Fourier Transform Table PDF

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Michael I. Miller...

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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 ) = F {f (t)} =

F (f )ej2πft df

⇐==⇒

f (t)

⇐==⇒

transform

F (f ) F (−f )

F

F ∗ (−f )

reversed conjugation

F

F ∗ (f )

complex conjugation

complex conjugation

f ∗ (t)

⇐==⇒

reversed conjugation

f ∗ (−t)

⇐==⇒

F (f ) = F ∗ (−f ) F (f ) = −F ∗ (−f )

f (t) = f ∗ (−t)

⇐==⇒

F

F (f ) is purely real

F

⇐==⇒

F (f ) is purely imaginary

F

F (f )e−j2πf t0

f (t − t0 )

⇐==⇒ ⇐==⇒

f (at)   1 f t |a| a

⇐==⇒

time scaling

linearity

F

⇐==⇒

F

F (f )G(f )

F

1

δ(t)

⇐==⇒

e−j2πft0 δ(f )

delta function

1

⇐==⇒

F

δ(f − f0 )

shifted delta function

a>0

⇐==⇒

F

ℜe{a} > 0

⇐==⇒

ℜe{a} > 0

F

⇐==⇒

2 e−πat

⇐==⇒

2a a2 +4π2 f 2 1 a+2πj f 1 a−2πj f πf 2 √1 e− a a

Gaussian function

Gaussian function

F

F

F

sine

sin (2πf0 t + φ)

⇐==⇒

cosine

cos (2πf0 t + φ)

⇐==⇒

sine modulation

f (t) sin (2πf0 t)

F

⇐==⇒

cosine modulation

f (t) cos (2πf0 t)

⇐==⇒

squared sine

sin2 (t)

⇐==⇒

squared cosine

cos2

triangular

triang

sinc

t  T

=

T  1

= 0

 1

−

|t| T

signum

sgn (t) =

1 (sgn(t) 2

n-th time derivative

⇐==⇒ ⇐==⇒

Dirac comb

P∞

n=0

F

F

⇐==⇒ F

sinc2π (B t)

F

1 −1

t>0 t0 t |t| 6 T |t| > T

0

squared sinc

constant

F

ej2πf0 t e−a|t|

t

frequency convolution frequency multiplication

⇐==⇒ F ⇐==⇒

constant

rect

frequency scaling

aF (f ) + bG(f )

⇐==⇒

δ(t − t0 )

rectangular

F (af )

F (f ) ∗ G(f )

f (t) ∗ g(t)

e−at u(t)

frequency shifting

F

⇐==⇒

e−at u(−t)

F (f −  f0) f 1 F |a| a

F

⇐==⇒

shifted delta function

exponential decay

F

f (t)g (t)

delta function

two-sided exponential decay

F

af (t) + bg(t)

time multiplication time convolution

even/symmetry odd/antisymmetry

−f ∗ (−t)

f (t)ej2πf0 t

time shifting

reversed exponential decay

F

⇐==⇒ ⇐==⇒

f (t) =

frequency reversal

F

f (t) is purely real f (t) is purely imaginary odd/antisymmetry

f (t)e−j2πf t dt

F

⇐==⇒

even/symmetry

−∞

F

f (−t)

time reversal

R +∞

⇐==⇒ F

⇐==⇒ F

⇐==⇒

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

  −j φ e δ (f + f0 ) − ej φ δ (f − f0 )   −j φ j φ e δ (f + f0 ) + e δ (f − f0 )

[F (f + f0 ) − F (f − f0 )]

[F (f + f0 ) + F (f − f0 )]     2δ(f ) − δ f − 1π − δ f +     1 2δ(f ) + δ f − π + δ f +

T sincπ (T f ) 2 (T f ) T sincπ   f 1 rect B = |B|   f 1 triang |B| B

⇐==⇒

1 1+t2

F ⇐= =⇒

δ(t − nf0 )

⇐==⇒

F



triangular

1 j πf

 + δ(f )

πe−2π|f | P∞ 1 f0

rectangular

signum

n (j2πf n) F (f ) j F (n) (f )  2π n j δ (n) (f ) 2π

F

1 (f ) 1 ] ,+ B |B| [− B 2 2

sgn (f )

F

tn

squared sinc

inverse

1 2

⇐==⇒ F ⇐= =⇒

sinc

1 j πf

F

⇐==⇒

1  π 1  π

k=−∞

δ(f −

n-th frequency derivative

k f0

)...