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Tables of Transform Pairs
2005 by Marc Stoecklin — marc a stoecklin.net — http://www.stoecklin.net/ — December 8, 2005 — version 1.5
Students and engineers in communications and mathematics are confronted with transformations such as the z-Transform, the Fourier transform, or the Laplace transform. Often it is
quite hard to quickly find the appropriate transform in a book or the Internet, much less to
have a good overview of transformation pairs and corresponding properties.
In this document I present a handy collection of the most common transform pairs and
properties of the
. continuous-time frequency Fourier transform (2⇡f ),
. continuous-time pulsation Fourier transform (!),
. z-Transform,
. discrete-time Fourier transform DTFT, and
. Laplace transform
arranged in a table and ordered by subject. The properties of each transformation are indicated
in the first part of each topic whereas specific transform pairs are listed afterwards.
Please note that, before including a transformation pair in the table, I verified their correctness. However, it is still possible that there might be some mistakes due to typos. I’d be
grateful to everyone for dropping me a line and indicating me erroneous formulas.
Some useful conventions and formulas
Sinc function
sinc (x) ⌘
Convolution
f ⇤ g(t) =
Parseval theorem
R +1
1
R +1
1
Real part
sin(x)
x
R +1
1
f (t)g ⇤ (t)dt =
|f (t)|2 dt =
<e{f (t)} =
1
2
Imaginary part
=m{f (t)} =
Sine / Cosine
sin (x) =
Geometric sequences
f (⌧ )g ⇤ (t
P1
k=0
1
2
1
F (f )G⇤ (f )df
|F (f )|2 df
[f (t)
f ⇤ (t)]
jx
cos (x) =
1
1 x
General case :
1
R +1
1
[f (t) + f ⇤ (t)]
ejx e
2j
xk =
R +1
⌧ )d⌧
Pn
k=0
Pn
k=m
ejx +e
2
xk =
xk =
jx
1 xn+1
1 x
xm xn+1
1 x
Marc Stoecklin : TABLES OF TRANSFORM PAIRS
2
Table of Continuous-time Frequency Fourier Transform Pairs
f (t) = F
1
{F (f )} =
R +1
1
(==)
f (t)
(==)
f ( t)
f ⇤ (t)
f (t) is purely real
f (t) is purely imaginary
even/symmetry
odd/antisymmetry
f (t) =
f ⇤(
t)
f ⇤ ( t)
f (t) =
time shifting
f (t
t0 )
f (t)ej2⇡f0 t
time scaling
f (af )
“ ”
1
f fa
|a|
af (t) + bg(t)
f (t)g(t)
f (t) ⇤ g(t)
(t)
(t
t0 )
1
ej2⇡f0 t
e
a|t|
a>0
e
2
⇡t
2
ej⇡t
sin (2⇡f0 t + )
cos (2⇡f0 t + )
f (t) sin (2⇡f0 t)
f (t) cos (2⇡f0 t)
sin2 (t)
cos2 (t)
rect
`t´
T
8
|t| 6 T2
|t| > T2
|t| 6 T
|t| > T
8
<1 t > 0
u(t) = 1[0,+1] (t) = :
0 t<0
8
<1
t>0
sgn (t) = :
1 t<0
<1
(t) = :
]
0
8
` t ´ <1 |t|
T
triang T = :
0
= 1[
T
2
,+ T
2
F
f (t)ej2⇡f t df
sinc (Bt)
sinc2 (Bt)
F (f ) = F {f (t)} =
R +1
1
F
F (f )
F
F ( f)
F
F ⇤( f )
F
F (f ) = F ⇤ ( f )
F
F (f ) =
F
F (f ) is purely real
F
F (f ) is purely imaginary
F
F (f )e
(==)
(==)
(==)
(==)
(==)
(==)
(==)
F
(==)
F
aF (f ) + bG(t)
(==)
F
(==)
F
(==)
F (af )
1
e
j2⇡f t0
F
(f )
F
(f
(==)
(==)
F
frequency scaling
F (f )G(f )
F
(==)
f0 )
F
2a
a2 +4⇡ 2 f 2
2
e ⇡f
F
ej⇡( 4
F
j
2
1
2
j
2
1
2
1
4
1
4
(==)
(==)
(==)
(==)
F
(==)
F
(==)
F
(==)
F
(==)
F
(==)
f 2)
1
ˆ
e
ˆ
e
j
(f + f0 )
ej
(f
f0 )
j
(f + f0 ) + ej
(f
f0 )
[F (f + f0 )
T sinc T f
F
T sinc2 T f
F
1
j2⇡f
F
1
j⇡f
F
1
B
1
B
(==)
(==)
(==)
(==)
F
(==)
dn
f (t)
dtn
tn f (t)
(==)
1
1+t2
(==)
F (f
+ (f )
rect
“
f
B
triang
“
”
f
B
=
”
1
1
B [
F
(j2⇡f )n F (f )
F
1
dn
F (f )
( j2⇡)n df n
2⇡|f
|
⇡e
(==)
F
f0 )]
[F (f + f0 ) + F (f f0 )]
ˆ
`
´
`
1
2 (f )
f
f+
⇡
ˆ
`
´
`
1
2 (f ) + f
+ f+
⇡
F
(==)
frequency shifting
F (f ) ⇤ G(f )
F
(==)
odd/antisymmetry
j2⇡f t0
F
(==)
even/symmetry
F ⇤( f )
F (f “ f0”)
1
F fa
|a|
F
(==)
j2⇡f t dt
f (t)e
B ,+ B ]
2
2
(f )
˜
˜
´˜
1
⇡
´˜
1
⇡
Marc Stoecklin : TABLES OF TRANSFORM PAIRS
3
Table of Continuous-time Pulsation Fourier Transform Pairs
x(t) = F! 1 {X(!)} =
R +1
1
!
