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s2t + a3H(s3)e
s3t URX3V
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x(t) =
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k
ake
skt URXNV
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∑
k
akH(sk)e
skt URXRyV
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2tTQM2M+B�Bb +QKTH2t�b ?�`KQMB+�K2Mi2 `2H�+BQM�/�b,
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k=−∞
ake
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b½Q Qb +Q2}+B2Mi2b /� bû`B2X
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/2 7Q`K� [m2 x∗(t) = x(t)- TQ/2@b2 Q#i2`,
x(t) =
∞∑
k=−∞
a∗ke
−jkω0t URXR8V
`2�HBx�M/Q mK� bm#biBimBϽQ /2 p�`B€p2Bb MQ bQK�iƦ`BQ /2 k T�`� −k,
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jkω0t URXReV
RXjX _2T`2b2Mi�ϽQ /2 bBM�Bb T2`BƦ/B+Qb �i`�pûb /2 aû`B2b /2 6Qm`B2` R8
[m2- +QKT�`�/� ¨ 1[X RXR9- BKTǤ2 � B;m�H/�/2 ak = a∗−k- Qm 2[mB@
p�H2Mi2K2Mi2,
a∗k = a−k URXRdV
*QK 2bi2 +QM?2+BK2MiQ û TQbbőp2H Q#i2` 7Q`K�b �Hi2`M�iBp�b T�`�
� aû`B2 /2 6Qm`B2`,
x(t) = a0 +
∞∑
k=1
[
ake
jkω0t + a−ke−jkω0t
] URXR3V
bm#biBimBM/Q a−k TQ` a∗k,
x(t) = a0 +
∞∑
k=1
[
ake
jkω0t + a∗ke
−jkω0t] URXRNV
LQi�M/Q [m2 Qb /QBb i2`KQb /2Mi`Q /Q bQK�iƦ`BQ b½Q +QKTH2tQb
+QMDm;�/Qb- TQ/2@b2 2b+`2p2`,
x(t) = a0 +
∞∑
k=1
2${akejkω0t} URXkyV
H2K#`�M/Q [m2 ak ∈ C TQ/2 b2` 2tT`2bbQ M� 7Q`K� TQH�`,
ak = Ake
jθk URXkRV
� 1[X RXky TQ/2 b2` `22b+`Bi� +QKQ,
x(t) = a0 +
∞∑
k=1
2$
{
Ake
j(kω0t+θk)
}
URXkkV
Qm b2D�,
x(t) = a0 + 2
∞∑
k=1
Ak +Qb (kω0t+ θk) URXkjV
� 1[X RXkj û mK� 7Q`K� +QKmK2Mi2 miBHBx�/� T�`� `2T`2b2Mi�`
bBM�Bb T2`BƦ/B+Qb MQ i2KTQ +QMiőMmQX lK� Qmi`� 7Q`K� TQ/2 b2`
Q#iB/� 2b+`2p2M/Q ak M� 7Q`K� `2i�M;mH�`,
ak = Bk + jCk URXk9V
Re *�TőimHQ RX aû`B2b /2 6Qm`B2`
+QK Bk 2 Ck `2�BbX �bbBK- � 1[X RXky TQ/2 b2` `22b+`Bi� +QKQ,
x(t) = a0 + 2
∞∑
k=1
[Bk +Qb(kω0t)− Ck bBM(kω0t)] URXk8V
*QM?2+B/� +QKQ � 7Q`K� i`B;QMQKûi`B+� /� aû`B2 /2 6Qm`B2`X
S�`� 7mMÏǤ2b T2`BƦ/B+�b `2�Bb- �b 7Q`K�b /2 `2T`2b2Mi�` � aû`B2 /2
6Qm`B2` M�b 1[bX RXR9- RXkj 2 RXk8 b½Q +QKTH2i�K2Mi2 2[mBp�H2Mi2bX
1K#Q`� �b /m�b ȹHiBK�b miBHBx2K 7mMÏǤ2b i`B;QMQKûi`B+�b U�T�`2M@
i2K2Mi2 K�Bb 7�KBHB�`2b 2 bBKTH2bV- � 7Q`K� 2tTQM2M+B�H +QKTH2t�
û T�`iB+mH�`K2Mi2 ȹiBH 2 +QMp2MB2Mi2 T�`� Qb T`QTƦbBiQb /� �M€HB@
b2b /2 bBM�Bb 2 bBbi2K�b MQ i2KTQ +QMiőMmQ- b2M/Q miBHBx�/� [m�b2
2t+HmbBp�K2Mi2X
amTQM/Q [m2 mK /�/Q bBM�H T2`BƦ/B+Q x(t) TQbb� b2` `2T`2b2M@
i�/Q �i`�pûb /� aû`B2 /2 6Qm`B2` /� 1[X RXR9X 6�x@b2 M2+2bb€`BQ mK
T`Q+2/BK2MiQ T�`� Q#i2` Qb +Q2}+B2Mi2b akX JmHiBTHB+�M/Q �K#Qb Qb
H�/Qb /� 1[X RXR9 TQ` e−jnω0t- Q#iûK@b2,
x(t)e−jnω0t =
∞∑
k=−∞
ake
jkω0te−jnω0t URXkeV
BMi2;`�M/Q i2KTQ`�HK2Mi2 /2 0 � T = 2pi/ω0- Qm b2D�- BMi2;`�M/Q
bQ#`2 Q T2`őQ/Q 7mM/�K2Mi�H,∫ T
0
x(t)e−jnω0t /t =
∫ T
0
∞∑
k=−∞
ake
jkω0te−jnω0t /t URXkdV
i`Q+�M/Q � Q`/2K /Q bQK�iƦ`BQ 2 /� BMi2;`�ϽQ,∫ T
0
x(t)e−jnω0t /t =
∞∑
k=−∞
ak
[∫ T
0
ej(k−n)ω0t /t
]
URXk3V
miBHBx�M/Q � `2H�ϽQ /2 1mH2` T�`� +�H+mH�` � BMi2;`�H 2Mi`2 +QH+?2i2b,∫ T
0
ej(k−n)ω0t /t =
∫ T
0
+Qb [(k − n)ω0t] /t+
j
∫ T
0
bBM [(k − n)ω0t] /t URXkNV
RX9X *QMp2`;āM+B� /� aû`B2 /2 6Qm`B2` Rd
ú TQbbőp2H Q#b2`p�` [m2- T�`� k %= n- i�MiQ +Qb [(k − n)ω0t]
[m�MiQ bBM [(k − n)ω0t] b½Q b2MƦB/2b T2`BƦ/B+�b 2K T - +mD� BMi2@
;`�ϽQ M2bi2 BMi2`p�HQ `2bmHi� 2K x2`QX S�`� k = n- Q BMi2;`�M/Q
iQ`M�@b2 mMBi€`BQ- Qm b2D�- � BMi2;`�H b2 `2bmK2 �,∫ T
0
ej(k−n)ω0t /t =
T, k = n0, k %= n URXjyV
/2 7Q`K� [m2 � 1[X RXk3 b2 `2/mx � Tan- �bbBK,
an =
1
T
∫ T
0
x(t)e−jnω0t /t URXjRV
.2 mK� 7Q`K� ;2`�H- 2bi2 `2bmHi�/Q û p€HB/Q M½Q �T2M�b T�`� Q
BMi2`p�HQ /2 0 � T - K�b T�`� [m�H[m2` BMi2`p�HQ /2 H�`;m`� T - /2
7Q`K� [m2 û K�Bb +QKmK /2}MB`,
an =
1
T
∫
T
x(t)e−jnω0t /t URXjkV
�bbBK- Q T�` /2 2[m�ÏǤ2b RXjk 2 RXR9 b½Q +QM?2+B/Qb +QKQ 1[m�@
ÏǤ2b /2 �M€HBb2 2 aőMi2b2 /� aû`B2 /2 6Qm`B2`- `2bT2+iBp�K2Mi2X
RX9 *QMp2`;āM+B� /� aû`B2 /2 6Qm`B2`
L2K iQ/Qb Qb bBM�Bb T2`BƦ/B+Qb MQ i2KTQ +QMiőMmQ TQbbm2K `2T`2@
b2Mi�ϽQ �i`�pûb /2 aû`B2b /2 6Qm`B2`X �BM/� bBK- �b aû`B2b /2 6Qm`B2`
TQ/2K b2` miBHBx�/�b T�`� `2T`2b2Mi�` mK� �KTH� +H�bb2 /2 bBM�Bb
T2`BƦ/B+Qb 2 +QMiőMmQb MQ i2KTQX
S�`� �p�HB�` � p�HB/�/2 /� `2T`2b2Mi�ϽQ TQ` aû`B2b /2 6Qm`B2`-
iQK�@b2 mK bBM�H T2`BƦ/B+Q x(t) +mD� �T`QtBK�ϽQ �iû � N @ûbBK�
?�`KƬMB+� û /2}MB/� +QKQ,
xN (t) =
N∑
k=−N
ake
jkω0t URXjjV
.2}MBM/Q eN (t) +QKQ Q 2``Q /� �T`QtBK�ϽQ,
eN (t) = x(t)− xN (t) URXj9V
R3 *�TőimHQ RX aû`B2b /2 6Qm`B2`
lK +`Biû`BQ +QKmK T�`� [m�MiB}+�` � [m�HB/�/2 /� �T`QtBK�ϽQ /2
bBM�Bb û � 2M2`;B� /Q 2``Q- TQBb 2H� /2bT`2x� Q bBM�H /Q 2``QX
EN =
∫
T
∣∣eN (t)∣∣2 /t URXj8V
L2bi� �M€HBb2- [m�MiQ K�Bb +QKTQM2Mi2b ?�`KƬMB+Qb b½Q miBHBx�@
/�b- K�Bb }2H û � `2T`Q/mϽQ /Q bBM�H- +QMb2[m2Mi2K2Mi2 K2MQ` û �
2M2`;B� /2 2``QX LQ +�bQ B/2�H /� aû`B2 /2 6Qm`B2` M½Q b2` i`mM+�/�
� mK p�HQ` }MBiQ UN →∞V- � 2M2`;B� /2 2``Q û MmH�X
�H;mMb bBM�Bb bBKTH2bK2Mi2 M½Q TQbbm2K mK� aû`B2 /2 6Qm`B2`
2[mBp�H2Mi2- Qm TQ`[m2 � BMi2;`�H /� 1[m�ϽQ /2 �M€HBb2 U1[X RXjkV
/Bp2`;2- Qm TQ`[m2 Qb +Q2}+B2Mi2b Q#iB/Qb M� K2bK�- [m�M/Q �THB@
+�/Qb M� 1[m�ϽQ /2 aőMi2b2 U1[X RXR9V- M½Q `2T`Q/mx2K Q bBM�H
Q`B;BM�HX
� +H�bb2 /2 bBM�Bb T2`BƦ/B+Qb `2T`2b2Mi€p2Bb TQ` aû`B2b /2 6Qm`B2`
+QMiûK bBM�Bb [m2 TQbbm2K 2M2`;B� }MBi� 2K mK T2`őQ/Q,∫
T
∣∣x(t)∣∣2 /t <∞ URXjeV
b2 2bi� +QM/BϽQ û b�iBb72Bi�- Qb +Q2}+B2Mi2b ak Q#iB/Qb M� 1[X RXjk b½Q
}MBiQbX h�K#ûK }+� ;�`�MiB/� [m2 � 2M2`;B� /2 2``Q /2 BMp2`b�K2Mi2
T`QTQ`+BQM�H �Q MȹK2`Q /2 ?�`KƬMB+�b miBHBx�/�b M� �T`QtBK�ϽQX
lK Qmi`Q +QMDmMiQ K�Bb �#`�M;2Mi2 /2 +QM/BÏǤ2b T�`� � 2[mB@
p�HāM+B� /2 mK bBM�H 2 bm� `2T`2b2Mi�ϽQ TQ` bû`B2b /2 6Qm`B2` 7QB
/2b2MpQHpB/Q TQ` CQ?�MM S2i2` :mbi�p G2D2mM2 .B`B+?H2ieX �b +?�@
K�/�b *QM/BÏǤ2b /2 .B`B+?H2i b½Q,
RX 1K [m�H[m2` T2`őQ/Q- x(t) /2p2 b2` �#bQHmi�K2Mi2 BMi2;`€p2H,∫
T
∣∣x(t)∣∣ /t <∞ URXjdV
TQBb,∣∣ak∣∣ ≤ 1
T
∫
T
∣∣x(t)e−jkω0t∣∣/t = ∫
T
∣∣x(t)∣∣ /t <∞ URXj3V
e UR3y8 Ĝ R38NV K�i2K€iB+Q �H2K½QX
RX8X S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2` RN
kX .�/Q [m�H[m2` BMi2`p�HQ }MBiQ /2 i2KTQ- x(t) TQbbmB p�`B�@
ϽQ HBKBi�/�- Qm b2D�- TQbbmB mK MȹK2`Q }MBiQ /2 K€tBKQb 2
KőMBKQbX
jX .�/Q [m�H[m2` BMi2`p�HQ }MBiQ /2 i2KTQ- x(t) TQbbmB mK Mȹ@
K2`Q }MBiQ /2 /2b+QMiBMmB/�/2b- b2M/Q +�/� mK� /2bi�b /2b@
+QMiBMmB/�/2b }MBi�bX
o�H2 `2bb�Hi�` [m2 iQ/Qb Qb bBM�Bb- T2`BƦ/B+Qb 2 +QMiőMmQb MQ
i2KTQ- }bB+�K2Mi2 `2�HBx€p2Bb `2bT2Bi�K �b +QM/BÏǤ2b /2 .B`B+?H2i-
TQbbmBM/Q `2T`2b2Mi�ϽQ p€HB/� �i`�pûb /2 aû`B2b /2 6Qm`B2`X
S�`� K2H?Q` 2t2KTHB}+�` +QKQ � +QMp2`;āM+B� /�b aû`B2b /2 6Qm@
`B2` 7mM+BQM� T�`� 7mMÏǤ2b +QK /2b+QMiBMmB/�/2b- iQK�@b2 Q i`�#�H?Q
/2 �H#2`i �#`�?�K JB+?2HbQMd- [m2 2K R3N3- +QMbi`mBm Q T`BK2B`Q
�M�HBb�/Q` ?�`KƬMB+Q [m2 +�H+mH�p� � �T`QtBK�ϽQ T�`� � aû`B2 /2
6Qm`B2` i`mM+�/� U1[X RXjjV T�`�- MQ K€tBKQ- N = ±80X
�Q �M�HBb�` Qb `2bmHi�/Qb Q#iB/Qb T�`� � �T`QtBK�ϽQ /2 mK�
QM/� [m�/`�/� T2`BƦ/B+�- � Q#i2MϽQ /2 `2bmHi�/Qb BM2bT2`�/Qb Q
H2pQm � #mb+�` �Dm/� /2 CQbB�? qBHH�`/ :B##b3- [m2 BMp2biB;Qm 2
Tm#HB+Qm � 2tTHB+�ϽQ 2K R3NNX
*QKQ TQ/2 b2` Q#b2`p�/Q M� 6B;m`� R- � �T`QtBK�ϽQ /2 mK�
QM/� [m�/`�/� TQ` mK� aû`B2 /2 6Qm`B2` i`mM+�/� 2K /Bp2`bQb p�@
HQ`2b /2 N �T`2b2Mi� mK bQ#`2bbBM�H T`ƦtBKQ ¨ /2b+QMiBMmB/�/2 [m2
M½Q /BKBMmB +QK Q �mK2MiQ /2 N X 1bi2 272BiQ û +?�K�/Q /2 72MƬ@
K2MQ /2 :B##b- 2K ?QK2M�;2K ¨ �miQ`B� /2 bm� 2tTHB+�ϽQ �M�Hő@
iB+�X
RX8 S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2`
� `2T`2b2Mi�ϽQ �i`�pûb /� aû`B2 /2 6Qm`B2` TQbbmB p€`B�b T`QT`B2@
/�/2b [m2 2tT�M/2K Q 2Mi2M/BK2MiQ /� `2T`2b2Mi�ϽQ- 2- �/B+BQM�H@
K2Mi2- �Dm/�K M� Q#i2MϽQ /� aû`B2 /2 6Qm`B2` T�`� �H;mMb bBM�BbX
d UR38k Ĝ RNjRV 7őbB+Q MQ`i2@�K2`B+�MQX
3 UR3jN Ĝ RNyjV K�i2K€iB+Q- 7őbB+Q 2 [mőKB+Q MQ`i2@�K2`B+�MQX
ky *�TőimHQ RX aû`B2b /2 6Qm`B2`
XX
t
X
xN (t)
X
N = 1
XX
t
X
xN (t)
X
N = 3
XX
t
X
xN (t)
X
N = 5
XX
t
X
xN (t)
X
N = 7
XX
t
X
xN (t)
X
N = 39
XX
t
X
xN (t)
X
N = 79
6B;m`� R Ĝ *QMp2`;āM+B� /� aû`B2 /2 6Qm`B2` i`mM+�/�- BHmbi`�M/Q Q
72MƬK2MQ /2 :B##bX
L� /2}MBϽQ /�b T`QT`B2/�/2b- û +QMp2MB2Mi2 /2}MB` mK� K�@
M2B`� bBKTH2b /2 BM/B+�` � aû`B2 /2 6Qm`B2` T�`� mK bBM�H T2`BƦ/B+Q
x(t)- +QK 7`2[māM+B� 7mM/�K2Mi�H ω0 = 2pi/T 2 Qb +Q2}+B2Mi2b /�
aû`B2 /2 6Qm`B2` ak- /� 7Q`K�,
x(t)
SF←−−−−−→ ak URXjNV
Qm- �Hi2`M�iBp�K2Mi2,
ak = SF
{
x(t)
}
x(t) = SF−1
{
ak
} URX9yV
[m2 BM/B+� [m2 Qb +Q2}+B2Mi2b ak 7Q`�K Q#iB/Qb /Q bBM�H x(t) � T�`iB`
/� 1[X RXjk- Qm �Hi2`M�iBp�K2Mi2- [m2 Q bBM�H x(t) TQ/2 b2` Q#iB/Q �
T�`iB` /Qb +Q2}+B2Mi2b ak �i`�pûb /� 1[X RXR9X P QT2`�/Q` SF
{ · }
/2MQi� � �THB+�ϽQ /� 1[m�ϽQ /2 �M€HBb2 2 SF−1{ ·} � �THB+�ϽQ
/� 1[m�ϽQ /2 aőMi2b2X
RX8X S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2` kR
RX8XR GBM2�`B/�/2
a2D�K x(t) 2 y(t) bBM�Bb T2`BƦ/B+Qb- +QK T2`őQ/Q T - 2 +QK +Q2}+B2Mi2b
/� aû`B2 /2 6Qm`B2` ak 2 bk- `2bT2+iBp�K2Mi2X
x(t)
SF←−−−−−→ ak
y(t)
SF←−−−−−→ bk
URX9RV
*QKQ x(t) 2 y(t) TQbbm2K Q K2bKQ T2`őQ/Q T - [m�H[m2` +QK#B@
M�ϽQ HBM2�` /2 �K#Qb i�K#ûK b2`€ T2`BƦ/B+� 2K T X Pb +Q2}+B2Mi2b
/� aû`B2 `2bmHi�Mi2 /� +QK#BM�ϽQ HBM2�` b½Q /�/Qb TQ`,
Ax(t) +By(t)
SF←−−−−−→ Aak +Bbk URX9kV
RX8Xk .2bHQ+�K2MiQ MQ i2KTQ
lK bBM�H T2`BƦ/B+Q- /2bHQ+�/Q /2 t0- K�MiûK bm� T2`BQ/B+B/�/2X �
T�`iB` /� /2}MBϽQ- � aû`B2 /2 6Qm`B2` /2 y = x(t− t0) û /�/� TQ`,
bk =
1
T
∫
T
x(t− t0)e−jkω0t /t URX9jV
7�x2M/Q τ = t− t0 2 bm#biBimBM/Q � p�`B€p2H /2 BMi2;`�ϽQ,
bk =
1
T
∫
T
x(τ)e−jkω0(τ+t0) /τ
= e−jkω0t0
1
T
∫
T
x(τ)e−jkω0τ /τ
= e−jkω0t0ak
URX99V
Qm b2D�- �bbmKBM/Q [m2,
x(t)
SF←−−−−−→ ak URX98V
2Mi½Q,
x(t− t0) SF←−−−−−→ e−jkω0t0ak URX9eV
o�H2 MQi�` [m2 Q /2bHQ+�K2MiQ MQ i2KTQ i`�x �T2M�b �Hi2`�ÏǤ2b
M� 7�b2 /Qb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2`- T2`K�M2+2M/Q BM�Hi2`�/�
� bm� K�;MBim/2X
kk *�TőimHQ RX aû`B2b /2 6Qm`B2`
RX8Xj _2~2t½Q MQ i2KTQ
P T2`őQ/Q T /2 mK bBM�H T2`BƦ/B+Q M½Q b2 �Hi2`� [m�M/Q Q K2bKQ û
`2~2iB/Q MQ i2KTQX hQK�M/Q Q bBM�H y(t) = x(−t) 2 � 1[X RXR9,
x(−t) =
∞∑
k=−∞
ake
−jkω0t URX9dV
bm#biBimBM/Q k = −m,
y(t) = x(−t) =
∞∑
m=−∞
a−mejmω0t URX93V
Q#b2`p�M/Q [m2 Q K2K#`Q /B`2BiQ /2bi� 2[m�ϽQ û � 1[m�ϽQ /2
aőMi2b2 T�`� x(−t),
bk = a−k URX9NV
Pm b2D�- b2,
x(t)
SF←−−−−−→ ak URX8yV
2Mi½Q,
x(−t) SF←−−−−−→ a−k URX8RV
.2bi� 7Q`K�- � `2~2t½Q MQ i2KTQ /Q bBM�H +�mb� mK� `2~2t½Q M�
b2[māM+B� /Qb +Q2}+B2Mi2bX SQ/2@b2 Q#b2`p�` [m2 b2 Q bBM�H x(t) 7Q`
T�` UBX2X x(t) = x(−t)V- 2Mi½Q � aû`B2 /2 6Qm`B2` i�K#ûK b2`€ T�`
Uak = a−kVX a2 Q bBM�H 7QB őKT�` UBX2X x(−t) = −x(t)V- 2Mi½Q � aû`B2
/2 6Qm`B2` i�K#ûK b2`€ őKT�` Ua−k = −akVX
RX8X9 Jm/�M� M� 2b+�H� /2 i2KTQ
� Km/�MÏ� M� 2b+�H� /Q i2KTQ �Hi2`� Q T2`őQ/Q /Q bBM�H Q`B;BM�HX
a2 x(t) û T2`BƦ/B+Q 2K T - +QK 7`2[māM+B� 7mM/�K2Mi�H ω0 = 2pi/T -
2Mi½Q x(αt)- b2M/Q α ∈ R+- b2`€ T2`BƦ/B+Q- +QK T2`őQ/Q T/α 2
7`2[māM+B� 7mM/�K2Mi�H αω0X lK� p2x [m2 � Km/�MÏ� /2 2b+�H�
û �THB+�/� 2K iQ/Qb Qb +QKTQM2Mi2b ?�`KƬMB+Qb /2 x(t)- TQ/2@b2
RX8X S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2` kj
+QM+HmB` [m2 iQ/Qb Qb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2` T2`K�M2+2K
BM�Hi2`�/Qb,
x(αt) =
∞∑
k=−∞
ake
jk(αω0)t URX8kV
o�H2 H2K#`�` [m2- 2K#Q`� Qb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2`
T2`K�M2Ï�K Qb K2bKQb- +QK � Km/�MÏ� /� 7`2[māM+B� 7mM/�K2Mi�H-
� aû`B2 /2 6Qm`B2`- T`QT`B�K2Mi2 /Bi�- Km/QmX
RX8X8 JmHiBTHB+�ϽQ
a2D�K x(t) 2 y(t) T2`BƦ/B+Qb 2K T - /2 7Q`K� [m2,
x(t)
SF←−−−−−→ ak
y(t)
SF←−−−−−→ bk
URX8jV
*QKQ Q T`Q/miQ x(t)y(t) i�K#ûK û T2`BƦ/B+Q 2K T - Q bBM�H `2@
bmHi�Mi2 TQ/2 b2` 2tT`2bbQ �i`�pûb /2 mK� aû`B2 /2 6Qm`B2` +mDQb
+Q2}+B2Mi2b b½Q /2}MB/Qb TQ`,
x(t)y(t)
SF←−−−−−→
∞∑
l=−∞
albk−l URX89V
RX8Xe *QMDm;�ϽQ 2 bBK2i`B� +QMDm;�/�
�THB+�ϽQ /� +QMDm;�ϽQ +QKTH2t� bQ#`2 x(t) `2bmHi� M� +QMDm;�ϽQ
2 `2~2t½Q MQ i2KTQ bQ#`2 Qb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2`- Qm b2D�-
�bbmKBM/Q,
x(t)
SF←−−−−−→ ak URX88V
2Mi½Q,
x∗(t) SF←−−−−−→ a∗−k URX8eV
1bi� T`QT`B2/�/2 T`Q/mx �H;mK�b +QMb2[māM+B� [m2 b½Q BKTQ`@
i�Mi2b /2 MQi�, b2 x(t) 7Q` `2�H- /2 7Q`K� [m2 x∗(t) = x(t)- 2Mi½Q Qb
+Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2` �T`2b2Mi�`½Q bBK2i`B� +QMm;�/�,
a−k = a∗k URX8dV
k9 *�TőimHQ RX aû`B2b /2 6Qm`B2`
� h�#2H� R +QMiûK Qmi`�b T`QT`B2/�/2b /� bBK2i`B� +QMDm;�/�
T�`� bBM�Bb `2�BbX
RX8Xd _2H�ϽQ /2 S�`b2p�H
� _2H�ϽQ /2 S�`b2p�H T�`� bBM�Bb T2`BƦ/B+Qb +QMiőMmQb MQ i2KTQ û
