❙✐♥❛✐s ✫
❙✐st❡♠❛s
Pr♦❢✳ ❊❞✉❛r❞♦ ●✳ ❇❡rt♦❣♥❛
❯❚❋P❘ ✴ ❉❆❊▲◆
❙✐♥❛✐s ❡ ❙✐st❡♠❛s
❈♦♥t❡ú❞♦s ❞❛ ❉✐s❝✐♣❧✐♥❛✿
• ❘❡✈✐sã♦ ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
• ❈♦♥❝❡✐t♦s ❡ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ❙✐♥❛✐s ❡ ❙✐st❡♠❛s
• ❈♦♥✈♦❧✉çã♦ ❈♦♥tí♥✉❛ ❡ ❉✐s❝r❡t❛
• ❙ér✐❡ ❊①♣♦♥❡♥❝✐❛❧ ❈♦♠♣❧❡①❛ ❡ ❚r✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r
• ❊s♣❡❝tr♦ ❞❡ ▼❛❣♥✐t✉❞❡ ❡ ❞❡ ❋❛s❡
• ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛
• ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛
• ❋✐❧tr♦s ❙❡❧❡t✐✈♦s ❞❡ ❋r❡q✉ê♥❝✐❛s
❇✐❜❧✐♦❣r❛✜❛ ❇ás✐❝❛✿
• ❙✐♥❛✐s ❡ ❙✐st❡♠❛s✳ ❍❆❨❑■◆✱ ❙✳❀ ❱❆◆ ❱❊❊◆✱ ❇✳ ✶➟❊❞✳✱ ❇♦✲
♦❦♠❛♥✱ ✷✵✵✶✳
• ❙✐♥❛✐s ❡ ❙✐st❡♠❛s✳ ❍❲❊■✱ P✳❍❙❯✳ ✶➟❊❞✳✱ ❇♦♦❦♠❛♥✱ ✷✵✵✹✳
• ❙✐♥❛✐s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✳ ▲❆❚❍■✱ ❇✳P✳ ✷➟❊❞✳ ❇♦♦❦♠❛♥✱
✷✵✵✼
• ❆♣♦st✐❧❛ ❡♠ ❤tt♣✿✴✴♣❡ss♦❛❧✳✉t❢♣r✳❡❞✉✳❜r✴❡❜❡rt♦♥❤❛✳
✶
❆♣❧✐❝❛çõ❡s ❞❛ ❚❡♦r✐❛ ❞❡ ❙✐♥❛✐s ❡ ❙✐st❡♠❛s
❆❧❣✉♠❛s ár❡❛s ❞❡ ❛♣❧✐❝❛çã♦✿
✲
✲
✲
✲
✲
❙✐st❡♠❛s ❞❡ ❈♦♠✉♥✐❝❛çã♦ ✭❚✐♣♦s ❞❡ ▼♦❞✉❧❛çã♦✮
❙✐st❡♠❛s ❞❡ ❈♦♥tr♦❧❡ ✭▼♦❞❡❧❛❣❡♠ ❡ ❙í♥t❡s❡✮
Pr♦❝❡ss❛♠❡♥t♦ ❞❡ ❙✐♥❛✐s ❇✐♦❧ó❣✐❝♦s❀
■♥str✉♠❡♥t❛çã♦ ✭▼♦❞❡❧❛❣❡♠ ❡ ❙í♥t❡s❡✮
Pr♦❝❡ss❛♠❡♥t♦ ❞❡ ❙✐♥❛✐s ✭❆✉❞✐♦✱ ❱✐❞❡♦✱ ❡t❝✮
❋♦r♠❛s ❞❡ ❆♣❧✐❝❛çã♦ ❞❛ ❚❡♦r✐❛
✶✳
❙í♥t❡s❡
✯ Pr♦❥❡t♦s ❞❡ ❙✐st❡♠❛s ❞❡ Pr♦❝❡ss❛♠❡♥t♦ ❞❡ ❙✐♥❛✐s
✲
✲
✲
✲
✲
❋✐❧tr❛❣❡♠ ❞❡ r✉í❞♦ ❡ ❡q✉❛❧✐③❛çã♦ ❞❡ s✐♥❛✐s❀
❈♦♠♣r❡ssã♦ ❞❡ s✐♥❛✐s❀
❈❧❛ss✐✜❝❛çã♦ ❞❡ ♣❛❞rõ❡s❀
❈♦♥tr♦❧❡ ❛❞❛♣t❛t✐✈♦❀
❈❛♥❝❡❧❛♠❡♥t♦ ❞❡ ❡❝♦ ❡♠ tr❛♥s♠✐ssã♦ ❞✐❣✐t❛❧✳
✷✳
❆♥á❧✐s❡
✯ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ❙✐st❡♠❛s
✲ ▼♦❞❡❧❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡❀
✲ ▼♦❞❡❧❛♠❡♥t♦ ❞❡ s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s✱ ❡①❡♠♣❧♦✿
❡st✉❞❛r ❝♦♠♦ ♦ s✐st❡♠❛ ❛✉❞✐t✐✈♦ r❡s♣♦♥❞❡ ❛♦s ✈ár✐♦s
t✐♣♦s ❞❡ ❡①❝✐t❛çõ❡s✳
✷
❘❡✈✐sã♦ ❞❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
◆ú♠❡r♦ ■♠❛❣✐♥ár✐♦ ♦✉ ❖♣❡r❛❞♦r ■♠❛❣✐♥ár✐♦
❖ ♥ú♠❡r♦ ♦✉ ♦♣❡r❛❞♦r ✐♠❛❣✐♥ár✐♦✱ ❞❡s✐❣♥❛❞♦ ♣♦r j ♦✉ i ✭❛q✉✐
❛❞♦t❛r❡♠♦s j ♣♦✐s i ❥á é ✉s❛❞♦ ♣❛r❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✮✱ é ❞❡✜♥✐❞♦
❝♦♠♦✿
P♦rt❛♥t♦✿
j=
√
j 2 = −1
−1✱ j 3 = j.j 2 = −j ✱
❡ ❛✐♥❞❛✿
j 4 = j 2 .j 2 = 1
◆ú♠❡r♦ ❈♦♠♣❧❡①♦
❖ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ é ❛q✉❡❧❡ q✉❡ s❡ ❝♦♥st✐t✉✐ ❞❡ ✉♠❛ ❝♦♠♣♦✲
♥❡♥t❡ r❡❛❧ ❡ ♦✉tr❛ ✐♠❛❣✐♥ár✐❛✱ ❞♦ t✐♣♦✿ A = a + jb✱ ♣♦❞❡♥❞♦ s❡r
r❡♣r❡s❡♥t❛❞♦ ❞❡ ✸ ❢♦r♠❛s✿
❋♦r♠❛ ❈❛rt❡s✐❛♥❛✿
❖♥❞❡✿
j=
√
A = a + jb
−1✱ a = Re(A)✱ b = Im(A)
A = C∠θ
p
C = |A| = a2 + b2 ✱
❡
a2 + b2 ✱
❡
❋♦r♠❛ P♦❧❛r✿
❖♥❞❡✿
❋♦r♠❛ ❊①♣♦♥❡♥❝✐❛❧✿
❖♥❞❡✿
C = |A| =
C =▼ó❞✉❧♦
❖❇❙✿
p
❞❡ ❆✱ ❡
θ = arctan( ab )
A = C.ejθ = C.cos(θ) + j.C.sen(θ)
θ = arctan( ab )
θ =❆r❣✉♠❡♥t♦
e = 2, 71828182...é
❞❡ ❆
❛ ❜❛s❡ ♥❛t✉r❛❧ ♦✉ ♥❡♣❡r✐❛♥❛✳
❈♦♠♣❧❡①♦ ❈♦♥❥✉❣❛❞♦
❖ ♥ú♠❡r♦
A∗ = a − j.b✱
é ❞❡♥♦♠✐♥❛❞♦ ❞❡ ❝♦♥❥✉❣❛❞♦ ❞❡ A✳
✸
P❧❛♥♦ ❈♦♠♣❧❡①♦✿
❉❛ ❢♦r♠❛ ❡①♣❧♦♥❡♥❝✐❛❧✿
s❡ ❛
A = C.ejθ = C.cos(θ) + j.C.sen(θ)✱
■❞❡♥t✐❞❛❞❡ ♦✉ ❋ór♠✉❧❛ ❞❡ ❊✉❧❡r
❝❤❡❣❛✲
✿
ejθ = cos(θ) + j.sen(θ)
❈♦♠✿
cos(θ) = 12 (ejθ + e−jθ )✱
❡✿
sen(θ) =
1
(ejθ
2j
− e−jθ )
❖♣❡r❛çõ❡s ❇ás✐❝❛s ❝♦♠ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❈♦♥s✐❞❡r❡ ✷ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❆ ❡ ❇✱ r❡♣r❡s❡♥t❛❞♦s ♣♦r✿
A = a + j.b = C.ejθ1 ✱
❙♦♠❛ ❆❧❣é❜r✐❝❛
✿
❉✐✈✐sã♦
✿
A
B
=
✿
B = c + j.d = D.ejθ2
A + B = (a + c) + j(b + d)
❙✉❜tr❛çã♦ ❆❧❣é❜r✐❝❛
▼✉❧t✐♣❧✐❝❛çã♦
❡
✿
A − B = (a − c) + j(b − d)
A.B = (ac − bd) + j.(bc + ad) = C.D.ej(θ1 +θ2 )
A.B ∗
B.B ∗
=
(a+j.b).(c−j.d)
c2 +d2
=
(ac+bd)+j(bc−ad)
c2 +d2
=
C j(θ1 −θ2 )
.e
D
✹
Pr♦♣r✐❡❞❛❞❡s ❞♦ ❈♦♠♣❧❡①♦ ❈♦♥❥✉❣❛❞♦
❈♦♥s✐❞❡r❡✿ ❆ ❡ ❆✯✱ ❡♥tã♦✿
• A + A∗ = 2.Re(A)
• A − A∗ = 2.Im(A)
• A.A∗ = (a + j.b).(a − j.b) = a2 − j 2.b2 = a2 + b2 ⇒ ◆➸ r❡❛❧ ✦✦✦
❊①❡r❝í❝✐♦s
✶✳ ❉❛❞♦s✿
x1 = 2 + j3✱ x2 = 4 − j3✱
❡
x3 = 5∠0, 927
❊❢❡t✉❛r ❛s ♦♣❡r❛çõ❡s ✐♥❞✐❝❛❞❛s✿
✭❛✮ x1 + x2
✭❜✮ x2 + x3
✭❝✮ x2 − x3
✭❞✮ x2.x3
✭❡✮ x2/x3
✭❢✮ Re(x3)
✷✳ ❊❢❡t✉❛r ❛s ♦♣❡r❛çõ❡s ✐♥❞✐❝❛❞❛s✿
✭❛✮ (6 + j7).(1 + j)
✭❜✮ (5 + j4).(1 − j) + (2 + j).j
✭❝✮
✭❣✮
✭❤✮
Im(x3 )
|x1 |
(1 + j2)2 − (3 + j4)
✸✳ ❉❡t❡r♠✐♥❡ ♦ ♠ó❞✉❧♦ ❡ ♦ ❛r❣✉♠❡♥t♦✱ ❝♦❧♦❝❛r ♥❛ ❢♦r♠❛ ♣♦❧❛r
❡ r❡♣r❡s❡♥t❛r ❣r❛✜❝❛♠❡♥t❡ ♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s✿
✭❛✮ 4
√
✭❜✮ 1 + j. 3
✭❝✮ j.3
√
√
✭❞✮ − 2 + j. 2
✭❡✮ −5
✭❢✮ −j.2
✺
✹✳ ❈❛❧❝✉❧❡ ♦ ♠ó❞✉❧♦ ❡ ♦ ❛r❣✉♠❡♥t♦ ❞♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s ❝♦♠✲
♣❧❡①♦s✿
✭❛✮ 3 − j.4
√
√
✭❜✮ 2 + j. 2
✭❝✮ 12 + j.5
✭❞✮ cos(θ) + jsen(θ)
✭❡✮ −3 + j.4
✭❢✮ 7 − j.2
✺✳ ❈♦❧♦❝❛r ♥❛ ❢♦r♠❛ ❝❛rt❡s✐❛♥❛ ♦✉ r❡t❛♥❣✉❧❛r ♦s s❡❣✉✐♥t❡s ♥ú✲
♠❡r♦s ❝♦♠♣❧❡①♦s✿
✭❛✮ 3.ejπ
✭❜✮ 4.ej11π/6
✭❝✮ 2.ejπ/4
✭❞✮ 5.ej3π/2
✭❡✮ 7.ejπ/2
✭❢✮ 4.ejπ/3
✻✳ ❯s❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ❝❛❧✉❧❛r ♦s ♠ó❞✉❧♦s ❞♦s s❡❣✉✐♥t❡s ♥ú♠❡✲
r♦s ❝♦♠♣❧❡①♦s✿
✭❛✮ (1 − j).(2 + j.2)
√ 6
✭❜✮ (1 + j. 3)
✭❝✮
3+j.3
1+j.2
✼✳ ❊s❝r❡✈❡r ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✿
1
1−j
−
1
j
♥❛ ❢♦r♠❛ ♣♦❧❛r✳
✽✳ ❘❡♣r❡s❡♥t❛r ♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥♦ ♣❧❛♥♦ ❝♦♠✲
♣❧❡①♦✱ ✐♥❞✐❝❛♥❞♦ ♦ s❡✉s ♠ó❞✉❧♦s ❡ ❛r❣✉♠❡♥t♦s✿
✭❛✮ 2 + j.5
✭❜✮ −3 + j.2
✭❝✮ −2 − j.3
✭❞✮ 1 − j.4
✻
❊st✉❞♦s ❞♦s ❙✐♥❛✐s
❉❡✜♥✐çõ❡s✿
❈♦♥❥✉♥t♦ ❞❡ ❉❛❞♦s✱ ♦✉ ■♥❢♦r♠❛çõ❡s✳
P♦❞❡♠ s❡r ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ♦✉ ♦✉tr❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳
❊①✳✿ ❙✐♥❛❧ ❞❡ ❉❡♥s✐❞❛❞❡ ❞❡ ❈❛r❣❛ ❊❧étr✐❝❛ ✲ ❋✉♥çã♦ ❞♦ ❊s♣❛ç♦✳
✲
✲
❈❧❛ss✐✜❝❛çã♦ ❞♦s ❙✐♥❛✐s
✶✳ ❙✐♥❛✐s ❈♦♥tí♥✉♦s ❡ ❉✐s❝r❡t♦s ♥♦ ❚❡♠♣♦
•
❙✐♥❛✐s ❞❡ ❚❡♠♣♦ ❈♦♥tí♥✉♦
❆ ❢✉♥çã♦ q✉❡ ♦s ❞❡s❝r❡✈❡ é ❞❡✜♥✐❞❛
♣❛r❛ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ s✉❛s
✈❛r✐á✈❡✐s✳
❊①❡♠♣❧♦✿
•
❙✐♥❛❧ ♣r♦✈❡♥✐❡♥t❡ ❞❡ ✉♠ s❡♥s♦r ❞❡ t❡♠♣❡r❛t✉r❛✳
❙✐♥❛✐s ❞❡ ❚❡♠♣♦ ❉✐s❝r❡t♦
❆ ❢✉♥çã♦ q✉❡ ♦s ❞❡s❝r❡✈❡ é ❞❡✜♥✐❞❛
❛♣❡♥❛s ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ s✉❛s
✈❛r✐á✈❡✐s✳
❂❃ ❙❡q✉ê♥❝✐❛ ❞❡ ❱❛❧♦r❡s✿
x = {..., x[−1], x[0], x[1], x[2], ...x[n]}
❊①❡♠♣❧♦✿
❖ ♠❡s♠♦ s✐♥❛❧ ❞❡ t❡♠♣❡r❛t✉r❛ ❛♠♦str❛❞♦✳
✼
✷✳ ❙✐♥❛✐s P❛r❡s ❡ ❮♠♣❛r❡s
• ❙✐♥❛✐s P❛r❡s✿ ❙✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ✈❡rt✐❝❛❧✱ ❡ à
♦r✐❣❡♠✳ ❉❡✈❡♠ s❛✜s❢❛③❡r ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
❊①✳ ❙✐♥❛❧ P❛r✿
❉♦♠í♥✐♦ ❈♦♥tí♥✉♦✿
x(t) = x(−t)
❉♦♠í♥✐♦ ❉✐s❝r❡t♦✿
x[n] = x[−n]
• ❙✐♥❛✐s ❮♠♣❛r❡s✯✿ ❆♥t✐ss✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠✳ ❉❡✲
✈❡♠ s❛t✐s❢❛③❡r ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
❉♦♠í♥✐♦ ❈♦♥tí♥✉♦✿
❊①✳ ❙✐♥❛❧ ❮♠♣❛r✿
x(−t) = −x(t)
❉♦♠í♥✐♦ ❉✐s❝r❡t♦✿
x[−n] = −x[n]
❖❜s✿ ❖ s✐♥❛❧ í♠♣❛r ❞❡✈❡ ♦❜❡❞❡❝❡r✿
x(0) = 0
♦✉
x[0] = 0
P❛rt❡s P❛r ❡ ❮♠♣❛r ❞❡ ✉♠ ❙✐♥❛❧ ❘❡❛❧
✲❚♦❞♦ s✐♥❛❧ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ s✉❛ ♣❛rt❡ P❛r
❝♦♠ s✉❛ ♣❛rt❡ ❮♠♣❛r✿
1 {x(t) + x(−t)}
P ar{x(t)} = 2
Impar{x(t)} = 1
2 {x(t) − x(−t)}, x(t)⇔ x[n]
❆s ❞❡✜♥✐çõ❡s ❛❝✐♠❛ só ✈❛❧❡♠ s❡ ♦ s✐♥❛❧ ♥ã♦ ❛♣r❡s❡♥t❛r
✈❛❧♦r ❝♦♠♣❧❡①♦✱ ❝❛s♦ ❝♦♥trár✐♦ ❞❡✈❡✲s❡ ❢❛❧❛r ❡♠ s✐♠❡tr✐❛ ❝♦♥❥✉✲
❣❛❞❛✿ ①✭✲t✮❂①✯✭t✮✱ ❝♦♠✿ ①✭t✮❂❛✭t✮✰❥❜✭t✮ ❡ ①✯✭t✮❂❛✭t✮✲❥❜✭t✮✳
❖❜s✿
✽
❊①❡♠♣❧♦✿
P❛rt❡ P❛r ❞❡ x(t) ⇒ x (t)✿
e
P❛rt❡ ❮♠♣❛r ❞❡ x(t) ⇒ x (t)✿
o
❙❡♥❞♦ q✉❡✿ x(t) ❂ x (t) ✰ x (t)✿
e
o
✾
✸✳ ❙✐♥❛✐s P❡r✐ó❞✐❝♦s ❡ ❆♣❡r✐ó❞✐❝♦s
•
❙✐♥❛✐s P❡r✐ó❞✐❝♦s
✕
❉♦♠í♥✐♦ ❈♦♥tí♥✉♦✿
x(t) = x(t + mT )✱ ∀t✱
P❡rí♦❞♦ ❋✉♥❞❛♠❡♥t❛❧
❡ ❝♦♠✿
m∈Z
(T0)✿
✲▼❡♥♦r ❱❛❧♦r ❞❡ ❚ ♣✴ ♦ q✉❛❧✿ x(t) = x(t+T ),∀t
✕
❉♦♠í♥✐♦ ❉✐s❝r❡t♦✿
x[n] = x[n + k.N ]✱
❝♦♠✿
P❡rí♦❞♦ ❋✉♥❞❛♠❡♥t❛❧
(N0)✿
k∈Z
❡
N ∈ Z+
✲▼❡♥♦r ❱❛❧♦r ❞❡ ◆ ♣✴ ♦ q✉❛❧✿ x[n] = x[n+N ],∀n
∗
•
❊①✿ ❙✐♥❛❧ ❝♦♠
N0 = 12✿
❙✐♥❛✐s ❆♣❡r✐ó❞✐❝♦s
✲ ❆q✉❡❧❡s q✉❡ ♥ã♦ sã♦ ♣❡ríó❞✐❝♦s s❡rã♦ ❆♣❡r✐ó❞✐❝♦s ✦
✶✵
✹✳ ❙✐♥❛✐s ❉❡t❡r♠✐♥íst✐❝♦s ❡ ❆❧❡❛tór✐♦s
•
P♦❞❡♠ s❡r ♠♦❞❡❧❛❞♦s ❝♦♠♦ ❢✉♥✲
çõ❡s ❞❡ t❡♠♣♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡s♣❡❝✐✜❝❛❞♦s✳ ◆ã♦ ❤á ✐♥❝❡r✲
t❡③❛ q✉❛♥t♦ ❛♦ s❡✉ ✈❛❧♦r ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✳
❙✐♥❛✐s ❉❡t❡r♠✐♥íst✐❝♦s✿
❊①❡♠♣❧♦✿
•
❖♥❞❛ q✉❛❞r❛❞❛✳
◆ã♦ ♣♦❞❡♠ s❡r ♠♦❞❡❧❛❞♦s ❝♦♠♦ ❢✉♥✲
çõ❡s ❞❡ t❡♠♣♦✱ ♣♦✐s ❤á ✐♥❝❡rt❡③❛s q✉❛♥t♦ ❛♦ s❡✉ ✈❛❧♦r ❡♠
q✉❛❧q✉❡r ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✳
❙✐♥❛✐s ❆❧❡❛tór✐♦s✿
❂❃ ❊st✉❞❛❞♦ ❛tr❛✈és ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s✳
❊①❡♠♣❧♦✿
❙✐♥❛❧ ❞❡ ❊❧❡tr♦❡♥❝❡❢❛❧♦❣r❛♠❛ ✲ ❊❊●
✺✳ ❙✐♥❛✐s ❞❡ ❊♥❡r❣✐❛ ❡ ❞❡ P♦tê♥❝✐❛
❈♦♥s✐❞❡r❡ ✉♠ r❡s✐st♦r R✱ ❛❧✐♠❡♥t❛❞♦ ♣♦r ✉♠❛ t❡♥sã♦ v(t)✱ ♣r♦✲
❞✉③✐♥❞♦ ✉♠❛ ❝♦rr❡♥t❡ i(t)✳ ❆ ♣♦tê♥❝✐❛ ✐♥st❛♥tâ♥❡❛ ❞✐ss✐♣❛❞❛ ♣♦r
❡st❡ r❡s✐st♦r s❡rá✿
v 2 (t)
p(t) =
= R.i2 (t)
R
❈♦♥s✐❞❡r❡ ❛❣♦r❛ q✉❡ R = 1ohm✱ ❡ x(t)r❡♣r❡s❡♥t❛♥❞♦ t❡♥sã♦ ♦✉
❝♦rr❡♥t❡✱ ♥❡st❡ ❝❛s♦ ❛ ♣♦tê♥❝✐❛ ✐♥st❛♥tâ♥❡❛✿
p(t) = x2 (t)
✶✶
❇❛s❡❛❞♦ ♥♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❊♥❡r❣✐❛ ❚♦t❛❧
❡ ❛ P♦tê♥❝✐❛ ▼é❞✐❛ ❞❡ ✉♠ s✐♥❛❧ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ ❝♦♠♦✿
❊♥❡r❣✐❛ ❚♦t❛❧ ❞❡ ✉♠ ❙✐♥❛❧ ❞❡ ❚❡♠♣♦ ❈♦♥tí♥✉♦✿
E = limT →∞
Z
T /2
2
x (t)dt =
−T /2
Z
∞
x2 (t)dt
−∞
P♦tê♥❝✐❛ ▼é❞✐❛ ❞❡ ✉♠ ❙✐♥❛❧ ❞❡ ❚❡♠♣♦ ❈♦♥tí♥✉♦✿
1
P = limT →∞
T
Z
T /2
x2 (t)dt
−T /2
❙❡ ♦ s✐♥❛❧ é P❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❚✱ ✜❝❛rá✿
1
P =
T
❊ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ P s❡rá ♦
♠❡❛♥✲sq✉❛r❡✮ ❞♦ s✐♥❛❧✳
Z
T /2
x2 (t)dt
−T /2
❱❛❧♦r ▼é❞✐♦ ◗✉❛❞rát✐❝♦ ✭r♠s❂r♦♦t
❊♥❡r❣✐❛ ❚♦t❛❧ ❡ P♦tê♥❝✐❛ ▼é❞✐❛ ❞❡ ✉♠ ❙✐♥❛❧ ♥♦
❚❡♠♣♦ ❉✐s❝r❡t♦✿
E=
∞
X
n=−∞
2
x [n]
N
1 X 2
P = limN →∞ {
x [n]}
2N
n=−N
❙❡ ♦ s✐♥❛❧ é P❡r✐ó❞✐❝♦ ❝♦♠ ♣❡rí♦❞♦ ❢✉♥❞❛♠❡♥t❛❧ ◆✱ ✜❝❛rá✿
N −1
1 X 2
x [n]
P =
N n=0
✶✷
✲ ❯♠ s✐♥❛❧ é ❞✐t♦ s❡r ❙✐♥❛❧ ❞❡ ❊♥❡r❣✐❛ s❡✿ 0 < E < ∞
✲ ❯♠ s✐♥❛❧ é ❞✐t♦ s❡r ❙✐♥❛❧ ❞❡ P♦tê♥❝✐❛ s❡✿ 0 < P < ∞
✲ ❊st❛s ❝❧❛ss✐✜❝❛çõ❡s ♠✉t✉❛♠❡♥t❡ ❡①❝❧✉s✐✈❛s✳
✲ ❙✐♥❛✐s ❛❧❡❛tór✐♦s ❡ ♣❡r✐ó❞✐❝♦s sã♦ ♥♦r♠❛❧♠❡♥t❡ ❙✐♥❛✐s ❞❡
P♦tê♥❝✐❛✳
✲ ❙✐♥❛✐s ❞❡t❡r♠✐♥íst✐❝♦s ❡ ♥ã♦✲♣❡r✐ó❞✐❝♦s sã♦ ❙✐♥❛✐s ❞❡ ❊♥❡r❣✐❛✳
✻✳ ❙✐♠❡tr✐❛ ❈♦♥❥✉❣❛❞❛ ❡♠ ❙✐♥❛✐s ❈♦♠♣❧❡①♦s
❉❡✜♥❡✲s❡ ♦ ❝♦♥❥✉❣❛❞♦ ❝♦♠♣❧❡①♦ ❝♦♠♦ s❡♥❞♦✿ x(−t) = x∗(t)
❖♥❞❡✿ x(t) = a(t) + jb(t)✱ ❡ x∗(t) = a(t) − jb(t)
❊①❡r❝í❝✐♦s✿
✶✳ ◗✉❛❧ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ ♣✉❧s♦
r❡t❛♥❣✉❧❛r ♠♦str❛❞♦ ❛♦ ❧❛❞♦ ❄
✷✳ ◗✉❛❧ ❛ ♣♦tê♥❝✐❛ ♠é❞✐❛ ❞❛ ♦♥❞❛
q✉❛❞r❛❞❛ ♠♦str❛❞❛ ❛♦ ❧❛❞♦ ❄
✸✳ ◗✉❛❧ ❛ ♣♦tê♥❝✐❛ ♠é❞✐❛ ❞❛ ♦♥❞❛
tr✐❛♥❣✉❧❛r ♠♦str❛❞❛ ❛♦ ❧❛❞♦ ❄
✹✳ ◗✉❛❧ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐♥❛❧ ❞❡
t❡♠♣♦ ❞✐s❝r❡t♦ ♠♦str❛❞♦ ❛♦ ❧❛❞♦ ❄
✺✳ ◗✉❛❧ ❛ ♣♦tê♥❝✐❛ ♠é❞✐❛ ❞♦ s✐♥❛❧
♣❡r✐ó❞✐❝♦ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ ♠♦s✲
tr❛❞♦ ❛❜❛✐①♦ ❄
✶✸
❖♣❡r❛çõ❡s ❇ás✐❝❛s ❡♠ ❙✐♥❛✐s
✶✳ ❖♣❡r❛çõ❡s ♥❛s ❱❛r✐á✈❡✐s ■♥❞❡♣❡♥❞❡♥t❡s
•
❘❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ✈❡rt✐❝❛❧
❉♦♠í♥✐♦ ❈♦♥tí♥✉♦✿ x(t) ⇒ x(−t)
❉♦♠í♥✐♦ ❉✐s❝r❡t♦✿ x[n] ⇒ x[−n]
❊①❡♠♣❧♦✿
•
❈♦♠♣r❡ssã♦✴❊①♣❛♥sã♦ ✭▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛
❞❡ ❚❡♠♣♦✮
❈♦♥tí♥✉♦✿ x(t) ⇒ x(k.t)✱ ❝♦♠ k > 0✱ ❡ k ∈ ℜ
❉✐s❝r❡t♦✿ x[n] ⇒ x[k.n]✱ ❝♦♠ k > 0✱ ❡ k ∈ Z
✭P❡r❞❡✲s❡ ❛♠♦str❛s✮
❊①❡♠♣❧♦✿
✶✹
• ❆tr❛s♦ ✭❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❚❡♠♣♦✮
x(t) ⇒ x(t − t0 )✱ ❝♦♠ t0 > 0✱ ❡ t0 ∈ ℜ
x[n] ⇒ x[n − n0 ]✱ ❝♦♠ n0 > 0✱ ❡ n0 ∈ Z
❊①❡♠♣❧♦✿
• Pr❡❝❡❞ê♥❝✐❛ ❡♥tr❡ ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❚❡♠♣♦
❡ ▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛
❆ ♦♣❡r❛çã♦ ❞❡ ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❚❡♠♣♦✱ ❞❡✈❡rá ♣r❡❝❡❞❡r ❛
♦♣❡r❛çã♦ ❞❡ ▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛ ✭❈♦♠♣r❡ssã♦✴❊①♣❛♥sã♦✮✳
❙❡ ♣♦r ❡①❡♠♣❧♦✿ y(t) = x(at − b)
❉❡s❧♦❝❛♠❡♥t♦ ♥♦ t❡♠♣♦ é ❡❢❡t✉❛❞♦ ♣r✐♠❡✐r♦✿ z(t) = x(t − b)
❙ó ❞❡♣♦✐s✱ ❛ ▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❚❡♠♣♦✿
y(t) = z(at) = x(at − b)
❊①❡♠♣❧♦✿
y(t) = x(2t + 1)
✶✺
✷✳ ❖♣❡r❛çõ❡s ♥❛s ❱❛r✐á✈❡✐s ❉❡♣❡♥❞❡♥t❡s
•
▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❆♠♣❧✐t✉❞❡
x(t) ⇒ k.x(t)✱
x[n] ⇒ k.x[n]✱
❝♦♠
k > 0✱
❡
❝♦♠
k > 0✱
❡
k∈ℜ
k∈ℜ
❊①❡♠♣❧♦s✿
•
✕
❆♠♣❧✐✜❝❛❞♦r ❊❧❡trô♥✐❝♦ ❝♦♠ ❣❛♥❤♦
✕
❘❡s✐st♦r✱ ♦♥❞❡ k✱ r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❞♦ r❡s✐st♦r ❡ k.x(t)✱
❛ t❡♥sã♦ ♥♦s s❡✉s t❡r♠✐♥❛✐s✱ s❡♥❞♦ x(t)✱ ❛ ❝♦rr❡♥t❡ q✉❡
♦ ❛tr❛✈❡ss❛✳
k
❆❞✐çã♦
x1 (t)✱ x2 (t) ⇒ y(t) = x1 (t) + x2 (t)
x1 [n]✱ x2 [n] ⇒ y[n] = x1 [n] + x2 [n]
❊①❡♠♣❧♦✿
✕
•
❆♠♣❧✐✜❝❛❞♦r ❙♦♠❛❞♦r ❞❡ ❣❛♥❤♦ ✉♥✐tár✐♦
▼✉❧t✐♣❧✐❝❛çã♦
x1 (t)✱ x2 (t) ⇒ y(t) = x1 (t).x2 (t)
x1 [n]✱ x2 [n] ⇒ y[n] = x1 [n].x2 [n]
❊①❡♠♣❧♦✿
✕
▼♦❞✉❧❛❞♦r ❆▼✱ ♦♥❞❡✿ x1[n]✱ x2[n]✱ sã♦ ♦s s✐♥❛✐s ♠♦❞✉✲
❧❛♥t❡ ❞❡ á✉❞✐♦✱ ❡ ✉♠❛ ♣♦rt❛❞♦r❛ t✐♣♦ s❡♥♦✐❞❛❧✳
✶✻
•
❉✐❢❡r❡♥❝✐❛çã♦
d x(t)
x(t) ⇒ dt
❊①❡♠♣❧♦✿
❯♠ ■♥❞✉t♦r r❡❛❧✐③❛ ❛ ❞✐❢❡r❡♥❝✐❛çã♦ ❞❛ ❝♦rr❡♥t❡ q✉❡ ♦ ❛tr❛✲
✈❡ss❛✱ ♦ q✉❡ s❡ tr❛❞✉③ ♥❛ t❡♥sã♦ ❡♥tr❡ s❡✉s t❡r♠✐♥❛✐s✳
d i(t)
v(t) = L dt
❖♥❞❡ ▲ é ❛ ✐♥❞✉tâ♥❝✐❛ ❞♦ ■♥❞✉t♦r
•
■♥t❡❣r❛çã♦
Rt
x(t) ⇒ −∞ x(τ )dτ
❊①❡♠♣❧♦✿
❯♠ ❈❛♣❛❝✐t♦r r❡❛❧✐③❛ ❛ ✐♥t❡❣r❛çã♦ ❞❛ ❝♦rr❡♥t❡ q✉❡ ♦ ❛tr❛✲
✈❡ss❛✱ ♦ q✉❡ s❡ tr❛❞✉③ ♥❛ t❡♥sã♦ ❡♥tr❡ s❡✉s t❡r♠✐♥❛✐s✳
Rt
1
v(t) = C −∞ i(τ )dτ
❖♥❞❡ ❈ é ❛ ❈❛♣❛❝✐tâ♥❝✐❛ ❞♦ ❈❛♣❛❝✐t♦r
✶✼
■♥t❡❣r❛✐s ❡ ❉❡r✐✈❛❞❛s ■♠♣♦rt❛♥t❡s
•
■♥t❡❣r❛çã♦
R
R
k.dx = k.x
R
f (x).dx
R
R
R
[f (x) ± g(x)].dx = f (x).dx ± g(x).dx
R n
xn+1
✱ ❝✴ n 6= −1
x .dx = n+1
R bx
e .dx = 1b ebx
R
sen(ax).dx = − 1a cos(x)
R
cos(ax).dx = 1a sen(x)
R
R
u.dv = u.v − v.du
R
R
u.dv = u.v − v.du
•
k.f (x).dx = k.
❉✐❢❡r❡♥❝✐❛çã♦
d
(k)
dx
=0
d
[k.u(x)]
dx
d
[u(x)
dx
= k.u′ (x)
± v(x)] = v ′ (x) ± u′ (x)
d
[u(x).v(x)]
dx
= u(x).v ′ (x) + u′ (x).v(x)
d
(xn )
dx
= n.xn−1
d
(ex )
dx
= ex
d
sen(x)
dx
= cos(x)
d
cos(x)
dx
= −sen(x)
✶✽
❙✐♥❛✐s ■♠♣♦rt❛♥t❡s
✶✳
❙✐♥❛❧ ❉❡❣r❛✉ ❯♥✐tár✐♦
❉❡❣r❛✉ ❯♥✐tár✐♦ ❈♦♥tí♥✉♦✿
u(t) =
❖❜s✿
0 , se : t < 0
1 , se : t > 0
u(t)
♥ã♦ é ❞❡✜♥✐❞♦ ♣❛r❛ t❂✵
❉❡❣r❛✉ ❯♥✐tár✐♦ ❉✐s❝r❡t♦✿
u[n] =
u(t)
u[n]
0 , se : n < 0
1 , se : n ≥ 0
❖♣❡r❛çõ❡s ❝♦♠ ♦ ❉❡❣r❛✉ ❯♥✐tár✐♦ ❈♦♥tí♥✉♦
✶✾
◆♦ ▼❛t❧❛❜ ♦ ❞❡❣r❛✉ ✉♥✐tár✐♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ✉s❛♥❞♦✲s❡ ♦ ♦♣❡✲
r❛❞♦r r❡❧❛❝✐♦♥❛❧ ✬❃❂✬✱ q✉❡ r❡t♦r♥❛ ✶ s❡ ✈❡r❞❛❞❡✐r♦✿
≫
≫
≫
✉❂✐♥❧✐♥❡✭✬✭t❃❂✵✮✬✱✬t✬✮
t❂✭✲✷✿✵✳✵✶✿✷✮
♣❧♦t✭t✱✉✭t✮✮
✷✳
❙✐♥❛❧ ❘❛♠♣❛ ❯♥✐tár✐❛
r(t)
❘❛♠♣❛ ❯♥✐tár✐❛ ❈♦♥tí♥✉❛✿
r(t) =
0 , se : t < 0
t , se : t > 0
❘❛♠♣❛ ❯♥✐tár✐❛ ❉✐s❝r❡t❛✿
r[n] =
•
r[n]
0 , se : n < 0
n , se : n ≥ 0
❘❡❧❛çã♦ ❡♥tr❡
u(t)
❡
r(t) =
r(t)✿
Z t
−∞
u(τ ).dτ
✷✵
✸✳ ❙✐♥❛❧ ■♠♣✉❧s♦ ❯♥✐tár✐♦
δ(t)
■♠♣✉❧s♦ ❯♥✐tár✐♦ ❈♦♥tí♥✉♦✿
Z +∞
−∞
δ(t).φ(t).dt = φ(0)⇒
✲ ❘❡❧❛çã♦ ❡♥tr❡
u(t)
❡
Z +∞
−∞
δ(t)✿
u(t) =
Z t
−∞
δ(τ ).dτ
■♠♣✉❧s♦ ❯♥✐tár✐♦ ❉✐s❝r❡t♦✿
δ[n] =
(
δ(t).dt = 1
δ[n]
1, n = 0
0, n =
6 0
✷✶
✲ ■♥t❡r♣r❡t❛çã♦ ❞♦ ■♠♣✉❧s♦ ❝♦♠♦ ✉♠ ❧✐♠✐t❡✿
(t)
1 , ✵❁t❁∆
=
δ∆(t) = du∆
dt
∆
⇒ δ(t) = lim△→0(δ∆(t))
♦✉ ❛✐♥❞❛✿
1 u(t) − 1 u(t−△)
δ∆(t) = △
△
⇒δ(t) = lim△→0(δ∆(t))
▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❋✉♥çã♦ ♣♦r ■♠♣✉❧s♦
Pr♦♣r✐❡❞❛❞❡ ❞❛ ❆♠♦str❛❣❡♠
✶✳ y(t) = x(t)δ(t) ⇒ y(t) = x(0)δ(t)
✷✳ y(t) = x(t)δ(t − t0 ) ⇒ y(t) = x(t0 )δ(t − t0 )
✸✳
Z +∞
−∞
x(t).δ(t − t0).dt =
Z +∞
−∞
x(t0).δ(t − t0).dt =x(t0)
✷✷
✹✳ ❙✐♥❛❧ ❊①♣♦♥❡♥❝✐❛❧
❙✐♥❛❧ ❊①♣♦♥❡♥❝✐❛❧ ❈♦♥tí♥✉♦
❈❛s♦ ●❡r❛❧✿
x(t) = K.est
s = σ + j.ω0
⇒ x(t) = K.e(σ+j.ω0).t = K.eσt.ejωt
❖♥❞❡✿
• s = σ + j.ω ✱ ❡ σ > 0
⇒ ❙❡♥ó✐❞❡ ❈r❡s❝❡♥t❡
• s = σ + j.ω ✱ ❡ σ < 0
⇒ ❙❡♥ó✐❞❡ ❉❡❝r❡s❝✳
✳
• s = j.ω ✱ ❡ σ = 0
⇒ ❙❡♥ó✐❞❡
✷✸
❈❛s♦s ❊s♣❡❝í✜❝♦s✿
s = σ✱
♦✉ s❡❥❛✿
s∈R
• a > 0✱ C > 0
⇒ ❊①♣✳ ❈r❡s❝❡♥t❡
• a < 0✱ C > 0
⇒ ❊①♣✳ ❉❡❝r❡s❝✳
• a = 0✱ C 6= 0
⇒ ❈♦♥st❛♥t❡
•
❋♦r♠❛ ❚r✐❣♦♥♦♠étr✐❝❛ ❞♦ ❙✐♥❛❧ ❊①♣♦♥❡♥❝✐❛❧
x(t) = K.e(σ+j.ω).t ⇒ x(t) = K.eσt.ejω.t
x(t) = K.eσt(cosωt + j.senωt)
✷✹
❊①❡r❝í❝✐♦s✿
✶✮ ❋❛ç❛ ✉♠ ❡s❜♦ç♦ ❞♦s s❡❣✉✐♥t❡s s✐♥❛✐s✿
❛✮ ①1 ✭t✮❂✲✷✳✉✭t✮
❜✮ ①2 ✭t✮❂✹✳✉✭t✲✷✮
❝✮ ①3 ✭t✮❂✷✉✭✲t✲✸✮
❞✮ ①4 ✭t✮❂✺✉✭t✰✷✮
❡✮ ①5 ✭t✮❂✲✸✉✭✲t✮
❢✮ ①6 ✭t✮❂✲✻✉✭✲t✰✷✮
❣✮ ①7 ✭t✮❂✉✭t✮✲✉✭t✲✷✮
❤✮ ①8 ✭t✮❂✸❬✉✭t✰✷✮✲✉✭t✲✷✮❪
✐✮ ①9 ✭t✮❂✉✭t✮✲✉✭t✲✶✮✰✉✭t✲✶✮✲✉✭t✲✷✮
❥✮ ①10 ✭t✮❂①7 ✭✷t✮
❦✮ ①11 ✭t✮❂①8 ✭✵✱✺t✮
❧✮ ①12 ✭t✮❂①7 ✭✷t✲✺✮
✷✮ ❋❛ç❛ ✉♠ ❡s❜♦ç♦ ❞♦s s❡❣✉✐♥t❡s s✐♥❛✐s✿
❛✮ ①1 ❬♥❪❂✷✳✉❬✲♥❪
❜✮ ①2 ❬♥❪❂✺✳✉❬♥✲✷❪
❝✮ ①3 ❬♥❪❂✲✸✉❬✲♥✲✸❪
❞✮ ①4 ❬♥❪❂✲✸✉❬♥✰✷❪
❡✮ ①5 ❬♥❪❂✲✷✉❬♥❪
❢✮ ①6 ❬♥❪❂✲✻✉❬✲♥✰✷❪
❣✮ ①7 ❬♥❪❂✷✉❬♥❪✲✷✉❬♥✲✸❪
❤✮ ①8 ❬♥❪❂✉❬♥✰✷❪✲✉❬♥✲✷❪
✐✮ ①9 ❬♥❪❂✉❬♥❪✲✉❬♥✲✶❪✰✉❬♥✲✶❪✲✉❬t✲✷❪
❥✮ ①10 ❬♥❪❂①7 ❬✷♥❪
❦✮ ①11 ❬♥❪❂①8 ❬✵✱✺♥❪
❧✮ ①12 ❬♥❪❂①7 ❬✷♥✲✺❪
✸✮ ❊s❜♦❝❡ ♦ s✐♥❛❧ x[n] ❛❜❛✐①♦ ❡ s❡✉s ❞❡r✐✈❛❞♦s✿

 1, −3 < n ≤ 0
−1, 0 < n < 4
❛✮ x[n] =
 0,
outros
❜✮
❝✮
❞✮
❡✮
❢✮ x5 [n] =
+∞
P
k = −∞
δ(n − 3k)
x1 [n] = 2x[2n]
x2 [n] = 3x[ 12 n]
x4 [n] = 2δ[n + 2] − 3δ[n] + 2δ[n − 2] + 3δ[n − 3] + δ[n − 4]
x3 [n] = 2x[n] − u[n − 2]
✷✺
❊①❡r❝í❝✐♦s ❈♦♠♣❧❡♠❡♥t❛r❡s✿
✶✮ ❊s❜♦❝❡ ♦s s❡❣✉✐♥t❡s s✐♥❛✐s✿
❛✮
❜✮
❝✮
❞✮
x(t) = 2ǫ−2t u(t)
x(t) = −2ǫ2t u(t)
x(t) = 2ǫ−2t u(−t)
x(t) = −2ǫ2t u(−t)
❡✮ x(t) = 2ǫ−2t u(t − 2)
❢✮ x(t) = 3ǫ−5t u(t)
❣✮ x(t) = −tu(t − 2)
❤✮ x(t) = (t2 − 4)[u(t + 2) − u(t − 2)]
✷✮ ❊s❜♦❝❡ ♦s s❡❣✉✐♥t❡s s✐♥❛✐s✿
❛✮ x(t) =
+∞
P
k = −∞
❜✮ x(t) =
δ(t − kt0 )✱ ♦♥❞❡ k ∈ Z ❡ t0 = T =♣❡rí♦❞♦
t + 2 ,t ≤ 0
−t + 2 , t ≥ 0
❝✮ x1 (t) = x(t)[u(t + 2) − u(t − 2)]
❞✮ x2 (t) = x1 (2t)
❡✮ x3 (t) = x1 ( 12 t)
❢✮ x4 (t) = x1 (t − 4)
❣✮ x[n] =
−n + 3 , n ≥ 0
n + 3 ,n ≤ 0
❤✮ x1 [n] = x[n]{u[n + 3] − u[n − 3]}
✐✮ x2 [n] = x1 [2n]
❥✮ x3 [n] = x1 [ 21 n]
❦✮ x4 [n] = x1 [n − 3]
✷✻
❊st✉❞♦ ❞♦s ❙✐st❡♠❛s
❊♥t✐❞❛❞❡ q✉❡ ♣r♦❝❡ss❛ ✉♠ ♦✉ ♠❛✐s s✐♥❛✐s ❞❡ ❡♥tr❛❞❛✱
♣❛r❛ r❡❛❧✐③❛r ✉♠❛ ❢✉♥çã♦✱ r❡s✉❧t❛♥❞♦ ❡♠ ♦✉tr♦s s✐♥❛✐s ❞❡ s❛í❞❛✳
❂❃ Pr♦❞✉③ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❡♠ ✉♠ s✐♥❛❧✳
✲ ❆ s❛í❞❛ ♣♦❞❡rá s❡r ❛ ♠♦❞✐✜❝❛çã♦ ❞❛ ❡♥tr❛❞❛ ♦✉ ❡①tr❛çã♦ ❞❡
✐♥❢♦r♠❛çõ❡s ❞❡st❛ ❡♥tr❛❞❛✳
❙✐st❡♠❛✿
✲ P♦❞❡ s❡r ❝♦♥str✉✐❞♦ ❝♦♠ ❝♦♠♣♦♥❡♥t❡s ❢ís✐❝♦s ✭r❡❛❧✐③❛çã♦ ❡♠
❤❛r❞✇❛r❡✮✱ ♦✉ ♣♦❞❡ s❡r ✉♠ ❛❧❣♦r✐t♠♦ ✭r❡❛❧✐③❛çã♦ ❡♠ s♦❢t✇❛r❡✮✳
✲ P♦❞❡ s❡r ❞♦ t✐♣♦ ❈♦♥tí♥✉♦ ♦✉ ❉✐s❝r❡t♦ ✳
➱ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ♦✉ ✐♥t❡r❝♦♥❡①ã♦
❞❡ ♦♣❡r❛çõ❡s q✉❡ ♠❛♣❡✐❛ ✉♠ s✐♥❛❧ ✭♦✉ ✉♠❛ s❡q✉ê♥❝✐❛✮ ❞❡ ❡♥✲
tr❛❞❛✿ x(t) ✭♦✉ x[n]✮ ❡♠ ✉♠ s✐♥❛❧ ✭♦✉ ✉♠❛ s❡q✉ê♥❝✐❛✮ ❞❡ s❛í❞❛
y(t) ✭♦✉ y[n]✮✳
▼❛t❡♠❛t✐❝❛♠❡♥t❡✿
✷✼
❙✐st❡♠❛s ❝♦♠♦ ■♥t❡r❝♦♥❡①ã♦ ❞❡ ❖♣❡r❛çõ❡s✿
❱✐st♦ ❝♦♠♦ ✉♠❛ ✐♥t❡r❝♦♥❡①ã♦ ❞❡ ♦♣❡r❛çõ❡s ♣♦❞❡✲s❡ ❝❤❡❣❛r ❛ ✉♠
♦♣❡r❛❞♦r ❣❧♦❜❛❧ ❞♦ s✐st❡♠❛✿ H ✱ ♦♥❞❡ ❡st❡ ♦♣❡r❛❞♦r tr❛♥s❢♦r♠❛rá
♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛ ❞❡ ❛❧❣✉♠ ♠♦❞♦✳
❊①❡♠♣❧♦✿
❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ ❝✉❥♦ s✐♥❛❧ ❞❡ s❛í❞❛ y[n]
q✉❡ s❡❥❛ ❛ ♠é❞✐❛ ❞♦s três ✈❛❧♦r❡s ♠❛✐s r❡❝❡♥t❡s ❞♦ s✐♥❛❧ ❞❡ ❡♥✲
tr❛❞❛ x[n]✱ ♦✉ s❡❥❛✿
y[n] = 31 (x[n] + x[n − 1] + x[n − 2])✳
❙❡ ♦ ♦♣❡r❛❞♦r S K ❞❡♥♦t❛r ✉♠ ❙✐st❡♠❛ q✉❡ ❞❡s❧♦❝❛ ❛ ❡♥tr❛❞❛ ♥♦
t❡♠♣♦ ❞❡ ❑ ✉♥✐❞❛❞❡s✱ ♣r♦❞✉③✐♥❞♦ ✉♠❛ s❛í❞❛ x[n − k]✱ s✐♠❜♦❧✐✲
❝❛♠❡♥t❡ ♦ ❖♣❡r❛❞♦r S K s❡r✐❛ r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦✿
❊♥tã♦ ♣♦❞❡✲s❡ ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦r ❣❧♦❜❛❧ ❞♦ s✐st❡♠❛ ❝♦♠♦✿
H = 31 (1 + S + S 2 )
❙✐♠❜♦❧✐❝❛♠❡♥t❡ ♦ ❖♣❡r❛❞♦r ▼é❞✐❛ ▼ó✈❡❧ s❡r✐❛ r❡♣r❡s❡♥t❛❞♦ ❞❡
✉♠❛ ❞❛s ❞✉❛s ❢♦r♠❛s ❛❜❛✐①♦✿
✷✽
❆❧❣✉♥s t✐♣♦s ❞❡ ❙✐st❡♠❛s✿
• ❙✐st❡♠❛s ❞❡ ❈♦♠✉♥✐❝❛çõ❡s
• ❙✐st❡♠❛s ❞❡ ❈♦♥tr♦❧❡
• ❙✐st❡♠❛s ❞❡ Pr♦❝❡ss❛♠❡♥t♦ ❞❡ ❙✐♥❛✐s
✶✳ ❙✐st❡♠❛ ❞❡ ❈♦♠✉♥✐❝❛çã♦
❯♠ s✐st❡♠❛ ❞❡ ❝♦♠✉♥✐❝❛çã♦ é ❝♦♥st✐t✉✐❞♦ ❞❡ ✸ ❜❧♦❝♦s ❜á✲
s✐❝♦s✿ ❚r❛♥s♠✐ss♦r✱ ❈❛♥❛❧ ❞❡ ❈♦♠✉♥✐❝❛çã♦✱ ❡ ❘❡❝❡♣t♦r✳
Pr♦♣♦r❝✐♦♥❛ ✉♠ ♠❡✐♦ ❢ís✐❝♦ ❡♥tr❡
♦ tr❛♥s♠✐ss♦r ❡ ♦ r❡❝❡♣t♦r✳ ➱ ♥❡st❡ ❝❛♥❛❧ q✉❡ ♦ s✐♥❛❧ ✭♦✉
♠❡♥s❛❣❡♠✱ ♦✉ ❛✐♥❞❛ ✐♥❢♦r♠❛çã♦✮ s♦❢r❡ ✐♥✢✉ê♥❝✐❛s ❞❡ ✐♥t❡r✲
❢❡rê♥❝✐❛s✱ ❡ r✉í❞♦s✳
❈❛♥❛❧ ❞❡ ❈♦♠✉♥✐❝❛çã♦✿
❚❡♠ ❛ ❢✉♥çã♦ ❞❡ Pr♦❝❡ss❛r ♦ s✐♥❛❧ ❛ s❡r
tr❛♥s♠✐t✐❞♦ ❡♠ ✉♠❛ ❢♦r♠❛ ❞❡ s✐♥❛❧ ❛♣r♦♣r✐❛❞♦ à tr❛♥s♠✐ssã♦
❛tr❛✈és ❞♦ ❝❛♥❛❧✳ ❊st❛ ♦♣❡r❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ♠♦❞✉❧❛çã♦✳
❚r❛♥s♠✐ss♦r✿
Pr♦❝❡ss❛ ♦ s✐♥❛❧ r❡❝❡❜✐❞♦ ❛tr❛✈és ❞♦ ❝❛♥❛❧✱ ❞❡
♠♦❞♦ ❛ ♣r♦❞✉③✐r ✉♠ s✐♥❛❧ q✉❡ s❡❥❛ ❞❡ ❛❧❣✉♠ ♠♦❞♦ ♣ró①✐♠♦
❞♦ s✐♥❛❧ ♦r✐❣✐♥❛❧ tr❛♥s♠✐t✐❞♦✳ ❊st❛ ♦♣❡r❛çã♦ é ❝❤❛♠❛❞❛ ❞❡
❞❡t❡❝çã♦ ♦✉ ❞❡♠♦❞✉❧❛çã♦✳
❘❡❝❡♣t♦r✿
✷✾
✷✳ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡
❯♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ é ✉♠ s✐st❡♠❛ r❡s♣♦♥sá✈❡❧ ♣♦r ❝♦♥tr♦❧❛r
✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❝♦♥tr♦❧❡ ❞❡ ✉♠ ♣r♦❝❡ss♦✱ ❡ ❡♠ ❣❡r❛❧ é r❡❛❧✐♠❡♥✲
t❛❞♦✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳
Pr♦♣♦r❝✐♦♥❛ ✉♠❛ ✐♥❢♦r♠❛çã♦ ❞♦ ♣♦♥t♦ ❞❡ ♦♣❡r❛çã♦
❞❡s❡❥❛❞♦ ♣❛r❛ ♦ s✐st❡♠❛✳
❙❡t✲P♦✐♥t✿
❚❡♠ ❛ ❢✉♥çã♦ ❞❡ Pr♦❝❡ss❛r ♦ s✐♥❛❧ ❞❡ ❡rr♦ ❛tr❛✈és
❞❡ ❛❧❣✉♠❛ ❡str❛té❣✐❛ ❞❡ ❝♦♥tr♦❧❡✱ ❡ ❛t✉❛r ♥❛ ♣❧❛♥t❛ ♣❛r❛ ❝♦rr✐❣✐r
♦ ❞❡s✈✐♦ ♦✉ ♣❡rt✉r❜❛çã♦✳
❈♦♥tr♦❧❛❞♦r✿
P❧❛♥t❛✿
Pr♦❝❡ss♦ ❛ s❡r ❝♦♥tr♦❧❛❞♦✳
❚r❛♥s❢♦r♠❛ ♦ s✐♥❛❧ ❞❡ s❛í❞❛ ❞❛ ♣❧❛♥t❛ ❡♠ ✉♠ s✐✲
♥❛❧ q✉❡ s❡rá ❡♥✈✐❛❞♦ ❛♦ ❝♦♠♣❛r❛❞♦r ❞❡ ❡rr♦ ♣❛r❛ ❣❡r❛r ✉♠ s✐♥❛❧
q✉❡ ♦ ❝♦♥tr♦❧❛❞♦r ✉s❛rá ♣❛r❛ ❝♦♥tr♦❧❛r ❛ ♣❧❛♥t❛✳
❙❡♥s♦r❡s✿
✸✵
Pr♦♣r✐❡❞❛❞❡s ❞♦s ❙✐st❡♠❛s
✲ ❊st❛❜✐❧✐❞❛❞❡
✲ ▼❡♠ór✐❛
✲ ❈❛✉s❛❧✐❞❛❞❡
✲ ■♥✈❡rt✐❜✐❧✐❞❛❞❡
✲ ■♥✈❛r✐❛♥❝✐❛ ♥♦ ❚❡♠♣♦
✲ ▲✐♥❡❛r✐❞❛❞❡
✶✳ ❙✐st❡♠❛ ❊stá✈❡❧ ❡ ■♥stá✈❡❧
❉✐③✲s❡ q✉❡ ✉♠ s✐st❡♠❛ é ❞♦ t✐♣♦ ❊♥tr❛❞❛✲▲✐♠✐t❛❞❛✲
❙❛í❞❛ ▲✐♠✐t❛❞❛ ✲ ❇■❇❖ ✭❇♦✉♥❞❡❞ ■♥♣✉t✴❇♦✉♥❞❡❞
❖✉t♣✉t✮✱ ❊stá✈❡❧ ✱ s❡ ♣❛r❛ ✉♠❛ ❊♥tr❛❞❛ ❧✐♠✐t❛❞❛ r❡✲
s✉❧t❛r ✉♠❛ s❛í❞❛ t❛♠❜é♠ ❧✐♠✐t❛❞❛✳
❂❃
❆ s❛í❞❛ ❞♦ s✐st❡♠❛ ♥ã♦ ❞✐✈❡r❣❡ s❡ ❛ ❡♥tr❛❞❛ ♥ã♦
❞✐✈❡r❣✐r✳
▼❛t❡♠❛t✐❝❛♠❡♥t❡✿
❙✐st❡♠❛ ❞❡ ❚❡♠♣♦ ❈♦♥tí♥✉♦✿
|y(t)| ≤ My < ∞ ⇒ |x(t)| ≤ Mx < ∞✱ ∀t
❙✐st❡♠❛ ❞❡ ❚❡♠♣♦ ❉✐s❝r❡t♦✿
|y[n]| ≤ My < ∞ ⇒ |x[n]| ≤ Mx < ∞✱ ∀n
❈♦♠
Mx , M y
s❡♥❞♦ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ✜♥✐t♦s✳
❂❃ ❖ s✐st❡♠❛ s❡rá
çã♦ ❛❝✐♠❛✳
■♥stá✈❡❧
s❡ ♥ã♦ ❛t❡♥❞❡r ❛ ❝♦♥❞✐✲
✸✶
✷✳ ❙✐st❡♠❛ ❝✴ ▼❡♠ór✐❛ ❡ s✴ ▼❡♠ór✐❛
▼❡♠ór✐❛ s❡ ❞❡♣❡♥❞❡r ❞❡ ✈❛❧♦r❡s ♣❛ss❛❞♦s
❖ s✐st❡♠❛ é ❞✐t♦ ❙❡♠ ▼❡♠♦r✐❛ s❡ ❛ s❛í❞❛ ❞❡♣❡♥❞❡r ❛♣❡♥❛s ❞♦
❯♠ s✐st❡♠❛ ♣♦ss✉✐
❞♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛✳
✈❛❧♦r ♣r❡s❡♥t❡ ❞❛ ❡♥tr❛❞❛✳
❆ ❡①t❡♥sã♦ ❞♦s ✈❛❧♦r❡s ♣❛ss❛❞♦s✱ ❞❡✜♥❡ q✉ã♦ ❧♦♥❣❡ ❛ ♠❡♠ór✐❛
❞♦ s✐st❡♠❛ s❡ ❡st❡♥❞❡ ♥♦ ♣❛ss❛❞♦✳
❊①❡♠♣❧♦ ✶✿ ❖ ❘❡s✐st♦r é s❡♠ ♠❡♠ór✐❛✱ ❥á q✉❡ i(t) ❞❡♣❡♥❞❡ ❞❛
t❡♥sã♦ v(t) ✐♥st❛♥tâ♥❡❛ ❛♣❧✐❝❛❞❛ ❛ ❡❧❡✱ ♦✉ s❡❥❛✿
1 v(t)
i(t) = R
❊①❡♠♣❧♦ ✷✿ ❖ ■♥❞✉t♦r é ❝♦♠ ♠❡♠ór✐❛✱ ❥á q✉❡ i(t) ❞❡♣❡♥❞❡ ❞❛
✐♥t❡❣r❛❧ ❞❛ t❡♥sã♦ v(t) ❛♣❧✐❝❛❞❛ ❛ ❡❧❡✱ ♦✉ s❡❥❛✿
Rt
1
i(t) = L −∞ v(τ )dτ
❆❧é♠ ❞✐st♦✱ ♦ ♣❛ss❛❞♦ ❞♦ ✐♥❞✉t♦r s❡ ❡st❡♥❞❡ ❛♦ ♣❛ss❛❞♦ ✐♥✜♥✐t♦✳
❊①❡♠♣❧♦ ✸✿ ❖ ❙✐st❡♠❛ ❞❡ ▼é❞✐❛ ▼ó✈❡❧ ❞❡s❝r✐t♦ ❛♥t❡s ♣♦r✿
y[n] = 1
3 (x[n] + x[n − 1] + x[n − 2]) t❡♠ ♠❡♠ór✐❛ ❡ s❡
❡st❡♥❞❡ ❛ ✷ ✈❛❧♦r❡s ♥♦ ♣❛ss❛❞♦✳
❊①❡♠♣❧♦ ✹✿ ❖ ❙✐st❡♠❛ ❞❡s❝r✐t♦ ♣♦r✿ y[n] = x2[n]✱ ♥ã♦ t❡♠
♠❡♠ór✐❛✱ ♣♦✐s ❛ s❛í❞❛ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞♦ t❡♠♣♦ ❛t✉❛❧ ♥✳
❊①❡♠♣❧♦ ✷✿ ❖ ❈❛♣❛❝✐t♦r é ❝♦♠ ♠❡♠ór✐❛✱ ❥á q✉❡ ❛ t❡♥sã♦ v(t)
♥♦s s❡✉s t❡r♠✐♥❛✐s ❞❡♣❡♥❞❡ ❞❛ ✐♥t❡❣r❛❧ ❞❛ ❝♦rr❡♥t❡ i(t) q✉❡ ♣♦r
❡❧❡ ❝✐r❝✉❧❛✱ ♦✉ s❡❥❛✿
1 Rt
v(t) = C
−∞ i(τ )dτ
✸✷
✸✳ ❙✐st❡♠❛ ❈❛✉s❛❧ ❡ ◆ã♦✲❈❛✉s❛❧
❯♠ s✐st❡♠❛ é
❈❛✉s❛❧ ♦✉ ◆ã♦✲❆♥t❡❝✐♣❛t✐✈♦
s❡ ❛ s❛í❞❛ ❞❡♣❡♥❞❡r
s♦♠❡♥t❡ ❞❡ ✈❛❧♦r❡s ♣r❡s❡♥t❡s ❡✴♦✉ ♣❛ss❛❞♦s ❞♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛✳
❯♠ s✐st❡♠❛ s❡rá
◆ã♦✲❈❛✉s❛❧ ♦✉ ❆♥t❡❝✐♣❛t✐✈♦
s❡ ❞❡♣❡♥❞❡r ❞❡
✈❛❧♦r❡s ❢✉t✉r♦s ❞♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛✳
❊①❡♠♣❧♦ ✶✿
❖ ❙✐st❡♠❛ ❞❡ ▼é❞✐❛ ▼ó✈❡❧✿
y[n] = 13 (x[n] + x[n − 1] + x[n − 2])
é ❝❛✉s❛❧✱ ❡ ❞❡♣❡♥❞❡ ❞❡ ✷ ✈❛❧♦r❡s ♥♦ ♣❛ss❛❞♦ ❡ ✉♠ ♥♦ ♣r❡s❡♥t❡✳
❊①❡♠♣❧♦ ✷✿
❖ ❙✐st❡♠❛✿
y[n] = x[n + 1] + x[n] + x[n − 1])
é ♥ã♦✲❝❛✉s❛❧✱ ❥á q✉❡ ❞❡♣❡♥❞❡ ❞❡ ✶ ✈❛❧♦r ♥♦ ❢✉t✉r♦✳
✹✳ ❙✐st❡♠❛ ■♥✈❡rtí✈❡❧
❯♠ s✐st❡♠❛ é
■♥✈❡rtí✈❡❧
s❡ ❛ ❡♥tr❛❞❛ ❛♣❧✐❝❛❞❛ ❛ ❡st❡ ♣♦❞❡ s❡r
r❡❝✉♣❡r❛❞❛ ❞❛ s❛í❞❛✳
▼❛t❡♠❛t✐❝❛♠❡♥t❡✿
⇒
❖ ❙✐st❡♠❛ é
■♥✈❡rtí✈❡❧
s❡✿
H −1H = I
❙❡♥❞♦✿
I ♦ ❖♣❡r❛❞♦r ■❞❡♥t✐❞❛❞❡✱ ♣r♦❞✉③ ❛ s❛í❞❛ ✐❞ê♥t✐❝❛ à ❡♥tr❛❞❛✳
H −1 é ♦ ❖♣❡r❛❞♦r ■♥✈❡rs♦✱ ❛ss♦❝✐❛❞♦ é ♦ ❙✐st❡♠❛ ■♥✈❡rs♦✳
❙✐♠❜♦❧✐❝❛♠❡♥t❡✿
❊①s✿
✶✲ ❊♠ ❙✐st❡♠❛s ❞❡ ❈♦♠✉♥✐❝❛çõ❡s✱ ▼♦❞✉❧❛çã♦ ① ❉❡♠♦❞✉❧❛çã♦✳
✷✲ y(t) = 4x(t) ⇒ z(t) = 1
4 y(t)
✸✸
✺✳ ❙✐st❡♠❛ ■♥✈❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦
❯♠ s✐st❡♠❛ é ■♥✈❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦ s❡ ✉♠ r❡t❛r❞♦ ♦✉ ❛✈❛♥ç♦
❞❡ t❡♠♣♦ ♥♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛✱ ❧❡✈❛r ❛ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ t❡♠♣♦
✐❞ê♥t✐❝♦ ♥❛ s❛í❞❛✳
■st♦ ✐♠♣❧✐❝❛ q✉❡ ✉♠ s✐st❡♠❛ ■♥✈❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦ s❡♠♣r❡ r❡❛❣❡
❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❛ ❡♥tr❛❞❛
s❡❥❛ ❛♣❧✐❝❛❞❛✳✱
❖✉ s❡❥❛✱ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ s✐st❡♠❛ ♥ã♦ s❡ ♠♦❞✐✜❝❛♠ ❝♦♠ ♦
t❡♠♣♦✱ ❝❛s♦ ❝♦♥trár✐♦ ❡st❡ s❡rá ❱❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦ ✳
❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡♣r❡s❡♥t❛❞♦ ♣♦r✿
y(t) = H{x(t)}
❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥tr❛❞❛ s❡❥❛ ❞❡s❧♦❝❛❞❛ ♥♦ t❡♠♣♦ ❞❡ t0 s❡❣✉♥❞♦s✿
x(t − t0) = S t0 {x(t)}
❖♥❞❡✿ S t0 r❡♣r❡s❡♥t❛ ♦ ♦♣❡r❛❞♦r ❞❡s❧♦❝❛♠❡♥t♦ ♥♦ t❡♠♣♦ ❞❡ t0
s❡❣✉♥❞♦s✳
❙❡ yi(t) ❢♦r ❛ s❛í❞❛ ❞♦ s✐st❡♠❛ ♣❛r❛ ❛ ❡♥tr❛❞❛ x(t − t0)✱ ❡♥tã♦
♣♦❞❡✲s❡ ❡s❝r❡✈❡r✿
yi(t) = H{x(t − t0)} = H{S t0 {x(t)} = H.S t0 {x(t)}
❙❡ y0(t) ❢♦r ❛ s❛í❞❛ ❞♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧ ❞❡s❧♦❝❛❞❛ ♥♦ t❡♠♣♦
s❡❣✉♥❞♦s✱ ❡♥tã♦ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r✿
y0(t) = S t0 {y(t)} = S t0 {H{x(t)}} = S t0 .H{x(t)}
❙❡ ♦ s✐st❡♠❛ ❢♦r ✐♥✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦✱ ❛s s❛í❞❛s
✐❣✉❛❧ ❛ s❛í❞❛ yi(t)✱ ♣❛r❛ q✉❛❧q✉❡r x(t)✱ ♦✉ s❡❥❛✿
y0 (t)
t0
❞❡✈❡ s❡r
H.S t0 = S t0 .H
✸✹
✻✳ ❙✐st❡♠❛ ▲✐♥❡❛r ❡ ◆ã♦✲▲✐♥❡❛r✿
❯♠ s✐st❡♠❛ é ▲✐♥❡❛r s❡ s❛t✐s✜③❡r ♦s ♣r✐♥❝í♣✐♦s ❞❛ ❤♦♠♦❣❡♥❡✐✲
❞❛❞❡ ❡ ❞❛ s✉♣❡r♣♦s✐çã♦✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ ❛ r❡s♣♦st❛ ❞♦ s✐st❡♠❛
❛ ✉♠❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❡ s✐♥❛✐s é ✐❣✉❛❧ ❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❛s
s❛í❞❛s ✐♥❞✐✈✐❞✉❛✐s ❞❡ ❝❛❞❛ ✉♠ ❞♦s s✐♥❛✐s✳
▼❛t❡♠❛t✐❝❛♠❡♥t❡✿
❙❡ ♦ s✐♥❛❧
❞❡ ❡♥tr❛❞❛ ❞♦ s✐st❡♠❛ ❢♦r r❡♣r❡s❡♥t❛❞♦ ♣♦r✿
PN
x(t) = i=1 aixi(t)✱ ♦✉ s❡❥❛ ✉♠❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❡ s✐♥❛✐s
❊ ♦ s✐♥❛❧ ❞❡ s❛í❞❛ ❞♦ s✐st❡♠❛
✿
P
y(t) = H{x(t)} = H{ N
i=1 ai xi (t)}
⇒ ❖ ❙✐st❡♠❛ é
▲✐♥❡❛r s❡✿
y(t) =
PN
i=1 ai yi (t)
❙❡♥❞♦✿ yi(t) ❛ s❛í❞❛ ❞♦ s✐st❡♠❛ ❡♠ r❡s♣♦st❛ à ❡♥tr❛❞❛ xi(t)✳
❉❡✈❡✲s❡ ♥♦t❛r q✉❡ s❡ ✉♠ s✐st❡♠❛ é ▲✐♥❡❛r ❛t❡♥❞❡ ❛s ♣r♦♣r✐❡❞❛✲
❞❡s ❞❛ ❤♦♠♦❣❡♥❡✐❞❛❞❡ ❡ s✉♣❡r♣♦s✐çã♦✱ ❡♥tã♦✿
x(t) = 0 ⇒ y(t) = 0
⇒ ❖ s✐st❡♠❛ s❡rá
❛❝✐♠❛✳
◆ã♦ ▲✐♥❡❛r
s❡ ♥ã♦ ❛t❡♥❞❡r ❛s ❝♦♥❞✐çõ❡s
❙✐♠❜♦❧✐❝❛♠❡♥t❡✿
✸✺
❘❡♣r❡s❡♥t❛çõ❡s ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ■♥✈❛r✐❛♥t❡s ♥♦
❚❡♠♣♦
❂❃ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
❂❃ ❙✐♥❛✐s ❞❡ ❊♥tr❛❞❛ ❡ ❙❛í❞❛ sã♦
❢✉♥çõ❡s ❞♦ t❡♠♣♦✳
❂❃ ▲✐♥❡❛r✐❞❛❞❡ ❡ ■♥✈❛r✐â♥❝✐❛ ♥♦ ❚❡♠♣♦
❂❃ Pr♦♣r✐❡❞❛❞❡s
♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦s s✐st❡♠❛s✳
❂❃
❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ■♥✈❛r✐❛♥t❡s ♥♦ ❚❡♠♣♦ ✭▲■❚✮
❂❃
❛❧✈♦ ❞❡ ❡st✉❞♦ ❞❡ ❙✐♥❛✐s ❡ ❙✐st❡♠❛s✳
❂❃ ❘❡♣r❡s❡♥t❛çõ❡s ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦❂❃
▼ét♦❞♦s
✉s❛❞♦s ♣✴ ❞❡s❝r❡✈❡r ✉♠ ❙✐st❡♠❛ ▲■❚ ❛tr❛✈és ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❛
❡♥tr❛❞❛ ❡ ❛ s❛í❞❛✳
❘❡s♣♦st❛ ❛♦ ■♠♣✉❧s♦ ♣✴ ❙✐st❡♠❛s ▲❚■
❂❃ ▼ét♦❞♦ ✉s❛❞♦ ♣✴ s❡ ❝❛r❛❝t❡r✐③❛r ❝♦♠♣❧❡t❛✴t❡ ♦ ❝♦♠♣♦r✲
t❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ▲❚■✳
❂❃ ❈♦♥❤❡❝❡♥❞♦✲s❡ ❛ ❘❡s♣♦st❛ ❛♦ ■♠♣✉❧s♦ ❞❡ ✉♠ ❙✐st❡♠❛✱ ♣♦❞❡✲
s❡ ❞❡t❡r♠✐♥❛r ❛ r❡s♣♦st❛ ♣❛r❛ ✉♠❛ ❡♥tr❛❞❛ qq✉❡r ❛r❜✐trár✐❛✳
❂❃ ❙❡ ❜❛s❡✐❛ ♥❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ■♠♣✉❧s♦ ♥❛ ❡♥tr❛❞❛ ❞♦ s✐st❡♠❛
♥♦ ✐♥st❛♥t❡
t = 0 ♦✉ n = 0✱
s❡♥❞♦ ❛ r❡s♣♦st❛ ❛ ❡st❡ ✐♠♣✉❧s♦ ❝♦♠♦
❛ s❡❣✉✐r✱ ♦♥❞❡ ❍ é ♦ ♦♣❡r❛❞♦r ❞♦ sst❡♠❛✿
h[n] = H{δ[n]}
❂❃ ◆♦ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♠♣✉❧s♦s é
❢❛❝✐❧♠❡♥t❡ ♦❜t✐❞❛✳
❂❃ ◆♦ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦✱ ♦ ✐♠♣✉❧s♦ ❞❡✈❡ s❡r ❝♦♥s✐❞❡✲
r❛❞♦ ❝♦♠♦ ✉♠ ♣✉❧s♦ ❞❡ ❣r❛♥❞❡ ❛♠♣❧✐t✉❞❡ ❡ ❧❛r❣✉r❛ ❡str❡✐t❛✳
✸✻
❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦ ❡ ■♥t❡❣r❛❧ ❞❡ ❈♦♥✈♦❧✉çã♦
❂❃ ❋❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s ♣❛r❛ s❡ ❞❡s❝♦❜r✐r ❛ r❡s♣♦st❛ ❞❡
✉♠ s✐st❡♠❛ ▲■❚ ❛ ✉♠❛ ❡♥tr❛❞❛ ❛r❜✐trár✐❛✳
❂❃ ◆♦ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ ❡st❛ ❢❡rr❛♠❡♥t❛ é ❛
❙♦♠❛
❞❡ ❈♦♥✈♦❧✉çã♦
❂❃ ◆♦ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ é ❛ ■♥t❡❣r❛❧
❞❡ ❈♦♥✈♦❧✉çã♦ ✳
❆ ❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦ ✭❚❡♠♣♦ ❉✐s❝r❡t♦✮
❂❃ ❆ ❡♥tr❛❞❛ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ♣♦❞❡ s❡r ❡①♣r❡ss❛ ❝♦♠♦
✉♠❛ s✉♣❡r♣♦s✐çã♦ ♣♦♥❞❡r❛❞❛ ❞❡ ✐♠♣✉❧s♦s ❞❡s❧♦❝❛❞♦s ♥♦ t❡♠♣♦✱
s❡♥❞♦ ❛ ♣♦♥❞❡r❛çã♦ ❞❛❞❛ ♣❡❧♦ ✈❛❧♦r ❞❛ ❡♥tr❛❞❛ ♥♦ ✐♥st❛♥t❡ ♦♥❞❡
♦ ✐♠♣✉❧s♦ ❞❡s❧♦❝❛❞♦ ♦❝♦rr❡✳
❙❛❜❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ x[n] ❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✐♠♣✉❧s♦s
δ[n]✱ r❡s✉❧t❛✿
x[n].δ[n] = x[0].δ[n]
❊♥tã♦✱ ❣❡♥❡r❛❧✐③❛♥❞♦ ♣✴ ♦ ♣r♦❞✉t♦ ❞❡ x[n] ❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡
✐♠♣✉❧s♦s ❞❡s❧♦❝❛❞♦s ♥♦ t❡♠♣♦ δ[n − k]✱ t❡r❡♠♦s✿
x[n].δ[n − k] = x[k].δ[n − k]
❂❃ x[n] r❡♣r❡s❡♥t❛ ✉♠ s✐♥❛❧ ♥♦ t❡♠♣♦ ❞✐s❝r❡t♦ ✭s❡q✉ê♥❝✐❛✮ ♦♥❞❡
n é ♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ ❝♦♥s✐❞❡r❛❞♦✳
❂❃ x[k] r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❞♦ s✐♥❛❧ ♥♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ k✳
❂❃ ❖ s✐♥❛❧ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ✉♠ ✐♠♣✉❧s♦ ❞❡s❧♦❝❛❞♦ ♥♦ t❡♠♣♦
x[n].δ[n − k]
❂❃ ❘❡s✉❧t❛ ✉♠ ■♠♣✉❧s♦ ❞❡s❧♦❝❛❞♦ ♥♦ t❡♠♣♦ ❝♦♠ ❛ ❛♠♣❧✐t✉❞❡
❞♦ s✐♥❛❧ ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ♦❝♦rr❡ ♦ ✐♠♣✉❧s♦✿
x[k].δ[n − k]
✸✼
❂❃ ❆ s♦♠❛tór✐❛ ❞❛ ❡①♣r❡ssã♦✿
k
❞❡s❧♦❝❛♠❡♥t♦
x[k].δ[n−k]✱
♣❛r❛ t♦❞♦ ❡ q✉❛❧q✉❡r
r❡s✉❧t❛ ❡♠✿
x[n] = ... + x[−2].δ[n + 2] + x[−1].δ[n + 1] + x[0].δ[n]+
+x[1].δ[n − 1] + x[2].δ[n − 2] + ...
❖✉ ❞❡ ❢♦r♠❛ ♠❛✐s ❝♦♥❝✐s❛✿
x[n] =
∞
X
k=−∞
x[k].δ[n − k]
❂❃ ❆ s❛í❞❛ ❞❡ ✉♠ s✐st❡♠❛ ▲■❚ ❛ ❡st❛ ❡♥tr❛❞❛ t❛♠❜é♠ s❡rá ✉♠❛
s✉♣❡r♣♦s✐çã♦ ♣♦♥❞❡r❛❞❛ ❞❛ r❡s♣♦st❛ ❞♦ s✐st❡♠❛ ❛ ❝❛❞❛ ✐♠♣✉❧s♦
♣♦♥❞❡r❛❞♦ ❡ ❞❡s❧♦❝❛❞♦ ♥♦ t❡♠♣♦✳ ❙❡ ♦ ♦♣❡r❛❞♦r
♦ s✐st❡♠❛✱ ❛ ❡♥tr❛❞❛
x[n]
H
r❡♣r❡s❡♥t❛r
♣r♦❞✉③✐rá ❛ s❛í❞❛✿
y[n] = H{x[n]} = H{
∞
X
k=−∞
x[k].δ[n − k]}
❆ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ♣❡r♠✐t❡ ✐♥t❡r❝❛♠❜✐❛r ♦ ♦♣❡r❛❞♦r
❝♦♠ ♦ s♦♠❛tór✐♦ ❞❡ ✈❛❧♦r❡s ❞❡
y[n] =
∞
X
k=−∞
❖♥❞❡✿
x[k]✱
❞❡ ♦♥❞❡ s❡ ♦❜té♠✿
x[k].H{δ[n − k]} =
h[n − k] = H{δ[n − k]}
✐♠♣✉❧s♦ ❞❡s❧♦❝❛❞♦ ♥♦ t❡♠♣♦ ❞❡
H
∞
X
k=−∞
x[k].h[n − k]
é ❛ r❡s♣♦st❛ ❞♦ s✐st❡♠❛ ❛ ✉♠
k✳
❈♦♥❝❧✉sã♦✿
❆ r❡s♣♦st❛ ❞❡ ✉♠ s✐st❡♠❛ ▲❚■ é ❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❛s r❡s♣♦st❛s
❛♦ ✐♠♣✉❧s♦ ❞❡s❧♦❝❛❞❛s ♥♦ t❡♠♣♦✳
❆ ❡①♣r❡ssã♦✿
y[n] = x[n] ∗ h[n] =
∞
X
k=−∞
x[k].h[n − k]
❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦ ♣❛r❛ ♦ ❚❡♠♣♦ ❉✐s✲
❞❡♥♦t❛❞❛ ♣❡❧♦ sí♠❜♦❧♦✿ ✑ ✯ ✑ ✳
é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦
❝r❡t♦ ✱
s❡♥❞♦
✸✽
❊①❡♠♣❧♦✿ ❙❡❥❛ ✉♠ s✐st❡♠❛ ▲❚■✱ ❝✉❥❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ s❡❥❛
❝♦♠♦ ♠♦str❛❞♦ ❛❜❛✐①♦✿

