(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 ...)
 
(Property of ROC)
 
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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
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.
+
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
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.
+
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
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)
+
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> with P(s),Q(s) polynomial,
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 +
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 14:59, 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.

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood