Tables of Common Transform Pairs

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Tables of Common Transform Pairs 2004-2014 by Marc Ph. Stoecklin — marc a stoecklin.net — http://www.stoecklin.net/ — 2014-01-10 — version v1.6.1

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. Nevertheless, 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

j 2 = −1

Complex conjugate

z = a + jb

Real part

<e {f (t)} =

Imaginary part

1 =m {f (t)} = 2j [f (t) − f ∗ (t)] ( 1, n = 0 δ[n] = 0, n 6= 0 ( 1, n ≥ 0 u[n] = 0, n < 0

Dirac delta/Unit impulse

Heaviside step/Unit step Sine/Cosine

sin (x) =

[f (t) + f ∗ (t)]

1 2

ejx −e−jx 2j

sinc (x) ≡

sin(x) x

sincπ (x)≡

sin(πx) πx

Sinc function

Rectangular function

( 1 t rect( T ) = 0 t T

z ∗ = a − jb

←→



ejx +e−jx 2

(unnormalized) = sinc (πx) T 2 T 2

if |t| 6 if |t| >

rect( Tt

(normalized)

)∗

rect( Tt

  1

Triangular function

triang

Convolution

continuous-time:

(f ∗ g)(t) =

R +∞

discrete-time:

(u ∗ v)[n] =

P∞

Parseval theorem

=

cos (x) =

general statement:

R +∞

continuous-time:

R +∞

Geometric series

P∞

k=0

xk =

n=−∞

1 1−x

k=m

1

|t| T

|t| 6 T |t| > T

f (τ ) g ∗ (t − τ )dτ

m=−∞

1−xn+1 1−x

xk =

xk =

xm −xn+1 1−x

u[m] v ∗ [n − m] R +∞ −∞

R +∞ −∞

|x[n]|2 =

Pn

k=0

Pn

in general:

−∞

|f (t)|2 dt =

P+∞

discrete-time:



0

f (t)g ∗ (t)dt =

−∞

−∞

)=

1 2π

F (f )G∗ (f )df

|F (f )|2 df R +π −π

|X(ejω )|2 dω

2

Marc Ph. Stoecklin — TABLES OF COMMON TRANSFORM PAIRS — v1.6.1

Table of Continuous-time Frequency Fourier Transform Pairs f (t) = F −1 {F (f )} =

R +∞ −∞

F

F (f )ej2πf t df

⇐==⇒

f (t)

⇐==⇒

transform

F

F

time reversal

f (−t)

⇐==⇒

complex conjugation

f ∗ (t)

⇐==⇒

reversed conjugation

f ∗ (−t)

⇐==⇒

f (t) is purely real

⇐==⇒

linearity

F (f ) = F ∗ (−f )

F

F (f ) = −F ∗ (−f )

f (t − t0 )

⇐==⇒

F

F

⇐==⇒ ⇐==⇒

af (t) + bg(t)

⇐==⇒

F F

⇐==⇒ F

F

f (t)g(t)

⇐==⇒

f (t) ∗ g(t)

⇐==⇒

δ(t)

⇐==⇒

δ(t − t0 )

⇐==⇒

shifted delta function

e−a|t|

F

F F F

⇐==⇒

e−πat

sin (2πf0 t + φ)

F

F

⇐==⇒

f (t) sin (2πf0 t)

⇐==⇒

cosine modulation

f (t) cos (2πf0 t)

⇐==⇒

sin2 (t)

⇐==⇒

cos2 (t)

F

rect

t T  

|t| 6 |t| > |t| 6 T |t| > T

= 0 |t|  1 − t T triang T =  0

triangular sinc

sincπ (Bt)

squared sinc

sinc2π (

1 sgn (t) = −1

signum

j 1 π t

inverse step

u(t) =

1 (sgn(t) 2

(

+ 1) =

n-th time derivative

polynomial

Dirac comb

(Bt)

t>0 t<0

P∞

n=0

1 0

t>0 t<0

F

F

F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

⇐==⇒ F

f (n) (t)

⇐==⇒

tn f (t)

⇐==⇒

F

F

tn

⇐==⇒

1 1+t2

⇐==⇒

δ(t − nf0 )

F

F

⇐==⇒

shifted delta function

πf 2

j 2 1 2 j 2 1 2 1 4 1 4

⇐==⇒

cos (2πf0 t + φ)

rectangular

delta function

F

sine modulation

T 2 T 2

δ(f )

√1 e− a a

cosine



constant

e−j2πf t0

F

⇐==⇒

frequency multiplication

1

2a a2 +4π 2 f 2 1 a+2πjf 1 a−2πjf

2

frequency convolution

F (f )G(f )

F

<e{a} > 0

 1

aF (f ) + bG(f ) F (f ) ∗ G(f )

⇐==⇒ ⇐==⇒

squared cosine

frequency scaling

a>0

F

frequency shifting

F (af )

δ(f − f0 )

<e{a} > 0

squared sine

F (f −  f0) 1 F fa |a|

⇐==⇒

e−at u(t)

sine

F (f )e−j2πf t0

⇐==⇒

e−at u(−t)

Gaussian function

F (f ) is purely real

1

F

even/symmetry odd/antisymmetry

F (f ) is purely imaginary

ej2πf0 t

constant

two-sided exponential decay

F

F

⇐==⇒

delta function

reversed exponential decay

F

f (at)  t 1 f |a| a

time convolution

exponential decay

complex conjugation

−f ∗ (−t)

time multiplication

frequency reversal

F ∗ (f )

f (t)ej2πf0 t time scaling

F (f ) F (−f )

F

⇐==⇒

time shifting

f (t)e−j2πf t dt

reversed conjugation

f (t) = f ∗ (−t) f (t) =

−∞

F ∗ (−f )

⇐==⇒

odd/antisymmetry

R +∞

F

f (t) is purely imaginary even/symmetry

F (f ) = F {f (t)} =

Gaussian function

 −jφ  e δ (f + f0 ) − ejφ δ (f − f0 )  −jφ  e δ (f + f0 ) + ejφ δ (f − f0 ) [F (f + f0 ) − F (f − f0 )] [F (f + f0 ) + F (f − f0 )]   2δ(f ) − δ f − π1 − δ f +   2δ(f ) + δ f − π1 + δ f +

1 π  1 π



T sincπ (T f )

sinc

T sinc2π (T f )   f 1 rect B = |B|   f 1 triang B |B|

squared sinc 1 1 (f ) ,+ B ] |B| [− B 2 2

triangular

1 jπf

inverse

sgn (f )  1 2

1 jπf

signum + δ(f )



n (j2πf  n) F (f ) j F (n) (f )  2π n j δ (n) (f ) 2π πe−2π|f | 1 f0

rectangular

P∞

k=−∞

δ(f −

n-th frequency derivative

k ) f0

3

Marc Ph. Stoecklin — TABLES OF COMMON TRANSFORM PAIRS — v1.6.1

Table of Continuous-time Pulsation Fourier Transform Pairs x(t) = Fω−1 {X(ω)} =

1 2π

R +∞

F

X(ω)ejωt dω

ω ⇐==⇒

x(t)

ω ⇐==⇒

time reversal

x(−t)

ω ⇐==⇒

complex conjugation

x∗ (t)

ω ⇐==⇒

reversed conjugation

x∗ (−t)

ω ⇐==⇒

x(t) is purely real

ω ⇐==⇒

−∞

transform

F F

F

X(ω) = X ∗ (−ω)



X(ω) = −X ∗ (−ω)

x(t − t0 )

ω ⇐==⇒

x(t)ejω0 t



⇐==⇒

x (af )   1 x fa |a|

ω ⇐==⇒

ax1 (t) + bx2 (t)

ω ⇐==⇒

x1 (t)x2 (t)

ω ⇐==⇒

time convolution

F

F

F F

ω ⇐==⇒

F

F



⇐==⇒

δ(t)

ω ⇐==⇒

δ(t − t0 )

ω ⇐==⇒

shifted delta function

e−a|t|

two-sided exponential decay

F

x1 (t) ∗ x2 (t)

delta function

F F



1

⇐==⇒

ejω0 t

ω ⇐==⇒

a>0

ω ⇐==⇒

F

F



e−at u(t)

<e{a} > 0

⇐==⇒

e−at u(−t)

<e{a} > 0

ω ⇐==⇒

<e{a} > 0

ω ⇐==⇒

sine

sin (ω0 t + φ)

ω ⇐==⇒

cosine

cos (ω0 t + φ)

ω ⇐==⇒

sine modulation

x(t) sin (ω0 t)

ω ⇐==⇒

exponential decay reversed exponential decay

2

e−πat

Gaussian function

cosine modulation

triangular

squared sinc

ω ⇐==⇒

t T  



1 u(t) = 1[0,+∞] (t) = 0

inverse (

signum

sgn (t) =

n-th time derivative n-th frequency derivative Dirac comb

F



cos2 (ω0 t)  1

(

step

F

⇐==⇒

= 0 |t|  1 − t T triang T =  0

sinc

F

ω ⇐==⇒

|t| 6 T2 |t| > T2 |t| 6 T |t| > T   sinc (T t) = sincπ Tπt   sinc2 (T t) = sinc2π Tπt

rect

F

sin2 (ω0 t)

squared cosine rectangular

F

x(t) cos (ω0 t)

squared sine

P∞

n=0

1 −1

t>0 t<0

frequency reversal complex conjugation

ω ⇐==⇒

time multiplication

X(−ω) X ∗ (ω)

x(t) = −x∗ (−t)

linearity

X(ω)

F

ω ⇐==⇒

time scaling

x(t)e−jωt dt

reversed conjugation

x(t) = x∗ (−t)

time shifting

−∞

X ∗ (−ω)

⇐==⇒

odd/antisymmetry

R +∞

F

x(t) is purely imaginary even/symmetry

X(ω) = Fω {x(t)} =

even/symmetry odd/antisymmetry

X(ω) is purely real X(ω) is purely imaginary X(ω)e−jωt0 X(ω − ω0 )  1 X ω |a| a

frequency shifting

X(aω)

frequency scaling

aX1 (ω) + bX2 (ω) 1 X (ω) 2π 1

∗ X2 (ω)

frequency convolution

X1 (ω)X2 (ω)

frequency multiplication

1 e−jωt0 2πδ(ω)

delta function

2πδ(ω − ω0 )

shifted delta function

2a a2 +ω 2 1 a+jω 1 a−jω ω2 √1 e− 4πa a

  jπ e−jφ δ (ω + ω0 ) − ejφ δ (ω − ω0 )  −jφ  π e δ (ω + ω0 ) + ejφ δ (ω − ω0 ) j 2 1 2

[X (ω + ω0 ) − X (ω − ω0 )] [X (ω + ω0 ) + X (ω − ω0 )]

F

π 2 [2δ(f ) − δ (ω − ω0 ) − δ (ω + ω0 )]

F

π 2 [2δ(ω) + δ (ω − ω0 ) + δ (ω + ω0 )]

F

ω ⇐==⇒

F

ω ⇐==⇒

T sinc

T sinc2

F

1 T

F

1 T

ω ⇐==⇒ ω ⇐==⇒



⇐==⇒ Fω

1 t

⇐==⇒

t>0 t<0

ω ⇐==⇒



rect

ωT 2





ωT 2

ω 2πT

triang

πδ(ω) +

F

(jω)n X(ω)



d j n df n X(ω) 1 P∞

ω ⇐==⇒

F

= T sinc2π =

 1 T 0



k=−∞





ωT 2π



sinc squared sinc rectangular triangular inverse signum

n

ω0

ωT 2π

|ω| 6 πT |ω| > πT

−jπ sgn (ω)

ω ⇐==⇒

δ(t − nω0 )





1 jω

2 jω

⇐==⇒



ω 2πT

F

dn x(t) dtn n t x(t)

= T sincπ

δ(ω −

2πk ) ω0

4

Marc Ph. Stoecklin — TABLES OF COMMON TRANSFORM PAIRS — v1.6.1

Table of z-Transform Pairs x[n] = Z −1 {X(z)} =

1 2πj

Z

P+∞

x[n]z −n

X(z)z n−1 dz

⇐==⇒

x[n]

⇐==⇒

X(z)

Rx

x[−n]

Z

⇐==⇒

X( z1 )

1 Rx

complex conjugation

x∗ [n]

Z

⇐==⇒

X ∗ (z ∗ )

Rx

reversed conjugation

x∗ [−n]

⇐==⇒

Z

X ∗ ( z1∗ )

1 Rx

<e{x[n]}

⇐==⇒

Z

1 [X(z) + X ∗ (z ∗ )] 2 1 [X(z) − X ∗ (z ∗ )] 2j

Rx

H

transform time reversal

real part

Z

Z

imaginary part

=m{x[n]}

⇐==⇒

time shifting

x[n − n0 ]

⇐==⇒

Z Z

an x[n]

⇐==⇒

downsampling by N/decimation

x[N n], N ∈ N0

⇐==⇒

linearity

ax1 [n] + bx2 [n]

⇐==⇒

scaling in Z

time multiplication time convolution

⇐==⇒

x1 [n] ∗ x2 [n]

⇐==⇒

δ[n]

⇐==⇒

δ[n − n0 ]

⇐==⇒

u[n]

⇐==⇒

shifted delta function step

−u[−n − 1] ramp

⇐==⇒ Z Z Z

⇐==⇒ Z Z Z

(−1)n

⇐==⇒

an u[n]

⇐==⇒

Z Z

⇐==⇒ Z

− 1]

⇐==⇒ ⇐==⇒

n2 an u[n]

⇐==⇒

e−an u[n]

Z

⇐==⇒

|a| < 1

⇐==⇒

n = 0, . . . , N − 1 otherwise

⇐==⇒

sin (ω0 n) u[n]

⇐==⇒

cos (ω0 n) u[n] an

Z

Z

Z

z sin(ω0 ) z 2 −2 cos(ω0 )z+1 z(z−cos(ω0 )) z 2 −2 cos(ω0 )z+1 za sin(ω0 ) z 2 −2a cos(ω0 )z+a2 z(z−a cos(ω0 )) z 2 −2a cos(ω0 )z+a2

Z

⇐==⇒ Z

⇐==⇒

nx[n]

⇐==⇒

accumulation Qm

Z

Z

Z

x[n] n

⇐==⇒

x[n] − x[n − 1] Pn k=−∞ x[n]

⇐==⇒

i=1 (n−i+1) am m!

am u[n]

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−a2 ) (z−a)(1−az)

Z

⇐==⇒

first difference

Rx ∩ Ry

1−aN z −N 1−az −1

sin (ω0 n) u[n]

integration in Z

X1 (z)X2 (z)

Z

an cos (ω0 n) u[n] differentiation in Z

Rx ∩ Ry

z z−1 z z−1 z (z−1)2 z(z+1) (z−1)3 z(z+1) (z−1)3 z(z 2 +4z+1) (z−1)4 z(z 2 +4z+1) (z−1)4 z z+1

Z

nan u[n]

cosine

Rx ∩ Ry

aX1 (z) + bX2 (z) H  −1 1 z X1 (u)X2 u u du 2πj

Z

⇐==⇒

sine

Rx

∀z

⇐==⇒

a|n|

Rx |a|Rx

∀z

−n3 u[−n − 1]

an−1 u[n

Rx

z −n0

⇐==⇒

− 1]

z −n0 X(z)  X az   1 −j 2ω k 1 PN −1 N ·e N k=0 X z N

ROC

1

⇐==⇒

− 1]

n=−∞

Z

nu[n]

−an u[−n

z 1 = z−1 1 − z −1

Z

n3 u[n]

exponential

Note:

Z

n2 u[n] −n2 u[−n

exp. interval

Z

Z

x1 [n]x2 [n]

delta function

 n a 0

Z

X(z) = Z {x[n]} =

Z

Z

⇐==⇒ Z

⇐==⇒

dX(z)

−z dz R X(z) − 0z z dz 1 )X(z) z z X(z) z−1 z (z−a)m+1

(1 −

|z| > 1 |z| < 1 |z| > 1 |z| > 1 |z| < 1 |z| > 1 |z| < 1 |z| < 1 |z| > |a| |z| < |a| |z| > |a| |z| > |a| |z| > |a| |z| > |e−a | |a| < z <

1 |a|

|z| > 0 |z| > 1 |z| > 1 |z| > a |z| > a Rx Rx Rx , z 6= 0 Rx

5

Marc Ph. Stoecklin — TABLES OF COMMON TRANSFORM PAIRS — v1.6.1

Table of Common Discrete Time Fourier Transform (DTFT) Pairs R +π

1 2π

x[n]e−jωn

X(ejω ) =

x[n] x[−n] x∗ [n] ∗ x [−n]

⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒

DT F T

X(ejω ) X(e−jω ) X ∗ (e−jω ) X ∗ (ejω )

x[n] is purely real x[n] is purely imaginary x[n] = x∗ [−n] x[n] = −x∗ [−n]

⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒

DT F T

X(ejω ) = X ∗ (e−jω ) X(ejω ) = −X ∗ (e−jω ) X(ejω ) is purely real X(ejω ) is purely imaginary

x[n − n0 ] x[n]ejω0 n

⇐==⇒ DT F T ⇐==⇒

DT F T

N ∈ N0

⇐==⇒

n = kN otherwise

⇐==⇒

X(ejω )e−jωn0 X(ej(ω−ω0 ) )  ω−2πk  j 1 PN −1 N k=0 X e N  X ejN ω

ax1 [n] + bx2 [n] x1 [n]x2 [n]

⇐==⇒ DT F T ⇐==⇒

x1 [n] ∗ x2 [n]

⇐==⇒

δ[n] δ[n − n0 ] 1 ejω0 n

⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒ DT F T ⇐==⇒

sin (ω0 n + φ)

⇐==⇒

time shifting

downsampling by N

x[N n] 

  x n N 0

upsampling by N linearity time multiplication time convolution delta function shifted delta function

sine cosine

cos (ω0 n + φ)

rectangular

P+∞

⇐==⇒

−π

transform time reversal complex conjugation reversed conjugation

even/symmetry odd/antisymmetry

DT F T

X(ejω )ejωn dω

x[n] =

rect

n 2M



 1

= 0

DT F T

DT F T

DT F T

X1 (ejω )X2 (ejω )

DT F T

DT F T

DT F T

⇐==⇒ ⇐==⇒

u[n]

⇐==⇒

DT F T

DT F T DT F T

(|a| < 1)

⇐==⇒

1)an u[n]

(|a| < 1)

⇐==⇒

sinc (ωc n)

⇐==⇒

decaying step (special)

(n +

sinc

sin(ωc n) πn

=

ωc π

frequency shifting

aX1 (ejω ) + bX2 (ejω ) X1 (ejω ) ∗ X2 (ejω ) = frequency convolution R +π 1 X1 (ej(ω−σ) )X2 (ejσ )dσ 2π −π

an u[n]

decaying step

even/symmetry odd/antisymmetry

DT F T

|n| 6 M otherwise

step

n=−∞

DT F T

DT F T

frequency multiplication

1 e−jωn0 ˜ δ(ω) ˜ − ω0 ) δ(ω

delta function shifted delta function

j −jφ ˜ [e δ (ω 2 1 −jφ ˜ [e δ (ω 2

+ ω0 + 2πk) − e+jφ δ˜ (ω − ω0 + 2πk)] + ω0 + 2πk) + e+jφ δ˜ (ω − ω0 + 2πk)]

sin[ω (M + 1 2 )] sin(ω/2) 1 ˜ + 12 δ(ω) 1−e−jω 1 1−ae−jω 1 (1−ae−jω )2    1 ω ˜ rect = ωc 0

|ω| < ωc ωc < |ω| < π



Moving avg. Moving avg.

rect rect



n M −1

1 0 6 n 6 M = 0 otherwise   1 0 6 n 6 M − 1 1 − 2 = 0 otherwise n M



1 2



derivation

nx[n] x[n] − x[n − 1] |a| < 1

difference

an sin[ω0 (n+1)] u[n] sin ω0

Note: ˜ δ(ω) =

+∞ X k=−∞

δ(ω + 2πk)

