∗
est z s
n
z
e → H (s)est , z n → H (z )z n , H (s)
∗
H (z ) s z
h(t)
x(t)
x(t) = est +∞
y(t) =
+∞
h(τ )x(t − τ )dτ = −∞
h(τ )es(t−τ ) dτ
−∞
es(t−τ )
est e−sτ
est +∞
y(t) = est
h(τ )e−sτ dτ
−∞
est y(t) = H (s)est H (s) s +∞
H (s) =
h(τ )e−sτ dτ
−∞
H (s)
s est
x[n] = z n
h[n] y[n] =
+∞
+∞
h[k]x[n − k] =
k=−∞
h[k]z n−k
k=−∞
z n−k
z n z −k
zn +∞
y[n] = z
n
h[k]z −k
k=−∞
zn
y[n] = H (z)z n H (s) s +∞
H (z) =
k=−∞
h[k]z −k
H (z)
z zn T x(t) = x(t + T )
t x(t)
x(t)
T s1 t
x(t) = a1 e
+ a2 e
s2 t
ω0 = 2π/T x(t)
s3 t
+ a3 e
x(t) = cos ω0 t
aes t → a1 H (s1 )es t , a2 es t → a2 H (s2 )es t , a3 es t → a3 H (s3 )es t , 1
1
2
2
3
3
x(t) = ejω t . 0
x(t) ω0 y(t) = a1 H (s1 )es t + a2 H (s2 )es t + a3 H (s3 )es 1
2
3
T = 2π/ω 0
t
φk (t) = ejkω
0
t
= ejk (2π/T )t ,
k = 0, ±1, ±2, . . . . ω0 T
|k | > 2 T x(t) =
φk (t)
ak es t , k
+∞
k
x(t) =
+∞ jkω 0 t
ak e
=
k=−∞
y(t) =
k=−∞
k
k
k = +1 ω0
k = 0 k = −1
k = +2 x[n] =
ak ejk (2π/T )t
T
ak H (sk )es t .
k = −2
ak zkn ,
k
k = +N y[n] =
ak H (zk )zkn .
k
x 2π +3
ak H (sk )
es
k
t
zkn
H (zk )
x(t) =
k =−3
ak ejk 2πt ,
k = −N
ak = Ak ejθ , k
a0 = 1 a1 = a−1 = a2 = a−2 =
1 4 1 2 1 3
a3 = a−3 =
∞
x(t) = a0 + 2
Re
Ak ej (kω
0
t+θk )
k=1
.
∞
x(t) = a0 + 2
Ak cos(kω0 t + θk ).
k=1
x(t) = 1 + 14 (ej 2πt + e−j 2πt ) + 12 (ej 4πt + e−j 4πt ) + 13 (ej 6πt + e−j 6πt ).
ak
x(t) ak = Bk + jC k ,
1 2 x(t) = 1 + cos2πt + cos 4πt + cos6πt. 2 3
Bk
x(t)
C k
∞ ∗
x (t) = x(t)
x(t) = a0 + 2
[Bk cos kω0 t − C k sin kω0 t] .
k=1
+∞
x(t) =
a∗k e−jkω t . 0
k=−∞
−k
k +∞
x(t) =
a∗−k ejkω t , 0
k=−∞
ak = a∗−k a∗k = a−k . +∞
ak
x(t) =
ak = a−k
+∞ jkω 0 t
ak e
=
k =−∞
ak e−jnω
∞
x(t) = a0 +
ak ejkω
0
t
+ a−k e−jkω
0
t
k=1
a∗k
a−k
0
ak ejk (2π/T )t ,
k =−∞
t
+∞ −jnω 0 t
x(t)e
=
ak ejkω t e−jnω t . 0
0
k =−∞
0 2π/ω0
∞
x(t) = a0 +
ak e
jkω 0 t
+
a∗k e−jkω 0 t
k=1
. T
−jnω 0 t
x(t)e 0
∞
x(t) = a0 +
2Re ak ejkω
k=1
ak
T =
0
t
.