/2}MB/� +QKQ,
1
T
∫
T
∣∣x(t)∣∣2 /t = ∞∑
k=−∞
∣∣ak∣∣2 URX83V
� _2H�ϽQ /2 S�`b2p�H BM/B+� � [m2 2M2`;B� /2 mK /�/Q bBM�H û �
K2bK�- BM/2T2M/2Mi2 b2 `2T`2b2Mi�/Q MQ i2KTQ x(t)- Qm TQ` aû`B2b
/2 6Qm`B2` akX
h�#2H� R Ĝ S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2`X
S`QT`B2/�/2 aBM�H aû`B2 /2 6Qm`B2`
Pb bBM�Bb x(t) 2 y(t) b½Q T2`BƦ/B+Qb 2K T U7`2[māM+B� 7mM/�K2M@
i�H ω0 = 2pi/T V +QK +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2` ak 2 bk-
`2bT2+iBp�K2Mi2X
GBM2�`B/�/2 Ax(t) +By(t) Aak +Bbk
.2bHQ+�K2MiQ MQ
i2KTQ x(t− t0) ake
−jkω0t0
.2bHQ+�K2MiQ M�
7`2[māM+B� e
jMω0tx(t) ak−M
*QMDm;�ϽQ x∗(t) a∗−k
_2~2t½Q MQ i2KTQ x(−t) a−k
Jm/�M� M� 2b+�H�
/Q i2KTQN x(αt) ak
*QMiBMm� M� T`ƦtBK� T€;BM�
N α > 0- T2`BƦ/B+� 2K T/α
RX8X S`QT`B2/�/2b /� aû`B2 /2 6Qm`B2` k8
S`QT`B2/�/2 aBM�H aû`B2 /2 6Qm`B2`
*QMpQHmϽQ
T2`BƦ/B+�
∫
T
x(τ)y(t− τ) /τ Takbk
JmHiBTHB+�ϽQ x(t)y(t)
∞∑
l=−∞
albk−l
.B72`2M+B�ϽQ /x(t)/t jkω0ak
AMi2;`�ϽQRy
∫ t
−∞
x(t) /t 1
jkω0
ak
aBK2i`B� +QMDm;�/�
T�`� bBM�Bb `2�Bb x(t) `2�H
ak = a∗−k
${ak} = ${a−k}
){ak} = −){a−k}∣∣ak∣∣ = ∣∣a−k∣∣
∠ak = −∠a−k
aBK2i`B� T�`�
bBM�Bb `2�Bb 2 T�`2b x(t) `2�H 2 T�` ak `2�H 2 T�`
aBK2i`B� T�`�
bBM�Bb `2�Bb 2
őKT�`2b
x(t) `2�H 2 őKT�` ak Tm`�K2Mi2 BK�;B@M€`BQ 2 őKT�`
.2+QKTQbBϽQ
T�`@őKT�` /2 bBM�Bb
`2�BbRR-Rk
xe(t) = E
{
x(t)
}
xo(t) = O
{
x(t)
} ${ak}
j){ak}
_2H�ϽQ /2
S�`b2p�H
1
T
∫
T
∣∣x(t)∣∣2 /t = ∞∑
k=−∞
∣∣ak∣∣2
Ry x(t) }MBi�- +QK a0 = 0
RR .2+QKTQbBϽQS�`, E{x(t)} = x(t)+x(−t)2
Rk .2+QKTQbBϽQ ŐKT�`, O{x(t)} = x(t)−x(−t)2
ke *�TőimHQ RX aû`B2b /2 6Qm`B2`
RXe aû`B2b /2 6Qm`B2` T�`� bBM�Bb T2`BƦ/B+Qb +QKmMbX
� h�#2H� k +QMiûK �H;mK�b aû`B2b /2 6Qm`B2` T�`� bBM�Bb T2`BƦ/B+Qb
+QKmMbX
h�#2H� k Ĝ aû`B2b /2 6Qm`B2` /2 bBM�Bb T2`BƦ/B+QbX
aBM�H aû`B2 /2 6Qm`B2`
∞∑
k=−∞
ake
jkω0t ak
ejω0t
ak = 1, k = 1ak = 0, k %= 1
+Qb(ω0t)
ak = 12 , k = ±1ak = 0, k %= ±1
bBM(ω0t)
ak = 1j2 , k = ±1ak = 0, k %= ±1
x(t) = 1
ak = 1, k = 0ak = 0, k %= 0
x(t)
1,
∣∣t∣∣ ≤ T1
0, T1 <
∣∣t∣∣ ≤ T/2
2 x(t+ T ) = x(t)
bBM(kω0T1)
kpi
∞∑
n=−∞
δ(t− nT ) ak = 1
T
- T�`� iQ/Q k
*�SŐhlGP k
h`�Mb7Q`K�/� /2
6Qm`B2`
1K#Q`� �b aû`B2b /2 6Qm`B2` b2D�K #�bi�Mi2 ȹi2Bb M� �M€HBb2 /Q +QM@
i2ȹ/Q 2K 7`2[māM+B� /2 bBM�Bb T2`BƦ/B+Qb MQ i2KTQ +QMiőMmQ- b2m
mMBp2`bQ /2 �THB+�ϽQ û #�bi�Mi2 `2/mxB/Q /2pB/Q ¨ HBKBi�ϽQ � bB@
M�Bb T2`BƦ/B+QbX
lK� `2T`2b2Mi�ϽQ 2K 7`2[māM+B� K�Bb �KTH� 2 ;2M2`�HBbi�- +QM@
i2KTH�M/Q bBM�Bb �T2`BƦ/B+Qb û /2b2D€p2HX � h`�Mb7Q`K�/� /2 6Qm@
`B2` û � `2T`2b2Mi�ϽQ /Q 2bT2+i`Q /Q bBM�H �T2`BƦ/B+Q +QMiőMmQ MQ
i2KTQX L2bi2 +�bQ- Q +QMi2ȹ/Q 2K 7`2[māM+B� M½Q û K�Bb HBKBi�/Q �
?�`KƬMB+�b /� 7`2[māM+B� 7mM/�K2Mi�H- K�b 2bi€ /Bbi`B#mő/Q BM}MB@
i2bBK�HK2Mi2 bQ#`2 iQ/Qb Qb p�HQ`2b /2 7`2[māM+B� UBX2X ω ∈ RVX
P /2b2MpQHpBK2MiQ /2 mK� `2T`2b2Mi�ϽQ 2K 7`2[māM+B� T�`�
bBM�Bb �T2`BƦ/B+Qb 7QB mK� /�b +QMi`B#mBÏǤ2b K�Bb BKTQ`i�Mi2b /2
6Qm`B2` M2bi2 +�KTQ /2 2bim/QX 6Qm`B2` Q#i2p2 � /2}MBϽQ /� h`�Mb@
7Q`K�/� /2 6Qm`B2` T�`� bBM�Bb �T2`BƦ/B+Qb �p�HB�M/Q � aû`B2 /2 6Qm@
`B2` /2 bBM�Bb T2`BƦ/B+Qb +QK Q T2`őQ/Q i2M/2M/Q �Q BM}MBiQ U272iB@
p�K2Mi2 iQ`M�M/Q Q bBM�H T2`BƦ/B+Q �T2`BƦ/B+QVX 1bi2 272BiQ û 7�+BH@
k3 *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
K2Mi2 Q#b2`p€p2H M�b aû`B2b /2 6Qm`B2`, [m�MiQ K�BQ` Q T2`őQ/Q T /2
mK bBM�H T2`BƦ/B+Q- K2MQ` bm� 7`2[māM+B� 7mM/�K2Mi�H ω0 = 2pi/T
2- +QMb2[m2Mi2K2Mi2- K�Bb T`ƦtBKQb b½Q Qb p�HQ`2b 2K 7`2[māM+B�
/�b +QKTQM2Mi2b ?�`KƬMB+�bX
kXR _2T`2b2Mi�ϽQ /2 bBM�Bb �T2`BƦ/B+Qb �i`�pûb /� h`�Mb@
7Q`K�/� /2 6Qm`B2`
kXRXR .2}MBϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`
S�`� BMB+B�` � �M€HBb2 /� Q#i2MϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2` /2 mK
bBM�H �T2`BƦ/B+Q- iQK�@b2 Q bBM�H T2`BƦ/B+Q QM/� [m�/`�/� /2}MB/Q
TQ`,
x(t) =
1,
∣∣t∣∣ ≤ T0
0, T0 <
∣∣t∣∣ < T/2 UkXRV
b2M/Q x(t) = x(t+nT )- +QK n ∈ Z- +QM7Q`K2 BHmbi`�/Q M� 6B;m`� kX
XX
t
X
x(t)
X
´2T
X
´T
X
´T0
X
T0
X
T
X
2T
6B;m`� k Ĝ PM/� [m�/`�/� T2`BƦ/B+�X
Pb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2` /2bi2 bBM�H b½Q,
ak =
2 bBM(kω0T0)
kω0T
UkXkV
QM/2 ω0 = 2pi/T X
