Which has fourier series such that ntthe fourier transform

300	Chapter 9

		t
	e	α	�		for t	�	0 , ( α	�	0
lim α→0	e	��		0	for t	�	0 , ( α	�	0

�	1	δ ( )	�	1	Sign
	2			2		(9.205)
�	1	δ ( )	�	2	1	(9.205)
	2			j	2 π f

	V e n	j	2π	n		(9.206)
	V e n	j	2π	T		(9.206)

�	� ∫��	v t e�				2
�		n��			V e n		j		2	π	n		t	e�		j	2	π ft
�		n��			V e n		j		2	π	T		t	e�		j	2	π ft
	� ∑		V n	∫ ��		e	j	2	π		n	t	e�			j	2	π ft
�	� ∑		V n	∫ ��		e	j	2	π		T	t	e�			j	2	π ft
			V n	� ∫��		e�			j	2	π	(	f	�	n	)	t	dt
�			V n	� ∫��		e�			j	2	π	(	f		T	)	t	dt	(9.207)

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Frequency Domain Circuit Analysis	301

	1	�	e�	j	2	πft	dt	�		(9.208)

�	�	j	2 π (	f	�	n	)	t				⎛			n	⎞
		j		f	�	T	)			�	δ	⎜⎜	f	�
∫ �												⎜⎜⎝			T	⎟⎟⎠

n��

⎜⎜⎝

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			⎞⎟⎟⎟⎟⎠		δ		⎛⎜	�	n
	n��	T		sinc		⎜⎜⎝	T			⎞⎟⎟⎟⎟⎠		δ		⎜⎜⎝		T	(9.211)
�	� ∑	V A	sinc		⎛⎜	n		⎞⎟⎟⎟⎟⎠	δ		⎛⎜⎜⎜⎝		f	�	n		(9.211)
	n��	3	sinc		⎜⎜⎝	3		⎞⎟⎟⎟⎟⎠	δ		⎛⎜⎜⎜⎝			�	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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