T
dt = 0
T
+∞
ak ejkω t e−jnω t dt. 0
0
k=−∞
x(t)
+∞
T
−jnω 0 t
x(t)e
dt =
0
T
ak
e
j (k−n)ω0 t
dt .
0
k=−∞
a0 =
x(t)dt, T
x(t)
T j (k−n)ω0 t T e dt = 0 cos(k − n)ω0 tdt 0 T + j 0 sin(k − n)ω0 tdt.
k = n cos(k −n)ω0 t
1 T
x(t) =
sin(k −n)ω0 t (T /|k − n|)
1, 0,
|t| < T 1 . T 1 < |t| < T /2 T
ω0 = 2π/T x(t)
T
k = n k=n 1 T
T
ej (k−n)ω t dt = 0
0
T, 0,
k=n , k= n
...
...
−2T
−T
−T −T 1 T 1 T 2 2
T x(t)
x(t) −T /2 ≤ t < T/2
t =0
T an an =
T
1 T
2T
T
x(t)e−jnω t dt, 0
0
k=0 a0 =
T
1 T
T 1
dt = −T 1
a0
T 1 an = T
x(t)e−jnω t dt. 0
T
1 ak = T
+∞
x(t) =
ak e
=
1 T
0
1 T
−jkω 0
e −T 1
jk (2π/T )t
ak e
k=−∞
x(t)e−jkω t dt = T
T 1
1 t dt = − e−jkω jkω0 T
2 ejkω ak = kω 0 T
+∞ jkω 0 t
k=−∞
ak =
k = 0
x(t)
x
2T 1 . T
0
T 1
− e−jkω 2 j
sin kω 0 T 1
x(t)e−jk (2π/T )t dt. T
ak =
0
T 1
0
t
T 1
, −T 1
.
ak 2sin(kω 0 T 1 ) sin(kω0 T 1 ) = , kω 0 T kπ
k = 0.
{ak } x(t) x(t) a0 x(t) k=0
x(t)
t
|x(t)|2 dt < ∞.
x(t)
T
ak xN (t)
N
xN (t) =
x(t)
ak ejkω t . 0
|k | ≤ N
k =−N
+N
eN (t)
xN (t) =
ak ejkω t . 0
k =−N N
eN (t) = x(t) − xN (t) = x(t) −
ak ejkω t . 0
k=−N
E N
0 +∞
e(t) = x(t) −
ak ejkω t . 0
k=−∞
|eN (t)|2 dt.
E N = T
|e(t)|2 dt = 0. T
x(t) ak =
1 T
x(t)e−jkω t dt.
+∞
0
T
ak ejkω t . 0
k =−∞
t
x(t)
N
x(t)
E N
t
x(t) E N
x(t)
N → ∞ x(t)
x(t) ak
|x(t)|dt < ∞. T
ak
x(t)
N
E N ∞
|ak | ≤
1 T
T
x(t)e−jkω
0
t
dt =
1 T
|x(t)|dt. T
x(t)
FS
x(t − t0 ) ←−→ e−jkω
F S
x(t) ←−→ ak
T
t
0 0
ak .
