ABSTRACT
U(t) ↔ u(− f )
Dirac delta functions:
δ(t) ↔ 1 exp (j2πf0t) ↔ δ(f − f0)
Unit length, unit amplitude pulse, centered on t = 0:
rect t f
f ( ) ↔ sinπ
π
u d U f j f
v t d V f d a
τ τ π
α α α α
τ ( ) ↔ ( )
( ) ↔ ( ) −
, ,
′( ) ↔ ( ) − ( ) ↔ ′( )
∂ ( ) ∂ ↔
∂ −( ) ∂
u t j fU f
j tu t U f
u t U f
2 2
π
π
α
α
α
α
, ,
u t v t U f V f
u t v t U f V f
( ) ( ) ↔ ( ) ( ) ( ) ( ) ↔ ( ) ( )
* *
u t T
TU f T⎛⎝ ⎞⎠ ↔ ( )
U f j ft df u t U f u t j ft dt( ) ( ) = ( ) ↔ ( ) = ( ) −( ) −∞
∞ ∫∫ exp exp2 2π π
Gaussian distribution:
exp (−πt2) ↔ exp (−πf 2)
Repeated and impulse (delta function) sampled waveforms:
Parseval’s lemma:
x x t st dtx = ( ) −( )∞∫ exp0
u t v t dt U f V f df
u t dt U f df
( ) ( ) = ( ) ( ) ( ) = ( )
* *
u t nT T
U f f n T
u t t nT T
U f n T
−( ) ↔ ⎛⎝ ⎞⎠ ( ) −⎛⎝ ⎞⎠ ( ) −( ) ↔ ⎛⎝ ⎞⎠ −⎛⎝ ⎞⎠
δ
δ
Convolution integral:
x x t d X s X s t
σ σ σ( ) −( ) → ( ) ( )∫