 1, n = ±1
2, n = 0
h[n] =
 0, outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞❡st❡ s✐st❡♠❛ ❡♠ r❡s♣♦st❛ à ❡♥tr❛❞❛ ♠♦str❛❞❛
❛❜❛✐①♦✿

2, n = 0


3, n = 1
x[n] =

 −2, n = 2
0, outros
❙♦❧✉çã♦✿
❊s❝r❡✈❡♥❞♦
x[n]
❝♦♠♦ ✉♠❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❡ ✐♠♣✉❧s♦s ♣♦♥❞❡✲
r❛❞♦s ❡ ❞❡s❧♦❝❛❞♦s ♥♦ t❡♠♣♦✿
x[n] = 2δ[n] + 3δ[n − 1] − 2δ[n − 2]
❊♥tã♦✱ ✈✐st♦ q✉❡ ♦ s✐st❡♠❛ é ▲✐♥❡❛r✱ ❛ s❛í❞❛ ❞❡st❡ ♣♦❞❡ s❡r
❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ s♦♠❛ ♣♦♥❞❡r❛❞❛ ❞❛s r❡s♣♦st❛s ❞♦s ✐♠♣✉❧s♦s
♣♦♥❞❡r❛❞♦s ❡ ❞❡s❧♦❝❛❞♦s ♥♦ t❡♠♣♦✱ ♦✉ s❡❥❛✿
y[n] =
∞
X
k=−∞
x[k].h[n − k] = 2h[n] + 3h[n − 1] − 2h[n − 2]
✸✾
❉❛ ❡①♣r❡ssã♦ ❞❡
y[n] =
∞
X
k=−∞
y[n]✿
x[k].h[n − k] = 2h[n] + 3h[n − 1] − 2h[n − 2]
●r❛✜❝❛♠❡♥t❡✿
✰
✰
❖ q✉❡ r❡s✉❧t❛✿