_22b+`2p2M/Q � 1[X kXk- /2 7Q`K� +QMp2MB2Mi2,
Tak =
2 bBM(ωT0)
ω
UkXjV
+QK ω = kω0X
kXRX _2T`2b2Mi�ϽQ /2 bBM�Bb �i`�pûb /� h`�Mb7Q`K�/� /2 6Qm`B2` kN
.2bi� 7Q`K�- �bbmKBM/Q [m2 ω û +QMiőMm�- � 7mMϽQ,
2 bBM(ωT0)
ω
UkX9V
`2T`2b2Mi� mK� 2MpQHiƦ`B� /Qb p�HQ`2b /2 TakX 6Bt�M/Q mK p�HQ`
T�`� T0- � 2MpQHiƦ`B� Tak û BM/2T2M/2Mi2 /2 T X � 6B;m`� j `2T`2@
b2Mi� /B72`2Mi2b aû`B2b /2 6Qm`B2` T�`� /B72`2Mi2b p�HQ`2b /2 T - bQ#
� K2bK� 2MpQHiƦ`B� TakX
XX
ω
X
0
X
U�V
X
Tak
X
2ω0
X
´2ω0
XX
ω
X
0
X
U#V
X
Tak
X
4ω0
X
´4ω0
XX
ω
X
0
X
U+V
X
Tak
X
8ω0
X
´8ω0
6B;m`� j Ĝ Pb +Q2}+B2Mi2b /� bû`B2 /2 6Qm`B2` 2 bm� 2MpQHiƦ`B�- T�`�
Q bBM�H QM/� [m�/`�/� T2`BƦ/B+Q- +QK, U�V T = 4T0c U#V
T = 8T0c U+V T = 16T0X
*QM7Q`K2 }+� +H�`Q T2H� �M€HBb2 /� 6B;m`� j- Q �mK2MiQ /Q T2@
jy *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
`őQ/Q U2 bm#b2[m2Mi2 /BKBMmBϽQ /� 7`2[māM+B� 7mM/�K2Mi�H ω0 =
2pi/T V 7�x +QK [m2 � 2MpQHiƦ`B� b2D� �KQbi`�/� K�Bb 7`2[m2Mi2K2Mi2X
S�`� mK T2`őQ/Q T →∞- � �KQbi`�;2K û 72Bi� i½Q 7`2[m2Mi2K2Mi2
[m2 b2 iQ`M� BM}MBi2bBK�HK2Mi2 b2T�`�/� ω0 → 0- Qm b2D�- b2 iQ`M�
� T`ƦT`B� 2MpQHiƦ`B�X
.2 mK� 7Q`K� ;2`�H- mK bBM�H x(t) �T2`BƦ/B+Q 2 }MBiQ UBX2X TQbbmB
p�HQ`2b M½Q@MmHQb /2Mi`Q /2 mK� 7�Bt� }MBi� /Q /QKőMBQV TQ/2 b2`
miBHBx�/Q T�`� +`B�` mK bBM�H T2`BƦ/B+Q x˜(t)- /Q [m�H x(t) +QKTǤ2 mK
T2`őQ/Q U6B;m`� 9VX *QM7Q`K2 �mK2Mi�@b2 �`#Bi`�`B�K2Mi2 Q T2`őQ/Q
T /Q bBM�H ;2`�/Q- x˜(t) 2 x(t) b½Q B;m�Bb 2K BMi2`p�HQb K�BQ`2b /2
i2KTQX LQ HBKBi2- +QK T →∞- x˜(t) ≡ x(t)X
XX
t
X
x(t)
X
´T0
X
T0
X
U�V
XX
t
X
x˜(t)
X
´2T
X
´T
X
´T0
X
T0
X
T
X
2T
X
U#V
6B;m`� 9 Ĝ U�V aBM�H �T2`BƦ/B+Q x(t) 2 U#V bBM�H T2`BƦ/B+Q x˜(t)- Q#iB/Q
� T�`iB` /2 x(t)X
�M�HBb�M/Q � `2T`2b2Mi�ϽQ TQ` aû`B2b /2 6Qm`B2` /Q bBM�H x˜(t)
kXRX _2T`2b2Mi�ϽQ /2 bBM�Bb �i`�pûb /� h`�Mb7Q`K�/� /2 6Qm`B2` jR
UTQ` +QMp2MBāM+B�- Q BMi2`p�HQ /2 �M€HBb2 b2`€ −T/2 < t ≤ T/2V,
x˜(t) =
∞∑
k=−∞
ake
jkω0t UkX8V
ak =
1
T
∫ T/2
−T/2
x˜(t)e−jkω0t /t UkXeV
+QK ω0 = 2pi/T X P#b2`p�M/Q [m2 x˜(t) = x(t) T�`� |t| ≤ T/2 2 [m2-
7Q`� /2bi2 BMi2`p�HQ x(t) = 0- TQ/2@b2 `22b+`2p2` � 1[X kXe +QKQ,
ak =
1
T
∫ T/2
−T/2
x(t)e−jkω0t /t
=
1
T
∫ ∞
−∞
x(t)e−jkω0t /t
UkXdV
�bbBK- � 2MpQHiƦ`B� Tak U6B;m`� jV- +?�K�/� /2 X(jω)- }+�
/2}MB/� TQ`,
X(jω) =
∫ ∞
−∞
x(t)e−jωt /t UkX3V
û TQbbőp2H Q#i2` Qb +Q2}+B2Mi2b ak /� aû`B2 /2 6Qm`B2` +QKQ,
ak =
1
T
X(jkω0) UkXNV
*QK#BM�M/Q �b 1[bX kXN 2 kX8- TQ/2@b2 Q#i2` x˜(t) � T�`iB` /2
X(jω) �i`�pûb /2,
x˜(t) =
∞∑
k=−∞
1
T
X(jkω0)e
jkω0t UkXRyV
+QKQ T = 2pi/ω0,
x˜(t) =
1
2pi
∞∑
k=−∞
X(jkω0)e
jkω0tω0 UkXRRV
*QM7Q`K2 T →∞- x˜(t) b2 �T`QtBK� /2 x(t) Q [m2 7�x +QK [m2- MQ
HBKBi2- � 1[X kXRR b2D� � /2}MBϽQ /2 x(t)X S�`�H2H�K2Mi2- +QM7Q`K2
T → ∞ ⇐⇒ ω0 → 0 2 Q K2K#`Q /B`2BiQ /� 1[X kXRR b2 iQ`M� mK�
BMi2;`�ϽQX *�/� i2`KQ /Q bQK�iƦ`BQ +Q``2bTQM/2 ¨ mK `2iM;mHQ
jk *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
/2 H�`;m`� ω0 2 �Him`� X(jω)ejω0tX *QM7Q`K2 ω0 → 0- Q bQK�iƦ`BQ
BM}MBi2bBK�H +QMp2`;2 T�`� mK� BMi2;`�H /2 X(jω)ejωt- /2 7Q`K�
[m2 û TQbbőp2H 2b+`2p2`,
x(t) =
1
2pi
∫ ∞
−∞
X(jω)ejωt /ω UkXRkV
2
X(jω) =
∫ ∞
−∞
x(t)e−jωt /t UkXRjV
�b 1[bX kXRk 2 kXRj b½Q +?�K�/�b /2 T�` i`�Mb7Q`K�/Q /2 6Qm@
`B2`- +QK � 1[X kXRj b2M/Q � h`�Mb7Q`K�/� /2 6Qm`B2` UQm BMi2;`�H
/2 6Qm`B2`V- i�K#ûK +?�K�/� /2 1[m�ϽQ /2 �M€HBb2X � 1[X kXRk û
+?�K�/� /2 h`�Mb7Q`K�/� AMp2`b� /2 6Qm`B2`- Qm 1[m�ϽQ /2 aőM@
i2b2X
� 7mMϽQ X(jω) û- 2Mi½Q- � h`�Mb7Q`K�/� /2 6Qm`B2` /Q bBM�H
x(t)X �Hi2`M�iBp�K2Mi2- � 7mMϽQ X(jω) û +?�K�/� /2 2bT2+i`Q /2
x(t)X � T�`iB` /Q T`ƦT`BQ /2b2MpQHpBK2MiQ /� /2}MBϽQ /� h`�Mb7Q`@
K�/� /2 6Qm`B2`- bm� BMi2`@`2H�ϽQ +QK � aû`B2 /2 6Qm`B2` /2 mK�
7mMϽQ T2`BƦ/B+� 2K T û /�/� TQ`,
ak =
1
T
X(jω)
∣∣∣∣∣
ω=kω0
UkXR9V
� �M€HBb2 /� 1[X kXR9 iQ`M� +H�`Q [m2 � aû`B2 /2 6Qm`B2` /Q bBM�H
x˜(t) û mK� �KQbi`�;2K- 2b+�H�/� TQ` 1/T - /� h`�Mb7Q`K�/� /2
6Qm`B2` /2 x(t)X
kXRXk *QMp2`;āM+B� /� h`�Mb7Q`K�/� /2 6Qm`B2`
P K2+�MBbKQ miBHBx�/Q M� /2}MBϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`
X(jω) � T�`iB` /2 mK bBM�H x(t) û p€HB/Q T�`� mK� �KTH� +H�bb2 /2
bBM�Bb- TQ`ûK M2K iQ/� 7mMϽQ K�i2K€iB+� x(t) TQbbmB`€ mK� `2T`2@
