Fourier and Laplace Transform Table

Basic Fourier-transform pairs

Table of Fourier Transform Pairs

Time-Domain: x(t) Frequency-Domain: X(f)

) 0 ( )

( >

− u t a

e ^at

f j a 2π

1 + )

0 ( )

(−t b>

u e^bt

f j a 2π

1

− 2 )

( 1 2 )

(t 1T u t T

u + − −

f fT) sin(

π π

t t f_b ) 2 sin(

π

π _{( ) ( )}

b

b u f f

f f

u + − −

)

δ(t 1

) (t−t_d

δ

e

^−j2πftd

) (t

u ^f _j π_f

δ

2 1 2

)

( +

t f₀ 2

cos π ( )

2 ) 1 2 (

1

0

0 f f

f

f − + δ +

δ t

f₀ 2

sin π ( )

2 ) 1 2 (

1

0

0 f f

f j

jδ f − − δ +

kt f j k

ke a ^2π0

∑

^∞

−∞

=

) (f kf₀ a

k

k −

∑

^∞

−∞

=

δ

∑

^∞

−∞

=

−

k

nT

t )

δ(

∑

^∞

−∞

=

−

k T

f k

T1 δ( )

Table of Laplace Transform Pairs

x(t) X(s)

1

¹_s ^s>0

tn

n an integer ! 0

1 >

+ s

s n

n

eat s a

a

s >

−1 bt

sin _{2 2} >0

+ s

b s

b

bt

cos _{2 2} >0

+ s

b s

s

) (t f

e^at F(s−a)

n att

e n an integer _s ⁿ_a n ^s>^a

− ⁺ )

(

!

1

bt

e^atsin _s−a^b _+b ^s>a

)

( ^{2 2}

bt

e^atcos _s−^s_a^−a_+b ^s>a

) (

) (

2 2

bt

tsin

(

^s22+bsb2

)

^{2 s>0}

bt

tcos

(

^{2 2}

)

^{2 0}

2

2 >

+

− s

b s

b s

∑

^∞

−∞

=

−

k

nT

t )

δ(

∑

^∞

−∞

=

−

k T

f k

T1 δ( )

Laplace transform operations

Operation )f(t F(s)

Addition f₁(t)± f₂(t) F₁(s)±F₂(s)

Scalar multiplication kf(t) kF(s)

Time Differentiation

dt t df( )

) 0 ( ) (s − f ⁻ sF

2 2 () dt

t f

d s²F(s)−sf(0⁻)− f′(0⁻)

3 3 () dt

t f

d s³F(s)−s²f(0⁻)−sf′(0⁻)− f′′(0⁻)

Integration

∫

^f(t)dt 1 ( )

s sF Convolution f₁(t)* f₂(t) F₁(s)F₂(s)

Time Shift f₁(t−a)u(t−a),a≥0 e^−asF(s)

Frequency Shift e^−atf(t) F(s+a)

Scaling f(at),a≥0

) 1 (

a F s a

Initial Value f(0⁺) limsF(s)

s→∞

Final Value f(∞) lim ( )

0sF s

s→ , all poles of sF(s)in LHS