(→Example) |
(→Example) |
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=<math>Y(jw)=H(jw)X(jw), H(jw)=\frac{Y(jw)}{X(jw)}</math> | =<math>Y(jw)=H(jw)X(jw), H(jw)=\frac{Y(jw)}{X(jw)}</math> | ||
== Example == | == Example == | ||
| − | Consider a LTI system that is chracterized by the differential equation | + | Consider a LTI system that is chracterized by the differential equation |
| + | |||
| + | |||
<math> \frac{d^2y(t)}{dt^2}+4\frac{dy(t)}{dt}+3y(t) = \frac{dx(t)}{dt}+2x(t)</math> | <math> \frac{d^2y(t)}{dt^2}+4\frac{dy(t)}{dt}+3y(t) = \frac{dx(t)}{dt}+2x(t)</math> | ||
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<math> H(jw)=\frac{(jw)+2}{(jw)^2+4(jw)+3}</math> | <math> H(jw)=\frac{(jw)+2}{(jw)^2+4(jw)+3}</math> | ||
| + | |||
| + | |||
| + | <math>H(jW)=\frac{jw+2}{(jw+1)(jw+3)}</math> | ||
Revision as of 17:34, 24 October 2008
System Characterized By Linear Constant-Coefficient Differential Equations
$ \sum_{k=0}^{N}a_k\frac {d^ky(t)}{dt^k} = \sum_{k=0}^{M}b_k\frac {d^kx(t)}{dt^k} $
=$ Y(jw)=H(jw)X(jw), H(jw)=\frac{Y(jw)}{X(jw)} $
Example
Consider a LTI system that is chracterized by the differential equation
$ \frac{d^2y(t)}{dt^2}+4\frac{dy(t)}{dt}+3y(t) = \frac{dx(t)}{dt}+2x(t) $
$ H(jw)=\frac{(jw)+2}{(jw)^2+4(jw)+3} $
$ H(jW)=\frac{jw+2}{(jw+1)(jw+3)} $
