FourierTransformPairs
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A short reference handout titled Signals & Systems - Reference Tables, apparently a downloaded copy kept in the Spectral Theory Book folder. It gives the Fourier transform and inverse definitions, property rules (shift, scaling, differentiation, integration) and transform pairs for delta, step, sgn, rect, tri, sinc, cosines, exponentials and Gaussians. It also lists trigonometric and complex exponential Fourier series, trig identities and common integrals. The text is symbol-garbled but the content is clear.
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Signals & Systems - Reference Tables1Table of Fourier Transform Pairs
Function, f(t) Fourier Transform, F( /g119)
Definition of Inverse Fourier Transform
/g242/g165
/g165/g45/g61 /g119 /g119/g112/g119de F tftj)(21)(Definition of Fourier Transform
/g242/g165
/g165/g45/g45/g61 dt etf Ftj/g119/g119 )( )(
) (0ttf/g45 0)(tje F/g119/g119/g45
tjetf0)(/g119 ) (0/g119/g119/g45 F
)(tf/g97)(1
/g97/g119
/g97F
)(tF )(2 /g119/g112/g45f
nn
dttfd )( )()( /g119 /g119 F jn
)()( tf jtn/g45
nn
dFd
/g119/g119)(
/g242
/g165/g45t
d f /g116/g116)( )()0()(/g119/g100 /g112/g119/g119FjF/g43
)(t/g100 1
tje0/g119 ) (20/g119/g119/g112/g100 /g45
(t)sgn
/g119j2
Signals & Systems - Reference Tables2tj/g1121 ) sgn(/g119
)(tu
/g119/g119/g112/g100j1)(/g43
/g229/g165
/g45/g165/g61nt jn
neF0/g119/g229/g165
/g45/g165/g61/g45
nn n F ) ( 20/g119 /g119/g100 /g112
)(/g116trect )2(/g119/g116/g116Sa
)2(2BtSaB
/g112)(Brect/g119
)(ttri)2(2/g119Sa
)2( )2cos(/g116 /g116/g112 trecttA2 2)2() cos(
/g119/g116/g112/g119/g116
/g116/g112
/g45A
) cos(0t/g119 /g91/g93 ) ( ) (0 0 /g119/g119/g100 /g119/g119/g100/g112 /g43 /g43 /g45
) sin(0t/g119/g91/g93 ) ( ) (0 0 /g119/g119/g100 /g119/g119/g100/g112/g43 /g45 /g45j
) cos()(0t tu /g119/g91/g932 2
00 0 ) ( ) (2 /g119 /g119/g119/g119/g119/g100 /g119/g119/g100/g112
/g45/g43 /g43 /g43 /g45j
) sin()(0t tu /g119/g91/g932 2
02
0 0 ) ( ) (2 /g119 /g119/g119/g119/g119/g100 /g119/g119/g100/g112
/g45/g43 /g43 /g45 /g45j
) cos( )(0t etut/g119/g97/g45
2 2
0 ) () (
/g119 /g97 /g119/g119 /g97
jj
/g43/g43/g43
Signals & Systems - Reference Tables3) sin( )(0t etut/g119/g97/g45
2 2
00
) ( /g119/g97 /g119/g119
j/g43/g43
te/g97/g45
2 22
/g119/g97/g97
/g43
)2/(2 2/g115 te/g45 2/22 2/g119/g115/g112/g115/g45e
tetu/g97/g45)(
/g119/g97 j/g431
ttetu/g97/g45)(
2) (1
/g119/g97 j/g43
/g216 Trigonometric Fourier Series
/g40/g41/g229/g165
/g61/g43 /g43/g61
10 0 0 ) sin( ) cos( )(
nn n nt b nt a a tf /g119 /g119
where
/g242/g242 /g242
/g61/g61 /g61
T
nTT
n
dtnt tfTbdtnt tfTa dttfTa
000000
) sin()(2 and, ) cos()(2 , )(1
/g119/g119
/g216 Complex Exponential Fourier Series
/g242 /g229/g45/g165
/g45/g165/g61/g61 /g61T
ntj
n
nntj
n dt etfTF eF tf
00 )(1 where, )(/g119 /g119
Signals & Systems - Reference Tables4Some Useful Mathematical Relationships
2) cos(jx jxe ex/g45/g43/g61
je exjx jx
2) sin(/g45/g45/g61
) sin() sin() cos() cos() cos( y x y x yx /g109 /g61/g177
) sin() cos() cos() sin() sin( y x y x yx /g177 /g61/g177
)(sin)( cos)2cos(2 2x x x /g45 /g61
) cos() sin(2)2sin( x x x/g61
)2cos(1)(cos22x x /g43/g61
)2cos(1)(sin22x x /g45/g61
1)(sin)( cos2 2/g61 /g43 x x
) cos() cos() cos() cos(2 yx yx y x /g43 /g43/g45 /g61
) cos() cos() sin() sin(2 yx yx y x /g43 /g45 /g45 /g61
) sin() sin() cos() sin(2 yx yx y x /g43 /g43/g45 /g61
Signals & Systems - Reference Tables5Useful Integrals
/g242dxx) cos( ) sin(x
/g242dxx) sin( ) cos( x /g45
/g242dxx x ) cos( ) sin( ) cos( x x x/g43
/g242dxx x ) sin( ) cos( ) sin( x x x/g45
/g242dxx x ) cos(2) sin()2 () cos(22x x x x /g45 /g43
/g242dxx x ) sin(2) cos()2 () sin(22x x x x /g45 /g45
/g242dxex/g97
aex/g97
/g242dx xex/g97
/g250/g251/g249
/g234/g235/g233/g4521
a axex/g97
/g242dxexx/g972
/g250
/g251/g249
/g234
/g235/g233/g45 /g453 222 2
a ax
axex/g97
/g242/g43xdx
/g98/g97x/g98/g97/g98/g43ln1
/g242/g4322 2xdx
/g98 /g97)( tan1 1
/g97/g98
/g97/g98x/g45