x(t) y(t) = −x(t)
x(t)
+∞
y(t) T
x(−t) =
ak
k =−∞
bk
k = −m
F S
x(t) ←−→ ak , FS
+∞
y(t) ←−→ bk . x(t)
y(t) = x(−t) =
y(t)
a−m ejm 2πt/T
m=−∞
T T
ck y(t) z(t) = Ax(t) + By(t)
x(t) x(t)
ak e−jk 2πt/T
x(−t) bk
y(t)
bk = a−k . FS
z(t) = Ax(t) + By(t) ←−→ ck = Aak + Bb k . F S
x(t) ←−→ ak , T F S
x(−t) ←−→ a−k . x(t)
T bk
y(t) = x(t − t0 )
|bk =
1 T
x(t − t0 )e−jkω t dt. 0
T
τ = t − t0
τ
x(t) ω0 = 2π/T
T
T x(αt)
α T /α
αω0 1 T
x(τ )e−jkω
0
(τ +t0 )
dτ = e−jkω
t
0 0
T
1 T
x(τ )e−jkω τ dτ 0
x(t)
T
x(t) −jkω 0 t0
e
ak = e
−jk (2π/T )t0
ak
ak ,
+∞
x(t)
x(αt) =
ak ejk (αω
0
)t
k=−∞ F S
x(t) ←−→ ak ,
x(αt)
x(t)
x(t)
y(t) T
ak
bk
x(t) x(t)
F S
x(t) ←−→ ak ,
T = 4 ak
FS
y(t) ←−→ bk .
|k| > 1
ak = 0
bk = e−jπk/2 a−k
x(t)y (t) T hk
1 4
+∞
F S
x(t)y(t) ←−→ hk =
4
|x(t)|2 dt = 1/2
al bk−l
l=−∞
x(t) x[n]
N
F S
x(t) ←−→ ak ,
x[n] = x[n + N ]. N ω0 =
F S
x∗ (t) ←−→ a∗−k .
2π/N ej (2π/N )n
N
∗
x(t)
x(t) = x (t)
N
a−k = a∗k . x(t)
φk [n] = ejkω
0
n
= ejk (2π/N )n ,
k = 0, ±1, ±2,...
a0 2π/N
|ak | = |a−k |. ak = a∗k
x(t) x(t)
N
2π
1 T
x[n] =
+∞
|x(t)|2 dt = T
|ak |2 ,
k =N
k=−∞
ak
ak φk [n] =
ak ejkω
k=N
n
=
ak ejk (2π/N )n .
k =N
k = 0, 1,...,N − 1
x(t)
T
0
ak 1 T
|ak |2 k x(t)
ak ejkω
T
0
t 2
dt =
1 T
|ak |2 dt = |ak |2 , T
x[n] N x[n] ak
N
n
x[n]
F S
x[n] ←−→ ak ,
x[0] =
ak
k =N
x[1] =
FS
x[n − n0 ] ←−→ e−jk (2π/N )n ak . 0
ak ej 2πk/N
k=N
F S
x[n] ←−→ ak , x[N − 1] =
ak ej 2πk (N −1)/N
k=N
F S
ejM (2π/N )n x[n] ←−→ ak−M .
N N
ak ak
ar =
1 N
F S
x[n]e−jr (2π/N )n
x[n] ←−→ ak ,
n=N FS
x[−n] ←−→ a−k . x[n] =
1 ak = N
n=N
ak ejkω
0
n
=
ak ejk (2π/N )n ,
k =N
−jkω 0 n
x[n]e
1 = N
F S
−jk (2π/N )n
x[n]e
x[n] ←−→ ak ,
.
F S
y[n] ←−→ bk ,
n=N
FS
x[n]y[n] ←−→ dk =
ak x[n]
al bk−l .
l=N
N
F S
x[n] ←−→ ak ,
FS
x[n] − x[n − 1] ←−→ (1 − e−jk (2π/N ) )ak .
F S
x[n] ←−→ ak . 1 N
|x[n]|2 =
n=N
|ak |2 .
n=N
F S
x[n] ←−→ ak ,
k
x[n] |ak |2 x[n]
F S
y[n] ←−→ bk ,
FS
Ax[n] + By[n] ←−→ Aak + Bb k .
