Tables of Common Transform Pairs - Chiara Daraio

Tables of Common Transform Pairs

2012 by Marc Ph. Stoecklin — marc a stoecklin.net — http://www.stoecklin.net/ — 2012-12-20 — version v1.5.3

Engineers and students 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 comprehensive overview
of transformation pairs and corresponding properties.

In this document I compiled a handy collection of the most common transform pairs and properties
of the

continuous-time frequency Fourier transform (2πf ),

continuous-time pulsation Fourier transform (ω),

z-Transform,

discrete-time Fourier transform DTFT, and

Laplace transform.

Please note that, before including a transformation pair in the table, I verified its correctness. Nev-
ertheless, it is still possible that you may find errors or typos. I am very grateful to everyone dropping
me a line and pointing out any concerns or typos.

Notation, Conventions, and Useful Formulas

Imaginary unit j2 = −1
Complex conjugate z = a + jb ←→ z∗ = a − jb
Real part
Imaginary part e {f (t)} = 1 [f (t) + f ∗(t)]
Dirac delta/Unit impulse 2
Heaviside step/Unit step
Sine/Cosine m {f (t)} = 1 [f (t) − f ∗(t)]
Sinc function 2j
Rectangular function
Triangular function 1, n = 0
Convolution δ[n] =

Parseval theorem 0, n = 0

Geometric series 1, n ≥ 0
u[n] =

0, n < 0

sin (x) = ej x −e−j x cos (x) = ej x +e−j x
2j 2

sinc (x) ≡ sin(x) (unnormalized)
x

rect( t ) = 1 if |t| T
T 0 2

if |t| > T
2

triang t = rect( t ) ∗ rect( t ) =  − |t| |t| T
T T T T |t| > T
1

0

continuous-time: (f ∗ g)(t) = +∞ f (τ ) g∗ (t − τ )dτ
discrete-time: −∞

(u ∗ v)[n] = ∞ u[m] v∗ [n − m]
m=−∞

general statement: +∞ f (t)g∗(t)dt = +∞ F (f )G∗(f )df
−∞ −∞

continuous-time: +∞ |f (t)|2dt = +∞ |F (f )|2 df
−∞ −∞

discrete-time: +∞ |x[n]|2 = 1 +π |X (ejω )|2 dω
n=−∞ 2π −π

∞ xk = 1 n xk = 1−xn+1
k=0 1−x k=0 1−x

in general: n xk = xm −xn+1
k=m 1−x

1

Marc Ph. Stoecklin — TABLES OF TRANSFORM PAIRS — v1.5.3 2

Table of Continuous-time Frequency Fourier Transform Pairs

f (t) = F −1 {F (f )} = +∞ f (t)ej2πf tdf ⇐=F=⇒ F (f ) = F {f (t)} = +∞ f (t)e−j2πf t dt
−∞ −∞
⇐=F=⇒
transform f (t) ⇐=F=⇒ F (f ) frequency reversal
time reversal ⇐=F=⇒ reversed conjugation
complex conjugation f (−t) ⇐=F=⇒ F (−f ) complex conjugation
reversed conjugation f ∗(t) F ∗(−f )
f ∗(−t) ⇐=F=⇒ F ∗(f )
⇐=F=⇒
f (t) is purely real ⇐=F=⇒ F (f ) = F ∗(−f ) even/symmetry
⇐=F=⇒ F (f ) = −F ∗(−f ) odd/antisymmetry
f (t) is purely imaginary
even/symmetry f (t) = f ∗(−t) ⇐=F=⇒ F (f ) is purely real
odd/antisymmetry ⇐=F=⇒
f (t) = −f ∗(−t) ⇐=F=⇒ F (f ) is purely imaginary
⇐=F=⇒
time shifting f (t − t0) F (f )e−j2πf t0
f (t)ej2πf0t ⇐=F=⇒
time scaling ⇐=F=⇒ F (f − f0) frequency shifting
⇐=F=⇒ frequency scaling
linearity f (af ) 1 F f
time multiplication ⇐=F=⇒ |a| a
frequency convolution ⇐=F=⇒
1 f f ⇐=F=⇒ F (af )
|a| a ⇐=F=⇒

af (t) + bg(t) ⇐=F=⇒ aF (f ) + bG(t) frequency convolution
f (t)g(t) ⇐=F=⇒ F (f ) ∗ G(f ) frequency multiplication
⇐=F=⇒ F (f )G(f )
f (t) ∗ g(t)
⇐=F=⇒
delta function δ(t) ⇐=F=⇒ 1 delta function
shifted delta function δ(t − t0) ⇐=F=⇒ e−j2πf t0 shifted delta function
⇐=F=⇒ δ(f )
two-sided exponential decay 1 ⇐=F=⇒ δ(f − f0)
ej 2πf0 t ⇐=F=⇒

e−a|t| a>0 ⇐=F=⇒ 2a
a2+4π2f 2
e−πt2 ⇐=F=⇒
ejπt2 e−πf 2
⇐=F=⇒
ejπ( 1 −f 2 )
⇐=F=⇒ 4

sine sin (2πf0t + φ) ⇐=F=⇒ j e−jφδ (f + f0) − ejφδ (f − f0)
cosine cos (2πf0t + φ) ⇐=F=⇒ 2
sine modulation f (t) sin (2πf0t)
cosine modulation f (t) cos (2πf0t) ⇐=F=⇒ 1 e−jφδ (f + f0) + ejφδ (f − f0)
squared sine ⇐=F=⇒ 2
squared cosine sin2 (t) ⇐=F=⇒
cos2 (t) j [F (f + f0) − F (f − f0)]
2

1 [F (f + f0) + F (f − f0)]
2

1 2δ(f ) − δ f − 1 −δ f + 1
4 π π

1 2δ(f ) + δ f − 1 +δ f + 1
4 π π

 |t| T
2
rectangular rect t = 1 T T sinc T f
triangular T 0 |t| > 2
step
signum triang t =  − |t| |t| T T sinc2 T f
sinc T T
squared sinc 1

0 |t| > T



u(t) = 1[0,+∞] (t) = 1 t0 1 + δ(f )
0 t<0 j2πf

 1
jπf
sgn (t) = 1 t 0
−1 t < 0

sinc (Bt) 1 rect f = 1 1[− B ,+ B ] (f )
B B B 2 2

sinc2 (Bt) 1 triang f
B B

n-th time derivative dn f (t) (j2πf )nF (f )
n-th frequency derivative dtn
dn
tnf (t) 1 df n F (f )
(−j2π)n

1 πe−2π|f |
1+t2

Marc Ph. Stoecklin — TABLES OF TRANSFORM PAIRS — v1.5.3 3

Table of Continuous-time Pulsation Fourier Transform Pairs

x(t) = Fω−1 {X(ω)} = +∞ x(t)ejωtdω ⇐F==ω⇒ X(ω) = Fω {x(t)} = +∞ x(t)e−jωtdt
−∞ −∞
⇐F==ω⇒
transform x(t) ⇐F==ω⇒ X (ω) frequency reversal
time reversal x(−t) ⇐F==ω⇒ X (−ω) reversed conjugation
complex conjugation x∗(t) ⇐F==ω⇒ X ∗ (−ω) complex conjugation
reversed conjugation x∗(−t) X ∗ (ω)
⇐F==ω⇒
x(t) is purely real ⇐F==ω⇒ X(f ) = X∗(−ω) even/symmetry
⇐F==ω⇒ X(f ) = −X∗(−ω) odd/antisymmetry
x(t) is purely imaginary ⇐F==ω⇒
even/symmetry x(t) = x∗(−t) X(ω) is purely real
odd/antisymmetry ⇐F==ω⇒
x(t) = −x∗(−t) ⇐F==ω⇒ X(ω) is purely imaginary
⇐F==ω⇒
time shifting x(t − t0) ⇐F==ω⇒ X (ω)e−j ωt0
x(t)ej ω0 t
time scaling ⇐F==ω⇒ X(ω − ω0) frequency shifting
⇐F==ω⇒ frequency scaling
linearity x (af ) ⇐F==ω⇒ 1 X ω
time multiplication |a| a
frequency convolution ⇐F==ω⇒
1 x f ⇐F==ω⇒ X (aω)
|a| a ⇐F==ω⇒
⇐F==ω⇒
ax1(t) + bx2(t) aX1(ω) + bX2(ω)
x1(t)x2(t) ⇐F==ω⇒
⇐F==ω⇒ 1 X1 (ω) ∗ X2 (ω) frequency convolution
x1(t) ∗ x2(t) ⇐F==ω⇒ 2π
⇐F==ω⇒
X1(ω)X2(ω) frequency multiplication
⇐F==ω⇒
delta function δ(t) ⇐F==ω⇒ 1
shifted delta function δ(t − t0) ⇐F==ω⇒ e−jωt0
⇐F==ω⇒ 2πδ(ω)
1 ⇐F==ω⇒ 2πδ(ω − ω0) delta function
ej ω0 t ⇐F==ω⇒ shifted delta function

two-sided exponential decay e−a|t| a > 0 ⇐F==ω⇒ 2a
exponential decay a2 +ω2
reversed exponential decay ⇐F==ω⇒
e−atu(t) {a} > 0 1
⇐F==ω⇒ a+jω

e−atu(−t) {a} > 0 ⇐F==ω⇒ 1

t2 ⇐F==ω⇒ a−jω
⇐F==ω⇒ σ√2πe−
e 2σ2 σ2ω2
⇐F==ω⇒ 2
⇐F==ω⇒
sine sin (ω0t + φ) ⇐F==ω⇒ jπ e−jφδ (ω + ω0) − ejφδ (ω − ω0)
cosine cos (ω0t + φ)
sine modulation x(t) sin (ω0t) π e−jφδ (ω + ω0) + ejφδ (ω − ω0)
cosine modulation x(t) cos (ω0t)
squared sine j [X (ω + ω0) − X (ω − ω0)]
squared cosine sin2 (ω0t) 2
cos2 (ω0t)
rectangular 1 [X (ω + ω0) + X (ω − ω0)]
2
triangular
π2 [2δ(f ) − δ (ω − ω0) − δ (ω + ω0)]
step
π2 [2δ(ω) + δ (ω − ω0) + δ (ω + ω0)]
signum
 |t| T
sinc 2
squared sinc rect t = 1 T T sinc ωT
T 0 |t| > 2 2

triang t =  − |t| |t| T T sinc2 ωT
T T 2
1

0 |t| > T



u(t) = 1[0,+∞] (t) = 1 t0 πδ(f ) + 1
0 t<0 jω

 2
jω
sgn (t) = 1 t 0
−1 t < 0

sinc (T t) 1 rect ω = 1 1[−πT ,+πT ](f )
sinc2 (T t) T 2πT T

1 triang ω
T 2πT

n-th time derivative dn f (t) (j ω)n X (ω)
n-th frequency derivative dtn
time inverse dn
tnf (t) j n df n X (ω)

1 −jπsgn(ω)
t

Marc Ph. Stoecklin — TABLES OF TRANSFORM PAIRS — v1.5.3 4

Table of z-Transform Pairs

x[n] = Z−1 {X(z)} = 1 X (z )z n−1 dz ⇐=Z=⇒ X(z) = Z {x[n]} = +∞ x[n]z−n ROC
2πj n=−∞
⇐=Z=⇒
transform x[n] ⇐=Z=⇒ X (z ) Rx
time reversal x[−n] ⇐=Z=⇒
complex conjugation x∗[n] ⇐=Z=⇒ X ( 1 ) 1
reversed conjugation x∗[−n] z Rx
⇐=Z=⇒
⇐=Z=⇒ X ∗ (z ∗ ) Rx

⇐=Z=⇒ X ∗ ( 1 ) 1
⇐=Z=⇒ z∗ Rx
⇐=Z=⇒
real part e{x[n]} 1 [X (z ) + X ∗ (z ∗ )] Rx
imaginary part m{x[n]} ⇐=Z=⇒ 2 Rx
⇐=Z=⇒
⇐=Z=⇒ 1 [X (z ) − X ∗ (z∗ )] Rx
2j |a|Rx
⇐=Z=⇒
time shifting x[n − n0] ⇐=Z=⇒ z−n0 X(z) Rx
scaling in Z anx[n]
downsampling by N ⇐=Z=⇒ X z Rx ∩ Ry
x[N n], N ∈ N0 ⇐=Z=⇒ a Rx ∩ Ry
⇐=Z=⇒ 1 j2ω Rx ∩ Ry
⇐=Z=⇒ 1 N −1 X WNk z N WN = e− N
⇐=Z=⇒ N k=0
⇐=Z=⇒
linearity ax1[n] + bx2[n] ⇐=Z=⇒ aX1(z) + bX2(z)
time multiplication x1[n]x2[n] ⇐=Z=⇒
frequency convolution 1 X1(u)X2 z u−1du
x1[n] ∗ x2[n] ⇐=Z=⇒ 2πj u
⇐=Z=⇒
⇐=Z=⇒ X1(z)X2(t)
⇐=Z=⇒
delta function δ[n] ⇐=Z=⇒ 1 ∀z
shifted delta function δ[n − n0] ⇐=Z=⇒ z−n0 ∀z

step u[n] ⇐=Z=⇒ z |z| > 1
ramp −u[−n − 1] z−1 |z| < 1
⇐=Z=⇒ |z| > 1
nu[n] ⇐=Z=⇒ z |z| > 1
n2u[n] ⇐=Z=⇒ z−1 |z| < 1
−n2u[−n − 1] ⇐=Z=⇒ |z| > 1
n3u[n] z |z| < 1
−n3u[−n − 1] ⇐=Z=⇒ (z−1)2 |z| < 1
(−1)n ⇐=Z=⇒
⇐=Z=⇒ z(z+1) |z| > |a|
exponential anu[n] (z−1)3 |z| < |a|
−anu[−n − 1] |z| > |a|
exp. interval an−1u[n − 1] z(z+1) |z| > |a|
sine (z−1)3 |z| > |a|
cosine nanu[n] z(z2 +4z+1) |z| > |e−a|
n2anu[n]
e−anu[n] (z−1)4 |z| > 0
z(z2 +4z+1)
an n = 0, . . . , N − 1 |z| > 1
0 otherwise (z−1)4 |z| > 1
|z| > a
sin (ω0n) u[n] z |z| > a
cos (ω0n) u[n] z+1
an sin (ω0n) u[n] Rx
an cos (ω0n) u[n] z Rx
z−a

z
z−a

1
z−a

az
(z−a)2

az(z+a
(z−a)3

z
z−e−a

1−aN z−N
1−az−1

z sin(ω0)
z2−2 cos(ω0)z+1

z(z−cos(ω0 ))
z2−2 cos(ω0)z+1

za sin(ω0)
z2−2a cos(ω0)z+a2

z(z−a cos(ω0))
z2−2a cos(ω0)z+a2

differentiation in Z nx[n] −z dX (z )
integration in Z dz

x[n] − z X (z ) dz
0 z
m (n−i+1) n
i=1 am z
u[n]
am m! (z−a)m+1

Note:
z1

z − 1 = 1 − z−1

Marc Ph. Stoecklin — TABLES OF TRANSFORM PAIRS — v1.5.3 5

Table of Common Discrete Time Fourier Transform (DTFT) Pairs

x[n] = 1 +π X (ejω )ejωn dω ⇐D=T=F⇒T X(ejω) = +∞ x[n]e−jωn
2π −π n=−∞
⇐D=T=F⇒T
transform x[n] ⇐D=T=F⇒T X (ejω )
time reversal ⇐D=T=F⇒T X(e−jω )
complex conjugation x[−n] ⇐D=T=F⇒T X∗(e−jω )
reversed conjugation x∗[n] X∗(ejω )
x∗[−n] ⇐D=T=F⇒T
⇐D=T=F⇒T
x[n] is purely real ⇐D=T=F⇒T X(ejω) = X∗(e−jω) even/symmetry
⇐D=T=F⇒T X(ejω) = −X∗(e−jω) odd/antisymmetry
x[n] is purely imaginary X(ejω) is purely real
even/symmetry x[n] = x∗[−n] ⇐D=T=F⇒T X(ejω) is purely imaginary
odd/antisymmetry ⇐D=T=F⇒T
x[n] = −x∗[−n]
⇐D=T=F⇒T
time shifting x[n − n0] ⇐D=T=F⇒T X(ejω )e−jωn0
x[n]ej ω0 n X (ej (ω−ω0 ) )
⇐D=T=F⇒T frequency shifting
⇐D=T=F⇒T
downsampling by N x[N n] N ∈ N0 1 N −1 X (ej ω−2πk )
upsampling by N ⇐D=T=F⇒T N k=0 N

linearity  n n = kN ⇐D=T=F⇒T X (ejN ω )
time multiplication N ⇐D=T=F⇒T
x ⇐D=T=F⇒T
⇐D=T=F⇒T
0 otherwise
⇐D=T=F⇒T
ax1[n] + bx2[n] ⇐D=T=F⇒T aX1(ejω) + bX2(ejω)
x1[n]x2[n]
⇐D=T=F⇒T X1(ejω) ∗ X2(ejω) = frequency convolution

⇐D=T=F⇒T 1 +π X1(ej(ω−σ))X2(ejσ )dσ
⇐D=T=F⇒T 2π −π
⇐D=T=F⇒T
frequency convolution x1[n] ∗ x2[n] X1(ejω )X2(ejω ) frequency multiplication
⇐D=T=F⇒T
delta function δ[n] 1
shifted delta function δ[n − n0] ⇐D=T=F⇒T e−jωn0
δ˜(ω)
1 ⇐D=T=F⇒T δ˜(ω − ω0) delta function
ej ω0 n shifted delta function
⇐D=T=F⇒T
sine sin (ω0n + φ) ⇐D=T=F⇒T j [e−jφδ˜ (ω + ω0 + 2πk) − e+jφδ˜ (ω − ω0 + 2πk)]
cosine cos (ω0n + φ) ⇐D=T=F⇒T 2

1 [e−jφδ˜ (ω + ω0 + 2πk) + e+jφδ˜ (ω − ω0 + 2πk)]
2

 |n| M 1
otherwise 2
rectangular rect n = 1 sin[ω(M + )]
M 0
sin(ω/2)

step u[n] 1 + 1 δ˜(ω)
1−e−jω 2

decaying step anu[n] (|a| < 1) 1

1−ae−jω

special decaying step (n + 1)anu[n] (|a| < 1) 1

(1−ae−jω )2

 |ω| < ωc
ωc < |ω| < π
sinc sin(ωc n) = ωc sinc (ωcn) re˜ct ω = 1
πn π ωc 0



MA rect n − 1 = 1 0nM sin[ω(M +1)/2] e−jωM/2
M 2 0 otherwise sin(ω/2)

 0 n M −1
otherwise
MA rect n − 1 = 1 sin[ωM/2] e−jω(M −1)/2
M −1 2 0 sin(ω/2)

derivation nx[n] j d X (ejω )
difference dω

x[n] − x[n − 1] (1 − e−jω)X(ejω)

an sin[ω0 (n+1)] u[n] |a| < 1 1
sin ω0
1−2a cos(ω0e−jω )+a2e−j2ω

Note: +∞

δ˜(ω) = +∞

δ(ω + 2πk) re˜ct(ω) = rect(ω + 2πk)

k=−∞ k=−∞

Parseval:

+∞ |x[n]|2 = 1 +π

|X (ejω )|2 dω
2π −π
n=−∞

Marc Ph. Stoecklin — TABLES OF TRANSFORM PAIRS — v1.5.3 6

Table of Laplace Transform Pairs

f (t) = L−1 {F (s)} = 1 limT →∞ c+jT F (s)estds ⇐=L=⇒ F (s) = L {f (t)} = +∞ f (t)e−stdt
2πj c−jT −∞
⇐=L=⇒
transform f (t) ⇐=L=⇒ F (s)
complex conjugation f ∗(t) F ∗(s∗)
⇐=L=⇒
time shifting f (t − a) t a>0 ⇐=L=⇒ a−asF (s)
time scaling e−atf (t) ⇐=L=⇒
F (s + a) frequency shifting
f (at) ⇐=L=⇒
⇐=L=⇒ 1 F ( s )
⇐=L=⇒ |a| a

linearity af1(t) + bf2(t) ⇐=L=⇒ aF1(s) + bF2(s)
time multiplication f1(t)f2(t) ⇐=L=⇒
time convolution ⇐=L=⇒ F1(s) ∗ F2(s) frequency convolution
f1(t) ∗ f2(t) ⇐=L=⇒
⇐=L=⇒ F1(s)F2(s) frequency product
⇐=L=⇒
delta function δ(t) 1 exponential decay
shifted delta function δ(t − a) ⇐=L=⇒
unit step ⇐=L=⇒ e−as
ramp u(t) ⇐=L=⇒
parabola tu(t) ⇐=L=⇒ 1
n-th power t2u(t) ⇐=L=⇒ s
1
exponential decay tn ⇐=L=⇒ s2
two-sided exponential decay ⇐=L=⇒ 2
e−at ⇐=L=⇒ s3
exponential approach e−a|t| ⇐=L=⇒
te−at ⇐=L=⇒ n!
(1 − at)e−at ⇐=L=⇒ sn+1
1 − e−at
⇐=L=⇒ 1
sine sin (ωt) ⇐=L=⇒ s+a
cosine cos (ωt)
hyperbolic sine sinh (ωt) ⇐=L=⇒ 2a
hyperbolic cosine cosh (ωt) ⇐=L=⇒ a2 −s2
exponentially decaying sine e−at sin (ωt) ⇐=L=⇒
exponentially decaying cosine e−at cos (ωt) 1
⇐=L=⇒ (s+a)2
frequency differentiation tf (t) ⇐=L=⇒
frequency n-th differentiation tnf (t) s
⇐=L=⇒ (s+a)2
⇐=L=⇒
a
s(s+a)

ω
s2 +ω2

s
s2 +ω2

ω
s2 −ω2

s
s2 −ω2

ω
(s+a)2 +ω2

s+a
(s+a)2 +ω2

−F (s)

(−1)nF (n)(s)

time differentiation f (t) = d f (t) sF (s) − f (0)
time 2nd differentiation dt s2F (s) − sf (0) − f (0)
time n-th differentiation d2 snF (s) − sn−1f (0) − . . . − f (n−1)(0)
f (t) = dt2 f (t)
time integration
frequency integration f (n)(t) = dn f (t)
dtn
time inverse
time differentiation t f (τ )dτ = (u ∗ f )(t) 1 F (s)
0 s

1 f (t) ∞ F (u)du
t s

f −1(t) F (s)−f −1
f −n(t)
s f −1(0) f −2(0) f −n(0)
sn sn−1 s
F (s) + + +...+
sn