(==)
x(t)
!
(==)
x( t)
x⇤ (t)
x(t) is purely real
x(t) is purely imaginary
even/symmetry
odd/antisymmetry
x(t) =
time shifting
x⇤ (
t)
x⇤ ( t)
x(t) =
x(t
t0 )
x(t)ej!0 t
time scaling
x (af )
“ ”
f
a
1
x
|a|
ax1 (t) + bx2 (t)
x1 (t)x2 (t)
x1 (t) ⇤ x2 (t)
(t)
(t
t0 )
1
ej!0 t
e
e
e
a|t|
at u(t)
at u(
t)
a>0
<{a} > 0
x(t) cos (!0 t)
sin2 (!0 t)
cos2 (!0 t)
rect
T
8
<1
|t| 6 T2
|t| > T2
|t| 6 T
|t| > T
8
<1 t > 0
u(t) = 1[0,+1] (t) = :
0 t<0
8
<1
t>0
sgn (t) = :
1 t<0
= 1[
(t) = :
0
8
` t ´ <1 |t|
T
triang T = :
0
T
2
,+ T
]
2
X( !)
F!
X ⇤ ( !)
F
X(f ) = X ⇤ ( !)
F!
X(f ) =
F!
X(!) is purely real
F
X(!) is purely imaginary
F
X(!)e
F!
X(!
(==)
!
(==)
(==)
(==)
!
(==)
!
(==)
(==)
F!
(==)
sinc (T t)
sinc2
(T t)
1
X
|a|
!0 )
`!´
a
aX1 (!) + bX2 (!)
F
1
X (!)
2⇡ 1
frequency scaling
⇤ X2 (!)
F!
X1 (!)X2 (!)
F
1
F
e
F!
2⇡ (!)
F!
2⇡ (!
(==)
!
(==)
!
(==)
(==)
(==)
F
!
(==)
F!
(==)
F
F
!
(==)
F
!
(==)
F
!
(==)
F!
(==)
j!t0
!0 )
2a
a2 +! 2
1
a+j!
1
a j!
p
F
2 !2
2
2⇡e
ˆ
˜
j⇡ e j (! + !0 ) ej (! !0 )
ˆ j
˜
⇡ e
(! + !0 ) + ej (! !0 )
j
2
1
2
[X (! + !0 )
X (!
!0 )]
[X (! + !0 ) + X (!
!0 )]
F
⇡ 2 [2 (f )
F
⇡ 2 [2 (!) + (!
F
T sinc
F
T sinc2
F
⇡ (f ) +
F
2
j!
F
1
T
1
T
!
(==)
!
(==)
!
(==)
!
(==)
!
(==)
!
(==)
!
(==)
F!
(==)
“
(!
!T
2
“
!T
2
1
j!
`
(j!)n X(!)
F!
d
j n df
n X(!)
!
(==)
1
t
!
(==)
(==)
F
(! + !0 )]
!0 ) + (! + !0 )]
”
´
!
= 1
2⇡T
` ! ´ T
triang 2⇡T
rect
!0 )
”
F
dn
f (t)
dtn
tn f (t)
odd/antisymmetry
frequency shifting
F
!
(==)
j!t dt
j!t0
X(a!)
!
(==)
x(t)e
even/symmetry
X ⇤ ( !)
F
!
(==)
!
(==)
x(t) sin (!0 t)
1
X(!)
e
cos (!0 t + )
R +1
F
!
(==)
!
(==)
t2
2 2
X(!) = F! {x(t)} =
F
<{a} > 0
sin (!0 t + )
`t´
F
x(t)ej!t d!
n
j⇡sgn(!)
1[
2⇡T,+2⇡T ] (f )
Marc Stoecklin : TABLES OF TRANSFORM PAIRS
4
Table of Z-Transform Pairs
x[n] = Z
1
{X(z)} =
1
2⇡j
H
X(z)z n
(==)
x[n]
(==)
x[ n]
x⇤ [n]
x⇤ [
n]
Z
1
N
Z
1]
Z
1
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
1]
an u[n]
1
sin (!0 n) u[n]
cos (!0 n) u[n]
sin (!0 n) u[n]
an cos (!0 n) u[n]
nx[n]
i=1 (n i+1)
am m!
Z
an u[n]
1]
x[n]
n
am u[n]
“
”
1
X WNk z N
Z
(==)
n
1
k=0
Z
Z
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
X1 (z)X2 (t)
n0
z
z
z
1
z
z
1
z
(z 1)2
z(z+1)
(z 1)3
z(z 2 +4z+1)
(z 1)4
z
z+1
z
z
a
z
z
a
1
z
a
az
(z a)2
az(z+a
(z a)3
z
z e a
Z
1 aN z N
1 az 1
Z
z sin(!0 )
z 2 2 cos(!0 )z+1
z(z cos(!0 ))
z 2 2 cos(!0 )z+1
za sin(!0 )
z 2 2a cos(!0 )z+a2
z(z a cos(!0 ))
z 2 2a cos(!0 )z+a2
(==)
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
Z
(==)
d
z dz
X(z)
R z X(z)
dz
0
z
(z
z
a)m+1
Rx
X ⇤ (z ⇤ )]
n0 X(z)
`z´
a
P
N
Rx
X ⇤ (z ⇤ )]
aX1 (z) + bX2 (z)
H
`z´
1
X1 (u)X2 u
u
2⇡j
Z
( 1)n
n = 0, . . . , N
otherwise
z 1
1 z 1
X
(==)
(==)
e
=
z
Z
[n]
n2 an u[n]
1
Z
(==)
nan u[n]
an
(==)
(==)
an 1 u[n
z
1
[X(z) +
2
1
[X(z)
2j
Z
x1 [n]x2 [n]
an u[
z
Z
x1 [n] ⇤ x2 [n]
n3 u[n]
Please note :
1
Rx
(==)
n2 u[n]
Qm
Rx
X ⇤ ( z1⇤ )
(==)
ax1 [n] + bx2 [n]
nu[n]
(
an
0
X ⇤ (z ⇤ )
Z
(==)
u[ n
ROC
Z
(==)
N 2 N0
u[n]
n
1
Rx
(==)
n0 ]
x[n]z
X( z1 )
n0 ]
an x[n]
[n
n= 1
X(z)
x[n
x[N n]
P+1
Z
(==)
(==)
time shifting
X(z) = Z {x[n]} =
Z
<e{x[n]}
=m{x[n]}
downsampling by N
Z
1 dz
Rx
Rx
WN = e
1 du
j2!
N
|a|Rx
Rx
Rx \ Ry
R x \ Ry
Rx \ Ry
8z
8z
|z| > 1
|z| < 1
|z| > 1
|z| > 1
|z| > 1
|z| < 1
|z| > |a|
|z| < |a|
|z| > |a|
|z| > |a|
|z| > |a|
|z| > |e
|z| > 0
|z| > 1
|z| > 1
|z| > a
|z| > a
Rx
Rx
a|
Marc Stoecklin : TABLES OF TRANSFORM PAIRS
5
Table of Discrete Time Fourier Transform (DTFT) Pairs
1
2⇡
R +⇡
DT F T
P+1
X(ej! )ej!n d!
(==)
X(ej! ) =
x[n]
x[ n]
x⇤ [n]
(==)
DT F T
(==)
DT F T
(==)
DT F T
X(ej! )
X(e j! )
X ⇤ (e j! )
x[n] is purely real
x[n] is purely imaginary
even/symmetry x[n] = x⇤ [ n]
odd/antisymmetry x[n] = x⇤ [ n]
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
X(ej! ) = X ⇤ (e j! ) even/symmetry
X(ej! ) = X ⇤ (e j! ) odd/antisymmetry
X(ej! ) is purely real
X(ej! ) is purely imaginary
time shifting
(==)
DT F T
(==)
DT F T
X(ej! )e j!n0
X(ej(! !0 ) ) frequency shifting
2⇡k
j! N
1 PN 1
)
k=0 X(e
N
x[n] =
⇡
x[n n0 ]
x[n]ej!0 n
downsampling by 8
N x[N n] N 2 N0
ˆ ˜
<x n
n = kN
upsampling by N : N
0
otherwise
ax1 [n] + bx2 [n]
x1 [n]x2 [n]
(|a| < 1)
(n + 1)an u[n]
sin (!0 n + )
cos (!0 n + )
= !c sinc (!c n)
8
<1 |n| 6 M
=:
0 otherwise
8
`n
´ <1 0 6 n 6 M
1
MA : rect M
=:
2
0 otherwise
8
“
” <1 0 6 n 6 M 1
n
1
MA : rect M 1
=:
2
0 otherwise
`
n
M
´
nx[n]
x[n]
an sin[!0 (n+1)]
u[n]
sin !0
x[n 1]
|a| < 1
j!n
X(ejN ! )
aX1 (ej! ) + bX2 (ej! )
R +⇡
1
j(!
