Which has fourier series such that ntthe fourier transform



t  

e  α  �  for t  �  0 , ( α  �  0  

e  �� 

for t  �  0 , ( α  �  0 
�  1  δ ( )  �  1  Sign  

2  2  (9.205)  
�  1  δ ( )  � 


2  j 

V e n  j  2π  n  (9.206)  

T 

� 

v t e�  j  2  π ft  

�  n��� 
V e n  j  2  π  n  t  e�  j  2  π ft  
T  
� 
V n  ∫ ���  e  j  2  π  n  t  e�  j  2  π ft  
�  T  
V n 

e�  j  2  π  (  f  �  n  )  t  dt  
�  T  (9.207) 
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Frequency Domain Circuit Analysis 


1  �  e�  j  2  πft  dt  �  (9.208) 

�  �  j  2 π (  f  �  n  )  t  ⎛  n 



j  f  �  T  )  �  δ  ⎜⎜  f  �  
∫ �  ⎜⎜⎝  T 


V  δ⎛⎜  f  �  n  (9.210)  

n���  n  ⎜⎜⎝  T 
Example 9.14
Determine the spectrum V ( f ) of the periodic voltage waveform, v ( t ) of Figure 9.35 with τ�T /3.Solution
From equations 9.117 and 9.210 we can write V ( f ) as follows:
V f ( ) �  � ∑ 
V A  τ  sinc  ⎛⎜  n  τ  ⎞⎟⎟⎟⎟⎠  δ  ⎛⎜  f  �  n  

n���  T  ⎜⎜⎝  T  ⎜⎜⎝  T  (9.211)  
�  � ∑ 
V A  sinc  ⎛⎜  n  ⎞⎟⎟⎟⎟⎠  δ  ⎛⎜⎜⎜⎝  f  �  n  
n���  3  ⎜⎜⎝  3  T 
9.3.6.6 Rayleigh’s Energy Theorem
This theorem states that the energy, Ex , of a signal x ( t ) can be calculated from its spectrum X( f ) according to the following equation:
E  x�  x t ( ) 2  dt    X f ( )  2  (9.212) 

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