(New page: == Linearity == A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems: :Suppose there are two in...) |
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:A linear system must satisfy the condition | :A linear system must satisfy the condition | ||
::<math>\,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\}</math> | ::<math>\,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\}</math> | ||
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| + | == Example of a Linear System == | ||
| + | ::<math>\,x1(t) = sin(t)</math> | ||
| + | ::<math>\,x2(t) = cos(t)</math> | ||
| + | |||
| + | ::<math>\,y1(t) = C\left\{x1(t)\right\} = C(sin(t))</math> | ||
| + | ::<math>\,y2(t) = C\left\{x2(t)\right\} = C(cos(t))</math> | ||
| + | |||
| + | ::<math>\,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\}</math> | ||
Revision as of 04:44, 9 September 2008
Linearity
A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems:
- Suppose there are two inputs
- $ \,x1(t) $
- $ \,x2(t) $
- with outputs
- $ \,y1(t) = C\left\{x1(t)\right\} $
- $ \,y2(t) = C\left\{x2(t)\right\} $
- A linear system must satisfy the condition
- $ \,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\} $
Example of a Linear System
- $ \,x1(t) = sin(t) $
- $ \,x2(t) = cos(t) $
- $ \,y1(t) = C\left\{x1(t)\right\} = C(sin(t)) $
- $ \,y2(t) = C\left\{x2(t)\right\} = C(cos(t)) $
- $ \,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\} $