0, n ≤ −2




2, n = −1



 7,
n=0
6,
n=1
y[n] =


−1, n = 2




−2, n = 3


0,
n≥4
❊st❛ ❛❜♦r❞❛❣❡♠ ❣rá✜❝❛ é ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞✐❞át✐❝❛✱ ❞❛♥❞♦ ✉♠❛
♥♦çã♦ ❞♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ❞❛ ❡①❡❝✉çã♦ ❞❛ ❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉✲
çã♦✱ ♥♦ ❡♥t❛♥t♦ ♥ã♦ é ❞❡ ❢♦r♠❛ ❛❧❣✉♠❛ ♣rát✐❝❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦
❞❛s ❙♦♠❛s ❞❡ ❈♦♥✈♦❧✉çã♦ ❡♠ ❣❡r❛❧✳
❉❡✈❡✲s❡ t❡r ❛❧❣✉♠ t✐♣♦ ❞❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣rát✐❝♦✱ ❝♦♠ ✈✐st❛s ❛
s❡r ❢❛❝✐❧♠❡♥t❡ ✐♠♣❧❡♠❡♥t❛❞♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦✳
✹✵
❆❜♦r❞❛❣❡♠ Prát✐❝❛
❙❡ ❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ é ✜♥✐t❛ ❡ ❝♦♥st✐t✉í❞❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛
❞❡ ❝✉rt❛ ❞✉r❛çã♦✱ ❛ ❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛
❞❡ ❢♦r♠❛ s✐♠♣❧❡s s❡ ❛♣❧✐❝❛❞♦s ♦s s❡❣✉✐♥t❡s ♣❛ss♦s✱ ♣♦❞❡♥❞♦ s❡r
❢❛❝✐❧♠❡♥t❡ ✐♠♣❧❡♠❡♥t❛❞❛ ❡♠ ✉♠ ❝♦♠♣✉t❛❞♦r✿
❛✮ ■♥✈❡rt❡r h[k] ✱ ♣r♦❞✉③✐♥❞♦ h[−k]✳
❜✮ ❉❡s❧♦❝❛r h[−k] ♣r♦❞✉③✐♥❞♦ h[−k + n]
❝✮ ▼✉❧t✐♣❧✐❝❛r h[−k + n] ♣♦r x[k] ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡ k
❞✮ ❙♦♠❛r t♦❞♦s ♦s t❡r♠♦s ♠✉❧t✐♣❧✐❝❛❞♦s ♣❛r❛ t♦❞♦ ✈❛❧♦r ❞❡ n
❊①❡♠♣❧♦ ✶
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞❡ ✉♠ s✐st❡♠❛ ▲❚■✱ q✉❡ t❡♥❤❛ r❡s♣♦st❛ ❛♦
✐♠♣✉❧s♦✱ ❡ ❡♥tr❛❞❛ ❝♦♥❢♦r♠❡ ❛❜❛✐①♦✿
h[n] =

 1/2, n = 0
2,
n=1
x[n] =
 0, outros
1, 0 ≤ n ≤ 2
0,
outros
❙♦❧✉çã♦✿
❛✮ ■♥✐❝✐❛❧♠❡♥t❡ ✐♥✈❡rt❡r h[k]✱ ♣r♦❞✉③✐♥❞♦ h[−k]
❜✮ ❉❡s❧♦❝❛r h[−k]✱ ♣r♦❞✉③✐♥❞♦ h[−k + n]
❝✱❞✮ ▼✉❧t✐♣❧✐❝❛r ❡ s♦♠❛r ❛tr❛✈és ❞❛ t❛❜❡❧❛ ❛❜❛✐①♦✿
❦
✲✸
✲✷
✲✶
✵
✶
✸
✹
♥❂✲✶
✶
✶
✶
✵
✶
✶
✶
✶
✶
✵ ✵ ✵
✵ ✵ ✵
✶
✵ ✵
✶
✶
✵
✵
✵
✵
✵
✵
♥❂✵
♥❂✶
♥❂✷
♥❂✸
♥❂✹
①❬❦❪
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✵
✵
✵✱✺
✷
✺
✵
✵
✵
✵
✶
✶
✶
✵
✵ ✶ ✶ ✶ ✵
✷
✵ ✵ ✵ ✵
②❬♥❪
②❬✲✶❪❂✵
②❬✵❪❂✵✱✺
②❬✶❪❂✷✱✺
②❬✷❪❂✷✱✺
②❬✸❪❂✷✱✵
②❬✹❪❂✵
①❬❦❪
✹✶
❊①❡♠♣❧♦ ✷
❙✉♣♦♥❤❛ ✉♠ s✐st❡♠❛
s❡❥❛ ❛ s❡❣✉✐♥t❡✿
h[n] =
H ❞♦ t✐♣♦ ▲❚■✱ ❝✉❥❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦
2, −2 ≤ n ≤ 4
0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞❡st❡ s✐st❡♠❛ ❡♠ r❡s♣♦st❛ à ❡♥tr❛❞❛✿
x[n] =
n, 2 ≤ n ≤ 6
0,
outros
❙♦❧✉çã♦✿
❛✮ ■♥✐❝✐❛❧♠❡♥t❡ ✐♥✈❡rt❡r h[k]✱ ♣r♦❞✉③✐♥❞♦ h[−k]
❜✮ ❉❡s❧♦❝❛r h[−k]✱ ♣r♦❞✉③✐♥❞♦ h[−k + n]✱ ✈❡r t❛❜❡❧❛ ❛❜❛✐①♦✿
♥❭❦
✲✺
✲✹
✲✸
✲✷
✲✶
✵
✶
✷
✸
✹
✺
✻
✼
✽
②❬♥❪
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✵
✷
✷
✷
✷
✷
✷
✷
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✵
✵
✵
✵
✵
✵
✹
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✷
✹✵
✵
✵
✵
✵
✵
✷
✷
✷
✷
✷
✷
✸✻
✵
✵
✵
✵
✷
✷
✷
✷
✷
✸✵
✵
✵
✵
✷
✷
✷
✷
✷✷
✵
✵
✷
✷
✷
✶✷
✵
✷
✷
✵
✷
✸
✹
✺
✻
✵
✵
①❬❦❪
✶
✷
✸
✹
✺
✻
✼
✽
✾
✶✵
✶✶
①❬❦❪
✶✵
✶✽
✷✽
✹✵
✹✵
✹✷
❊①❡r❝í❝✐♦ ✸
❙✉♣♦♥❤❛ ✉♠ s✐st❡♠❛
s❡❥❛ ❛ s❡❣✉✐♥t❡✿
h[n] =
H ❞♦ t✐♣♦ ▲❚■✱ ❝✉❥❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦
1, 0 ≤ n ≤ 2
0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞❡st❡ s✐st❡♠❛ ❡♠ r❡s♣♦st❛ à ❡♥tr❛❞❛✿

 1, 0 ≤ n ≤ 2
1, 5 ≤ n ≤ 6
x[n] =
 0,
outros
❙♦❧✉çã♦✿
❛✮ ■♥✐❝✐❛❧♠❡♥t❡ ✐♥✈❡rt❡r h[k]✱ ♣r♦❞✉③✐♥❞♦ h[−k]
❜✮ ❉❡s❧♦❝❛r h[−k]✱ ♣r♦❞✉③✐♥❞♦ h[−k + n]✱ ✈❡r t❛❜❡❧❛ ❛❜❛✐①♦✿
♥❭❦
✲✹
✲✸
✲✷
✲✶
✵
✶
✷
✸
✹
✺
✻
✼
✽
✾
②❬♥❪
✲✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✶
✶
✶
✵
✵
✶
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✶
✵
✶
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✵
✶
✶
✵
✵
✵
①❬❦❪
✵
✶
✷
✸
✹
✺
✻
✼
✽
✾
①❬❦❪
✶
✷
✸
✷
✶
✶
✷
✷
✶
✹✸
❖ ❡♥❢♦q✉❡ ❞❛ s♦♠❛ ❞❡ t♦❞❛s ❛s r❡s♣♦st❛s ❛♦ ✐♠♣✉❧s♦ ♣♦♥❞❡r❛❞❛s
❡ ❞❡s❧♦❝❛❞❛s ♥♦ t❡♠♣♦ ♣❛r❛ ❦✱ só é ✈✐á✈❡❧ s❡ ❛ ❡♥tr❛❞❛ é ❞❡ ❝✉rt❛
❞✉r❛çã♦✱ ✈✐❛❜✐❧✐③❛♥❞♦ ❛ss✐♠ ❛ s♦♠❛✳ P♦ré♠ q✉❛♥❞♦ ❛ ❡♥tr❛❞❛
t❡♠ ✉♠❛ ❞✉r❛çã♦ ❧♦♥❣❛✱ ❡st❡ ❡♥❢♦q✉❡ ❥á ♥ã♦ é ♠❛✐s ♣rát✐❝♦✱
❛ss✐♠ ✉♠❛ ♥♦✈❛ ❛❧t❡r♥❛t✐✈❛ s❡rá ❛♣r❡s❡♥t❛❞❛✳
❆❜♦r❞❛❣❡♠ ❆❧t❡r♥❛t✐✈❛ ♣❛r❛ ❆✈❛❧✐❛r ❛ ❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦
♥♦ ❚❡♠♣♦ ❉✐s❝r❡t♦
❈♦♥s✐❞❡r❡ ❛✈❛❧✐❛r ❛ s❛í❞❛ ♥✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ ✜①♦
y[n0 ] =
∞
X
n0 ✿
v[n0 ]
k=−∞
❙✉♣♦♥❤❛ ♦ s✐♥❛❧✿ wn0 [k]✱ r❡♣r❡s❡♥t❛♥❞♦ ♦s ✈❛❧♦r❡s ❡♠ n = n0
❝♦♠♦ ❢✉♥çã♦ ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ k✱ ♦✉ s❡❥❛✿ wn [k] = vk [n0]
0
❆ s❛í❞❛ ❡♠
n = n0
s❡rá ♦❜t✐❞❛ ❛✈❛❧✐❛♥❞♦✲s❡✿
y[n0 ] =
∞
X
wn0 [k]
k=−∞
❉❡✜♥❛♠♦s ❡♥tã♦✱ ❛ s❡q✉ê♥❝✐❛ ✐♥t❡r♠❡❞✐ár✐❛✿
wn [k] = vk [n] = x[k].h[n − k]✱
❖♥❞❡ ❛❣♦r❛ k é ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❡
✉♠❛ ❝♦♥st❛♥t❡✳
n
é tr❛t❛❞♦ ❝♦♠♦
❖❜s❡r✈❡ q✉❡✿ h[n−k] = h[−(k−n)]✱ é ✉♠❛ ✈❡rsã♦ ❞❡ h[k] r❡✢❡t✐❞❛
❡ ❞❡s❧♦❝❛❞❛ ♥♦ t❡♠♣♦ ❞❡ −n✱ ♦♥❞❡ ❡st❡ ❞❡s❧♦❝❛♠❡♥t♦ ♥♦ t❡♠♣♦
❞❡t❡r♠✐♥❛ ♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ ❡♠ q✉❡ y[n] s❡rá ❛✈❛❧✐❛❞❛✱ ❞❡
❛❝♦r❞♦ ❝♦♠✿
y[n] =
∞
X
wn [k]
k=−∞
✹✹
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲❚■ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ r❡♣r❡✲
s❡♥t❛❞♦ ♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❛❜❛✐①♦✿
3
h[n] = ( )n u[n]
4
❙❡ ❛ ❡♥tr❛❞❛ ❞♦ s✐st❡♠❛ ❢♦r✿
❡♠✿
n = −5✱ n = 5,
❡
x[n] = u[n]✱
❞❡t❡r♠✐♥❡ ❛ s❛í❞❛
n = 10✳
❙♦❧✉çã♦✿
❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❜❛✐①♦✿
y[n] =
P∞
k=−∞ wn [k]
❉❡✈❡♠♦s ❛❝❤❛r ❛s ❢✉♥çõ❡s
❞❡
n
❞❡ ✐♥t❡r❡ss❡✳
wn [k] = x[k].h[n − k]
♣❛r❛ ♦s ✈❛❧♦r❡s
❱❡♠♦s q✉❡✿
3
h[n − k] = ( )n−k .u[n − k] =
4
( 43 )n−k , k ≤ n
0,
outros
n = −5 ⇒ w−5 [k] = u[k].h[−5 − k] = 0 ⇒
u[k] = 0
∞
X
y[−5] =
w−5 [k] = 0
❙❡✿
P♦✐s✿
k ≤ −5 ⇒
k=−∞
❙❡✿
n = 5 ⇒ w5 [k] = u[k].h[5 − k] =
P♦✐s✿
0 ≤ k ≤ 5 ⇒ u[k] = 1
( 34 )5−k , 0 ≤ k ≤ 5
0,
outros
⇒
5
5
4
X
3 5X 4 k
3 5 1 − ( 3 )6
3 5−k
y[5] =
=( )
w5 [k] =
( ) =( ) .
( )
4
4
4
3
4
)
1
−
(
3
k=−∞
k=0
k=0
3
( 4 )10−k , 0 ≤ k ≤ 10
❙❡✿ n = 10 ⇒ w10 [k] =
⇒ P♦✐s✿ 0 ≤ k ≤
0,
outros
10 ⇒ u[k] = 1
∞
X
10
10
4
X
3 10 X 4 k
3 10 1 − ( 3 )11
3 10−k
( ) =( ) .
=( )
y[10] =
w10 [k] =
( )
4
4
3
4
1 − ( 34 )
k=−∞
k=0
k=0
∞
X
✹✺
■♥t❡❣r❛❧ ❞❡ ❈♦♥✈♦❧✉çã♦ ✲ ❈♦♥✈♦❧✉çã♦ ❈♦♥tí♥✉❛
❆ ❙♦♠❛ ❞❡ ❈♦♥✈♦❧✉çã♦ ♣❛r❛ ♦s ❙✐st❡♠❛s ❞❡ ❚❡♠♣♦ ❉✐s❝r❡t♦✱
t❡♠ ❛ s✉❛ ❝♦♥tr❛♣❛rt✐❞❛ ♣❛r❛ ♦s ❙✐st❡♠❛s ❞❡ ❚❡♠♣♦ ❈♦♥tí♥✉♦✱
q✉❡ é ❛ ■♥t❡❣r❛❧ ❞❡ ❈♦♥✈♦❧✉çã♦✱ ❡♠ q✉❡ ❛ s♦♠❛tór✐❛ ❞❡ t♦❞❛s ❛s
r❡s♣♦st❛s ❛♦ ✐♠♣✉❧s♦ ♣♦♥❞❡r❛❞❛s ❡ ❞❡s❧♦❝❛❞❛s ♥♦ t❡♠♣♦ ♣❛r❛ ❦✱
❛❣♦r❛ s❡ ❝♦♥✈❡rt❡ ❡♠ ✉♠❛ ✐♥t❡❣r❛❧ ❡♠ τ ✱ ❞❡ −∞ ❛ ✰∞✱ ❝♦♥❢♦r♠❡
♠♦str❛❞♦ ❛ s❡❣✉✐r✿
y(t) = x(t) ∗ h(t) =
+∞
Z
−∞
x(τ )h(t − τ )dτ
❈♦♠✿ x(t) r❡♣r❡s❡♥t❛♥❞♦ ❛ ❡♥tr❛❞❛ ❛r❜✐trár✐❛ ❡ h(t) ❛ r❡s♣♦st❛
❛♦ ✐♠♣✉❧s♦ ❞♦ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦✳
❊st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❝♦♠♣r♦✈❛❞♦ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ✉♠ s✐st❡♠❛
❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ ❝✉❥♦ ♦♣❡r❛❞♦r ❞♦ s✐st❡♠❛ é H ✱ ❛♣r❡s❡♥t❛♥❞♦
✉♠❛ s❛í❞❛ y(t) q✉❛♥❞♦ ❛ ❡♥tr❛❞❛ x(t) ❛ ❡st❡ é ❛♣❧✐❝❛❞❛✱ s❛❜❡♥❞♦✲
s❡ ❛✐♥❞❛ q✉❡ q✉❛❧q✉❡r
s✐♥❛❧ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥✲
R +∞
t❛❞♦ ❝♦♠♦✿ x(t) = −∞ x(τ )δ(t − τ )dτ ✱ ❡♥tã♦ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r✿
+∞
Z
x(τ )δ(t − τ )dτ }
y(t) = H{x(t)} = H{
−∞
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♣❛r❛ ❛ ❝♦♥✈♦❧✉çã♦ ❞✐s❝r❡t❛✱ ❛ ❈♦♥✈♦❧✉çã♦
❈♦♥tí♥✉❛ ❜❛s❡✐❛✲s❡ ❡♠ ✹ ♦♣❡r❛çõ❡s✿✳
❛✮ ■♥✈❡rt❡r h(τ )✱ ♦❜t❡♥❞♦✲s❡ h(−τ )✳
❜✮ ❉❡s❧♦❝❛r h(−τ )✱ ♦❜t❡♥❞♦✲s❡ h(−τ + t)✳
❝✮ ▼✉❧t✐♣❧✐❝❛r x(τ ) ♣♦r h(t − τ ) ♣❛r❛ t♦❞♦ τ ✳
❞✮ ■♥t❡❣r❛r ✭❝❛❧❝✉❧❛r ❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛✮ ♣❛r❛ t♦❞♦ ✈❛❧♦r ❞❡ τ ✳
✹✻
❊①❡♠♣❧♦ ✶✿
❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡✲
♣r❡s❡♥t❛❞♦ ♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❛❜❛✐①♦✿

 1, 0 < t < 1
−1, 1 < t < 2
h(t) =
 0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞♦ s✐st❡♠❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡♥tr❛❞❛✿
x(t) =
1, −1 < t < 1
0,
outros
❙♦❧✉çã♦✿
❛✮ ■♥✈❡rt❡r h(τ )✱ ♣❛r❛ s❡ ♦❜t❡r h(−τ )✿
❜✮ ❉❡s❧♦❝❛r h(−τ )✱ ♣❛r❛ s❡ ♦❜t❡r h(t − τ )✿
❝✱❞✮ ▼✉❧t✐♣❧✐❝❛r ❡ ■♥t❡❣r❛r x(τ ).h(t−τ )✱ ♣❛r❛ ♦s s❡❣✉✐♥t❡s tr❡❝❤♦s✿
✶➸❚r❡❝❤♦✿ −∞ < τ < −1✱ ❡ t < −1
⇒ y(t) = 0✱ ♣♦✐s x(τ ).h(t − τ ) = 0
✹✼
✷➸❚r❡❝❤♦✿
< τ < t✱ ❡ −1 < t R< 0
R −1
+∞
t
⇒ y(t) = −∞ x(τ )h(t − τ )dτ = −1 1.1dτ = τ |t−1 = t + 1
✸➸❚r❡❝❤♦✿
< τ < t✱ ❡ 0R< t < 1
R −1
t−1
t
t
⇒ y(t) = −1 (−1).1dτ + t−1 1.1dτ = −τ |t−1
−1 + τ |t−1 = −t + 1
✹➸❚r❡❝❤♦✿
− 2 < τ < 1✱ ❡R 1 < t < 2
R tt−1
1
1
⇒ y(t) = t−2 (−1).1dτ + t−1 1.1dτ = −τ |t−1
t−2 + τ |t−1 = −t + 1
✹✽
✺➸❚r❡❝❤♦✿
t<3
R t1 − 2 < τ < 1✱ ❡ 2 <
⇒ y(t) = t−2 (−1).1dτ = −τ |1t−2 = t − 3
❘❡s♣♦st❛ ✜♥❛❧✿

 t + 1, −1 ≤ t ≤ 0
−t + 1, 0 ≤ t ≤ 2
y(t) =
 t − 3,
2≤t≤3
✹✾
❊①❡♠♣❧♦ ✷✿
❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡✲
♣r❡s❡♥t❛❞♦ ♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❛❜❛✐①♦✿
h(t) =
t, 0 < t < 2T
0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞♦ s✐st❡♠❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡♥tr❛❞❛✿
x(t) =
1, 0 < t < T
0,
outros
❙♦❧✉çã♦✿
❛✮ ■♥✈❡rt❡r h(τ )✱ ♣❛r❛ s❡ ♦❜t❡r h(−τ )✿
❜✮ ❉❡s❧♦❝❛r h(−τ )✱ ♣❛r❛ s❡ ♦❜t❡r h(t − τ )✿
❝✱❞✮ ▼✉❧t✐♣❧✐❝❛r ❡ ■♥t❡❣r❛r x(τ ).h(t−τ )✱ ♣❛r❛ ♦s s❡❣✉✐♥t❡s tr❡❝❤♦s✿
✶➸❚r❡❝❤♦✿ −∞ < τ < 0✱ ❡ t < 0 ⇒ y(t) = 0✱ ♣♦✐s x(τ ).h(t − τ ) = 0
✺✵
R +∞
✷➸❚r❡❝❤♦✿ 0 < τ < t✱ ❡ 0 < t < T ⇒ y(t) =
x(τ )h(t − τ )dτ
−∞
Rt
Rt
1.(t − τ )dτ = 0 1.(t − τ )dτ = t.τ − τ 2 /2|t0 = t2 − t2 /2 = t2 /2
0
✸➸❚r❡❝❤♦✿
0 < τ < T ✱ ❡ T < t < 2T
⇒ y(t) = t.τ − τ 2 /2|T0 = tT − T 2 /2
✹➸❚r❡❝❤♦✿
t − 2T < τ < T ✱ ❡ 2T < t < 3T
⇒ y(t) = t.τ − τ 2 /2|Tt−2T = −t2 /2 + tT − 3T 2 /2
✺✶
=
❊①❡r❝✐❝✐♦ Pr♦♣♦st♦s
✶✮ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡♣r❡s❡♥t❛❞♦
♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❛❜❛✐①♦✿

t,
−1 < t < 1


−1, −2 < t < −1
h(t) =
1<t<2

 1,
0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞♦ s✐st❡♠❛
♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡♥tr❛❞❛✿
x(t) =
1, 0 < t < 4
0,
outros
✷✮ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡♣r❡s❡♥t❛❞♦
♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❛❜❛✐①♦✿

 2t,
0<t<1
−2t, −1 < t < 0
h(t) =
 0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞♦ s✐st❡♠❛
♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡♥tr❛❞❛✿
x(t) =
2, 0 < t < 2
0,
outros
✸✮ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡♣r❡s❡♥t❛❞♦
♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❤✭t✮ ❛❜❛✐①♦✳
s✐st❡♠❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡♥tr❛❞❛ ①✭t✮✳
h(t) =
t, 0 < t < 1
0,
outros
❉❡t❡r♠✐♥❡ ❛ s❛í❞❛ ❞♦

 −1, 0 < t < 1
1, −1 < t < 0
x(t) =
 0,
outros
✹✮ ❈♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ r❡♣r❡s❡♥t❛❞♦
♣❡❧❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ h(t) = e−2t .u(t)✱ ❛❝❤❡ ❛ r❡s♣♦st❛ ❞❡st❡
s✐st❡♠❛ à ❡♥tr❛❞❛ x(t) = e−t .u(t)✳
✺✷
❘❡s♦❧✉çã♦ ❞♦ ❊①❡r❝✐❝✐♦ ✹ ♥♦ ▼❛t❧❛❜✴❖❝t❛✈❡
❞t❂✵✳✵✵✶❀
t❂✵✿❞t✿✼❀
①❂✭❡①♣✭✲t✮✳✯✭t❃❂✵✮✮❀
♣❧♦t✭t✱①✮❀
♣❛✉s❡❀
❤❂✭❡①♣✭✲✷✯t✮✳✯✭t❃❂✵✮✮❀
♣❧♦t✭t✱❤✮❀
♣❛✉s❡❀
②❂❝♦♥✈✭①✱❤✮✯❞t❀
♣❧♦t✭t✱②✭✶✿❧❡♥❣t❤✭t✮✮✮❀
t✐t❧❡✭✬②✭t✮❂❭✐♥t ①✭❭t❛✉✮✳❤✭t✲❭t❛✉✮❞❭t❛✉✬✮❀
①❧❛❜❡❧✭✬t✬✮❀
②❧❛❜❡❧✭✬②✭t✮✬✮❀
✺✸
❙✐st❡♠❛s ■♥t❡r❝♦♥❡❝t❛❞♦s
■♥t❡r❝♦♥❡①ã♦ P❛r❛❧❡❧♦
■♥t❡r❝♦♥❡①ã♦ ❙ér✐❡ ✭❈❛s❝❛t❛✮
❘❡s♣♦st❛ ❛♦ ❉❡❣r❛✉
g(t) =
Z t
h(τ )dτ
Z t
g(τ )dτ
−∞
❘❡s♣♦st❛ à ❘❛♠♣❛
f (t) =
−∞
✺✹
Pr♦♣r✐❡❞❛❞❡s ❞❛ ❈♦♥✈♦❧✉çã♦
•
Pr♦♣r✐❡❞❛❞❡ ❈♦♠✉t❛t✐✈❛
•
Pr♦♣r✐❡❞❛❞❡ ❉✐str✐❜✉t✐✈❛
•
Pr♦♣r✐❡❞❛❞❡ ❆ss♦❝✐❛t✐✈❛
•
Pr♦♣r✐❡❞❛❞❡ ❞❡ ❉❡s❧♦❝❛♠❡♥t♦
•
❈♦♥✈♦❧✉çã♦ ❝♦♠ ■♠♣✉❧s♦
•
Pr♦♣r✐❡❞❛❞❡ ❞❛ ▲❛r❣✉r❛
x(t) ∗ h(t) = h(t) ∗ x(t)
x(t) ∗ [y(t) + z(t)] = x(t) ∗ y(t) + x(t) ∗ z(t)
x(t) ∗ [y(t) ∗ z(t)] = [x(t) ∗ y(t)] ∗ z(t)
x(t) ∗ y(t) = c(t)
❊♥tã♦✿ x(t) ∗ y(t − t0 ) = x(t − t0 ) ∗ y(t) = c(t − t0 )
❙❡✿
x(t) ∗ δ(t) = x(t)
x(t) t❡♠ ❧❛r❣✉r❛ T1❡ y(t) ❧❛r❣✉r❛ T2✱
❊♥tã♦✿ x(t) ∗ y(t) t❡rá ❧❛r❣✉r❛ T = T1 + T2
❙❡✿
✺✺
❆♥á❧✐s❡ ❞❡ ❋♦✉r✐❡r
❘❡♣r❡s❡♥t❛çã♦ ❡♠ ❙ér✐❡ ❞❡ ❋♦✉r✐❡r ❞❡ ❙✐♥❛✐s P❡✲
r✐ó❞✐❝♦s
❈♦♠♦ ✈✐st♦ ✉♠ s✐♥❛❧ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ ♣❡r✐ó❞✐❝♦ é ❞♦ t✐♣♦✿
x(t) = x(t + mT )✱ ∀t✱
❡ ❝♦♠✿
m∈Z
❙❡♥❞♦ ♦ P❡rí♦❞♦ ❋✉♥❞❛♠❡♥t❛❧
q✉❛❧✿ x(t) = x(t + T ),∀t
(T0 )✱
♦ ♠❡♥♦r ✈❛❧♦r ❞❡ ❚ ♣✴ ♦
❉♦✐s ❡①❡♠♣❧♦s ❞❡ s✐♥❛✐s ♣❡r✐ó❞✐❝♦s ❜ás✐❝♦s sã♦✿
x(t) = ejω0 t
x(t) = cos(ω0 t + φ)
❖♥❞❡✿
ω0 = 2π/T0
é ❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ❢✉♥❞❛♠❡♥t❛❧
❙ér✐❡ ❊①♣♦♥❡♥❝✐❛❧ ❈♦♠♣❧❡①❛ ❞❡ ❋♦✉r✐❡r
P♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ✉♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) ❝♦♠ ♣❡rí♦❞♦ T0 ❛tr❛✲
✈és ❞❛ s♦♠❛tór✐❛ ❞❡ ❡①♣♦♥❡♥❝✐❛✐s ❝♦♠♣❧❡①❛s✱ ❝♦♠ ❛❜❛✐①♦✿
x(t) =
∞
X
ck .ejkω0t
k=−∞
❙❡♥❞♦✿ ω0 = 2π/T0✱ ❡ ck ♦s ❝♦❡✜❝✐ê♥t❡s ❞❡ ❋♦✉r✐❡r ❝♦♠♣❧❡①♦s✱
❞❛❞♦s ♣♦r✿
1
x(t).e−jkω0tdt
ck =
T0 T0
Z
◆❛ ✐♥t❡❣r❛❧ T0 ❞❡♥♦t❛ ♦ ❝á❧❝✉❧♦ ❞❡st❛ ✐♥t❡❣r❛❧ ❡♠ ✉♠ ♣❡rí♦❞♦✱
✉s❛♥❞♦✲s❡ ♣♦r ❡①❡♠♣❧♦✿ ✵ ❛ T0 ✱ ♦✉ −T 0/2 ❛ T0/2
✺✻
❙❡ ✜③❡r♠♦s k = 0 ♥❛ ❡①♣r❡ssã♦ ❞❡ ck ✱ ♦ ✈❛❧♦r c0✱ ✐♥❞✐❝❛rá ♦
✈❛❧♦r ♠é❞✐♦ ❞❡ x(t) ❡♠ ✉♠ ♣❡rí♦❞♦✱ ♦✉ s❡❥❛✿
1
c0 =
T0
Z
x(t)dt
T0
❉❡t❡r♠✐♥❡ ❛ sér✐❡ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠♣❧❡①❛ ❞❡ ❋♦✉r✐❡r
❞♦ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) ♠♦str❛❞♦ ❛❜❛✐①♦✿
❊①❡♠♣❧♦✿
❙♦❧✉çã♦✿
1
ck =
T0
Z
ω0 = 2π/T = 2π/2π = 1✭r❛❞✴s✮
1
x(t).e−jkω0 t dt =
2π
T0
Z
π/2
1.e−jkt dt =
−π/2
−1
π/2
.e−jkt |−π/2 =
2π.j.k
1 ejkπ/2 − e−jkπ/2
1
−1
−jkπ/2
jkπ/2
=
.(e
−e
)=
(
)=
sen(kπ/2)
2π.j.k
k.π
2j
k.π
ck =
0,
k = par
±1/kπ, k = ±1, ±5, ±9, ...
=
c0 = a0 /2 = 1/2
 ∓1/kπ, k = ±3, ±7, ±11, ...
P∞
P∞
1
jkω
0t
.sen(kπ/2).ejkt ]
x(t) = k=−∞ ck .e
= k=−∞ [ kπ
sen(kπ/2)
k.π
❊♥tã♦✿


