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System --> z(t) = x(2t) | System --> z(t) = x(2t) | ||
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| + | ---------------------------- | ||
X1(t) --> Y1(t) = 4X1(t) | X1(t) --> Y1(t) = 4X1(t) | ||
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X2(t) --> Y2(t) = 3X2(t) | X2(t) --> Y2(t) = 3X2(t) | ||
| − | W(t) = Y1(t) + Y2(t) = 4X1(t) + 3X2(t) --> System --> Z(t) = 4X1(2t) + 3X2(2t) | + | W(t) = Y1(t) + Y2(t) = 4X1(t) + 3X2(t) --> System --> Z(t) = 4X1(2t) + 3X2(2t) ============== (1) |
---------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------- | ||
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| − | Z(t) = Y1 + Y2 = 4Y1(2t) + 3Y2(2t) | + | Z(t) = Y1 + Y2 = 4Y1(2t) + 3Y2(2t) ============ (2) |
Equations (1) and (2) are the same so therefore it is a linear system. | Equations (1) and (2) are the same so therefore it is a linear system. | ||
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| + | == Example of non-Linear System == | ||
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| + | System --> Z(t) = sqrt[x(t)] | ||
| + | |||
| + | -------------------------------- | ||
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| + | X1(t) --> Y1(t) = 4X1(t) | ||
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| + | X2(t) --> Y2(t) = 9X2(t) | ||
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| + | W(t) = Y1(t) + Y2(t) = 4X1(t) + 9X2(t) --> System --> 2sqrt[X1(t)] + 3sqrt[X2(t)] =============================== (1) | ||
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| + | --------------------------------------------------------------------------------------- | ||
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| + | X1(t) --> System --> Y1(t) = sqrt[X1(t)] --> multiply by 4 --> W1(t) = 4sqrt[X1(t)] | ||
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| + | X2(t) --> System --> Y2(t) = sqrt[X2(t)] --> multiply by 9 --> W2(t) = 9sqrt[X2(t)] | ||
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| + | Z(t) = W1(t) + W2(t) = 4sqrt[X1(t)] + 9sqrt[X2(t)] ================================ (2) | ||
| + | |||
| + | Equations (1) and (2) are not equal so there for the system is not linear. | ||
Latest revision as of 12:50, 11 September 2008
A linear system is a system for which if you can add two functions and multiply them by scalars then pass them through the system, it is equivalent to passing the two signals through the system and then adding them and multiplying them by scalars.
Example of Linear System
System --> z(t) = x(2t)
X1(t) --> Y1(t) = 4X1(t)
X2(t) --> Y2(t) = 3X2(t)
W(t) = Y1(t) + Y2(t) = 4X1(t) + 3X2(t) --> System --> Z(t) = 4X1(2t) + 3X2(2t) ============== (1)
X1(t) --> System --> Y1(2t) --> multiply by 4 --> 4Y1(2t)
X2(t) --> System --> Y2(2t) --> multiply by 3 --> 3Y2(2t)
Z(t) = Y1 + Y2 = 4Y1(2t) + 3Y2(2t) ============ (2)
Equations (1) and (2) are the same so therefore it is a linear system.
Example of non-Linear System
System --> Z(t) = sqrt[x(t)]
X1(t) --> Y1(t) = 4X1(t)
X2(t) --> Y2(t) = 9X2(t)
W(t) = Y1(t) + Y2(t) = 4X1(t) + 9X2(t) --> System --> 2sqrt[X1(t)] + 3sqrt[X2(t)] =============================== (1)
X1(t) --> System --> Y1(t) = sqrt[X1(t)] --> multiply by 4 --> W1(t) = 4sqrt[X1(t)]
X2(t) --> System --> Y2(t) = sqrt[X2(t)] --> multiply by 9 --> W2(t) = 9sqrt[X2(t)]
Z(t) = W1(t) + W2(t) = 4sqrt[X1(t)] + 9sqrt[X2(t)] ================================ (2)
Equations (1) and (2) are not equal so there for the system is not linear.