N
x[n] x[n]
ejωt ejωn
N = 6
5 n=0 x[n]
=2
7 n n=2 (−1) x[n]
x(t)
=1
x[n] +∞
x(t) =
ak ejkω t . 0
k =−∞
h(t)
+∞
y(t) =
ak H ( jkω0 ) ejkω t . 0
k =−∞
y(t) x(t)
x(t) = est y (t) = H (s)est +∞
H (s) =
{ak } x(t)
{ak H ( jk ω0 )} y(t)
h(τ )e−sτ dτ,
−∞
h(t)
x[n]
x[n] = z
n
y[n] = H (z)z n
x[n] =
+∞
H (z) =
ak ejk (2π/N )n .
k=N
h[k]z −k ,
k=−∞
H (s)
h[n]
h[n] H (z) s
z y[n] =
Re{s}
=0 ejωt ω s = jω H ( jω)
est
s = jω
ak H e
jk (2π/N )
k =N
ejk (2π/N )n .
y[n] x[n]
ω
{ak H ejk (2π/N ) } y [t] +∞
H ( jω) =
h(t)e−jωt dt.
−∞
z
|z | = 1 ejωn ω H (ejω )
jω
z=e
z
n
z = ejω ω +3
x(t) =
+∞
H (ejω ) =
n=−∞
h[n]e−jωn .
k =−3
ak ejk 2πt ,
{ak } x[t]
a0 = 1 a1 = a−1 = a2 = a−2 = a3 = a−3 =
1 4 1 2 1 3
h(t) = e−t u(t). y(t) ∞
H ( jω) =
e−τ e−jωτ dτ
0
H ( jω) =
1 . 1 + jω
ω0 = 2π +3
y(t) =
bk ejk 2πt ,
k =−3
bk = ak H ( jk2π) b0 1 b1 = 4 1 b2 = 2 1 b3 = 3
1 1 + j2π 1 1 + j4π 1 1 + j6π
= 1
,
b−1
,
b−2
,
b−3
1 = 4 1 = 2 1 = 3
1 1 − j2π 1 1 − j4π 1 1 − j6π
, , . vs (t)
vc (t)
dvc (t) RC + vc (t) = vs (t). dt y(t) = d x(t)/dt x(t) y (t) = jωejωt
x(t) = ejωt
+
vr (t)
− +
− vs (t)
+
H ( jω) = jω.
vc (t)
−
ejωt ω
H ( jω)
vs (t) = ejωt
vc (t) = H ( jω)ejωt .
s(t)
1
d RC H ( jω)ejωt + H ( jω)ejωt = ejωt . dt
1−
1 e
t
τ
1 H ( jω) = . 1 + RCjω H ( jω ) ω=0
|H ( jω )| ≈ 1
ω
|H ( jω )| |ω | vc (t)
dvr (t) dvs (t) RC + vr (t) = RC . dt dt
|H (ω)| 1
vs (t) = ejωt vr (t) = G( jω)ejωt .
G( jω)
−1/RC 0
ω
1/RC
∠H (ω)
dG( jω)ejωt dejωt jωt RC + G( jω)e = RC . dt dt
π/2 π/4 1/RC
ω
−1/RC
G( jω) =
−π/4
jωRC . 1 + ωRC G( jω )
−π/2 |H (ω)| 1 h(t) =
1 −t/RC e u(t), RC
−1/RC 0
1/RC
ω
∠H (ω)
π/2
s(t) = 1 − e−t/RC u(t),
π/4 −1/RC τ =
1/RC
RC
ω
−π/4
h(t)
−π/2 1 τ
s(t) = e−t/RC u(t),
1 τe
τ
t
τ = RC
s(t)
1 1 e
t
τ = RC
y[n] − ay[n − 1] = x[n] x[n] = ejωn
y[n] = H (ejω )ejωn
H (ejω )
H (ejω )ejωn − aH (ejω )ejω (n−1) = ejωn ,
[1 − ae−jω ]H (ejω )ejωn = ejωn ,
H (ejω ) =
1 1 − ae−jω a
ω
ω=0 ω=π
a
ω=π a<1 a > −1
|a| |a|
M
y[n] =
bk x[n − k].
k =−N
1)
y[n] x[n]
(N +M + x[n − M ] bk
x[n + N ]
y[n] n0
n x[n]
n0
