formule ss lucrarea 2

1 Formule SS lucrarea 2

x(t) X(ω)

X (ω )=∫−∞

∞

x (t)e− jωt dt ;∀ t ∊ R

x (t )= 12π

∫−∞

∞

X (ω) e jωt dω;∀ t ∊ R

x (t )=F−1 {X (ω)}

X (ω )=F {x ( t )}

∫−∞

∞

¿¿¿

∑k

ck xk (t) ∑k

ckX k (ω ) ;ck∊ R¿

F {x (n) ( t ) }=( jω )n X (ω)

F ¿{∫−∞

∞

x (τ )dτ }= X (ω )jω

+π δ (ω )F ¿)

F {x (t ) }=X (ω )

F {X (ω) }=2πx (−ω)

F−1 {X ' (ω ) }=− jtx (t)

F−1{∫−∞

ω

X ( λ )dλ }=−x (t )jt

x ( ta ) |a|X (ωa );a ∊ R¿

x (t−t 0) e− jωt0 X (ω)

X (ω−ω0) e jω0 t x (t)

x (t )∗y (t) X (ω )Y (ω)

x (t ) y ( t ) 12π

[X (ω)Y (ω) ]

pa ( t )={ 1 ; t ∊ [−a ;a ]0 ; t ∊ (−∞ ;−a )∪ (a;∞ )

pa(t ) 2asinc (ωa)

πasinc (at) pa(ω)

e−αt σ (t) 1α+ jω

x (t )=x (−t ); X (ω ) ∊ R

ℱ

ℱ

ℱ

ℱℱℱ

ℱ

ℱℱℱ


x (t )=−x (−t ); X (ω) ∊ j R

F (ω )=VP∫−∞

∞

f (t )e− jωtdt

VP∫−∞

∞x (t)tdt=lim

ε❑→0 [∫

−∞

εx (t )tdt+∫

ε

∞x (t)tdt ]

∫−∞

∞

δ ( t )dt=1

δ (t ) 1ω

∫−∞

∞

f (t ) δ (t−t 0 )dt=f ( t0 )

1t 2π δ (−ω )=2πδ (ω )

F−1 {𝜹(ω)} = 12π

A ∊ R¿ A 2π δ(ω)

e jω0 t 2π δ ¿)

cos (ω0t ) π (δ (ω−ω0 )+δ (ω+ω0 ))

sin(ω0 t) − jπ(δ (ω−ω0 )−δ (ω+ω0 ))

sgn(t) 2jω

𝜎(t) π δ ( t )+ 1jω

|t| −2ω2

f(t) F(ω)

g ( t )=∫−t

t

f (τ )dτ

g(t) F (ω )jω

+π δ (ω )F (0)

∫−∞

∞

x (t ) y ( t )dt=¿ 12π

∫−∞

∞

X (ω )Y ¿ (ω)dω¿

f ( t )=∑k ∊Zf (t−kT ) f (t )∊ Sam(T )

f(t) F(ω)

f (t−kT ) F (ω)e− jkTω

f T (t ) ∑k∊ ZF (ω)e− jkTω

~f T (t )=∑

k ∊ Zak e

jkω0 t

ℱ

ℱ

ℱℱ

ℱℱ

ℱℱ

ℱ

ℱ

ℱ

ℱℱ

ℱ


ak=AFCk=1TF(ω) ω=k ω0

a0=AFC0=1TF (ω) ω=0

x(t) X(s)

X(s)=LB{x(t)}=∫−∞

∞

x (t)e−st dt ; ∫−∞

∞

|x (t )|e−st dt<∞

f ( t )=L1loc ; supp (f )=[0 ;∞ );

F ( s )=L {f ( t ) }=∫0

∞

f (t)e−st dt

(∀ ) s0 ∊C es0 t F (s−s0)

(∀ ) s0>0 f (t−t 0) e−s t 0F (s)

(∀ )a>0 f (at) 1aF ( sa )

k=0 ;n−1 limt⟶0f ( k ) (t )=f ( k ) ¿

f (n ) ( t ) sn F ( s)−sn−1 f ¿

(−t )n f (t ) dn

d snF(s)

∫0

t

f (τ )dτ F (s)s

f ( t)t

∫s

∞

F ( λ )dλ

∑k

ak xk (t) ∑k

ak Xk (s )ak ∊ N

f 1(t) F1 ( s)α>c1

f 2(t) F2 ( s)α>c2

( f 1∗f 2 )(t) F1 ( s) F2(s)

LB

L

L

L

L

L

L

L

L

L

L

L

L


f 1 ( t ) f 2 ( t ) 12πj

[F1 ( s)∗F2 (s ) ]

α>c1+c2

f❑ (t ) F❑ ( s)α>c❑ supp ( f ( t ) )⊂ [0;T ]

f sp (t )=∑k∊ Nf (t−kT )

F(s)1−e−st

𝜹(t) 1; 𝛼>0δ (n )(t) sn

𝜎(t) 1s

e−at σ (t) 1s+a

sin (ω0 t )σ ( t ) ω0s2+ω0

2

cos (ω0 t )σ (t) s

s2+ω02

t n

n !σ (t)

1

sn+1

Rezz= z0 {f ( z ) }= 1(k−1 )!

limz⟶ z0

dk−1

d zk−1[ ( z−z0 ) f (z )]

f (t )={ 0 ; t<0∑

polii luiF ( s)Rezs=sk {F ( s )est }

; t>0=( ∑polii lui F ( s)

Rezs=sk {F ( s )est }) σ (t)

limt⟶0f (t )= lim

s⟶∞sF ( s )

limt⟶∞f ( t )= lim

s⟶0sF (s )

1

s2 ( s+1 ) [ (t−1 ) e−t ]σ (t )

1

θn1

s (s+ 1θ )n [1−∑

k=0

n−1t k

k !e

−tθ ( 1θ )

k ] σ ( t ) ;θ ∊ R¿ , n ∊ N ¿

e−sT

s (s+1) (1−e−(t−T ) )σ (t−T )

c<0 ;α>c ;F ( jω )=F L(s) s=jω

L

L

L

L

L

L

L

L

L−1

lim|s|⟶∞

F (s )=0

L−1

L−1


c=0; F ( jω )=FL (s) s=jω + ∑polii lui FL (s )de peaxa jω

ak πδ (ω−ωk ) ;ak=Rezs= j ωk {F L (s )}

c>0 ;α>c ∄ tr F ,∄tr L

f ( t )∗δ ( t )=f ( t )

f ( t )∗δ (t−t 0 )=f (t−t0 )

sh ( x )= ex−e−x

2

ch ( x )= ex+e− x

2

( 1x )n

=(−1 )nn ! 1xn+1

sin x=¿ ejx−e− jx

2 j¿

cos x= ejx+e− jx

2

formule ss lucrarea 2

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