b2Mi�ϽQX(jω) p€HB/�X 1bT2+B}+�K2Mi2- +QMbB/2`�@b2 � +QMp2`;āM+B�
/� h`�Mb7Q`K�/� /2 6Qm`B2` b�iBb72Bi� b2 � 7mMϽQ X(jω)- Q#iB/� /2
kXRX _2T`2b2Mi�ϽQ /2 bBM�Bb �i`�pûb /� h`�Mb7Q`K�/� /2 6Qm`B2` jj
x(t) �i`�pûb /� 1[m�ϽQ /2 �M€HBb2 U1[X kXRjV- [m�M/Q �THB+�/� ¨
1[m�ϽQ /2 aőMi2b2 U1[X kXRkV T`Q/mx mK bBM�H xˆ(t)- QM/2 2bi2 bBM�H
û 2[mBp�H2Mi2 �Q bBM�H Q`B;BM�H x(t) UBX2X xˆ(t) ≡ x(t)VX
a2 x(t) û mK bBM�H /2 2M2`;B� }MBi�- Qm b2D�,∫ ∞
−∞
∣∣x(t)∣∣2 /t <∞ UkXR8V
b�#2@b2 2Mi½Q [m2 X(jω) b2`€ }MBiQ UQm b2D�- � 1[m�ϽQ /2 �M€HBb2
+QMp2`;2VX �bbmKBM/Q mK bBM�H /2 2``Q e(t) = xˆ(t)−x(t)- b2 � 2M2`;B�
/2 2``Q û MmH�, ∫ ∞
−∞
∣∣e(t)∣∣2 /t = 0 UkXReV
}+� ;�`�MiB/� � 2[mBp�HāM+B� 2Mi`2 x(t) 2 xˆ(t)X 1K Qmi`�b T�H�p`�b-
M½Q û 2bi`Bi�K2Mi2 M2+2bb€`B� [m2 xˆ(t) = x(t)- �T2M�b [m2 bm�b /B@
72`2MÏ�b M½Q TQbbm�K 2M2`;B�X
�M�HQ;�K2Mi2 �Q [m2 �+QMi2+2 M� aû`B2 /2 6Qm`B2`- 2tBbi2 mK
+QMDmMiQ �Hi2`M�iBpQ /2 +QM/BÏǤ2b [m2 ;�`�Mi2K � 2[mBp�HāM+B� /2
xˆ(t) 2 x(t)- 2t+2iQ 2K /2b+QMiBMmB/�/2b-QM/2 Q p�HQ` /� 7mMϽQ û
B;m�H ¨ Kû/B� /� /2b+QMiBMmB/�/2X 1bi�b +QM/BÏǤ2b b½Q- MQp�K2Mi2-
+?�K�/�b /2 +QM/BÏǤ2b /2 .B`B+?H2i,
RX � 7mMϽQ x(t) /2p2 b2` �#bQHmi�K2Mi2 BMi2;`€p2H,∫ ∞
−∞
∣∣x(t)∣∣ /t <∞
kX � 7mMϽQ x(t) /2p2 TQbbmB` mK MȹK2`Q }MBiQ /2 K€tBKQb 2
KőMBKQb 2K [m�H[m2` BMi2`p�HQ }MBiQX
jX � 7mMϽQ x(t) /2p2 TQbbmB` mK MȹK2`Q }MBiQ /2 /2b+QMiBMmB@
/�/2b 2K [m�H[m2` BMi2`p�HQ }MBiQX 1bi�b /2b+QMiBMmB/�/2b /2@
p2K b2` }MBi�bX
�T2b�` /Qb /QBb +QMDmMiQb /2 +QM/BÏǤ2b b2`2K bm}+B2Mi2b T�`�
;�`�MiB` [m2 mK bBM�H TQbbm� h`�Mb7Q`K�/� /2 6Qm`B2` p€HB/�- ?€
mK� +H�bb2 BKTQ`i�Mi2 /2 bBM�Bb [m2 M½Q b½Q �#bQHmi�K2Mi2 BMi2;`€@
p2Bb- M2K [m�/`�iB+�K2Mi2 BMi2;`€p2Bb- bQ#`2 mK BMi2`p�HQ BM}MBiQX
j9 *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
�H;mMb bBM�Bb T2`BƦ/B+Qb BKTQ`i�Mi2b- +QKQ Q b2MQ 2 Q +Qbb2MQ- 2bi½Q
M2bi� +QM/BϽQ- K�b TQbbm2K `2T`2b2Mi�ϽQ /2 6Qm`B2` b2 7mMÏǤ2b
BKTmHbQ mMBi€`BQ Uδ(t) ě /2Hi� /2 .B`�+V 7Q`2K T2`KBiB/�b M� bm�
`2T`2b2Mi�ϽQX
1bi� �#Q`/�;2K i`�x mK� BMi2`b2+ϽQ +QKmK 2Mi`2 �b aû`B2b /2
6Qm`B2` 2 � h`�Mb7Q`K�/� /2 6Qm`B2` [m2 û ȹiBH M� �M€HBb2 /2 �H;mMb
bBM�BbX
kXk h`�Mb7Q`K�/� /2 6Qm`B2` T�`� bBM�Bb T2`BƦ/B+Qb
JmBiQ 2K#Q`� � KQiBp�ϽQ T�`� Q /2b2MpQHpBK2MiQ /� h`�Mb7Q`K�/�
/2 6Qm`B2` b2D� � `2T`2b2Mi�ϽQ 2K 7`2[māM+B� /2 bBM�Bb �T2`BƦ/B+Qb- �
`2T`2b2Mi�ϽQ /2 bBM�Bb T2`BƦ/B+Qb i�K#ûK �i`�pûb /� h`�Mb7Q`K�/�
/2 6Qm`B2` û ȹiBH TQBb i`�x � �M€HBb2 2K 7`2[māM+B� /2Mi`Q /2 mK
+QMi2tiQ mMB}+�/QX
� i`�Mb7Q`K�/� `2bmHi�Mi2 /2 mK bBM�H T2`BƦ/B+Q +QMbBbi2 /2 mK
i`2K /2 BKTmHbQb mMBi€`BQb U/2Hi�b /2 .B`�+V +mD�b €`2�b b½Q MmK2@
`B+�K2Mi2 B;m�Bb �Qb +Q2}+B2Mi2b /� aû`B2 /2 6Qm`B2` /Q bBM�HX
6�x2M/Q � 2M;2M?�`B� `2p2`b� /2bi2 `2bmHi�/Q- �bbmKBM/Q mK
bBM�H x(t)- +mD� h`�Mb7Q`K�/� /2 6Qm`B2` b2D� X(jω)- b2M/Q 2bi� mK
BKTmHbQ /2 €`2� 2pi M� 7`2[māM+B� ω = ω0,
X(jω) = 2piδ(ω − ω0) UkXRdV
�THB+�M/Q � h`�Mb7Q`K�/� AMp2`b� /2 6Qm`B2` T�`� Q#i2` x(t),
x(t) =
1
2pi
∫ ∞
−∞
2piδ(ω − ω0)ejωt /ω
= ejω0t
UkXR3V
.2 mK� 7Q`K� ;2`�H- �bbmKBM/Q X(jω) +QKQ mK� +QK#BM�ϽQ
HBM2�` /2 BKTmHbQb mMB7Q`K2K2Mi2 2bT�Ï�/Qb M� 7`2[māM+B�,
X(jω) =
∞∑
k=−∞
2piakδ(ω − kω0) UkXRNV
kXjX S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ +QMiőMmQ j8
2Mi½Q Q bBM�H x(t) Q#iB/Q �i`�pûb /� h`�Mb7Q`K�/� AMp2`b� /2 6Qm`B2`
b2`€ /� 7Q`K�,
x(t) =
∞∑
k=−∞
ake
jkω0t UkXkyV
QM/2 b2 Q#b2`p� [m2 � 1[X kXky +Q``2bTQM/2 2t�i�K2Mi2 ¨ `2T`2b2M@
i�ϽQ TQ` aû`B2b /2 6Qm`B2` /Q bBM�H x(t)X
.2bi� 7Q`K� � h`�Mb7Q`K�/� /2 6Qm`B2` /2 mK bBM�H T2`BƦ/B+Q
+QK Q +QMDmMiQ /2 +Q2}+B2Mi2b /� aď`B2 /2 6Qm`B2` {ak} TQ/2 b2`
BMi2`T`2i�/� +QKQ mK i`2K /2 BKTmHbQb mMBi€`BQb- Q+Q``2M/Q M�b
7`2[māM+B� ?�`KQMB+�K2Mi2 `2H�+BQM�/�b- /2 €`2� 2piak T�`� � k@
ûbBK� ?�`KƬMB+� U7`2[māM+B� ω = kω0VX
kXj S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ
+QMiőMmQ
lK 2bim/Q /�b T`QT`B2/�/2b ;2`�Bb /� h`�Mb7Q`K�/� /2 6Qm`B2` û
ȹiBH TQBb 2bi�b T`QT`B2/�/2b MQ`K�HK2Mi2 `2/mx2K /2 7Q`K� bB;MB}@
+�iBp� � [m�MiB/�/2 /2 +€H+mHQb M2+2bb€`B� M� bQHmϽQ /2 T`Q#H2K�b
K�Bb +QKTH2tQbX