⇐==⇒

DT F T

sin[ω(M +1)/2] −jωM/2 e sin(ω/2)

DT F T

sin[ωM/2] −jω(M −1)/2 e sin(ω/2)

DT F T

d j dω X(ejω )

⇐==⇒ ⇐==⇒ DT F T

⇐==⇒ DT F T ⇐==⇒

(1 − e−jω )X(ejω ) 1 1−2a cos(ω0 e−jω )+a2 e−j2ω

˜ rect(ω) =

+∞ X k=−∞

rect(ω + 2πk)

6

Marc Ph. Stoecklin — TABLES OF COMMON TRANSFORM PAIRS — v1.6.1

Table of Laplace Transform Pairs f (t) = L−1 {F (s)} =

1 2πj

limT →∞

R c+jT

⇐==⇒

f (t)

⇐==⇒

f ∗ (t)

⇐==⇒

t>a>0

⇐==⇒

e−at f (t)

L

⇐==⇒

F (s + a)

f (at)

⇐==⇒

L

1 F ( as ) |a|

af1 (t) + bf2 (t)

⇐==⇒

f1 (t)f2 (t)

⇐==⇒

f1 (t) ∗ f2 (t)

⇐==⇒

δ(t)

⇐==⇒

δ(t − a)

⇐==⇒

L

e−as

L

1 s 1 s2 2 s3 n! sn+1

transform complex conjugation f (t − a)

time shifting time scaling linearity time multiplication time convolution

L

F (s)est ds

c−jT

delta function shifted delta function unit step ramp parabola n-th power exponential decay

L

F ∗ (s∗ )

L

e−as F (s)

L L

L

L

u(t)

⇐==⇒

tu(t)

⇐==⇒

t2 u(t)

⇐==⇒

tn

⇐==⇒

e−at

⇐==⇒

L

L

L L

L

⇐==⇒

te−at

⇐==⇒

(1 − at)e−at

⇐==⇒

1 − e−at

⇐==⇒

sin (ωt)

⇐==⇒

exponential approach sine

L L L

L

L

cos (ωt)

⇐==⇒

hyperbolic sine

sinh (ωt)

⇐==⇒

hyperbolic cosine

cosh (ωt)

⇐==⇒

cosine

L

L

L

exponentially decaying sine

e−at sin (ωt)

⇐==⇒

exponentially decaying cosine

e−at cos (ωt)

⇐==⇒

tf (t)

⇐==⇒

frequency differentiation

⇐==⇒

f (τ )dτ = (u ∗ f )(t)

⇐==⇒

time n-th differentiation

f (n) (t) =

time integration

Rt 0

=

L

frequency product

exponential decay

1 s+a 2a a2 −s2 1 (s+a)2 s (s+a)2 a s(s+a) ω s2 +ω 2 s s2 +ω 2 ω s2 −ω 2 s s2 −ω 2 ω (s+a)2 +ω 2 s+a (s+a)2 +ω 2

sF (s) − f (0)

L

s2 F (s) − sf (0) − f 0 (0)

L

sn F (s) − sn−1 f (0) − . . . − f (n−1) (0)

L

1 F (s) Rs ∞ s F (u)du

⇐==⇒

L

1 f (t) t

⇐==⇒

time inverse

f −1 (t)

⇐==⇒

time differentiation

f −n (t)

⇐==⇒

frequency integration

frequency convolution

1

(−1)n F (n) (s)

⇐==⇒

time 2nd differentiation

F1 (s) ∗ F2 (s)

−F 0 (s)

d f (t) dt d2 f (t) 2 dt dn f (t) dtn

f 0 (t) = f 00 (t)

f (t)e−st dt

frequency shifting

F1 (s)F2 (s)

L

⇐==⇒

−∞

aF1 (s) + bF2 (s)

L

tn f (t)

frequency n-th differentiation time differentiation

L

R +∞

F (s)

L

e−a|t|

two-sided exponential decay

F (s) = L {f (t)} =

L L

F (s)−f −1 s F (s) f −1 (0) + sn sn

+

f −2 (0) sn−1

+ ... +

f −n (0) s

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