formule ss lucrarea 2
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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
2 Formule SS lucrarea 2
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
3 Formule SS lucrarea 2
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
4 Formule SS lucrarea 2
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
5 Formule SS lucrarea 2
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
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