math> ====> ytot = y1[n] + y2[n] = <math>4n^2 + n^4</math> | ||
| + | |||
| + | <math>4n^2 + 4n^3 + n^4</math> != <math>4n^2 + n^4</math> | ||
| + | |||
| + | Since the output of the two is not the same, the system is nonlinear. | ||
Latest revision as of 08:10, 11 September 2008
Linearity
Linearity- A system is linear if a constant that multiplies an input to a system is also present in the output. Adding any number of linear combinations of complex numbers and functions of time together does not affect the linearity of the system.
A*x1(t) + B*x2(t) => A*y1(t) + B*y2(t) .... extendable for any amount of complex numbers (A, B, C...) and functions (x1, x2, x3...)
Linear System: y[n] = 4 * x[n] Let x1[n] = $ 2n $ x2[n] = $ n^2 $ x1[n] + x2[n] = xtot[n] = $ 2n + n^2 $ ====> ytot[n] = 4* xtot[n] = $ 8n + 4n^2 $ x1[n] => y1[n] = 8n x2[n] => y2[n] = $ 4n^2 $ ====> ytot = y1[n] + y2[n] = $ 8n + 4n^2 $ $ 8n + 4n^2 $ = $ 8n + 4n^2 $ Since the output of the two is the same, the system is linear.
Nonlinear System: y[n] = x[n]$ ^2 $ Let x1[n] = $ 2n $ x2[n] = $ n^2 $ x1[n] + x2[n] = xtot[n] = $ 2n + n^2 $ ====> ytot[n] = xtot[n]$ ^2 $ = $ 4n^2 + 4n^3 + n^4 $ x1[n] => y1[n] = 4n^2 x2[n] => y2[n] = $ n^4 $ ====> ytot = y1[n] + y2[n] = $ 4n^2 + n^4 $ $ 4n^2 + 4n^3 + n^4 $ != $ 4n^2 + n^4 $ Since the output of the two is not the same, the system is nonlinear.
