(New page: The signal is f(t) = sin(t) and t1=0 and t2=2pi == Energy of sin(t) == The energy expended from t1 to t2 is: <math>E=\int_{t0}^{t2}{|f(t)|^2dt}</math> Therefore for our signal: <mat...)
 
 
Line 6: Line 6:
  
  
<math>E=\int_{t0}^{t2}{|f(t)|^2dt}</math>
+
<math>E=\int_{t1}^{t2}{|f(t)|^2dt}</math>
  
  
Line 20: Line 20:
  
  
<math>=\frac{1}{2}(2\pi-0+0-0)</math>
+
<math>=\frac{1}{2}(2\pi)</math>
  
 
Therefore the total energy for the signal sin(t) from t1 to t2 is:
 
Therefore the total energy for the signal sin(t) from t1 to t2 is:
 
<math>E=\pi</math>
 
<math>E=\pi</math>
 +
 +
== Average Power in time interval [t1, t2] ==
 +
 +
The average power for a signal is given by:
 +
 +
 +
<math>P=\frac{1}{t2-t1}\int_{t1}^{t2}{|f(t)|^2dt}</math>
 +
 +
Therefore for our signal sin(t) from t1=0 to t2=2pi:
 +
 +
 +
<math>P=\frac{1}{2\pi-0}\int_{0}^{2\pi}{|sin(t)|^2dt}</math>
 +
 +
 +
<math>=\frac{1}{2\pi}\frac{1}{2}\int_0^{2\pi}(1-cos(2t))dt</math>
 +
 +
 +
<math>=\frac{1}{4\pi}(t-\frac{1}{2}sin(2t))|_{t=0}^{t=2\pi}</math>
 +
 +
 +
<math>=\frac{1}{4\pi}(2\pi)</math>
 +
 +
Therefore the average power for the signal sin(t) from t1 to t2 is:
 +
<math>P=\frac{1}{2}</math>

Latest revision as of 14:45, 5 September 2008

The signal is f(t) = sin(t) and t1=0 and t2=2pi

Energy of sin(t)

The energy expended from t1 to t2 is:


$ E=\int_{t1}^{t2}{|f(t)|^2dt} $


Therefore for our signal:

$ E=\int_0^{2\pi}{|sin(t)|^2dt} $


$ =\frac{1}{2}\int_0^{2\pi}(1-cos(2t))dt $


$ =\frac{1}{2}(t-\frac{1}{2}sin(2t))|_{t=0}^{t=2\pi} $


$ =\frac{1}{2}(2\pi) $

Therefore the total energy for the signal sin(t) from t1 to t2 is: $ E=\pi $

Average Power in time interval [t1, t2]

The average power for a signal is given by:


$ P=\frac{1}{t2-t1}\int_{t1}^{t2}{|f(t)|^2dt} $

Therefore for our signal sin(t) from t1=0 to t2=2pi:


$ P=\frac{1}{2\pi-0}\int_{0}^{2\pi}{|sin(t)|^2dt} $


$ =\frac{1}{2\pi}\frac{1}{2}\int_0^{2\pi}(1-cos(2t))dt $


$ =\frac{1}{4\pi}(t-\frac{1}{2}sin(2t))|_{t=0}^{t=2\pi} $


$ =\frac{1}{4\pi}(2\pi) $

Therefore the average power for the signal sin(t) from t1 to t2 is: $ P=\frac{1}{2} $

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett