Which has fourier series such that ntthe fourier transform
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t | ||||||||||
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e | α | � | for t | � | 0 , ( α | � | 0 | |||
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e | �� |
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for t | � | 0 , ( α | � | 0 |
� | 1 | δ ( ) | � | 1 | Sign | |||
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2 | 2 | (9.205) | ||||||
� | 1 | δ ( ) | � |
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2 | j |
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V e n | j | 2π | n | (9.206) | ||
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T |
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� |
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v t e� | j | 2 | π ft | |||||||||||||||||
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� | 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 |
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e� | j | 2 | π | ( | f | � | n | ) | t | dt | |||||||||||
� | T | (9.207) |
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Frequency Domain Circuit Analysis |
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1 | � | e� | j | 2 | πft | dt | � | (9.208) |
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� | � | j | 2 π ( | f | � | n | ) | t | ⎛ | n |
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j | f | � | T | ) | � | δ | ⎜⎜ | f | � | ||||||||
∫ � | ⎜⎜⎝ | T |
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V | δ⎛⎜ | f | � | n | (9.210) | |||
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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 | |||||||
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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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