fourier transform pairs
TRANSCRIPT
Signals & Systems - Reference Tables 1
Table of Fourier Transform Pairs
Function, f(t) Fourier Transform, F(�)
Definition of Inverse Fourier Transform
��
��
� ��
�
� deFtf tj)(21)(
Definition of Fourier Transform
��
��
�
� dtetfF tj�� )()(
)( 0ttf � 0)( tjeF �
��
tjetf 0)( � )( 0�� �F
)( tf �)(1
�
�
�
F
)(tF )(2 �� �f
n
n
dttfd )( )()( �� Fj n
)()( tfjt n�
n
n
dFd�
�)(
���
t
df �� )( )()0()(���
�
� FjF
�
)(t� 1
tje 0� )(2 0���� �
(t)sgn�j2
Signals & Systems - Reference Tables 2
tj�
1 )sgn(�
)(tu�
���j1)( �
��
���n
tjnneF 0�
��
���
�
nn nF )(2 0����
)(�
trect )2
(���Sa
)2
(2
BtSaB�
)(B
rect �
)(ttri)
2(2 �Sa
)2
()2
cos(��
� trecttA22)2(
)cos(�
�
�
��
�
�
�
A
)cos( 0t� � �)()( 00 ������� ���
)sin( 0t�
� �)()( 00 �������
���
j
)cos()( 0ttu �
� �22
000 )()(
2 ��
�������
�
�
����j
)sin()( 0ttu �
� �22
0
2
00 )()(2 ��
�������
�
�
����
j
)cos()( 0tetu t�
��
220 )(
)(���
��
jj��
�
Signals & Systems - Reference Tables 3
)sin()( 0tetu t�
��
220
0
)( ���
�
j��
te ��
222��
�
�
)2/( 22�te� 2/22
2 ��
���e
tetu ��)(�� j�
1
ttetu ��)(2)(
1�� j�
� Trigonometric Fourier Series
� ���
�
���
1000 )sin()cos()(
nnn ntbntaatf ��
where
�
��
�
��
T
n
TTn
dtnttfT
b
dtnttfT
adttfT
a
00
0000
)sin()(2
and, )cos()(2 , )(1
�
�
� Complex Exponential Fourier Series
�� �
�
���
��
Tntj
nn
ntjn dtetf
TFeFtf
0
0)(1 where, )( ��
Signals & Systems - Reference Tables 4
Some Useful Mathematical Relationships
2)cos(
jxjx eex�
��
jeex
jxjx
2)sin(
���
)sin()sin()cos()cos()cos( yxyxyx ���
)sin()cos()cos()sin()sin( yxyxyx ���
)(sin)(cos)2cos( 22 xxx ��
)cos()sin(2)2sin( xxx �
)2cos(1)(cos2 2 xx ��
)2cos(1)(sin2 2 xx ��
1)(sin)(cos 22�� xx
)cos()cos()cos()cos(2 yxyxyx ����
)cos()cos()sin()sin(2 yxyxyx ����
)sin()sin()cos()sin(2 yxyxyx ����
Signals & Systems - Reference Tables 5
Useful Integrals
� dxx)cos( )sin(x
� dxx)sin( )cos(x�
� dxxx )cos( )sin()cos( xxx �
� dxxx )sin( )cos()sin( xxx �
� dxxx )cos(2 )sin()2()cos(2 2 xxxx ��
� dxxx )sin(2 )cos()2()sin(2 2 xxxx ��
� dxe x�
ae x�
� dxxe x�
��
���
� 2
1aa
xe x�
� dxex x�2
��
���
�� 32
2 22aa
xaxe x�
�� xdx��
x���
�ln1
��
222 xdx��
)(tan1 1
�
�
��
x