(New page: == Property of ROC == ---- Property 1 The ROC of the Laplace Transformation consists of vertical strips in the complex plane. It could be empty or the entire plane. ---- Property 2 If ...) |
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Property 2 | Property 2 | ||
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If x(t) is of "Finite duration", i.e. there exists a <math>t_m</math> such that x(t)=0 when <math>|t|>t_m</math>, | If x(t) is of "Finite duration", i.e. there exists a <math>t_m</math> such that x(t)=0 when <math>|t|>t_m</math>, | ||
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Property 3 | Property 3 | ||
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| − | then whenever a vertical line is in the ROC, the half plane left of that line is also in the ROC. | + | If x(t) is "left sided", i.e. there exists a <math>t_m</math> such that x(t)=0 when <math>t>t_m</math>, then whenever a vertical line is in the ROC, the half plane left of that line is also in the ROC. |
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Property 4 | Property 4 | ||
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| − | then whenever a vertical line is in the ROC, the half plane right of that line is also in the ROC. | + | If x(t) is "right sided", i.e. there exists a <math>t_M</math> such that x(t)=0 when <math>t<t_M</math>, then whenever a vertical line is in the ROC, the half plane right of that line is also in the ROC. |
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Property 5 | Property 5 | ||
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| − | then the ROC is either empty of it is a strip in the complex plane. (only one strip) | + | If x(t) is "two sided", i.e. there exists no <math>t_m</math> such that x(t)=0 for <math>t>t_m</math> and no <math>t_M</math> such that x(t)=0 for <math>t<t_M</math>, then the ROC is either empty of it is a strip in the complex plane. (only one strip) |
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Property 6 | Property 6 | ||
| − | If X(s) is rational, i.e. <math>X(s)=\frac {P(s)}{Q(s)}</math> | + | |
| + | If X(s) is rational, i.e. | ||
| + | <math>X(s)=\frac {P(s)}{Q(s)}</math> | ||
| + | where P(s),Q(s) are polynomial, | ||
Then the ROC does not contain any zero of Q(s), i.e. the poles of X(s). | Then the ROC does not contain any zero of Q(s), i.e. the poles of X(s). | ||
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Property 7 | Property 7 | ||
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If <math>X(s)=\frac {P(s)}{Q(s)}</math> and x(t) is right sided, | If <math>X(s)=\frac {P(s)}{Q(s)}</math> and x(t) is right sided, | ||
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Property 8 | Property 8 | ||
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If <math>X(s)=\frac {P(s)}{Q(s)}</math>, | If <math>X(s)=\frac {P(s)}{Q(s)}</math>, | ||
ROC is either bounded by poles or extends to infinity or -infinity. | ROC is either bounded by poles or extends to infinity or -infinity. | ||
Latest revision as of 13:59, 24 November 2008
Property of ROC
Property 1
The ROC of the Laplace Transformation consists of vertical strips in the complex plane. It could be empty or the entire plane.
Property 2
If x(t) is of "Finite duration", i.e. there exists a $ t_m $ such that x(t)=0 when $ |t|>t_m $,
and if $ \int_{-\infty}^\infty|x(t)|^2dt $ is finite for all values of s,
Then the ROC is the entire complex plane.
Property 3
If x(t) is "left sided", i.e. there exists a $ t_m $ such that x(t)=0 when $ t>t_m $, then whenever a vertical line is in the ROC, the half plane left of that line is also in the ROC.
Property 4
If x(t) is "right sided", i.e. there exists a $ t_M $ such that x(t)=0 when $ t<t_M $, then whenever a vertical line is in the ROC, the half plane right of that line is also in the ROC.
Property 5
If x(t) is "two sided", i.e. there exists no $ t_m $ such that x(t)=0 for $ t>t_m $ and no $ t_M $ such that x(t)=0 for $ t<t_M $, then the ROC is either empty of it is a strip in the complex plane. (only one strip)
Property 6
If X(s) is rational, i.e. $ X(s)=\frac {P(s)}{Q(s)} $ where P(s),Q(s) are polynomial,
Then the ROC does not contain any zero of Q(s), i.e. the poles of X(s).
Property 7
If $ X(s)=\frac {P(s)}{Q(s)} $ and x(t) is right sided,
Then the ROC is the half plane starting from the vertical line through the pole with the largest real part and extending to infinity.
If $ X(s)=\frac {P(s)}{Q(s)} $ and x(t) is left sided,
Then the ROC is the half plane starting from the vertical line through the pole with the smallest real part and extending to -infinity.
Property 8
If $ X(s)=\frac {P(s)}{Q(s)} $,
ROC is either bounded by poles or extends to infinity or -infinity.