x(t) = ...+
1 −j5t 1 −j3t
1 −jt 1
1 jt 1 j3t
1 j5t
e
−
e
+
e + +
e −
e +
e +...
5.π
3.π
1.π
2 1.π
3.π
5.π
❆♣❧✐❝❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❊✉❧❡r ♣❛r❛ ♦ ❝♦ss❡♥♦✿
x(t) =
1
2
1
1
1
+ [cost − cos3t + cos5t − cos7t + ...]
2
π
3
5
7
✺✼
❙ér✐❡ ❚r✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r
P♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ✉♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) ❝♦♠ ♣❡rí♦❞♦ T0 ❛tr❛✲
✈és ❞❛ s♦♠❛tór✐❛ ❞❡ s❡♥♦s ❡ ❝♦ss❡♥♦s✱ ❝♦♠ ❛❜❛✐①♦✿
∞
X
a0
x(t) =
+
[ak .cos(kω0t) + bk .sen(kω0t)]
2
k=1
❙❡♥❞♦✿
ω0 = 2π/T0 ✱
❡ ak ❡ bk ♦s ❝♦❡✜❝✐ê♥t❡s ❞❡ ❋♦✉r✐❡r✿
2
x(t).cos(kω0t)dt
ak =
T0 T0
Z
2
x(t).sen(kω0t)dt
bk =
T0 T0
Z
❖s ❝♦❡✜❝✐ê♥t❡s ak ❡ bk s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ♦s ❝♦❡✜❝✐ê♥t❡s ❞❡
❋♦✉r✐❡r ❝♦♠♣❧❡①♦s ♣♦r✿
a0
c0 =
2
1
ck = .(ak − jbk )
2
1
c−k = .(ak + jbk )
2
◗✉❛♥❞♦ x(t) ❢♦r r❡❛❧✱ ❡♥tã♦ ak ❡ bk s❡rã♦ r❡❛✐s✱ ❡♥tã♦ t❡r❡♠♦s✿
ak = 2.Re[ck ]
bk = −2.Im[ck ]
✺✽
❙ér✐❡ ❚r✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r ❞❡ ❙✐♥❛✐s P❡r✐ó✲
❞✐❝♦s P❛r❡s ❡ ❮♠♣❛r❡s
❙❡ ♦ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) ❝♦♠ ♣❡rí♦❞♦
③❡r♦✱ ❡ ❛ sér✐❡ ❞❡ ❋♦✉r✐❡r ✜❝❛✿
T0
P❛r✱ ❡♥tã♦ bk s❡rá
❢♦r
∞
X
a0
[ak .cos(kω0 t)]
x(t) =
+
2
ω0 = 2π/T0
k=1
❙❡ ♦ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) ❝♦♠ ♣❡rí♦❞♦
s❡rá ③❡r♦✱ ❡ ❛ sér✐❡ ❞❡ ❋♦✉r✐❡r ✜❝❛✿
x(t) =
∞
X
T0
❢♦r
[bk .sen(kω0 t)]
❮♠♣❛r✱
❡♥tã♦ ak
ω0 = 2π/T0
k=1
❙ér✐❡ ❞❡ ❋♦✉r✐❡r ❡♠ ❋♦r♠❛ ❍❛r♠ô♥✐❝❛
❖✉tr❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❡♠ sér✐❡ ❞❡ ❋♦✉❡✐❡r ❞❡ ✉♠ s✐♥❛❧
♣❡r✐ó❞✐❝♦ x(t) ❝♦♠ ♣❡rí♦❞♦ T0é✿
x(t) = ck +
∞
X
k=1
2
ak =
T0
Z
[ck .cos(kω0 t − θk )]
x(t).cos(kω0 t)dt
T0
2
bk =
T0
ω0 = 2π/T0
Z
x(t).sen(kω0 t)dt
T0
❖s ❝♦❡✜❝✐ê♥t❡s ck ❡ θk s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ak ❡ bk ♣♦r✿
a0
c0 =
2
p
ck = a 2 k + b 2 k
θk = arctg(
bk
)
ak
✺✾
❈♦♥✈❡r❣ê♥❝✐❛ ❞❛ ❙ér✐❡ ❞❡ ❋♦✉r✐❡r
❯♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) t❡rá ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡♠ sér✐❡ ❞❡ ❋♦✉✲
r✐❡r s❡ ❛t❡♥❞❡r às ❝♦♥❞✐çõ❡s ❞❡ ❉✐r✐❝❤❧❡t
✶✳
❙❡ x(t) é ✐♥t❡❣rá✈❡❧ ❡♠ ♠ó❞✉❧♦ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ✉♠ ♣❡rí♦❞♦
q✉❛❧q✉❡r✿
Z
T0
|x(t)|dt < ∞
✷✳
❙❡ x(t) t❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❡♥tr♦
❞❡ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ✜♥✐t♦ t
✸✳
❙❡ x(t) t❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡s ❞❡♥tr♦
❞❡ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ✜♥✐t♦ t ❡ ❝❛❞❛ ✉♠❛ ❞❡❧❛s é ✜♥✐t❛
❊s♣❡❝tr♦ ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❋❛s❡ ❞❡ ❙✐♥❛✐s P❡r✐ó❞✐✲
❝♦s
❆❞♠✐t✐♥❞♦✲s❡ q✉❡ ♦s ❝♦❡✜❝✐ê♥t❡s ❝♦♠♣❧❡①♦s ❞❡ ❋♦✉r✐❡r ck ❞❡ ✉♠
s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t) s❡❥❛✿
ck = |ck |.ejθk
❖ ❣rá✜❝♦ ❞❡ |ck | ✈❡rs✉s ❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω é ❝❤❛♠❛❞♦ ❞❡
❊s♣❡❝tr♦ ❞❡ ❆♠♣❧✐t✉❞❡ ❞♦ s✐♥❛❧ x(t)✱ ❡ ♦ ❣rá✜❝♦ ❞❡ θk ✈❡rs✉s
❛ ❢r❡q✉ê♥❝✐❛ ❛♥❣✉❧❛r ω é ❝❤❛♠❛❞♦ ❞❡ ❊s♣❡❝tr♦ ❞❡ ❋❛s❡ ❞♦ s✐♥❛❧
x(t)✳
❈♦♠♦ ♦s ✈❛❧♦r❡s ❞❡ k sã♦ ✐♥t❡✐r♦s ♦s ❣rá✜❝♦s ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❋❛s❡
♥ã♦ sã♦ ❝✉r✈❛s ❝♦♥tí♥✉❛s✱ ♠❛s ♦❝♦rr❡♠ s♦♠❡♥t❡ ♥❛s ❢r❡q✉ê♥❝✐❛s
❞✐s❝r❡t❛s kω0 ✱ s❡♥❞♦ ♣♦r ❡st❛ r❛③ã♦ ❝❤❛♠❛❞♦s ❞❡ ❊s♣❡❝tr♦s
❉✐s❝r❡t♦s ❞❡ ❋r❡q✉ê♥❝✐❛✱ ♦✉ ❊s♣❡❝tr♦s ❞❡ ▲✐♥❤❛✳
✻✵
❉❡t❡r♠✐♥❡ ❛ sér✐❡ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r ❞♦ s✐♥❛❧
♣❡r✐ó❞✐❝♦ x(t)✿
❊①❡♠♣❧♦✿
❙♦❧✉çã♦✿ ω0 = 2π/T = 2π/2π = 1✱ ❡ ♦s ❝♦❡✜❝✐ê♥t❡s ❞❛ sér✐❡
s❡rã♦ ❝❛❧❝✉❧❛❞♦s ❝♦♠♦ ❛❜❛✐①♦✱ s❡♥❞♦ ❛ ✐♥t❡❣r❛çã♦ ❞❡ −π/2 ❛
π/2✿
1
T0
1
2π
R π/2
π/2
1 π
1
x(t)dt =
.t|−π/2 = 2π
.[ 2 −(− π2 )] = 1/2
1.dt = 2π
T0
−π/2
R
2 π/2
⇒ bk = π −π/2 sen(kt)dt = 0✱ ✉♠❛ ✈❡③ q✉❡✿ x(t) é ♣❛r✳
R
R π/2
2
⇒ ak = T20 T0 x(t).cos(kω0 t)dt = 2π
1.cos(kt)dt =
−π/2
⇒ a20
=
=
R
π/2
1
sen(kt)|
−π/2
kπ
2
1
).[sen( k.π
) − sen(− k.π
)] = ( k.π
).sen( k.π
)
= ( k.π
2
2
2
❙❛❜❡♥❞♦✲s❡ q✉❡✿


