TABLE 3.1
Short Table of Fourier Transforms
g(t)
G(f)
e-atu(t)
a + {21Cf
2
eatu(-t)
3
e-altl
a - j21Cf
2a
4
te-at u(t)
5
tne-at u(t)
6
7
8
B(t)
1
ej21rfot
9 ' cos2Jrfot
10 sin 2Jrfot
11 u(t)
12 sgnt
(a + j21Cf)n+l
8(f)
8(f - fo)
0.5 [8(f + fo)+ 8(f - fo)]
jO.5 [8(/ + fo) -8(/ - fo)]
(f
1
218 )+ji;j
-2
- fo) + B(f + fo)] +
14 sin 21Cfot
u(t)
1
4][B(f
- fo) - B(f + fo)] +
n(f)
18 2B sinc (21TBt)
19 .:l (f)
20 B sinc2(]fBt)
21
21rfo
(a + j21rf)2+ 41r2fi
a + j21rf
(a + j2nf)2 + 4n2f6
-rsinc(nf-r)
n (fB)
2sinc2(~)
Ll(fB)
E~-oo 8(1- nT: to L~-oo 8(/ - nfO)
22 e-t2/2O"2
a>O
(a + j21Cf)2
n!
j21rf
![B(f
16 e-at cas 2nfot u(t)
a>O
~+~)2
cos 21Tfotu(t)
15 e-at sin 2nfot u(t)
a>O
u./i"iie-2(=ff
j21Cf
(27ffo)2
- (27ff)2
27ffo
(21rfO)2 (21tf)2
-
TABLE 3.2
Propertiesof Fourier TransformOperations
Operation
g(t)
G(f)
Superposition
Scalarmultiplication
Duality
Time scaling
gl (t) + gZ(t)
kg(t)
G(t)
Gl (f) + ~(f)
kG(f)
g(-f)
Time shifting
g(at)
g(t - to)
Frequencyshifting
Time convolution
Frequencyconvolution
g(t)ejZ1ffot
gl (t) * gZ(t)
gl (t)gZ(t)
Gl(f)G2(f)
Gl (f) * G2(f)
Time differentiation
~ dt
(j27rf)nG(f)
Time integration
f~oo g(x) dx
~G (~1
G(f)e-j27rfto
G(f - fo)
~
127rf
+ 1G(O)FJ(f)
2"