fourier1
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Collection of downloaded reference tables, apparently from several sources: a Signals & Systems table set, a UBC M267 Fourier transform table, and a Wikibooks Engineering Tables page. It lists transform pairs (rect, sinc, triangle, Gaussian, step, delta, damped sinusoids), properties such as shift, modulation, derivative, convolution and Parseval, Fourier series formulas, trig identities, and basic integrals. It sits in the PDF downloads of the Spectral Theory Book folder.
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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
Fourier Transform Table
UBC M267 Resources for 2005
F(t) bF(!) Notes (0)
f(t)Z1
−1f(t)e−i!tdt De nition. (1)
1
2Z1
−1bf(!)ei!td! bf(!) Inversion formula. (2)
bf(−t) 2f(!) Duality property. (3)
e−atu(t)1
a+i!aconstant,<e(a)>0 (4)
e−ajtj 2a
a2+!2aconstant,<e(a)>0 (5)
(t)=
1;ifjtj<1,
0;ifjtj>12s i n c (!)=2sin(!)
!Boxcar in time. (6)
1
sinc(t) (!) Boxcar in frequency. (7)
f0(t) i!bf(!) Derivative in time. (8)
f00(t) (i!)2bf(!) Higher derivatives similar. (9)
tf(t) id
d!bf(!) Derivative in frequency. (10)
t2f(t) i2d2
d!2bf(!) Higher derivatives similar. (11)
ei!0tf(t) bf(!−!0) Modulation property. (12)
ft−t0
k
ke−i!t0bf(k!) Time shift and squeeze. (13)
(fg)(t) bf(!)bg(!) Convolution in time. (14)
u(t)=
0;ift<0
1;ift>01
i!+(!) Heaviside step function. (15)
(t−t0)f(t) e−i!t0f(t0) Assumesfcontinuous at t0.(16)
ei!0t2(!−!0) Useful for sin( !0t), cos(!0t).(17)
Convolution: (fg)(t)=Z1
−1f(t−u)g(u)du=Z1
−1f(u)g(t−u)du.
Parseval:Z1
−1jf(t)j2dt=1
2Z1
−1 bf(!) 2
d!.
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
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Engineering Tables/Fourier Transform Table 2
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Signal Fourier transform
unitary, angular frequency Fourier transform
unitary, ordinary frequency Remarks
10
The rectangular pulse and the normalized sinc function
11
Dual of rule 10. The rectangular function is an idealized
low-pass filter, and the sinc function is the non -causal
impulse response of such a filter.
12
tri is the triangular function
13
Dual of rule 12.
14
Shows that the Gaussian function exp( - at2) is its own
Fourier transform. For this to be integrable we must have
Re(a) > 0 .
common in optics
a>0
the transform is the function itself
J0(t) is the Bessel function of first kind of order 0, rect is
the rectangular function
it's the generalization of the previous transform; Tn (t) is the
Chebyshev polynomial of the first kind.
Un (t) is the Chebyshev polynomial of the second kind
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Category : Engineering Tables
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