sen( k.π
) = −sen(− k.π
)
2
2
0,
k = par
2/kπ,
k = 1, 5, 9, ...
❛k =
 −2/kπ, k = 3, 7, 11, ...
P∞ 1
2
1
)cos(kt)]
❊♥tã♦✿ x(t) = 2 + π k=1[ k .sen( k.π
2
x(t) =
2
1
1
1
1
+ [cos(t) − cos(3t) + cos(5t) − cos(7t) + ...]
2
π
3
5
7
❊s♣❡❝tr♦s ❞❡ ❆♠♣❧✐t✉❞❡
❊s♣❡❝tr♦s ❞❡ ❋❛s❡
bk
0
θ k = arctan( ) = arctan( ) = 0
ak
ak
✻✶
❉❡t❡r♠✐♥❡ ❛ sér✐❡ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r ❞♦ s✐♥❛❧
♣❡r✐ó❞✐❝♦ x(t) ♠♦str❛❞♦✿
❊①❡♠♣❧♦✿
❙♦❧✉çã♦✿
ω0 = 2π/T = 2π/2π = 1✱
sér✐❡ s❡rã♦ ❝❛❧❝✉❧❛❞♦s ❝♦♠♦✿
❡♥tã♦ ♦s ❝♦❡✜❝✐ê♥t❡s ❞❛
Rπ
1
1
1
.t|π0 = 2π
.[π − 0] = 1/2
1.dt = 2π
x(t)dt
=
T0
2π 0
R
Rπ
2
2
⇒ ak = T0 T0 x(t).cos(kω0 t)dt = 2π 0 1.cos(kt)dt =
⇒ a20 =
1
T0
R
1
sen(kt)|π0
kπ
1
).[sen(kπ) − sen(0)] = 0
= ( k.π
R
Rπ
2
2 π
.cos(kt)|π0 =
⇒ bk = π 0 sen(kt)dt = 2π 0 1.sen(kt)dt = −1
kπ
0,
k = par
1
.[cos(kπ)
−
cos0]
=
.[1
−
cos(kπ)]
=
= −1
kπ
kπ
2/kπ, k = ı́mpar
P
1
❊♥tã♦✿ x(t) = 21 + π1 ∞
k=1 { k [1 − cos(kπ)](1 − cos(kπ).sen(kt)}
=
x(t) =
1
2
1
1
+ [sen(t) + sen(3t) + sen(5t) + ...]
2
π
3
5
❊s♣❡❝tr♦s ❞❡ ❆♠♣❧✐t✉❞❡
❊s♣❡❝tr♦s ❞❡ ❋❛s❡
θ k = arctan( abkk ) = arctan( b0k ) = π/2
✻✷
❉❡t❡r♠✐♥❡ ❛ sér✐❡ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r ❞♦ s✐♥❛❧
♣❡r✐ó❞✐❝♦ x(t) ♠♦str❛❞♦✿
❊①❡♠♣❧♦✿
❊s❝♦❧❤❡♥❞♦ ♦ ✐♥t❡r✈❛❧♦ ❞❡ ✐♥t❡❣r❛çã♦ ❞❡ ✵ ❛ π✱ ❡ s❛❜❡♥❞♦ q✉❡
ω0 = 2π/T = 2π/π = 2(rd/s)✱ ❡♥tã♦ ♦s ❝♦❡✜❝✐ê♥t❡s ❞❛ sér✐❡
s❡rã♦ ❝❛❧❝✉❧❛❞♦s ❝♦♠♦✿
1
a0
=
2
π
Z
π
0
1
e−t/2 dt = .(−2)
π
Z
π
0
2
e−t/2 .(−1/2)dt = − e−t/2 |π0 =
π
2
= − .(0, 2079 − 1) = 0, 504
π
2
ak =
π
Z
2
bk =
π
Z
❊♥tã♦✿
π
e−t/2 .cos(2kt)dt = 0, 504.(
2
)
1 + 16.k2
e−t/2 .sen(2kt)dt = 0, 504.(
8.k
)
1 + 16.k2
0
π
0
∞
X
a0
x(t) =
[ak .cos(kω0 t) + bk .sen(kω0 t) =
+
2
k=1
= 0, 504[1 +
∞
X
[(
k=1
= 0, 504[1 +
8.k
2
).cos(kω
t)
+
(
)sen(kω0 t)] =
0
2
2
1 + 16.k
1 + 16.k
∞
X
k=1
[(
2
).(cos(kω0 t) + 4k.sen(kω0 t))]
1 + 16.k2
✻✸
❊①❡r❝✐❝✐♦s Pr♦♣♦st♦s
❉❡t❡r♠✐♥❡ ❛ sér✐❡ ❚r✐❣♦♥♦♠étr✐❝❛ ❞❡ ❋♦✉r✐❡r ❞♦s s✐♥❛✐s ♣❡r✐ó❞✐✲
❝♦s x(t) ❛❜❛✐①♦✱ ❡s❜♦ç❛♥❞♦ ♦s ❡s♣❡❝tr♦s ❞❡ ❛♠♣❧✐t✉❞❡ ❡ ❢❛s❡✿
❛✮
❜✮
❝✮
❈♦♥t❡ú❞♦ ❞❡ P♦tê♥❝✐❛ ❞❡ ✉♠ ❙✐♥❛❧ P❡r✐ó❞✐❝♦
❈♦♠♦ ✈✐st♦ ❛♥t❡s ❛ P♦tê♥❝✐❛ ▼é❞✐❛ ❞❡ ✉♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x(t)
❡♠ ✉♠ ♣❡rí♦❞♦ q✉❛❧q✉❡r é ❞❛❞♦ ♣♦r✿
1
P =
T0
Z
T0
|x(t)|2 dt
❙❡ x(t) ❢♦r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ sér✐❡ ❊①♣♦♥❡♥❝✐❛❧ ❈♦♠♣❧❡①❛ ❞❡
❋♦✉r✐❡r✱ ❡♥tã♦ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡✿
1
P =
T0
Z
2
T0
|x(t)| dt =
∞
X
k=−∞
|ck |2
❙❡♥❞♦ ❡st❛ r❡❧❛çã♦ ❝❤❛♠❛❞❛ ❞❡ ❘❡❧❛çã♦ ♦✉ ❚❡♦r❡♠❛ ❞❡ P❛rs❡✈❛❧
❞❛ sér✐❡ ❞❡ ❋♦✉r✐❡r✳
✻✹
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r
❙❡❥❛ ✉♠ s✐♥❛❧ ♥ã♦✲♣❡r✐ó❞✐❝♦ x(t) ❞❡ ❞✉r❛çã♦ ✜♥✐t❛✱ ♦✉ s❡❥❛✿
x(t) = 0✱ ♣❛r❛ |t| > T1 ✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✿
❙❡❥❛ ❛❣♦r❛ ✉♠ s✐♥❛❧ ♣❡r✐ó❞✐❝♦ x′(t) ❢♦r♠❛❞♦ ♣❡❧❛ r❡♣❡t✐çã♦ ❞❡
x(t)✱ ❝♦♠ ♣❡rí♦❞♦ ❢✉♥❞❛♠❡♥t❛❧ T0 ✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ❛❜❛✐①♦✿
❙❡✿
T0 → ∞✱
t❡r❡♠♦s✿
lim x′(t) = x(t)
T0 →∞
❊ ❛ sér✐❡ ❊①♣♦♥❡♥❝✐❛❧ ❈♦♠♣❧❡①❛ ❞❡ ❋♦✉r✐❡r ❞❡
x′(t) =
∞
X
x′ (t)✱
s❡rá✿
ck .ejkω0t
k=−∞
❙❡♥❞♦✿ ω0 = 2π/T0✱ ❡ ck ♦s ❝♦❡✜❝✐ê♥t❡s ❞❡ ❋♦✉r✐❡r ❝♦♠♣❧❡①♦s✱
❞❛❞♦s ♣♦r✿
1 T0/2 ′
ck =
x (t).e−jkω0tdt
T0 −T 0/2
Z
✻✺
▼❛s ❝♦♠♦✿ x′(t) = x(t)✱ ♣❛r❛
❞❡st❡ ✐♥t❡r✈❛❧♦✱ ❡♥tã♦✿
1
ck =
T0
❱❛♠♦s ❞❡✜♥✐r
Z
|t| < T0 /2✱
T0 /2
1
x′ (t).e−jkω0 t dt =
T0
−T 0 /2
X(ω)
Z
❡ ❝♦♠♦
∞
x(t) = 0
❢♦r❛
x(t).e−jkω0 t dt
−∞
❝♦♠♦ s❡♥❞♦✿
X(ω) =
Z
∞
x(t).e−jωt dt
−∞
❊♥tã♦✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡①♣r❡ssã♦ ❞❡ ck ✜❝❛✿ ck =
❉❛s ❡①♣r❡ssõ❡s✿ ck = T1 X(kω0) ✱ ❡ x′(t) =
t✐t✉✐♥❞♦ ❛ ♣r✐♠❡✐r❛ ♥❛ s❡❣✉♥❞❛ ✈❡♠✿
0
P∞
1 X(kω )
0
T0
k=−∞ ck .e
jkω0 t
✱ s✉❜s✲
∞
∞
X
X
ω
1
0
X(kω 0 ).ejkω0 t
X(kω 0 ).ejkω0 t =
x′ (t) =
T0
2π
k=−∞
k=−∞
◗✉❛♥❞♦ T0 → ∞✱ ω0 = (2π/T0) → 0✱ ❡♥tã♦ ❢❛③❡♥❞♦ ω0 = ∆ω✱
❡ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ ∆ω → 0✱ ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r s❡
t♦r♥❛✿
′
x (t)|T0 →∞
∞
1 X
X(k∆ω).ejk∆ωt ∆ω} = x(t)
= lim x (t) = lim {
T0 →∞
△ω→0 2π
′
k=−∞
▼❛s✱
Z ∞
∞
1 X
1
X(ω).ejωt dω
lim {
X(k∆ω).ejk∆ωt ∆ω} =
△ω→0 2π
2π −∞
k=−∞
❘❡s✉❧t❛♥❞♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❋♦✉r✐❡r ❞❡ ✉♠ s✐♥❛❧ ♥ã♦ ♣❡r✐ó✲
❞✐❝♦✿
1
x(t) =
2π
Z
∞
X(ω).ejωt dω
−∞
✻✻
P❛r❡s ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ❋♦✉r✐❡r
❆ ❢✉♥çã♦
X(ω)
é ❝❤❛♠❛❞❛ ❞❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡
x(t)✱
❡ ❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r é ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ❋♦✉r✐❡r✱ ❡
❢♦r♠❛♠ ♦ ♣❛r ❞❡ tr❛♥s❢♦r♠❛❞❛s ❞❡ ❋♦✉r✐❡r✱ s❡♥❞♦ ❞❡♥♦t❛❞♦ ♣♦r✿
x(t) ↔ X(ω)✳
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡
X(ω) = F {x(t)} =
Z
x(t)✿
∞
x(t).e−jωt dt
−∞
❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ❋♦✉r✐❡r ❞❡
1
x(t) = F −1 {X(ω)} =
2π
❊①❡♠♣❧♦ ✶✿
❝♦♠♦
∞
X(ω).ejωt dω
−∞
❖❜t❡♥❤❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞♦ ♣✉❧s♦ r❡✲
t❛♥❣✉❧❛r ❞❛❞♦ ♣♦r
❙♦❧✉çã♦✿
Z
x(t)✿
x(t) = ret(t/τ )✱
❱❛♠♦s ❝❛❧❝✉❧❛r✿
ret( τt ) =
♠♦str❛❞♦ ❛♦ ❛❜❛✐①♦✿
X(ω) =
R∞
t
ret(
) · e−jωt dt
−∞
τ
R τ /2
1, | t |< τ /2
⇒ X(ω) = −τ /2 1 · e−jωt dt
0, | t |> τ /2
✻✼
♠❛s✿
R
e−ax dx = − 1a · eax ✱
X(ω) = −
❛ss✐♠ t❡r❡♠♦s✿
1
1
τ /2
· e−jωt |−τ /2 = −
· (e−jωτ /2 − ejωτ /2 )
jω
jω
❯s❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❊✉❧❡r ♣❛r❛ ♦ s❡♥♦✱
sen(x) =
ejx −e−jx
2j
✱ ✜❝❛✿
−2 e−jωτ /2 − ejωτ /2
2sen(ωτ /2)
X(ω) =
·(
)=
ω
2j
ω
▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ ❞✐✈✐❞✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r ♣♦r
X(ω) = τ ·
τ /2
t❡r❡♠♦s✿
sen(ωτ /2)
= τ · sinc(ωτ /2)
(ωτ /2)
❊st❛ ❢✉♥çã♦ ❞♦ t✐♣♦ sen(x)/x✱ ❝✉❥♦ ❡s❜♦ç♦ é ♠♦str❛❞♦ ❛❜❛✐①♦✱
é ❝❤❛♠❛❞❛ ❞❡ sinc(x)✱ s❡♥❞♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛✱ ♣♦r ❡s✲
t❛r r❡❧❛❝✐♦♥❛❞❛ ❡♠ ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ❞❛ t❡♦r✐❛ ❞❡ ❙✐♥❛✐s ❡
❙✐st❡♠❛s ❡ Pr♦❝❡ss❛♠❡♥t♦ ❉✐❣✐t❛✐s ❞❡ ❙✐♥❛✐s✳
✻✽
P❛r❡s ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ■♠♣♦rt❛♥t❡s
x(t)
↔
X(ω)
❈♦♥❞✐çã♦
e−atu(t)
↔
1
a+jω
a>0
eatu(−t)
↔
1
a−jω
a>0
e−a|t|
↔
a2 +ω 2
2a
a>0
t.e−atu(t)
↔
1
(a+jω)2
δ(t)
↔
1
δ(t − t0)
↔
e−jωt0
1
↔
2πδ(ω)
ejω0t
↔
2πδ(ω − ω0)
cos(ω0t)
↔
π[δ(ω − ω0) + δ(ω + ω0)]
sen(ω0t)
↔
jπ[δ(ω + ω0) − δ(ω − ω0)]
u(t)
↔
1
πδ(ω) + jω
u(−t)
↔
1
πδ(ω) − jω
a>0
✻✾
Pr♦♣r✐❡❞❛❞❡s ❞❛ ❚r❛♥s❢✳ ❞❡ ❋♦✉r✐❡r
✶✳ ▲✐♥❡❛r✐❞❛❞❡
a1 .x1 (t) + a2 .x2 (t) ↔ a1 .X1 (ω) + a2 .X2 (ω)
✷✳ ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❚❡♠♣♦
x(t − t0 ) ↔ e−jωt0 .X(ω)
✸✳ ❉❡s❧♦❝❛♠❡♥t♦ ❞❡ ❋r❡q✉ê♥❝✐❛
e−jω0 t .x(t) ↔ X(ω − ω0 )
✹✳ ▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❚❡♠♣♦
x(a.t) ↔
ω
1
.X( )
|a|
a
✺✳ ■♥✈❡rsã♦ ❞❡ ❚❡♠♣♦
x(−t) ↔ X(−ω)
✻✳ ❉✉❛❧✐❞❛❞❡ ♦✉ ❙✐♠❡tr✐❛
X(t) ↔ 2π.x(−ω)
✼✳ ❉✐❢❡r❡♥❝✐❛çã♦ ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
d
x(t) ↔ jω.X(ω)
dt
✽✳ ❉✐❢❡r❡♥❝✐❛çã♦ ♥♦ ❉♦♠í♥✐♦ ❞❛ ❋r❡q✉ê♥❝✐❛
(−jt).x(t) ↔
d
X(ω)
dω
✼✵
✾✳ ■♥t❡❣r❛çã♦ ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
Z
t
−∞
x(τ )dτ ↔ π.X(0).δ(ω) +
1
.X(ω)
jω
✶✵✳ ❈♦♥✈♦❧✉çã♦
x1 (t) ∗ x2 (t) ↔ X1 (ω).X2 (ω)
✶✶✳ ▼✉❧t✐♣❧✐❝❛çã♦
x1 (t).x2 (t) ↔
1
.X 1 (ω) ∗ X2 (ω)
2π
x(t)
✶✷✳ P❛rt❡s P❛r ❡ ❮♠♣❛r ❞❡
❙❡
x(t)
❢♦r r❡❛❧✱ s❡❥❛✿
x(t) = xe (t) + xo (t)✱
x(t) ↔ X(ω) = A(ω) + jB(ω)✱
❡
X(−ω) = X ∗ (ω)
❛✐♥❞❛✿
❊♥tã♦✿
xe (t) ↔ Re{X(ω)} = A(ω)
xo (t) ↔ j.Im{X(ω)} = j.B(ω)
✶✸✳ ❘❡❧❛çõ❡s ❞❡ P❛rs❡✈❛❧
Z
Z
∞
x1 (λ).X2 (λ)dλ =
−∞
∞
1
x1 (t).x2 (t)dt =
2π
−∞
Z
∞
Z
Z
1
|x(t)|2 dt =
2π
−∞
∞
X1 (λ).x2 (λ)dλ
−∞
∞
X1 (ω)X2 (−ω)dω
−∞
Z
∞
−∞
|X(ω)|2 dω
✼✶
❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ ❞❡ ❙✐st❡♠❛s ❞❡ ❚❡♠♣♦
❈♦♥tí♥✉♦ ▲■❚
❯♠ s✐st❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦✱ ❝✴ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦
h(t)✱ t❡rá ✉♠❛ s❛í❞❛ y(t) ❡♠ r❡s♣♦st❛ ❛ x(t)✿
y(t) = x(t) ∗ h(t)
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❈♦♥✈♦❧✉çã♦ ♣✴ ❛ ❚r❛♥s❢✳ ❞❡ ❋♦✉✲
r✐❡r✿
Y (ω) = X(ω).H(ω)
❖♥❞❡✿ Y (ω)✱ X(ω)✱ ❡ H(ω)✱ sã♦ ❛s ❚r❛♥s❢✳ ❞❡ ❋♦✉r✐❡r ❞❡ y(t)✱
x(t)✱ ❡ h(t)✳
■s♦❧❛♥❞♦ H(ω)✱ q✉❡ é ❛ ❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ ❞♦ s✐st❡♠❛✿
H(ω) =
Y (ω)
= |H(ω)|.ejθH (ω)
X(ω)
❙❡♥❞♦✿ |H(ω)|✱ ❛ ❘❡s♣♦st❛ ❞❡ ▼❛❣♥✐t✉❞❡ ✭♦✉ ❆♠♣❧✐t✉❞❡✮✱ ❡
θH (ω) ❛ ❘❡s♣♦st❛ ❞❡ ❋❛s❡ ❞♦ s✐st❡♠❛✳
❆ ❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ H(ω) ❝❛r❛❝t❡r✐③❛ t♦t❛❧♠❡♥t❡ ♦ s✐s✲
t❡♠❛ ▲■❚ ❞❡ t❡♠♣♦ ❈♦♥tí♥✉♦✳
❈♦♥❝❧✉sõ❡s✿
• ❖ ❊s♣❡❝tr♦ ❞❡ ❆♠♣❧✐t✉❞❡ ❞❛ s❛í❞❛ ❞❡ ✉♠ s✐st❡♠❛ ▲■❚ ✲
|Y (ω)|✱ é ♦ ♣r♦❞✉t♦ ❞♦ ❊s♣❡❝tr♦ ❞❡ ❆♠♣❧✐t✉❞❡ ❞❛ ❡♥tr❛❞❛
|X(ω)|✱ ♣❡❧❛ ❘❡s♣♦st❛ ❞❡ ❆♠♣❧✐t✉❞❡ ❞♦ s✐st❡♠❛ |H(ω)|✳
• ❖ ❊s♣❡❝tr♦ ❞❡ ❋❛s❡ ❞❛ s❛í❞❛✱ é ❛ s♦♠❛ ❞♦ ❊s♣❡❝tr♦ ❞❡
❋❛s❡ ❞❛ ❡♥tr❛❞❛ θx (ω)✱ ❝♦♠ ❛ ❘❡s♣♦st❛ ❞❡ ❋❛s❡ ❞♦ s✐st❡♠❛
θH (ω)✳
✼✷
❚r❛♥s♠✐ssã♦ s❡♠ ❉✐st♦rçã♦
P❛r❛ q✉❡ ❛ tr❛♥s♠✐ssã♦ ❛tr❛✈és ❞❡ ✉♠ s✐st❡♠❛ ▲■❚ ♥ã♦ ♣r♦❞✉③❛
❞✐st♦rçã♦✱ ♦ s✐♥❛❧ ❞❡ s❛í❞❛ ❞❡✈❡ t❡r ♦ ♠❡s♠♦ ❢♦r♠❛t♦ ❞♦ s✐♥❛❧ ❞❡
❡♥tr❛❞❛✱ ❡①❝❡t♦ ♣❡❧❛ ❛♠♣❧✐t✉❞❡✱ ❡ ❡st❛r ❛tr❛s❛❞❛ ♥♦ t❡♠♣♦✳
P♦rt❛♥t♦ s❡ x(t) r❡♣r❡s❡♥t❛r ♦ s✐♥❛❧ ❞❡ ❡♥tr❛❞❛ ❞♦ s✐st❡♠❛✱ ❡ y(t)
❛ s✉❛ s❛í❞❛✱ ❡st❛ s❛í❞❛ s❡rá ❞♦ t✐♣♦✿
y(t) = K.x(t − t0 )
❆♣❧✐❝❛♥❞♦✲s❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r✱
✜❝❛✿
Y (ω) = K.e−jωt0 .X(ω)
▼❛s ❝♦♠♦ ✈✐st♦ ❛♥t❡s✿ H(ω) = Y (ω)/X(ω) = K.e−jωt
❉❛ ❢♦r♠❛ ♣♦❧❛r ♣❛r❛ H(ω)✿ H(ω) = |H(ω)|.ejθ(ω) =
♦❜té♠✲s❡✿
0
K.e−jωt0 ✱
|H(ω)| = K
θ(ω) = −jωt0
✭✶✮ ♦ ❡s♣❡❝tr♦ ❞❡ ❛♠♣❧✐t✉❞❡ ❞❡ H(ω) ❞❡✈❡ s❡r ❝♦♥s✲
t❛♥t❡ ❞❡♥tr♦ ❞❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛s ❞❡ ✐♥t❡r❡ss❡✱ ❡ ✭✷✮ ❛ ❢❛s❡
θ(ω) ❞❡✈❡ s❡r ❧✐♥❡❛r ❝♦♠ ❛ ❢r❡q✉ê♥❝✐❛✳
❈♦♥❝❧✉sã♦✿
✼✸
❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ ❞❡ ❈✐r❝✉✐t♦s ❘❈
❯s❛♥❞♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛s ✭❋P❇✮ t✐♣♦ ❘❈ ❞❡ ✶➟ ❖r❞❡♠
❯♠ ✜❧tr♦ ❞♦ t✐♣♦ P❛ss❛✲❇❛✐①❛s ❝❛✉s❛❧ ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦
❝♦♠♦ ♦ ❝✐r❝✉✐t♦ ❘❈ ♠♦str❛❞♦ ❛❜❛✐①♦✱ ♦♥❞❡ x(t) r❡♣r❡s❡♥t❛ ❛
❡♥tr❛❞❛ ❞♦ ✜❧tr♦ ❡ y(t) ❛ s✉❛ s❛í❞❛✿
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ▲❈❑ ♥♦ ♥ó ❆✱ ✈❡♠
q✉❡✿
dy(t)
iR = iC ⇒ x(t)−y(t)
=
C
R
dt
❉❡ ♦♥❞❡ r❡s✉❧t❛✿ RC dy(t)
dt + y(t) = x(t)
❊♥tã♦✱ ♦❜t❡♥❞♦✲s❡ ❛ ❚r❛♥❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱
♦♥❞❡ s❡ ❛♣❧✐❝♦✉ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦✱ ✈❡♠ q✉❡✿
RC.jω.Y (ω) + Y (ω) = X(ω)
Y (ω)
1
= 1+jωRC
= H(ω)
Y (ω).(RC.jω + 1) = X(ω) ⇒ X(ω)
❋❛③❡♥❞♦✲s❡
ω0 =
1
RC
♥❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r✱ ✜❝❛✿
1
⇒ ❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ ❞♦ ❋P❇ ❘❡❛❧✳
H(ω) = 1+jω/ω
0
❊♥tã♦✱ ❛ ❘❡s♣♦st❛ ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❞❡ ❋❛s❡ s❡rã♦ ❞❛❞❛s ♣♦r✿
1
1
p
|H(ω)| = |1+jω/ω
=
0|
1+(ω/ω0 )2
θ(ω) = − arctan(ω/ω0)
✼✹
❋✐❧tr♦ P❛ss❛✲❆❧t❛s ✭❋P❆✮ t✐♣♦ ❘❈ ❞❡ ✶➟ ❖r❞❡♠
❯♠ ✜❧tr♦ ❞♦ t✐♣♦ P❛ss❛✲❆❧t❛s ❝❛✉s❛❧ ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦
❝♦♠♦ ♦ ❝✐r❝✉✐t♦ ❘❈ ♠♦str❛❞♦ ❛❜❛✐①♦✱ ♦♥❞❡ x(t) r❡♣r❡s❡♥t❛ ❛
❡♥tr❛❞❛ ❞♦ ✜❧tr♦ ❡ y(t) ❛ s✉❛ s❛í❞❛✿
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ▲❈❑ ♥♦ ♥ó ❆✱ ✈❡♠
q✉❡✿
y(t)
iR = iC ⇒ C d[x(t)−y(t)]
=
dt
R
= y(t)
❉❡ ♦♥❞❡ r❡s✉❧t❛✿ R.C. d[x(t)−y(t)]
dt
❊♥tã♦✱ ♦❜t❡♥❞♦✲s❡ ❛ ❚r❛♥❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱
♦♥❞❡ s❡ ❛♣❧✐❝♦✉ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦✱ ✈❡♠ q✉❡✿
R.C.jω.[X(ω) − Y (ω)] = Y (ω)
R.C.jω.X(ω) = Y (ω).(R.C.jω + 1)
Y (ω)
RCjω
1
=
=
jωRC+1
X(ω)
1+1/(jωRC)
Y (ω)
1
=
= H(ω)
X(ω)
1+1/(jωRC)
❋❛③❡♥❞♦✲s❡
ω0 =
1
RC
♥❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r✱ ✜❝❛✿
1
H(ω) = 1+1/(jω/ω
⇒ ❘❡s♣♦st❛ ❡♠ ❋r❡q✉ê♥❝✐❛ ❞♦ ❋P❆
0)
❘❡❛❧✳
❊♥tã♦✱ ❛ ❘❡s♣♦st❛ ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❞❡ ❋❛s❡ s❡rã♦ ❞❛❞❛s ♣♦r✿
1
1
p
|H(ω)| = |1+1/(jω/ω
=
0 )|
1+1/(ω/ω0 )2
θ(ω) = − arctan(1/(ω/ω0))
✼✺
❆♥á❧✐s❡ ❞❡ ❙✐st❡♠❛s ✉s❛♥❞♦ ❛
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❇✐❞✐r❡❝✐♦♥❛❧ ❞❡ ✉♠ s✐♥❛❧
t❡♠♣♦ ❝♦♥tí♥✉♦✱ é ❞❡✜♥✐❞❛ ❝♦♠♦✿
L{x(t)} = X(S) =
Z∞
x(t)
❞❡
x(t).e−st dt
−∞
❖♥❞❡ ❡♠ ❣❡r❛❧ ❛ ✈❛r✐á✈❡❧
❝♦♠♦✿ s = σ + .jω✳
s
t❡♠ ✈❛❧♦r ❝♦♠♣❧❡①♦ ❡ é ❡①♣r❡ss❛
❊♠ ❝♦♥tr❛st❡ à ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❇✐❞✐r❡❝✐♦♥❛❧ ❛ ❚r❛♥s✲
❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❯♥✐❞✐r❡❝✐♦♥❛❧ é ❞❡✜♥✐❞❛ ❝♦♠♦✿
L{x(t)} = XI (S) =
Z∞
x(t).e−st dt
0−
❡ ❛s ❞✉❛s tr❛♥s❢♦r♠❛❞❛s s❡rã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♣❡♥❛s s❡
♣❛r❛ t < 0✳
x(t) = 0
❉❛í r❡s✉❧t❛ q✉❡ x(t) s❡rá ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡
X(s)✱ s❡♥❞♦ ❡①♣r❡ss❛ ♣♦r✿
L−1 {X(s)} = x(t) =
1
2πj
c+j∞
Z
x(s).est ds
c−j∞
❆ ✐♥t❡❣r❛❧ ❛❝✐♠❛ é ❝❛❧❝✉❧❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡
❞❡ −∞ ❛ +∞✳
c + jω
❝♦♠
ω
✈❛r✐❛♥❞♦
❉✐③✲s❡ q✉❡ ♦ s✐♥❛❧ x(t) ❡ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ X(S)
❢♦r♠❛♠ ✉♠ ♣❛r ❞❡ tr❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ s✐♠❜♦❧✐③❛❞♦ ♣♦r✿
x(t) ↔ X(s)✳
✼✻
❘❡❣✐ã♦ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛
❞❡ ▲❛♣❧❛❝❡ ✭❘❉❈✮
❖ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛ s ♣❛r❛ ♦ q✉❛❧ ❛
✐♥t❡❣r❛❧ ❞❛ ❡①♣r❡ssã♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❝♦♥✈❡r❣❡ é
❝❤❛♠❛❞♦ ❞❡ ❘❡❣✐ã♦ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ✭❘❉❈✮✳
❊①❡♠♣❧♦ ✶✿
❈♦♥s✐❞❡r❡♠♦s ♦ s✐♥❛❧ x(t) = e−at u(t)✱ ❝♦♠ a ∈ ℜ
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ x(t)✱ ✈❡♠ q✉❡✿
X(S) =
Z∞
e−at u(t).e−st dt =
−∞
X(S) = −
Z∞
0
e−at .e−st dt =
Z∞
e−(s+a)t dt
0
1
1
1
∞
e−(s+a)t |0 = −
(0 − 1) =
s+a
s+a
s+a
■st♦ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ q✉❡✿
limt→∞ e−(s+a)t = 0
❖ q✉❡ ♦❝♦rr❡ s❡ ❡ s♦♠❡♥t❡ s❡✿
ℜ{s + a} > 0
♦✉ s❡❥❛✿ ℜ{σ + jω + a} > 0
♦✉ ❛✐♥❞❛✿ σ + a > 0
❊♥tã♦✿ σ > −a✳
✼✼
❊①❡♠♣❧♦ ✷✿
❈♦♥s✐❞❡r❡♠♦s ♦ s✐♥❛❧
x(t) = −e−at u(−t)✱
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡
X(S) =
Z∞
(−e−at )u(t).e−st dt = −
−∞
−∞
X(S) =
Z0
x(t)✱
❝♦♠
a∈ℜ
✈❡♠ q✉❡✿
e−at .e−st dt = −
Z0
e−(s+a)t dt
−∞
1
1
1
0
e−(s+a)t |−∞ =
(1 − 0) =
s+a
s+a
s+a
■st♦ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ q✉❡✿
limt→−∞ e−(s+a)t = 0
❖ q✉❡ ♦❝♦rr❡ s❡ ❡ s♦♠❡♥t❡ s❡✿
e−(s+a)(−∞) = e(s+a)(∞) = 0
❖✉ s❡❥❛✿
ℜ{s + a} < 0
ℜ{σ + jω + a} < 0 → σ + a < 0
❊♥tã♦✿
σ < −a✳
✼✽
❈♦♥❝❧✉sã♦ s♦❜r❡ ❛ ❘❉❈✿
❈♦♠♦ ♦s s✐♥❛✐s✿ e−atu(t)✱ ❝❛✉s❛❧✱ ❡ −e−atu(−t)✱ ♥ã♦✲❝❛✉s❛❧✱ ♣♦s✲
s✉❡♠ ❛ ♠❡s♠❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ♣♦ré♠ ❝♦♠ r❡❣✐õ❡s ❞❡
❝♦♥✈❡r❣ê♥❝✐❛ ❞✐st✐♥t❛s✱ ❛ ❝♦♥❝❧✉sã♦ q✉❡ s❡ t✐r❛ é✿
P❛r❛ q✉❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ s❡❥❛ ú♥✐❝❛ ❛ ❘❉❈
❞❡✈❡ s❡r ❡s♣❡❝✐✜❝❛❞❛ ❝♦♠♦ ♣❛rt❡ ❞❛ tr❛♥s❢♦r♠❛❞❛✳
❘❡❣✐ã♦ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❙✐♥❛✐s ❞❡
❉✉r❛çã♦ ❋✐♥✐t❛
❈♦♥s✐❞❡r❡ ✉♠ s✐♥❛❧
t1 ❡ t2 ✜♥✐t♦s✳
x(t)
♥ã♦ ♥✉❧♦ s♦♠❡♥t❡ ♣❛r❛
t1 ≤ t ≤ t2
❝♦♠
P❛r❛ ❡st❡ s✐♥❛❧ ✜♥✐t♦✱ ❛❜s♦❧✉t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ❛ ❘❉❈ s❡rá t♦❞♦
♦ ♣❧❛♥♦ s✳
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❯♥✐❞✐r❡❝✐♦♥❛❧
❈♦♠♦ ✈✐st♦ ❛♥t❡s s❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ é ❞♦ t✐♣♦ ❇✐❞✐✲
r❡❝✐♦♥❛❧ ❡❧❛ s❡ ❛♣❧✐❝❛ ❛ s✐♥❛✐s ❝❛✉s❛✐s ❡ ♥ã♦✲❝❛✉s❛✐s✳
P❛r❛ s✐♥❛✐s ❝❛✉s❛✐s✱ ❡st❛ ❛♠❜✐❣✉✐❞❛❞❡ ❞❡s❛♣❛r❡❝❡✱ ❡ ❡①✐st✐rá ✉♠❛
r❡❧❛çã♦ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦ s✐♥❛❧ ❡ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱
s❡♥❞♦ ❞❡s♥❡❝❡ssár✐♦ ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❛ ❘❉❈✳
❈♦♥✈é♠ ❧❡♠❜r❛r q✉❡ ♦s s✐♥❛✐s q✉❡ sã♦ tr❛t❛❞♦s ♥❛ ♣rát✐❝❛ s❡rã♦
s❡♠♣r❡ ❞♦ t✐♣♦ ❝❛✉s❛❧✳
✼✾
❉❛í ♣♦❞❡♠♦s t❡r ✉♠❛ t❛❜❡❧❛ ❞❡ ♣❛r❡s ❞❡ tr❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛✲
♣❧❛❝❡✱ ❝♦♠♦ t❡♠♦s ♣❛r❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳
P❛r❡s ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ■♠♣♦rt❛♥t❡s
x(t)
↔
X(s)
δ(t)
↔
1
u(t)
↔
1
s
t.u(t)
↔
1
s2
tn.u(t)
↔
sn+1
e−at.u(t)
↔
1
s+a
t.e−at.u(t)
↔
1
(s+a)2
cosω0t.u(t)
↔
s2 +ω02
senω0t.u(t)
↔
ω0
s2 +ω02
e−at.cosω0t.u(t)
↔
s+a
(s+a)2 +ω02
e−at.senω0t.u(t)
↔
ω0
(s+a)2 +ω02
n!
s
✽✵
Pr♦♣r✐❡❞❛❞❡s ❞❛ ❚r❛♥s❢✳ ❞❡ ▲❛♣❧❛❝❡
✶✲ ▲✐♥❡❛r✐❞❛❞❡ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
❙❡ t❡♠♦s✿
❊♥tã♦✿
x1 (t) ↔ X1 (s)
❡
x2 (t) ↔ X2 (s)
a1 x1 (t) + a2 x2 (t) ↔ a1 X1 (s) + a2 X2 (s)
✷✲ ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❚❡♠♣♦
x(t − t0 ) ↔ e−st0 .X(s)
✸✲ ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ❉♦♠í♥✐♦ s
es0 t .x(t) ↔ X(s − s0 )
✹✲ ▼✉❞❛♥ç❛ ❞❡ ❊s❝❛❧❛ ❞❡ ❚❡♠♣♦
x(a.t) ↔
1
X( as )
|a|
✺✲ ■♥✈❡rsã♦ ❞❡ ❚❡♠♣♦
x(−t) ↔ X(−s)
✻✲ ❉✐❢❡r❡♥❝✐❛çã♦ ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
d
x(t)
dt
↔ s.X(s)
✼✲ ❉✐❢❡r❡♥❝✐❛çã♦ ♥♦ ❉♦♠í♥✐♦ s
−t.x(t) ↔
d
X(s)
ds
✽✲ ■♥t❡❣r❛çã♦ ♥♦ ❉♦♠í♥✐♦ ❞♦ ❚❡♠♣♦
Rt
1
x(τ
)dτ
↔
.X(s)
−∞
s
✽✶
✾✲ ❈♦♥✈♦❧✉çã♦
❙❡ t❡♠♦s✿
❊♥tã♦✿
x1 (t) ↔ X1 (s)
❡
x2 (t) ↔ X2 (s)
x1 (t) ∗ x2 (t) ↔ X1 (s).X2 (s)
Pó❧♦s ❡ ❩❡r♦s
❯s✉❛❧♠❡♥t❡
X(s)
s❡rá ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❞♦ t✐♣♦✿
a0 (s − z1 )...(s − zm )
a0 sm + a1 sm−1 + ... + am
=
X(s) =
b0 sn + b1 sn−1 + ... + bn
b0 (s − p1 )...(s − pn )
❖♥❞❡✿ ak ❡ bk sã♦ ❝♦♥st❛♥t❡s r❡❛✐s✱ ❡ m ❡ n sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳
❆s r❛í③❡s ❞♦ ♥✉♠❡r❛❞♦r zk sã♦ ❝❤❛♠❛❞♦s ③ér♦s ❞❡ X(s)✱ ♣♦r
❧❡✈❛r❡♠ X(s) ❛ ③❡r♦✱ ❡ s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠ o✳
❆s r❛í③❡s ❞♦ ❞❡♥♦♠✐♥❛❞♦r pk sã♦ ❝❤❛♠❛❞♦s ♣ó❧♦s ❞❡ X(s) ♣♦r
❧❡✈❛r❡♠ X(s) ❛♦ ✐♥✜♥✐t♦✱ ❡ s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠ x✳
P♦rt❛♥t♦ ♦s ♣ó❧♦s ❞❡ X(s) ✜❝❛rã♦ ❢♦r❛ ❞❛ ❘❉❈ ♣♦rq✉❡ X(s) ♥ã♦
❝♦♥✈❡r❣❡ ♥♦s ♣ó❧♦s✳
❖s ③❡r♦s ♣♦r s✉❛ ✈❡③ ♣♦❞❡♠ ✜❝❛r ❞❡♥tr♦ ♦✉ ❢♦r❛ ❞❛ r❡❣✐ã♦ ❞❡
❝♦♥✈❡r❣ê♥❝✐❛✳
2s+4
s+2
❊①❡♠♣❧♦✿ X(s) = s +4s+3
= 2 (s+1)(s+3)
✳
2
✽✷
▼ét♦❞♦s ❞❡ ■♥✈❡rsã♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛
■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡
❊①✐st❡♠ ✈ár✐♦s ♠ét♦❞♦s ♣❛r❛ s❡ ❛❝❤❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛
❞❡ ▲❛♣❧❛❝❡✱ ♦s q✉❛✐s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛ s❡❣✉✐r✿
✶✲ ❯s❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❛ ■♥✈❡rsã♦
❈♦♥❢♦r♠❡ ✈✐st♦ ❛♥t❡s✱ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛ é ✉♠❛
♦♣❡r❛çã♦ q✉❡ ❧❡✈❛ ❛
x(t)
❛ ♣❛rt✐r ❞❡
L−1 {X(s)} = x(t) =
X(s)✱
1
2πj
s❡♥❞♦ ❡①♣r❡ss❛ ❝♦♠♦✿
c+j∞
Z
x(s).est ds
c−j∞
❊st❛ ✐♥t❡❣r❛❧ ❞❡ ❧✐♥❤❛ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ s❡♥❞♦ ❞❡ ❞✐❢í❝✐❧ r❡s♦✲
❧✉çã♦✳
✷✲ ❯s❛♥❞♦ ♦s P❛r❡s ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡
❯♠ ♠ét♦❞♦ ✈✐á✈❡❧ ❞❡ ✐♥✈❡rsã♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ é
t❡♥t❛r ❡①♣r❡ss❛r
X(s)
❝♦♠♦ ✉♠❛ s♦♠❛ ❞♦ t✐♣♦✿
X(s) = X1 (s) + X2 (s) + ... + Xn (s)
❉❡ ♦♥❞❡ ❡♠ s❡ ❝♦♥❤❡❝❡♥❞♦ ❛s tr❛♥s❢♦r♠❛❞❛s ✐♥✈❡rs❛s✿
x1 (t), x2 (t), ...xn (t)✱
❞❡
X1 (s), X2 (s), ...Xn (s)
P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ♣♦❞❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ❞❡t❡r♠✐♥❛r
x(t)✱
❝♦♠♦ s❡♥❞♦✿
x(t) = x1 (t) + x2 (t) + ... + xn (t)
✽✸
✸✲ ❊①♣❛♥sã♦ ❡♠ ❋r❛çõ❡s P❛r❝✐❛✐s
❙❡ X(s) ❢♦r ❡①♣r❡ss❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❞❛ ❢♦r♠❛✿
X(s) =
N (s)
(s − z1 )...(s − zm )
= k.
D(s)
(s − p1 )...(s − pn )
✶➸❈❛s♦✿ ◗✉❛♥❞♦ X(s) ❢♦r ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ♣ró♣r✐❛ ♦✉
s❡❥❛✿ m < n
✭❛✮ Pó❧♦s ❙✐♠♣❧❡s✿ ❙❡ ♦s ♣ó❧♦s ❞❡ X(s) ❢♦r❡♠ s✐♠♣❧❡s ✭❞✐st✐♥✲
t♦s✮ ❡♥tã♦✿
X(s) =
c2
cn
c1
+
+ ... +
s − p1
s − p2
s − pn
❖♥❞❡✿
ck = (s − pk )X(s)|s=pk
❊①❡♠♣❧♦✿
❊♥❝♦♥tr❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛ ❞❡ X(s) =
❙♦❧✉çã♦✿ X(s) =
2s+4
s2 +4s+3
s+2
= 2 s2 +4s+3
=
c1
s+1
+
c2
s+3
ck = (s − pk )X(s)|s=pk ⇒ c1 = (s + 1)X(s)|s=−1 = 2
c2 = (s + 3)X(s)|s=−3 = 2
2s+4
s +4s+3
2
s+2
|s=−1 = 1
s+3
s+2
|s=−3 = 1
s+1
▲♦❣♦✿
X(s) =
1
s+1
+
1
s+3
⇒ x(t) = e−t u(t) + e−3t u(t)✳
✽✹
❙❡ ♦s ♣ó❧♦s ❞❡ X(s) ❢♦r❡♠ ♠ú❧t✐♣❧♦s✱ ♦✉
s❡❥❛ ❝♦♥t❡r ❢❛t♦r❡s (s − pi)r ✱ ♦♥❞❡ ♦ ♣ó❧♦ pi t❡♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ r✱
❡♥tã♦ ❛ ❡①♣❛♥sã♦ ❡♠ ❢r❛çõ❡s ♣❛r❝✐❛✐s ✜❝❛rá✿
✭❜✮ Pó❧♦s ▼ú❧t✐♣❧♦s✿
c1
c2
cn
+
+
...
+
s − pi
(s − pi )2
(s − pi )r
❊①❡♠♣❧♦✿
s +2s+5
❊♥❝♦♥tr❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛ ❞❡ X(s) = (s+3)(s+5)
2
2
❙♦❧✉çã♦✿
X(s) =
s2 +2s+5
(s+3)(s+5)2
=
A
s+3
+
A
s+3
+
B
s+5
+
C
(s+5)2
❊♥tã♦✿
X(s) =
+
B
s+5
C
(s+5)2
=
A(s+5)2 +B(s+3)(s+5)+C(s+3)
(S+3)(S+5)2
⇒ s2 + 2s + 5 = A(s + 5)2 + B(s + 3)(s + 5) + C(s + 3)
⇒ s2 + 2s + 5 = (A + B)s2 + (10A + 8B + C)s + 25A + 15B + 3C
❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✿