X1 (ej! ) ⇤ X2 (ej! ) = 2⇡
⇡ X1 (e
DT F T
1
e j!n0
˜(!) = P+1
k= 1 (! + 2⇡k)
˜(! !0 ) = P+1
!0 + 2⇡k)
k= 1 (!
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
DT F T
(==)
1
+ 12 ˜(!)
1 e j!
1
1 ae j!
1
(1 ae j! )2
j
[e j
2
1
[e j
2
˜
rect
“
˜ (! + !0 + 2⇡k)
e+j ˜ (!
˜ (! + !0 + 2⇡k) + e+j ˜ (!
!
!c
8
<1
”
=:
0
sin[! (M + 1
)]
2
|!| < !c
!c < |!| < ⇡
sin(!/2)
DT F T
sin[!(M +1)/2]
e j!M/2
sin(!/2)
DT F T
sin[!M/2]
e j!(M
sin(!/2)
DT F T
d
j d!
X(ej! )
(==)
(==)
(==)
DT F T
(==)
DT F T
(==)
(1
e
1)/2
j! )X(ej! )
1
1 2a cos(!0 e j! )+a2 e j2!
Some remarks
˜(!) =
Parseval :
+1
X
n= 1
+1
X
(! + 2⇡k)
˜
rect(!)
=
k= 1
|x[n]|2 =
+1
X
k= 1
1
2⇡
Z
+⇡
⇡
|X(ej! )|2 d!
) )X (ej
2
X1 (ej! )X2 (ej! )
(==)
DT F T
(==)
DT F T
(==)
an u[n]
x[n]e
DT F T
[n]
n0 ]
1
u[n]
rect
DT F T
(==)
DT F T
(==)
(==)
ej!0 n
Window :
DT F T
(==)
x1 [n] ⇤ x2 [n]
[n
sin(!c n)
n
DT F T
(==)
n= 1
rect(! + 2⇡k)
!0 + 2⇡k)]
!0 + 2⇡k)]
)d
Marc Stoecklin : TABLES OF TRANSFORM PAIRS
6
Table of Laplace Transform Pairs
f (t) =
1
{F (s)} =
1
2⇡j
R c+j1
c j1
f (t
a)
(==)
f (t)
(==)
t>a>0
(==)
at f (t)
e
f (at)
a>0
af1 (t) + bf2 (t)
f1 (t)f2 (t)
f1 (t) ⇤ f2 (t)
(t)
1
t
e
at
te
at
e
at
1
e
a
t
a
1
1
a
`
1
e
´
at
sin (!t)
F (s)
L
a
L
F (s + a)
L
1
F ( as )
a
L
aF1 (s) + bF2 (s)
(==)
(==)
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
(==)
L
cos (!t)
(==)
(==)
cosh (!t)
(==)
L
L
L
e
at
sin (!t)
e
at
(==)
cos (!t)
(==)
tn
(==)
f 00 (t) =
f (n) (t)
=
Rt
0
L
L
f
n (t)
1
s+a
1
(s+a)2
a
s(s+a)
1
1+as
1
s+a
!
s2 +! 2
s
s2 +! 2
!
s2 ! 2
s
s2 ! 2
!
(s+a)2 +! 2
s+a
(s+a)2 +! 2
sf (0)
L
sn F (s)
sn 1 f (0)
L
1
F (s)
Rs 1
s F (u)du
(==)
1 (t)
1
s
1
s2
s2 F (s)
f (u)du
f
1
sF (s)
(==)
L
(==)
L
(==)
L
(==)
st dt
F1 (s)F2 (s)
L
(==)
f (t)e
F1 (s) ⇤ F2 (s)
L
(==)
1
as F (s)
n!
s+n+1
( 1)n F (n) (s)
(==)
1
f (t)
t
R +1
L
d
f (t)
dt
2
d
f (t)
dt2
dn
f (t)
dtn
f 0 (t) =
F (s) = L {f (t)} =
L
sinh (!t)
tn f (t)
In general :
L
F (s)est ds
f (0)
F (s) f 1
s
F (s)
f 1 (0)
+ sn
n
s
+
f 0 (0)
...
f 2 (0)
sn 1
f (n
+ ... +
1) (0)
f
n
s
(0)
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