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