A+B =1
10A + 8B + C = 2
 25A + 15B + 3C = 5
▲♦❣♦✿
X(s) =
2
s+3
−
1
s+5
r❡s✉❧t❛ ❡♠✿
−
A=2
B = −1
C = −10
✳
10
(s+5)2
❊ ❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ✜❝❛✿
x(t) = (2e−3t − e−5t − 10te−5t )u(t)
✽✺
✷➸ ❈❛s♦✿ ◗✉❛♥❞♦
♦✉ s❡❥❛✿ m ≥ n
X(s)
❢♦r ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ✐♠♣ró♣r✐❛
◆❡st❡ ❝❛s♦ ♣♦r ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r X(s)
R(s)
N (s)
=
Q(s)
+
❝♦♠♦✿ X(s) =
D(s)
D(s)
❖♥❞❡✿
Q(s) é ♦ ♣♦❧✐♥ô♠✐♦ q✉♦❝✐ê♥t❡✱ ❞❡ ❣r❛✉ m − n
R(s) é ♦ ♣♦❧✐♥ô♠✐♦ r❡st♦ ❞❡ ❣r❛✉ ♠❡♥♦r q✉❡ n✳
P♦❞❡✲s❡ ❛❣♦r❛ ❛❝❤❛r ❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ❞❡ Q(s)✱ ❛tr❛✈és
❞❛ t❛❜❡❧❛ ❞❡ ♣❛r❡s ❞❡ tr❛♥s❢♦r♠❛❞❛✱ ❡ R(s)/D(s) s❡♥❞♦ ✉♠❛
❢✉♥çã♦ r❛❝✐♦♥❛❧ ♣ró♣r✐❛ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ❝♦♠♦ ♥♦ ✐t❡♠ ✭❛✮✱ ♦✉
s❡❥❛ ♣♦r ❢r❛çõ❡s ♣❛r❝✐❛✐s✳
❊①❡♠♣❧♦✿
X(s) =
❆❝❤❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛ x(t) ❞❡
s2 +1
s2 −s−2
❙♦❧✉çã♦✿
X(s) =
s2 +1
s −s−2
2
X(s) = 1 +
=1+
5/3
s−2
❊①❡r❝í❝✐♦s✿
X(s) ❛❜❛✐①♦✿
✭❛✮ X(s) =
✭❜✮ X(s) =
✭❝✮ X(s) =
−
2/3
s+1
s+3
s2 −s−2
=1+
s+3
(s−2)(s+1)
=1+
A
s−2
+
B
s+1
⇒ x(t) = δ(t) + (5/3)e2t − (2/3)e−t
❆❝❤❡ ❛s ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛s x(t) ❞❡
2s+1
s+2
10s+15
(s−1)(s+2)s2
s3 +s2 +s+1
s+1
x(t) = 2δ(t) − 3e−2t u(t)✳
x(t) = ( 25
et +
3
x(t) =
d2 δ(t)
dt
5 −2t
e
12
+ δ(t)
−
35
4
−
15
t)u(t)✳
2
✽✻
✸➸ ❈❛s♦✿ ◗✉❛♥❞♦
❝♦♠♣❧❡①♦s✿
❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡
X(s)
❢♦r ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❝♦♠ ♣ó❧♦s
σ − jω ✱
❡
σ + jω ✱
s❡❥❛♠ ✉♠ ♣❛r ❞❡ ♣ó❧♦s ❝♦♠✲
♣❧❡①♦s ❝♦♥❥✉❣❛❞♦s✱ ♥❡st❡ ❝❛s♦ ❛ ❡①♣❛♥sã♦ ♣♦r ❢r❛çõ❡s ♣❛r❝✐❛✐s ❞❡
X(s)
♣❡r♠✐t❡ ❡s❝r❡✈❡r ♦s s❡❣✉✐♥t❡s t❡r♠♦s ❛ss♦❝✐❛❞♦s ❛♦ ♣❛r ❞❡
♣ó❧♦s✿
B1 s+B2
C1
B1 s+B2
C1
=
+
=
s−σ−jω0
s−σ+jω0
(s−σ−jω0 )(s−σ+jω0 )
(s−σ)2 +ω02
B1 s + B2
Aω0
B(s − σ)
=
+
(s − σ)2 + ω02
(s − σ)2 + ω02
(s − σ)2 + ω02
P♦❞❡✲s❡ ❛❣♦r❛ ❛❝❤❛r ❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ❞❡
X(s)✱
❛tr❛✈és
X(s) =
s+1
s2 +s+1)
❞♦s ♣❛r❡s ❞❡ tr❛♥s❢♦r♠❛❞❛s✿
A(s − σ)
↔ Aeσt cos(ω0 t).u(t)
2
2
(s − σ) + ω0
Bω0
σt
↔
Be
sen(ω0 t).u(t)
(s − σ)2 + ω02
✳
❊①❡♠♣❧♦✿
❆❝❤❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛
x(t)
❞❡
❙♦❧✉çã♦✿
X(s)
1
t❡♠ ✉♠ ♣❛r ❞❡ ♣ó❧♦s ❝♦♠♣❧❡①♦s ❝♦♥❥✉❣❛❞♦s✱ −
2
❊♥tã♦ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r✿
X(s) =
±j
√
3
✳
2
√ s+1
√
(s+1/2+j 3/2)(s+1/2−j 3/2)
❉❡ ♦♥❞❡ ✈❡♠ q✉❡✿
√
B( 3/2)
A(s + 1/2)
+
X(s) =
(s + 1/2)2 + 3/4
(s + 1/2)2 + 3/4
✽✼
❆ss✐♠ ♦ ♥✉♠❡r❛❞♦r ❞❡
X(s)
✜❝❛rá✿
√
N (s) = A(s + 1/2) + B( 3/2) = s + 1
√
N (s) = As + A/2 + B( 3/2) = s + 1
❊♥tã♦✿
A = 1,
❡
A/2 + B
√
3
2
√
= 1 ⇒ B = 1/ 3
❊ ♥❛s ❢r❛çõ❡s ♣❛r❝✐❛✐s t❡r❡♠♦s✿
√
√
(1/ 3)( 3/2)
s+1/2
X(s) = (s+1/2)2+3/4 + (s+1/2)2+3/4
❉❡ ♦♥❞❡ r❡s✉❧t❛ ❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛❞❛s ✐♥✈❡rs❛s✿
√
√
x(t) = e−t/2cos( 23 t)u(t) + √1 e−t/2sen( 23 t)u(t)
3
❊①❡r❝í❝✐♦s✿
X(s)
❛❜❛✐①♦✿
✭❛✮
X(s) =
❆❝❤❡ ❛s ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ✐♥✈❡rs❛s
x(t)
❞❡
2s+12
s2 +2s+5
x(t) = 5e−t sen(2t).u(t) + 2e−t cos(2t).u(t)
✭❜✮
x(t) =
✭❝✮
4s2 +6
(s−1)(s2 +2s+2)
2et u(t) + 2e−t cos(t).u(t)
X(s) =
− 4e−t sen(t).u(t)
s2 +s−2
s3 +3s2 +5s+3
√
−e−t u(t) + 2e−t cos( 2t).u(t)
X(s) =
x(t) =
−
√1 e−t sen(
2
√
2t).u(t)
◆♦ ▼❛t❧❛❜✴❖❝t❛✈❡ ❡①✐st❡ ❛ ❢✉♥çã♦✿ ❬r✱♣✱❦❪❂r❡s✐❞✉❡✭♥s✱❞s✮
q✉❡ ❞❡✈♦❧✈❡ ❝♦♠♦ r❡s✉❧t❛❞♦ ♦s r❡sí❞✉♦s ✭r✮✱ ♣ó❧♦s ✭♣✮ ❡ ❝♦♥st❛♥✲
t❡s ✭❦✮ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧✿ ◆✭s✮✴❉✭s✮✱ s❡♥❞♦ ♥s ❡ ❞s ♦s ✈❡t♦r❡s
❝♦♥st✐t✉✐❞♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ◆✭s✮ ❡ ❉✭s✮✳
❖❜s✿
✽✽
❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛ ✭❋✳❚✳✮
❈♦♠♦ ✈✐st♦ ❛♥t❡s ❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ h(t) ❝❛r❛❝t❡r✐③❛ ❝♦♠♣❧❡✲
t❛♠❡♥t❡ ♦ s✐st❡♠❛✱ ❧♦❣♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ r❡s♣♦st❛
❛♦ ✐♠♣✉❧s♦ H(s) ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛ ✭♦✉
❋✉♥çã♦ ❙✐st❡♠❛✮ t❛♠❜é♠ ❝❛r❛❝t❡r✐③❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ s✐st❡♠❛✳
❊sq✉❡♠❛t✐❝❛♠❡♥t❡✿
H(s) =
Y (s)
X(s)
❋✐❧tr❛❣❡♠
❖♣❡r❛çã♦ ❜ás✐❝❛ ❡♠ q✉❛❧q✉❡r s✐st❡♠❛ ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ s✐✲
♥❛✐s✱ ♣❡r♠✐t❡ ❛❧t❡r❛r✴❡❧✐♠✐♥❛r ❝♦♠♣♦♥❡♥t❡s ❞❡ ❢r❡q✉ê♥❝✐❛ ✐♥❞❡✲
s❡❥á✈❡✐s ❞❡ ✉♠ s✐♥❛❧✳
❋✐❧tr♦s ❞♦ t✐♣♦ P❛ss❛✲❇❛✐①❛s sã♦ r❡q✉❡r✐❞♦s ❡♠ s✐st❡♠❛s ❞❡ ❛q✉✐✲
s✐çã♦ ❞❡ ❞❛❞♦s✱ ♥❛ ❡t❛♣❛ ❛♥t❡r✐♦r à ❛♠♦str❛❣❡♠ ❞♦ s✐♥❛❧ ❞❡
❡♥tr❛❞❛✱ ❞❡ ♠♦❞♦ ❛ ❧✐♠✐t❛r ❞❛ ❧❛r❣✉r❛ ❞❡ ❢❛✐①❛ ❞♦ ❡s♣❡❝tr♦ ❞❡
❢r❡q✉ê♥❝✐❛s ❞❡st❡ s✐♥❛❧✱ ❡✈✐t❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ s♦❜r❡♣♦s✐çã♦ ❞❡
❡s♣❡❝tr♦s✳
❋✐❧tr♦s P❛ss❛✲❆❧t❛s sã♦ r❡q✉❡r✐❞♦s ❡♠ s✐st❡♠❛s ❞❡ ❛q✉✐s✐çã♦ ❞❡
❞❛❞♦s✱ ♥❛ ❡t❛♣❛ ♣♦st❡r✐♦r à ❝♦♥✈❡rsã♦ ❞❡ ❞✐❣✐t❛❧ ♣❛r❛ ❛♥❛❧ó❣✐❝♦✱
❞❡ ♠♦❞♦ ❛ s✉❛✈✐③❛r ♦ s✐♥❛❧ ❞❡ s❛í❞❛ ❛♥❛❧ó❣✐❝♦✳
✽✾
❋✐❧tr♦s ❙❡❧❡t✐✈♦s ❞❡ ❋r❡q✉ê♥❝✐❛s ■❞❡❛✐s
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛s ■❞❡❛❧ ✭❋P❇✮
|H(ω)| =
1, |ω| < ωc
0, |ω| > ωc
❋✐❧tr♦ P❛ss❛✲❆❧t❛s ■❞❡❛❧ ✭❋P❆✮
|H(ω)| =
0, |ω| < ωc
1, |ω| > ωc
❋✐❧tr♦ P❛ss❛✲❋❛✐①❛ ■❞❡❛❧ ✭❋P❋✮
|H(ω)| =
1, ω1 < |ω| < ω2
0,
outros
❋✐❧tr♦ ❘❡❥❡✐t❛✲❋❛✐①❛ ■❞❡❛❧ ✭❋❘❋✮
|H(ω)| =
0, ω1 < |ω| < ω2
1,
outros
✾✵
❘❡s♣♦st❛s ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❋❛s❡
❉❡✜♥✐çõ❡s ■♠♣♦rt❛♥t❡s
✶✳ ▲❛r❣✉r❛ ❞❡ ❋❛✐①❛ ❞♦ ❋✐❧tr♦ ❇❲ ✭❇❛♥❞✇✐❞t❤✮
❆ ❧❛r❣✉r❛ ❞❡ ❢❛✐①❛ ❇❲ ❞❡ ✉♠ ✜❧tr♦ ✐❞❡❛❧ é ❞❡✜♥✐❞♦ ♣❛r❛ ❝❛❞❛
t✐♣♦ ❞❡ ✜❧tr♦ ❝♦♠♦✿
BW = ωC
✿ BW = ω2 − ω1
P❛r❛ ♦ ❋P❋ s❡ BW ≪ ω0 =
❋P❇✿
❋P❋
tr❡✐t❛✳
❋P❆ ❡
❋❘❋
✿ ◆ã♦ s❡ ❞❡✜♥❡
ω2 −ω1
2
✱ ❡♥tã♦ ❡st❡ s❡rá ❞❡ ❢❛✐①❛ ❡s✲
BW
✷✳ ▲❛r❣✉r❛ ❞❡ ❋❛✐①❛ ❞❡ ✸❞❇
P❛r❛ ✜❧tr♦ ❝❛✉s❛✐s ✭♦✉ ♣rát✐❝♦s✮ ❞❡✜♥❡✲s❡ ω3dB √❝♦♠♦ s❡♥❞♦ ❛
❢r❡q✉ê♥❝✐❛ ♣❛r❛ ❛ q✉❛❧ |H(ω)| ❝❛✐ ♣❛r❛ |H(0)|/ 2 ❡♠ i(t) ♦✉
v(t)✳ ❖✉ s❡❥❛✱ ❝♦rr❡s♣♦♥❞❡ ❛ ❛t❡♥✉❛r ♣♦tê♥❝✐❛ à ♠❡t❛❞❡✳
✸✳ ▲❛r❣✉r❛ ❞❡ ❋❛✐①❛ ❞♦ ❙✐♥❛❧
❆ ❧❛r❣✉r❛ ❞❡ ❢❛✐①❛ ❞♦ s✐♥❛❧ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✐♥t❡r✈❛❧♦ ❞❡ ❢r❡q✉ê♥✲
❝✐❛s ❞❡♥tr♦ ❞♦ q✉❛❧ s❡ s✐t✉❛ ❛ ♠❛✐♦r ♣❛rt❡ ❞❛ ❡♥❡r❣✐❛ ❞♦ s✐♥❛❧✳
✹✳ ▲❛r❣✉r❛ ❞❡ ❋❛✐①❛ ❞♦ ❙✐♥❛❧ ♣✴ ✸❞❇
❉❛ ♠❡s♠❛ q✉❡ ❞❡✜♥✐❞♦ ♣❛r❛ ♦ ✜❧tr♦✱ ❞❡✜♥❡✲s❡ ❛ ❧❛r❣✉r❛ ❞❡ ✸❞❇
♣❛r❛ ✉♠ s✐♥❛❧
√ ❝♦♠♦ s❡♥❞♦ ❛ ❢r❡q✉ê♥❝✐❛ ♣❛r❛ ❛ q✉❛❧ |X(ω)| ❝❛✐
♣❛r❛ |X(0)|/ 2 ♣r♦✈♦❝❛♥❞♦ ❛t❡♥✉❛çã♦ ❞❛ ♣♦tê♥❝✐❛ ♣✴ ♠❡t❛❞❡✳
✾✶
✺✳ ❙✐♥❛❧ ❞❡ ❋❛✐①❛ ▲✐♠✐t❛❞❛
❯♠ s✐♥❛❧ t❡rá s✉❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛ ❛
ωM
s❡✿
|X(ω)| = 0, |ω| > ωM
✻✳ ❋❛✐①❛s ❞❡ P❛ss❛❣❡♠✱ ❚r❛♥s✐çã♦ ❡ ❈♦rt❡
✼✳ ❙❡❧❡t✐✈✐❞❛❞❡ ❞♦s ❋P❋ ❡ ❋❘❋ ✲ ❋❛t♦r ❞❡ ◗✉❛✲
❧✐❞❛❞❡
❉❡✜♥❡✲s❡ ❋❛t♦r ❞❡ ◗✉❛❧✐❞❛❞❡ ❞❡ ✉♠ ❋✐❧tr♦ P❛ss❛✲❋❛✐①❛ ♦✉ ❘❡✲
❥❡✐t❛ ❋❛✐①❛ ❝♦♠♦ s❡♥❞♦✿
Q0 =
❈♦♠✿ ω0 = ω2 − ω1
◗✉❛♥t♦ ♠❛✐♦r ♦ ✈❛❧♦r ❞❡
Q0
ω0
BW
♠❛✐♦r ❛ s❡❧❡t✐✈✐❞❛❞❡ ❞♦ ✜❧tr♦✳
✾✷
❋✐❧tr♦s ❙❡❧❡t✐✈♦s ❞❡ ❋r❡q✉ê♥❝✐❛s ◆ã♦✲■❞❡❛✐s
✭❈❛✉s❛✐s ♦✉ Prát✐❝♦s✮
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛s ✭❋P❇✮ t✐♣♦ ❘❈ ❞❡ ✶➟ ❖r❞❡♠
❯♠ ✜❧tr♦ ❞♦ t✐♣♦ P❛ss❛✲❇❛✐①❛s ❝❛✉s❛❧ ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦
❝♦♠♦ ♦ ❝✐r❝✉✐t♦ ❘❈ ♠♦str❛❞♦ ❛❜❛✐①♦✱ ♦♥❞❡ x(t) r❡♣r❡s❡♥t❛ ❛
❡♥tr❛❞❛ ❞♦ ✜❧tr♦ ❡ y(t) ❛ s✉❛ s❛í❞❛✿
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ▲❈❑ ♥♦ ♥ó ❆✱ ✈❡♠
q✉❡✿
dy(t)
iR = iC ⇒ x(t)−y(t)
=
C
R
dt
❉❡ ♦♥❞❡ r❡s✉❧t❛✿ RC dy(t)
dt + y(t) = x(t)
❊♥tã♦✱ ♦❜t❡♥❞♦✲s❡ ❛ ❚r❛♥❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱
♦♥❞❡ s❡ ❛♣❧✐❝♦✉ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦✱ ✈❡♠ q✉❡✿
R.C.s.Y (s) + Y (s) = X(s)
Y (s)
1
= 1+sRC
= H(s)
Y (s).(R.C.s + 1) = X(s) ⇒ X(s)
❋❛③❡♥❞♦✲s❡
ω0 =
1
RC
♥❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r✱ ✜❝❛✿
1
⇒ ❋✳ ❚✳ ❞♦ ❋P❇ ❘❈ ✶➟❖r❞❡♠✳
H(s) = 1+s/ω
0
❊♥tã♦✱ ❛ ❘❡s♣♦st❛ ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❞❡ ❋❛s❡ s❡rã♦ ❞❛❞❛s ♣♦r✿
1
1
p
|H(s)|s=jω = |1+jω/ω
=
0|
1+(ω/ω0 )2
θ(ω) = − arctan(ω/ω0)
✾✸
❋✐❧tr♦ P❛ss❛✲❆❧t❛s ✭❋P❆✮ t✐♣♦ ❘❈ ❞❡ ✶➟ ❖r❞❡♠
❯♠ ✜❧tr♦ ❞♦ t✐♣♦ P❛ss❛✲❆❧t❛s ❝❛✉s❛❧ ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦
❝♦♠♦ ♦ ❝✐r❝✉✐t♦ ❘❈ ♠♦str❛❞♦ ❛❜❛✐①♦✱ ♦♥❞❡ x(t) r❡♣r❡s❡♥t❛ ❛
❡♥tr❛❞❛ ❞♦ ✜❧tr♦ ❡ y(t) ❛ s✉❛ s❛í❞❛✿
❆♣❧✐❝❛♥❞♦✲s❡ ❛ ▲❈❑ ♥♦ ♥ó ❆✱ ✈❡♠
q✉❡✿
y(t)
=
iR = iC ⇒ C d[x(t)−y(t)]
dt
R
= y(t)
❉❡ ♦♥❞❡ r❡s✉❧t❛✿ R.C. d[x(t)−y(t)]
dt
❊♥tã♦✱ ♦❜t❡♥❞♦✲s❡ ❛ ❚r❛♥❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱
♦♥❞❡ s❡ ❛♣❧✐❝♦✉ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦✱ ✈❡♠ q✉❡✿
R.C.s.[X(s) − Y (s)] = Y (s)
R.C.s.X(ω) = Y (s).(R.C.s + 1)
Y (s)
RCs =
1
=
sRC+1
X(s)
1+1/(sRC)
Y (s)
1
=
= H(s)
X(s)
1+1/(sRC)
❋❛③❡♥❞♦✲s❡
ω0 =
1
RC
♥❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r✱ ✜❝❛✿
1
H(s) = 1+1/(s/ω
⇒ ❋✳ ❚✳ ❞♦ ❋P❆ ❘❈ ✶➟❖r❞❡♠✳
0)
❊♥tã♦✱ ❛ ❘❡s♣♦st❛ ❞❡ ❆♠♣❧✐t✉❞❡ ❡ ❞❡ ❋❛s❡ s❡rã♦ ❞❛❞❛s ♣♦r✿
1
1
p
|H(s)|s=jω = |1+1/(jω/ω
=
0 )|
1+1/(ω/ω0 )2
θ(ω) = − arctan(1/(ω/ω0))
✾✹
❋✐❧tr♦s ❘▲❈ ❞❡ ✷➸ ❖r❞❡♠
❆ ♣❛rt✐r ❞❛ r❡s♣♦st❛ ❛♦ ✐♠♣✉❧s♦ ❞♦s ✜❧tr♦s ♣♦❞❡✲s❡ ♦❜t❡r ❛s ❝❛✲
r❛❝t❡ríst✐❝❛s ❞❡st❡s✱ ❜❡♠ ❝♦♠♦ s✉❛ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥✲
❝✐❛✱ ♣ó❧♦s✱ ❡ ③❡r♦s✳
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛ ❞❡ ✷➟❖r❞❡♠
❈♦♥s✐❞❡r❡ ♦ ❝✐r❝✉✐t♦ ❘▲❈ ❛❜❛✐①♦✿
P❛r❛ s❡ ♦❜t❡r ❍✭s✮ ❞❡st❡ ❝✐r❝✉✐t♦✱ ❛♣❧✐❝❛✲s❡ ♦ ✐♠♣✉❧s♦ ♥❛ ❡♥tr❛❞❛
x(t)✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ X(s) = 1✱ ❡♥tã♦✿
Y (s)
= Y (s)
H(s) = X(s)
❈♦♥s✐❞❡r❛♥❞♦✲s❡ ❛s ✐♠♣❡❞â♥❝✐❛s ❡♠ ❘✱ ▲✱ ❡ ❈✱ ❝♦♠♦ ZR = R✱
ZL = SL✱ ❡ ZC = 1/SC ✱ ❡ s❛❜❡♥❞♦✲s❡ q✉❡ H(s) = Y (s) = V C (s)✱
t❡r❡♠♦s✿
i (s).ZC
H(s) = Z V+Z
=
C
L +ZR
1.(1/SC)
1
SC +SL+R
❊st❛ ❡①♣r❡ssã♦ ❛♣ós ❛❧❣✉♠ ❛❧❣❡❜r✐s♠♦ ❝♦♥❞✉③ à✿
H(s) =
1/LC
1
S +S RL + LC
2
◗✉❡ á ❛ ❡①♣r❡ssã♦ ❞❡ ✉♠ ✜❧tr♦ P❛ss❛✲❇❛✐①❛s ❞❡ ✷➟ ❖r❞❡♠✱ ♣♦✲
❞❡♥❞♦ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
ωc2
H(s) = 2
s + 2ξωc s + ωc2
✾✺
❖♥❞❡✿
ξ
é ♦ ❢❛t♦r ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❞♦ ✜❧tr♦❀
ωc
❛ ❢r❡q✳ ❞❡ ❝♦rt❡✳
❆❧é♠ ❞✐st♦✱ s❡ ♥♦t❛ q✉❡✿
√
ωc = 1/ LC ✱ 2ξωc = R/L✱
s❡♥❞♦ ❛✐♥❞❛ ♦ ❢❛t♦r
Q = 1/2ξ
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ❞❡st❡ ✜❧tr♦ s❡rá ❞❛❞❛ ♣♦r
H(ω) ♦❜t✐❞❛ ❝♦♠♦✿
H(ω) = H(s)|s=jω = |H(jω)|
❊♥tã♦✿
H(jω) =
1/LC
1
−ω 2 +jω RL + LC
=
1/LC
1
( LC
−ω 2 )+jω RL
⇒H(ω) = √
(
1
LC
1/LC
−ω 2 )+(ω RL )2
◗✉❡ s❡ ♥♦r♠❛❧✐③❛❞♦ ♣❛r❛
❘❂▲❂❈❂✶✱ ✜❝❛rá✿
1
H(ω) = q
(1 − ω 2) + ω 2
✾✻
❋✐❧tr♦ P❛ss❛✲❆❧t❛s ❞❡ ✷➟❖r❞❡♠
❈♦♥s✐❞❡r❡ ♦ ❝✐r❝✉✐t♦ ❘▲❈ ❛❜❛✐①♦✿
❙❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❍✭s✮ ♣❛r❛ ❡st❡
❝✐r❝✉✐t♦✱ s❡rá✿
Y (s)
H(s) = X(s)
= Y (s)
❈♦♥s✐❞❡r❛♥❞♦✲s❡ ♥♦✈❛♠❡♥t❡ ❛s ✐♠♣❡❞â♥❝✐❛s ❡♠ ❘✱ ▲✱ ❡ ❈✱ ❝♦♠♦
ZR = R✱ ZL = SL✱ ❡ ZC = 1/SC ✱ t❡r❡♠♦s✿
i (s).ZL
=
H(s) = Z V+Z
C
L +ZR
1.SL
S 2L
=
1
1
S 2 L+SR+ C
SC +SL+R
❊st❛ ❡①♣r❡ssã♦ ❛♣ós ❛❧❣✉♠ ❛❧❣❡❜r✐s♠♦ ❝♦♥❞✉③ à✿
H(s) =
S2
1
S +S RL + LC
2
◗✉❡ á ❛ ❡①♣r❡ssã♦ ❞❡ ✉♠ ✜❧tr♦ P❛ss❛✲❆❧t❛s ❞❡ ✷➟ ❖r❞❡♠✱ ♣♦✲
❞❡♥❞♦ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
2
S
H(s) = s2+2ξω
c s+ωc2
❖♥❞❡✿
√
ωc = 1/ LC ✱ 2ξωc = R/L✱
❡
Q = 1/2ξ
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ❞❡st❡ ✜❧tr♦ s❡rá ❞❛❞❛ ♣♦r
H(ω) ♦❜t✐❞❛ ❝♦♠♦✿
H(ω) = H(s)|s=jω = |H(jω)|
✾✼
❊♥tã♦✿
H(jω) =
−ω 2
1
−ω 2 +jω RL + LC
=
−ω 2
1
( LC −ω 2 )+jω RL
⇒H(ω) = √
√
(−ω 2 )2
1
( LC
−ω 2 )+(ω RL )2
◗✉❡ s❡ ♥♦r♠❛❧✐③❛❞♦ ♣❛r❛ ❘❂▲❂❈❂✶
❋✐❝❛rá✿
ω2
H(ω) = p
(1 − ω 2 ) + ω 2
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ❞❡st❡ ✜❧tr♦ ♣♦❞❡ s❡r ❡s❜♦ç❛❞❛
❝♦♠♦✿
❋✐❧tr♦ P❛ss❛✲❋❛✐①❛ ❞❡ ✷➟❖r❞❡♠
❈♦♥s✐❞❡r❡ ♦ ❝✐r❝✉✐t♦ ❘▲❈ ❛❜❛✐①♦✿
❙❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❍✭s✮ ♣❛r❛ ❡st❡
❝✐r❝✉✐t♦✱ s❡rá✿
Y (s)
H(s) = X(s)
= Y (s)
✾✽
❈♦♥s✐❞❡r❛♥❞♦✲s❡ ♥♦✈❛♠❡♥t❡ ❛s ✐♠♣❡❞â♥❝✐❛s ❡♠ ❘✱ ▲✱ ❡ ❈✱ ❝♦♠♦
ZR = R✱ ZL = SL✱ ❡ ZC = 1/SC ✱ t❡r❡♠♦s✿
i (s).R
=
H(s) = Z V+Z
C
L +ZR
R
SR
=
1
1
S 2 L+SR+ C
SC +SL+R
