(→CT Signal and its Fourier Series Coefficients) |
(→CT Signal and its Fourier Series Coefficients) |
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Line 2: | Line 2: | ||
Let the signal be | Let the signal be | ||
− | <math>\ x(t) = \cos( | + | <math>\ x(t) = \cos(2t\pi /3) \sin(2t \pi /9) </math> |
− | which has a fundamental frequency of | + | which has a fundamental frequency of 9 |
Now computing its coefficients: | Now computing its coefficients: | ||
− | <math>\ x(t)= (\frac{e^{ | + | <math>\ x(t)= (\frac{e^{j2t\pi /3} + e^{-j 2t\pi/ 3}}{2}) (\frac{e^{j 2t\pi /9} - e^{-j2t \pi /9}}{2j}) </math> |
− | <math>\ x(t)= \frac{e^{ | + | <math>\ x(t)= \frac{e^{-j4t\pi /9} - e^{-j8t\pi /9} + e^{j8t\pi /9} - e^{j4t\pi /9}}{4j} </math> |
Now rearranging the terms: | Now rearranging the terms: | ||
− | <math>\ x(t)= \frac{e^{ | + | <math>\ x(t)= \frac{-e^{j4t\pi /9} + e^{-j4t\pi /9} + e^{j8t\pi /9} - e^{j8t\pi /9}}{4j} </math> |
+ | |||
+ | <math>\ x(t)= \frac{-e^{j(2)2t\pi /9} + e^{-j(2)2t\pi /9} + e^{j(4)2t\pi /9} - e^{j(4)2t\pi /9}}{4j} </math> | ||
+ | |||
+ | Now, the Fourier coefficients can clearly be seen: | ||
+ | |||
+ | <math>\ a_{2}= \frac{-1}{4j} = \frac{j}{4}</math> | ||
+ | |||
+ | <math>a_{-2}= \frac{1}{4j} = \frac{-j}{4}</math> | ||
+ | |||
+ | <math>a_{4}= \frac{1}{4j} = \frac{-j}{4}</math> | ||
+ | |||
+ | <math>a_{-4}= \frac{-1}{4j} = \frac{j}{4}</math> |
Revision as of 14:46, 26 September 2008
CT Signal and its Fourier Series Coefficients
Let the signal be $ \ x(t) = \cos(2t\pi /3) \sin(2t \pi /9) $
which has a fundamental frequency of 9 Now computing its coefficients:
$ \ x(t)= (\frac{e^{j2t\pi /3} + e^{-j 2t\pi/ 3}}{2}) (\frac{e^{j 2t\pi /9} - e^{-j2t \pi /9}}{2j}) $
$ \ x(t)= \frac{e^{-j4t\pi /9} - e^{-j8t\pi /9} + e^{j8t\pi /9} - e^{j4t\pi /9}}{4j} $
Now rearranging the terms:
$ \ x(t)= \frac{-e^{j4t\pi /9} + e^{-j4t\pi /9} + e^{j8t\pi /9} - e^{j8t\pi /9}}{4j} $
$ \ x(t)= \frac{-e^{j(2)2t\pi /9} + e^{-j(2)2t\pi /9} + e^{j(4)2t\pi /9} - e^{j(4)2t\pi /9}}{4j} $
Now, the Fourier coefficients can clearly be seen:
$ \ a_{2}= \frac{-1}{4j} = \frac{j}{4} $
$ a_{-2}= \frac{1}{4j} = \frac{-j}{4} $
$ a_{4}= \frac{1}{4j} = \frac{-j}{4} $
$ a_{-4}= \frac{-1}{4j} = \frac{j}{4} $