lK� Qmi`� miBHB/�/2 û [m2- +QKQ ?€ mK� BMi2`@`2H�ϽQ 2Mi`2 �
h`�Mb7Q`K�/� /2 6Qm`B2` 2 � aû`B2 /2 6Qm`B2` /2 mK bBM�H T2`BƦ/B+Q-
Q 2bim/Q /�b T`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` `2bmHi� 2K
�H;mK�b T`QT`B2/�/2b /�b aû`B2b /2 6Qm`B2` [m2 M½Q b½Q /2b+`Bi�b MQ
i2tiQ /� a2ϽQ RX8- K�b };m`�K M� h�#2H� RX
.2 K�M2B`� �M€HQ;� ¨b aû`B2b /2 6Qm`B2`- û +QMp2MB2Mi2 2tT`2bb�`
� h`�Mb7Q`K�/� /2 6Qm`B2` �i`�pûb /2 mK� MQi�ϽQ K�Bb +QKT�+i�
[m2 � /2}MBϽQ /� K2bK�- /2 7Q`K� [m2 b2 2bi�#2H2+2 [m2,
x(t)
F←−−−→ X(jω) UkXkRV
Qm- �Hi2`M�iBp�K2Mi2,
X(jω) = F
{
x(t)
}
x(t) = F−1
{
X(jω)
} UkXkkV
je *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
QM/2 Q QT2`�/Q` F{ · } /2MQi� � �THB+�ϽQ /� h`�Mb7Q`K�/� /2
6Qm`B2` 2 F−1{ ·} � �THB+�ϽQ /� h`�Mb7Q`K�/� AMp2`b� /2 6Qm`B2`X
�b 7mMÏǤ2b x(t) 2 X(jω) }+�K +?�K�/�b /2 T�` i`�Mb7Q`K�/Q /2
6Qm`B2`X
kXjXR GBM2�`B/�/2
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkXkjV
2 [m2,
y(t)
F←−−−→ Y (jω) UkXk9V
2Mi½Q,
ax(t) + by(t)
F←−−−→ aX(jω) + bY (jω) UkXk8V
QM/2 a 2 b b½Q +QMbi�Mi2b [m�Bb[m2`X 1bi� T`QT`B2/�/2 û 7�+BHK2Mi2
T`Qp�/� � T�`iB` /� /2}MBϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2` 2 T`QMi�@
K2Mi2 2tT�Mbőp2H T�`� mK� +QK#BM�ϽQ HBM2�` /2 i�K�M?Q �`#Bi`€@
`BQX
kXjXk .2bHQ+�K2MiQ MQ i2KTQ
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkXkeV
2Mi½Q TQ/2@b2 T`Qp�` [m2,
x(t− t0) F←−−−→ e−jωt0X(jω) UkXkdV
1bi� T`QT`B2/�/2 û 7�+BHK2Mi2 T`Qp�/� 7�x2M/Q � bm#biBimBϽQ /2
p�`B€p2Bb t→ t− t0- /� /2}MBϽQ /� 1[m�ϽQ /2 aőMi2b2X
kXjXj *QMDm;�ϽQ 2 bBK2i`B� +QMDm;�/�
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkXk3V
kXjX S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ +QMiőMmQ jd
2Mi½Q,
x∗(t) F←−−−→ X∗(−jω) UkXkNV
1bi� T`QT`B2/�/2 TQ/2 b2` T`Qp�/� �i`�pûb /� �THB+�ϽQ /B`2i�
/� +QMDm;�ϽQ bQ#`2 � 1[m�ϽQ /2 �M€HBb2X
� T�`iB` /2bi� T`QT`B2/�/2- û TQbbőp2H T`Qp�` [m2 b2 x(t) 7Q` `2�H
UBX2X x(t) = x∗(t)V- 2Mi½Q X(jω) �T`2b2Mi�`€ bBK2i`B� +QMDm;�/�- Qm
b2D�,
b2 x(t) `2�H → X(−jω) = X∗(jω) UkXjyV
a�#2M/Q /BbbQ- û TQbbőp2H +QM+HmB` [m2,
RX .2}MBM/Q X(jω) = ${X(jω)} + j){X(jω)}- b2 x(t) û `2�H
2Mi½Q,
�V � T�`i2 `2�H /� h`�Mb7Q`K�/� /2 6Qm`B2` û 7mMϽQ T�`X
${X(jω)} = ${X(−jω)}
#V � T�`i2 BK�;BM€`B� /� h`�Mb7Q`K�/� /2 6Qm`B2` û 7mMϽQ
őKT�`
){X(jω)} = −){X(−jω)}
kX .2}MBM/Q X(jω) = ∣∣X(jω)∣∣ej∠X(jω)- T�`� x(t) `2�H,
�V � K�;MBim/2 /� h`�Mb7Q`K�/� /2 6Qm`B2` û 7mMϽQ T�`X∣∣X(jω)∣∣ = ∣∣X(−jω)∣∣
#V � 7�b2 /� h`�Mb7Q`K�/� /2 6Qm`B2` û 7mMϽQ őKT�`X
∠X(jω) = −∠X(−jω)
�/B+BQM�HK2Mi2- TQ/2@b2 T`Qp�` [m2 b2 x(t) 7Q` 7mMϽQ `2�H 2 T�`-
X(jω) b2`€ `2�HX a2 x(t) 7Q` `2�H 2 őKT�`- X(jω) b2`€ Tm`�K2Mi2
BK�;BM€`B�X
j3 *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
kXjX9 .B72`2M+B�ϽQ 2 BMi2;`�ϽQ
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkXjRV
/B72`2M+B�M/Q MQ i2KTQ � 1[m�ϽQ /2 aőMi2b2- û TQbbőp2H T`Qp�` [m2,
/x(t)
/t
F←−−−→ jωX(jω) UkXjkV
1bi� T`QT`B2/�/2 û /2 T�`iB+mH�` BKTQ`iM+B� MQ mbQ /� �M€HBb2
/2 6Qm`B2` 2K bBbi2K�b GAh- bm#biBimBM/Q � �#Q`/�;2K /2 2[m�ÏǤ2b
/B72`2M+B�Bb MQ i2KTQ TQ` 2[m�ÏǤ2b �H;û#`B+�b M� 7`2[māM+B�X
L½Q /2p2 b2` bm`T`2b� [m2- b2 � /B72`2M+B�ϽQ MQ i2KTQ `2bmHi�
2K mK� KmHiBTHB+�ϽQ TQ` jω M� 7`2[māM+B�- 2Mi½Q � BMi2;`�ϽQ MQ
i2KTQ /2p2 `2bmHi�` 2K mK� /BpBb½Q TQ` jω M� 7`2[māM+B�X L� p2`@
/�/2- BbiQ û �T2M�b mK� T�`i2 /� `2H�ϽQ- [m2 M� őMi2;`� û 2tT`2bb�
+QKQ, ∫ t
−∞
x(τ) /τ F←−−−→ 1
jω
X(jω) + piX(0)δ(ω) UkXjjV
QM/2 Q BKTmHbQ mMBi€`BQ 2K 7`2[māM+B� +QMi2KTH� Q p�HQ` Kû/BQ
`2bmHi�Mi2 /� BMi2;`�ϽQX
kXjX8 Jm/�MÏ� M� 2b+�H� /2 i2KTQ 2 M� 7`2[māM+B�
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkXj9V
2Mi½Q,
x(at)
F←−−−→ 1∣∣a∣∣X
(
jω
a
)
UkXj8V
T�`� a [m�H[m2` `2�H M½Q@MmHQX 1bi� T`QT`B2/�/2 û /2`Bp�/� /B`2i�@
K2Mi2 /� /2}MBϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`,
F
{
x(at)
}
=
∫ ∞
−∞
x(at)e−jωt /t UkXjeV
kXjX S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ +QMiőMmQ jN
7�x2M/Q � bm#biBimBϽQ /2 p�`B€p2Bb τ = at,
F
{
x(at)
}
=
 1a
∫∞
−∞ x(τ)e
−j(ω/a)τ /τ, a > 0
− 1a
∫∞
−∞ x(τ)e
−j(ω/a)τ /τ, a < 0
UkXjdV