❊st❛ ❡①♣r❡ssã♦ ❛♣ós ❛❧❣✉♠ ❛❧❣❡❜r✐s♠♦ ❝♦♥❞✉③ à✿
H(s) =
S.R/L
1
S +S RL + LC
2
◗✉❡ á ❛ ❡①♣r❡ssã♦ ❞❡ ✉♠ ✜❧tr♦ P❛ss❛✲❋❛✐①❛ ❞❡ ✷➟ ❖r❞❡♠✱ ♣♦✲
❞❡♥❞♦ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
S.2ξωc
H(s) = s2+2ξω
c s+ωc2
√
❖♥❞❡✿ ωc = 1/ LC ✱ 2ξωc = R/L✱ ❡ Q = 1/2ξ
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ❞❡st❡ ✜❧tr♦ s❡rá ❞❛❞❛ ♣♦r
H(ω) ♦❜t✐❞❛ ❝♦♠♦✿
H(ω) = H(s)|s=jω = |H(jω)|
❊♥tã♦✿
H(jω) =
jωR/L
1
2
−ω +jω RL + LC
=
jωR/L
1
( LC −ω 2 )+jω RL
⇒H(ω) = √
◗✉❡ s❡ ♥♦r♠❛❧✐③❛❞♦ ♣❛r❛ ❘❂▲❂❈❂✶
❋✐❝❛rá✿
√
(ωR/L)2
1
−ω 2 )+(ω RL )2
( LC
ω
H(ω) = p
(1 − ω 2 ) + ω 2
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ❞❡st❡ ✜❧tr♦ ♣♦❞❡ s❡r ❡s❜♦ç❛❞❛
❝♦♠♦✿
✾✾
❙í♥t❡s❡ ❞❡ ❋✐❧tr♦s
P❛r❛ s❡ ♣♦❞❡r s✐♥t❡t✐③❛r ✉♠ ✜❧tr♦ P❛ss❛✲❇❛✐①❛s ❞❡✈❡✲s❡ ♣❛rt✐r ❞❛
❘❡s♣♦st❛ ❡♠ ▼❛❣♥✐t✉❞❡ ◗✉❛❞rát✐❝❛✿ A2(ω)✱ ❡ s❛❜❡♥❞♦✲s❡ q✉❡✿
A2 (ω) = |H(jω)| = H(jω).H ∗ (jω) = H(s).H(−s)|s=jω
❞❡r✐✈❛✲s❡ ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛
❊①✿
❉❡t❡r♠✐♥❡
A2 (ω)
H(s)✳
s❡ ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ é✿
s2 + 1
H(s) = 2
s + 4s + 2
❙♦❧✉çã♦✿
s2 + 1
H(−s) = 2
s − 4s + 2)
❊♥tã♦✿
H(s).H(−s) =
❊ ✜♥❛❧♠❡♥t❡✿
s2 +1
s2 +1
.
s2 +4s+2 s2 −4s+2)
=
s4 +2s2 +1
s4 −12s2 +4
A2 (ω) = H(s).H(−s)|s=jω =
ω 4 −2ω 2 +1
ω 4 +12ω 2 +4
❉❡ ♦✉tr❛ ♠❛♥❡✐r❛✱ s❡ ❛ ❡①♣r❡ssã♦ ❞❡ A2(ω) é ❝♦♥❤❡❝✐❞❛✱ ♣♦❞❡✲s❡
♦❜t❡r H(s) s✉❜st✐t✉✐♥❞♦✲s❡ (jω)2 = −ω2 = s2✳
❊①✿
❉❡t❡r♠✐♥❡
é✿
A2 (ω)
H(s)
s❡ ❛ r❡s♣♦st❛ ❡♠ ♠❛❣♥✐t✉❞❡ q✉❛❞rát✐❝❛
16(−ω 2 + 1)
A (ω) =
(ω 2 + 4)(ω 2 + 9)
2
❙♦❧✉çã♦✿
H(s).H(−s) = A2 (ω)|ω2 =−s2 =
16(s2 +1)
(−s2 +4)(−s2 +9)
❊st❛ ❡①♣r❡ssã♦ ♣♦ss✉✐ ③❡r♦s ❡♠✿
s = ±j ✱
❡ ♣ó❧♦s ❡♠✿
s = ±2,±3✳
✶✵✵
❊♥tã♦ ♣♦r ✉♠❛ q✉❡stã♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ H(s) ❞❡✈❡rá ❝♦♥t❡r ❛♣❡✲
♥❛s ♦s ♣ó❧♦s s✐t✉❛❞♦s ♥♦ s❡♠✐✲♣❧❛♥♦ ❡sq✉❡r❞♦ ❞♦ ♣❧❛♥♦ s✱ ♦✉
s❡❥❛✿ s = −2, ❡ s = −3✳
▲♦❣♦✿
K.(s2 + 1)
H(s) =
(s + 2)(s + 3)
❆❧é♠ ❞✐st♦✱
√ ♣❛r❛ s❡ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ K ✱ ❞❡✈❡✲s❡ t❡r H(0) =
A(0) = 4/ 36 = 2/3
❖✉ s❡❥❛✿
❈♦♠♦✿
A2 (0) = 16/36 = 4/9⇒A(0) = 2/3
H(0) = K/6 ⇒K/6 = 2/3 ⇒K = 4
❋✐♥❛❧♠❡♥t❡✿
4.(s2 + 1)
H(s) =
(s + 2)(s + 3)
❙í♥t❡s❡ ❞❡ ❋P❇ ❞❡ ❇✉tt❡r✇♦rt❤
❆ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ♠❛❣♥✐t✉❞❡ ❞❡ ✉♠ ✜❧tr♦ ❞❡ ❇✉tt❡r✇♦t❤
é ❞❛❞❛ ♣♦r✿
A
|H(ω)| = q
1 + ( ωωc )2k
❖♥❞❡✿ ωc é ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡✱ A é ♦❣❛♥❤♦✱ ❡
♣♦s✐t✐✈♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ♦r❞❡♠ ❞♦ ✜❧tr♦✳
◆♦r♠❛❧✐③❛♥❞♦✲s❡ ♣❛r❛
ω c = 1✱
✜❝❛✿
A
|HN (ω)| = p
1 + ω 2k
k
é ✉♠ ✐♥t❡✐r♦
❆s ✈ár✐❛s ❝✉r✈❛s ❞❡ r❡s♣♦st❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ k ♥❛ ❡①♣r❡ssã♦
❛♥t❡r✐♦r sã♦ ♠♦str❛❞♦s ❛❜❛✐①♦✿
❚♦❞♦s ♦s ✜❧tr♦s ❞❡ ❇✉tt❡r✇♦rt❤ ❝♦♠ s✉❛ r❡s♣♦st❛ ♥♦r♠❛❧✐③❛❞❛
✭ωc = 1 r❛❞✴s✮ t❡rá ♦s ♣ó❧♦s ❞❛ s✉❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ H(s)
s♦❜r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✉♥✐tár✐♦ ♥♦ ♣❧❛♥♦ s✱ ✜❝❛♥❞♦
s❡♣❛r❛❞♦s ❞❡ π/k✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ❡①❡♠♣❧♦ ❞❡ ❝♦♠♦ ✜❝❛♠ ♦s
✻ ♣ó❧♦s ❞❡ H(s) ❞❡ ♦r❞❡♠ ✸✳
◗✉❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ t❡♠ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ♣ó✲
❧♦s✱ s❡♠♣r❡ ❤❛✈❡rá ✉♠ ♣ó❧♦ ❡♠ s❂✲✶✳ ❆❧é♠ ❞✐st♦✱ t♦❞♦s ♦s ♣ó❧♦s
❡st❛rã♦ ❡♠ s✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ jω✳ P♦r q✉❡stõ❡s ❞❡ ❡st❛✲
❜✐❧✐❞❛❞❡ ♥❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ✜❧tr♦ s♦♠❡♥t❡ ♦s ♣ó❧♦s ❧♦❝❛❧✐③❛❞♦s
♥♦ s❡♠✐♣❧❛♥♦ ❡sq✉❡r❞♦ ❢♦r♠❛rã♦ ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛
✶✵✶
P❛r❛ ✉♠ ❋P❇ ❇✉tt❡r✇♦rt❤ ❞❡ ✸➟ ♦r❞❡♠✱ t❡♠✲s❡ k =
3✱ ❡ ❛ ❡①♣r❡ssã♦ ❞❛ r❡s♣♦st❛ ❡♠ ❢r❡q✉ê♥❝✐❛ ♥♦r♠❛❧✐③❛❞❛ s❡rá✿
❊①❡♠♣❧♦✿
HN (ω) = p
1
1 + ω6
❊♥tã♦✱ ♣❛rt✐♥❞♦✲s❡ ❞❛ r❡s♣♦st❛ ❡♠ ♠❛❣♥✐t✉❞❡ q✉❛❞rát✐❝❛✿
1
A2 (ω) = 1+ω
6 ✱ ♣❛r❛ s❡ ❛❝❤❛r ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡st❡ ✜❧tr♦
❢❛③✲s❡✿ ω2 = −s2✱ ❡♠✿ H(s).H(−s) = A2(ω)|ω2=−s2 ✱ ♦❜t❡♥❞♦✲s❡✿
H(s).H(−s) =
❈✉❥♦s ♣ó❧♦s s❡ ❧♦❝❛❧✐③❛♠ ❡♠✿
1
1 − s6
s = ±1, s =
− 21
±j
√
3
2
✱s=
1
2
±j
√
3
2
✳
P♦ré♠✱ ❝♦♠♦ ❡①♣❧✐❝❛❞♦ ❛♥t❡s s♦♠❡♥t❡ ♦s ♣ó❧♦s ❞♦ s❡♠✐✲♣❧❛♥♦
❡sq✉❡r❞♦ ❡st❛rã♦ ❡♠ H(s)✿
H(s) =
K
(s + 1)(s +
❖ q✉❡ r❡s✉❧t❛ ❡♠✿
H(s) =
❙❡♥❞♦ ♦ ❣❛♥❤♦
K
1
2
−j
√
3
)(s
2
+
1
2
+j
√
3
)
2
K
s3 + 2s2 + 2s + 1
❝❛❧❝✉❧❛❞♦ ❞❡
A2 (0) = H(0) =⇒1 = K
❋✐♥❛❧♠❡♥t❡✿
H(s) =
1
s3 + 2s2 + 2s + 1
✶✵✷
❉❡ ❢♦r♠❛ s✐st❡♠❛t✐③❛❞❛✱ ♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ❇✉tt❡r✇♦t❤ ♥♦r♠❛❧✐③❛✲
❞♦s✱ q✉❡ ❝♦♥st✐t✉❡♠ ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❛ ❋✳❚✳✱ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s
❝♦♠ ❛ ❡①♣r❡ssã♦✿
Bk (s) = P (s).
❙❡♥❞♦✿
P (s) =
Y
(s2 + 2cos(θN )s + 1)
1,
k = par
s + 1, k = ı́mpar
❝♦♠✿ 2.cos(θ) = 2ξ = 1/Q
❖♥❞❡✿ k =❖r❞❡♠ ❞♦ ✜❧tr♦✱ θ =➶♥❣✉❧♦ ❡♥tr❡♦ ♣ó❧♦ ❡ ♦ ❡✐①♦ r❡❛❧✱
ξ =❈♦❡✜❝✐ê♥t❡ ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❞♦ ✜❧tr♦✳
• ❙❡ k =❮♠♣❛r ⇒ ❡①✐st✐rá ✉♠❛ r❛✐③ ❡♠ θ = 0➸❀
• ❙❡ k =P❛r ⇒ θ = ±90➸/k❀
• ❖s ♣ó❧♦s ❡st❛rã♦ s❡♠♣r❡ s❡♣❛r❛❞♦s ❞❡ π/k✳
P♦r ❡①❡♠♣❧♦✱ ♣❛r❛
k =✶✱
✷✱ ✸ ❡ ✹✱ ♦s ♣♦❧✐♥ô♠✐♦s ✜❝❛r✐❛♠✿
k = 1 ⇒ B1 (s) = s + 1
❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂
= ± π/2
= ± π4
k = 2 ⇒ ❦ é P❛r ⇒ θ1,2 = ± π/2
k
2
⇒ B2 (s) =
s2
+ 2cos(π/4) + 1 =
s2
+
√
2s + 1
❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂
k = 3 ⇒ ❦ é ❮♠♣❛r ⇒ θ1 = 0✱ θ2,3 = π/3
⇒ B3 (s) = (s + 1)(s2 + 2cos(π/3) + 1) = (s + 1)(s2 + s + 1)
❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂
= ± π8 ✱ θ3,4 = ± π/2
+ πk = ± 3π
k = 4 ⇒θ1,2 = ± π/2
k
k
8
⇒ B4 (s) = (s2 + 2cos(3π/8).s + 1).(s2 + 2cos(π/8).s + 1)
⇒ B4 (s) = (s2 + 0, 7654.s + 1).(s2 + 1, 8478.s + 1)
✶✵✸
◆❛ t❛❜❡❧❛ ❛ s❡❣✉✐r✱ ❡stã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♣♦❧✐♥ô♠✐♦s ❞♦ ❞❡✲
♥♦♠✐♥❛❞♦r ❞❛ ❋❚ ♣❛r❛ ♦s ❋P❇s ❞❡ ❇✉tt❡r✇♦t❤ ♥♦r♠❛❧✐③❛❞♦s
✭ωc = 1rad/s✮ ❞❡ ✶➟ ❛ ✼➟ ♦r❞❡♠✿
❦
P♦❧✐♥ô♠✐♦s ❞♦ ❉❡♥♦♠✐♥❛❞♦r ❞❛ ❋✳❚✳
✶
✷
✸
✹
✺
✻
✼
s+1
s2 + 1, 4142s + 1
(s + 1)(s2 + s + 1)
(s2 + 0, 7654s + 1)(s2 + 1, 8478s + 1)
(s + 1)(s2 + 0, 6180s + 1)(s2 + 1, 6180s + 1)
(s2 0, 5176s + 1)(s2 + 1.4142s + 1)(s2 + 1, 9318s + 1)
(s + 1)(s2 + 0, 4449s + 1)(s2 + 1, 2465s + 1)(s2 + 1, 8022s + 1)
P❛r❛ q✉❡ s❡ ♣♦ss❛ ❛❝❤❛r ❛ ❋✳❚✳ ♣❛r❛ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ωc
q✉❛❧q✉❡r✱ ❛ ♣❛rt✐r ❞❛ ❋✳❚✳ ♥♦r♠❛❧✐③❛❞❛ HN (s) ❞❡✈❡✲s❡ ❢❛③❡r ✉♠
r❡❡s❝❛❧❛♠❡♥t♦✿
H(s) = HN (
s
)
ωc
P♦r ❡①❡♠♣❧♦✱ s❡❥❛ ❛ ❋✳❚✳ ❞❡ ✉♠ ✜❧tr♦ P❛ss❛✲❇❛✐①❛s ❇✉tt❡r✇♦t❤
❝♦♠♦ ❛❜❛✐①♦✿
1
(s2 + 0, 7654s + 1)(s2 + 1, 8478s + 1)
HN (s) =
❙❡ ❞❡s❡❥❛✲s❡ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ❞❡ ✺r❛❞✴s✱ ♦ r❡❡s❝❛❧❛♠❡♥t♦
s❡r✐❛ ❝♦♠♦ s❡❣✉❡✿
H(s) =
H(s) =
1
2
s
+
( 25
0,7654s
5
2
s
+ 1)( 25
+
1,8478s
5
+ 1)
=
625
(s2 + 3, 827s + 25)(s2 + 9, 239s + 25)
❖ ❝♦♥❞✉③✐rá ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✿
H(s) =
625
(s4 + 13, 066s3 + 85, 385s2 + 326, 650s + 625)
✶✵✹
Pr♦❥❡t♦ ❞❡ ❋✐❧tr♦s ❆t✐✈♦s
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛s t✐♣♦ ❇✉tt❡r✇♦t❤ ❞❡ ✶➟❖r❞❡♠
P❛r❛ ♦ ✜❧tr♦ ❛♣r❡s❡♥t❛❞♦ ♦ ❣❛♥❤♦ ✭❆✮ s❡rá ❞❡✿
A=1+
R3
R2
✭✶✮
P❛r❛ s❡ ♠✐♥✐♠✐③❛r ❛ t❡♥sã♦ ❞❡ ♦✛s❡t✱ ❞❡✈❡♠♦s t❡r✿
R1 =
R2.R3
R2 + R3
✭✷✮
❊ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ✭fC ✮ ❡ R1 s❡rá✿
1
R1 =
2.π.fC .C
✭✸✮
✶✵✺
■s♦❧❛♥❞♦ R3 ❡♠ ✭✶✮✱ ❡ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✷✮✱ ♦❜té♠✲s❡✿
R2 = (
A
).R1
A−1
R3 = A.R1
❙❡✿
✭✹✮
✭✺✮
A = 1✱ R 2 = ∞ ✱ ❡ R 3 = 0
❖ ✈❛❧♦r ❞❡ C ❞❡✈❡rá s❡r✿
C = 10
fc ✱ ✭❡♠ µF ✮
Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ♣r♦❥❡t♦✿
✶✲ ❊st❛❜❡❧❡❝❡r ♦ ❣❛♥❤♦ A❀
✷✲ ❊st❛❜❡❧❡❝❡r ❛ ❋r❡q✉❡♥❝✐❛ ❞❡ ❈♦rt❡ fC ❀
✸✲ ❉❡t❡r♠✐♥❛r R1 ❛tr❛✈és ❞❛ ❡q✉❛çã♦ ✭✸✮✱ s❛❜❡♥❞♦✲s❡
q✉❡ C = 10
f ✱ ✭❡♠ µF ✮❀
c
✹✲ ❉❡t❡r♠✐♥❛r R2 ❛tr❛✈és ❞❛ ❡q✉❛çã♦ ✭✹✮❀
✺✲ ❉❡t❡r♠✐♥❛r R3 ❛tr❛✈és ❞❛ ❡q✉❛çã♦ ✭✺✮❀
✻✲ ❆❥✉st❛r ♥❛ ♣rát✐❝❛ ♦ ❣❛♥❤♦ A ❛tr❛✈és ❞❡ R2 ♦✉ R3❀
✼✲ ❆❥✉st❛r ♥❛ ♣rát✐❝❛ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ❞❡ −3dB
❛tr❛✈és ❞❡ R1✳
✶✵✻
❊①❡♠♣❧♦ ❞❡ ♣r♦❥❡t♦ ❞❡ ✜❧tr♦ ❛t✐✈♦ ❞❡ ✶➟❖r❞❡♠
✶✮ Pr♦❥❡t❛r ✉♠ ✜❧tr♦ P❛ss❛✲❇❛✐①❛s ❞❡ ✶➟❖r❞❡♠ t✐♣♦
❇✉tt❡r✇♦t❤✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ❞❡ ✶✵❦❍③✳
❙♦❧✉çã♦✿
✶✲ ❆r❜✐tr❛♥❞♦ ✉♠ ❣❛♥❤♦ ❞❡
✷✲ ❋r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡
✸✲ ❈♦♠✿
A = 2❀
fc = 10kHz ❀
10 = 10−3 µF = 1nF
=
C = 10
fc
10k
❙❡ ♦❜té♠✿
1
1
R1 = 2.π.f
=
= 15, 9kΩ
c .C
2.π.104 .10−9
✹✲ ❊♥tã♦✿
A ).R = ( 2 ).15, 9k = 31, 8kΩ
R2 = ( A−1
1
2−1
✺✲ ❆✐♥❞❛✿
R3 = 2.R1 = 31, 8kΩ
✻✱✼✲ Pr♦❝❡❞✐♠❡♥t♦ ♣rát✐❝♦ ❞❡ ❛❥✉st❡ ❞♦ ❣❛♥❤♦ ❡ ❞❛
❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡✳
✶✵✼
Pr♦❥❡t♦s ❞❡ ❋✐❧tr♦s ❆t✐✈♦s ✉s❛♥❞♦ ❛ ❊str✉t✉r❛
❙❛❧❧❡♥✲❑❡②
❆ ❡str✉t✉r❛ ❙❛❧❧❡♥✲❑❡② s❡ ❞❡✈❡ ❛♦s s❡✉s ✐❞❡❛❧✐③❛❞♦r❡s✱ ♦s ♣❡s✲
q✉✐s❛❞♦r❡s ❞♦ ▲✐♥❝♦♥ ▲❛❜♦r❛t♦r② ❞♦ ▼■❚✱ ❘✳P✳❙❛❧❧❡♥ ❡ ❊✳▲✳❑❡②✳
❆ ❡str✉t✉r❛ ❣❡r❛❧ ❝♦♠ ❣❛♥❤♦ ✉♥✐tár✐♦ ❡ ❝♦♥✜❣✉r❛çã♦ ✐♥✈❡rs♦r❛ é
♠♦str❛❞❛ ❛❜❛✐①♦✿
❊ ❝✉❥❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ é ❛ s❡❣✉✐♥t❡✿
H(s) =
Z3 (s).Z4 (s)
Z1 (s).Z2 (s) + Z4 (s)[Z1 (s) + Z2 (s)] + Z3 (s).Z4 (s)
❊st❛ ❡str✉t✉r❛ ❛♣r❡s❡♥t❛ ✉♠❛ ✐♠♣❡❞â♥❝✐❛ ❞❡ ❡♥tr❛❞❛ ♣r❛t✐❝❛✲
♠❡♥t❡ ✐♥✜♥✐t❛ ❡ ✐♠♣❡❞â♥❝✐❛ ❞❡ s❛í❞❛ ♣r❛t✐❝❛♠❡♥t❡ ③❡r♦✱ ♣♦✲
❞❡♥❞♦ s❡r ✉s❛❞❛ ♣❛r❛ s❡ ✐♠♣❧❡♠❡♥t❛r ✜❧tr♦s ❞❡ ✷➟ ♦r❞❡♠ ❞♦s
t✐♣♦s P❛ss❛✲❆❧t❛s✱ P❛ss❛✲❇❛✐①❛s ❡ P❛ss❛✲❋❛✐①❛✱ ❝♦♠♦ s❡rá ✈✐st♦
❛ s❡❣✉✐r✳
✶✵✽
❋✐❧tr♦ P❛ss❛✲❇❛✐①❛s t✐♣♦ ❇✉tt❤❡r✇♦t❤ ❞❡ ✷➟❖r❞❡♠
❙❛❧❧❡♥✲❑❡②
ωc 2
H(s) = 2
s + ωQc + ωc2
P❛r❛ ♦ ✜❧tr♦ s❡r ❞♦ t✐♣♦ ❇✉tt❡r✇♦rt❤ ♦ ❣❛♥❤♦ ✭❆✮ ❞❡✈❡rá s❡r ❞❡✿
A = 1, 586 ⇒ A(dB) = 20.log(1, 586) = 4
▲♦❣♦✿
A=1+
R
RB
= 1, 586 ⇒ B = 0, 586
RA
RA
❊♥tã♦✱ ♣❛r❛✿ RR = 0, 586✱ s❡ ♣♦❞❡ ❛tr✐❜✉✐r ♦s ✈❛❧♦r❡s ♣rát✐❝♦s✿
RA = 47kΩ✱ ❡ RB = 27kΩ✱ q✉❡ s❡ ♦❜té♠ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦✳
B
A
✶✵✾
❊ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ✭fC ✮ s❡rá✿
fc =
❖♥❞❡ ❢❛③❡♥❞♦✲s❡✿
1
2.π. R1.R2.C1.C2
√
R1 = R2 ✱
❡
C1 = C2 ✱
fc =
1
2.π.R.C
✭✻✮
r❡s✉❧t❛✿
✭✼✮
Pr♦❥❡t❛r ✉♠ ✜❧tr♦ ❆t✐✈♦ P❛ss❛✲❇❛✐①❛s ❞❡ ✷➟❖r❞❡♠ t✐♣♦
❇✉tt❡r✇♦t❤✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ❞❡ ✼✵✵❍③✳
❊①✿
❙♦❧✉çã♦✿
✶✲ ❆r❜✐tr❛♥❞♦ ✉♠ ❝❛♣❛❝✐t♦r ❞❡ C = 3, 3nF ❀
✷✲ P❛r❛ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ fc = 700Hz ♦❜té♠✲s❡✿ R =
1
1
2.π.fc .C = 2.π.700.3,3.10−9 = 68, 9kΩ
✶✶✵
❋✐❧tr♦ P❛ss❛✲❆❧t❛s t✐♣♦ ❇✉tt❤❡r✇♦t❤ ❞❡ ✷➟❖r❞❡♠
❙❛❧❧❡♥✲❑❡②
s2
H(s) = 2
s + ωQc + ωc2
▼❛✐s ✉♠❛ ✈❡③✱ ♣❛r❛ ♦ ✜❧tr♦ s❡r ❞♦ t✐♣♦ ❇✉tt❡r✇♦rt❤ ♦ ❣❛♥❤♦ ✭❆✮
❞❡✈❡rá s❡r ❞❡✿
A = 1, 586
❊ ❝♦♠♦ ✈✐st♦ ❛♥t❡s✿ RR = 0, 586✱ ❝♦♠ ♦s ✈❛❧♦r❡s✿ RA = 47kΩ✱ ❡
RB = 27kΩ✳
❆ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ✭fC ✮ s❡rá t❛♠❜é♠ ✐❞ê♥t✐❝❛✱ ❝♦♠ R1 = R2
❡ C1 = C2 ✿
B
A
1
fc =
2.π.R.C
✭✽✮
✶✶✶
❋✐❧tr♦ P❛ss❛✲❋❛✐①❛ t✐♣♦ ❇✉tt❤❡r✇♦t❤ ❞❡ ✷➟❖r❞❡♠
❙❛❧❧❡♥✲❑❡②
◆❡st❡ ❝❛s♦ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝♦rt❡ ✭fC ✮ s❡rá ❞❛❞❛ ♣♦r✿
fc =
1
2.π.C
s
(R1 + R2 )
R1 .R2 .R3
✭✾✮
❙✐♠♣❧✐✜❝❛♥❞♦✲s❡✿
Q
2.π.f0 .A0 .C
✭✶✵✮
Q
2.π.f0 .C.(2Q2 − A0 )
✭✶✶✮
Q
π.f0 .C
✭✶✷✮
R1 =
R2 =
R3 =
✶✶✷
❈♦♠❜✐♥❛♥❞♦✲s❡ ❛s ❡q✉❛çõ❡s ✭✶✵✮✱ ❡ ✭✶✶✮✱ ♦❜té♠✲s❡ ♦ ❣❛♥❤♦ ♥❛
❢r❡q✉ê♥❝✐❛ ❝❡♥tr❛❧✿
A0 =
❚❡♥❞♦ ❝♦♠♦ r❡str✐çã♦✿
Q>
p
R3
2.R1
✭✶✸✮
A0 /2
Pr♦❥❡t❛r ✉♠ ✜❧tr♦ ❆t✐✈♦ P❛ss❛✲❋❛✐①❛ ❞❡ ✷➟❖r❞❡♠ t✐♣♦ ❇✉t✲
t❡r✇♦t❤✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❝❡♥tr❛❧ ❞❡ ✼✺✵❍③✱ ❣❛♥❤♦ ♥❡st❛ ❢r❡q✉ê♥✲
❝✐❛ ❞❡ ✶✱✸✷✱ ❡ ❢❛t♦r ❞❡ q✉❛❧✐❞❛❞❡ ❞❡ ✹✱✷✳
❙♦❧✉çã♦✿
❊①✿
✶✲ ❆r❜✐tr❛♥❞♦ ✉♠ ❝❛♣❛❝✐t♦r ❞❡
C = 0, 01µF
❆tr❛✈és ❞❛ ❡q✉❛çã♦ ✭✶✵✮✿
R1 =
Q
2.π.f0 .A0 .C
= 67, 6kΩ ⇒R1 = 68kΩ
✷✲ ❊ ❡♥tã♦✱ ✉s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✶✸✮✿
R3 = 2.A0 .R1 = 2.1, 32.67.6k = 178kΩ ⇒ R3 = 180kΩ
✸✲ ❋✐♥❛❧♠❡♥t❡✱ ❛ ❡q✉❛çã♦ ✭✶✶✮ ❧❡✈❛ ❛✿
R2 =
Q
2.π.f0 .C.(2Q2 −A0 )
= 2, 6kΩ ⇒ R2 = 2, 7kΩ
✶✶✸