�bbBK- ¨ T�`i2 /� 2b+�H� 1/∣∣a∣∣- mK 7�iQ` /2 Km/�MÏ� /2 2b+�H�
/2 a MQ i2KTQ- +Q``2bTQM/2`€ � mK� Km/�MÏ� /2 1/a M� 7`2[māM+B�-
2 pB+2@p2`b�X
S�`� Q +�bQ 2bT2+ő}+Q /2 a = −1- /2`Bp�@b2 � T`QT`B2/�/2 /�
`2p2`b½Q MQ i2KTQ,
x(−t) F←−−−→ X(−jω) UkXj3V
Qm b2D�- mK� `2p2`b½Q MQ i2KTQ 2[mBp�H2 � mK� `2p2`b½Q M� 7`2[māM@
+B�X
kXjXe .m�HB/�/2
*QKT�`�M/Q �b 1[bX /2 �M€HBb2 U1[X kXRjV 2 /2 aőMi2b2 U1[X kXRkV
/� h`�Mb7Q`K�/� /2 6Qm`B2`- û TQbbőp2H MQi�` �H;mK�b bBKBH�`B/�@
/2b 2K bm� 2bi`mim`�X 1bi� bBK2i`B� MQi€p2H H2p� ¨b T`QT`B2/�/2b /2
/m�HB/�/2 /� h`�Mb7Q`K�/� /2 6Qm`B2`X
� /m�HB/�/2 û K2H?Q` 2Mi2M/B/� �i`�pûb /2 mK 2t2KTHQ, iQ@
K�M/Q Q bBM�H H�TH�+B�MQ e−
∣∣t∣∣- +mDQ T�` i`�Mb7Q`K�/Q û,
x(t) = e−
∣∣t∣∣ F←−−−→ X(jω) = 2
1 + ω2
UkXjNV
hQK�M/Q �;Q`� mK� 7mMϽQ y(t) /� 7Q`K�,
y(t) =
2
1 + t2
UkX9yV
� Q#i2MϽQ /2 Y (jω) � T�`iB` /� /2}MBϽQ û- MQ KőMBKQ-#�bi�Mi2
2MpQHpB/�X SQ`ûK- � /m�HB/�/2 Q72`2+2 mK +�KBM?Q #2K K�Bb bBK@
TH2bX P#b2`p�M/Q � bBKBH�`B/�/2 M� 7Q`K� /2 y(t) 2 X(jω)- TQ/2@b2
miBHBx�` � 1[m�ϽQ /2 aőMi2b2 /Q T�` x(t) F←−−−→ X(jω) /� 7Q`K�,
e−
∣∣t∣∣ = 1
2pi
∫ ∞
−∞
(
2
1 + ω2
)
ejωt /ω UkX9RV
9y *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
KmHiBTHB+�M/Q Qb /QBb H�/Qb /� B;m�H/�/2 TQ` 2pi- 2 bm#biBimBM/Q �
p�`B€p2H t TQ` −t,
2pie−
∣∣t∣∣ = ∫ ∞
−∞
(
2
1 + ω2
)
e−jωt /ω UkX9kV
}M�HK2Mi2- bm#biBimBM/Q �b p�`B€p2Bb t TQ` ω 2 pB+2@p2`b�,
2pie−
∣∣ω∣∣ = ∫ ∞
−∞
(
2
1 + t2
)
e−jωt /t UkX9jV
Q#b2`p�@b2 [m2 Q T�`i2 ¨ /B`2Bi� /� B;m�H/�/2 û Dmbi�K2Mi2 � 1[m�@
ϽQ /2 �M€HBb2 /� 7mMϽQ y(t)- /2 7Q`K� [m2 � T�`i2 ¨ 2b[m2`/�
/� B;m�H/�/2 û- 2Mi½Q- Y (jω)X �bbBK- �i`�pûb /� /m�HB/�/2- TƬ/2@b2
Q#i2` Q T�` i`�Mb7Q`K�/Q,
y(t) =
2
1 + t2
F←−−−→ Y (jω) = 2pie−
∣∣ω∣∣ UkX99V
lK� �M€HBb2 +mB/�/Qb� KQbi`� [m2 � /m�HB/�/2- 2K ;2`�H +QM@
bBbi2 /� bBKTH2b bm#biBimBϽQ /2 t TQ` ω- 2 pB+2@p2`b�- #2K +QKQ Q
2b+�H�K2MiQ /2 2piX
� /m�HB/�/2 i�K#ûK TQ/2 b2` �THB+�/� T�`� Q#i2` MQp�b T`Q@
T`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2`- +?�K�/�b /2 T`QT`B2/�/2b
/m�Bb /2 T`QT`B2/�/2b D€ /2}MB/�bX � b2;mB` b½Q 2tTQbi�b �H;mK�b
/2bi�b,
Ç .2bHQ+�K2MiQ M� 7`2[māM+B� U/m�H /� T`QT`B2/�/2 /2 /2bHQ+�@
K2MiQ MQ i2KTQV,
ejω0tx(t)
F←−−−→ X (j(ω − ω0)) UkX98V
Ç .B72`2M+B�ϽQ M� 7`2[māM+B� UT`QT`B2/�/2 /m�H /� /B72`2M+B�@
ϽQ MQ i2KTQV,
−jtx(t) F←−−−→ /X(jω)/ω UkX9eV
kXjX S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ +QMiőMmQ 9R
Ç AMi2;`�ϽQ M� 7`2[māM+B� UT`QT`B2/�/2 /m�H /� BMi2;`�ϽQ MQ
i2KTQV,
− 1
jt
x(t) + pix(0)δ(t)
F←−−−→
∫ ω
−∞
X(jν) /ν UkX9dV
kXjXd _2H�ϽQ /2 S�`b2p�H
�bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkX93V
2Mi½Q TQ/2@b2 T`Qp�` [m2,∫ ∞
−∞
∣∣x(t)∣∣2 /t = 1
2pi
∫ ∞
−∞
∣∣X(jω)∣∣2 /ω UkX9NV
� `2H�ϽQ /2 S�`b2p�H TQ/2 b2` Q#iB/� � T�`iB` /� T`ƦT`B� /2}MB@
ϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`- Q#b2`p�M/Q �T2M�b [m2 ∣∣x(t)∣∣2 =
x(t)x∗(t)- 2 bm#biBimBM/Q x∗(t) TQ` bm� 1[m�ϽQ /2 aőMi2b2- /2Mi`Q
/� 1[m�ϽQ /2 �M€HBb2 /2 x(t)x∗(t)X
� _2H�ϽQ /2 S�`b2p�H bBKTH2bK2Mi2 /2KQMbi`� � 2[mBp�HāM+B�
/� 2M2`;B� /Q bBM�H- b2D� TQ` bm� `2T`2b2Mi�ϽQ MQ i2KTQ- b2D� TQ`
bm� `2T`2b2Mi�ϽQ 2K 7`2[māM+B�X � 7mMϽQ ∣∣X(jω)∣∣2 û +?�K�/� /2
/2MbB/�/2 /2 2M2`;B� /Q bBM�H x(t)X
kXjX3 S`QT`B2/�/2 /� +QMpQHmϽQ
aBbi2K�b GAh b½Q +QKTH2i�K2Mi2 /2}MB/Qb TQ` bm� `2bTQbi� BKTmH@
bBp� h(t)- /2 7Q`K� [m2 û TQbbőp2H Q#i2` � b�ő/� y(t) T�`� [m�H[m2`
2Mi`�/� x(t)- +QKQ � +QMpQHmϽQ y(t) = h(t) ∗ x(t)X �bbmKBM/Q [m2,
x(t)
F←−−−→ X(jω) UkX8yV
2- /�/� � BMi2;`�H /2 +QMpQHmϽQ,
y(t) =
∫ ∞
−∞
x(τ)h(t− τ) /τ UkX8RV
9k *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
2Mi½Q- � T�`iB` /� 1[m�ϽQ /� �M€HBb2- TQ/2@b2 2b+`2p2`,
Y (jω) =
∫ ∞
−∞
∫ ∞
−∞
x(τ)h(t− τ) /τ e−jωt /t UkX8kV
am#biBimBM/Q � Q`/2K /2 BMi2;`�ϽQ 2 Q#b2`p�M/Q [m2 x(τ) û
BM/2T2M/2Mi2 /2 t,
Y (jω) =
∫ ∞
−∞
x(τ)
[∫ ∞
−∞
h(t− τ)e−jωt /t
]
/τ UkX8jV
/� T`QT`B2/�/2 /Q /2bHQ+�K2MiQ MQ i2KTQ- Q i2`KQ 2Mi`2 +QH+?2i2b
b2 B;m�H� � e−jωτH(jω)- /2 7Q`K� [m2,
Y (jω) =
∫ ∞
−∞
x(τ)e−jωτH(jω) /τ
= H(jω)
∫ ∞
−∞
x(τ)e−jωτ /τ
UkX89V
Q `2bmHi�/Q /� BMi2;`�H û X(jω)- �bbBK,
Y (jω) = H(jω)X(jω) UkX88V
.2bi� 7Q`K�- }+� 2bi�#2H2+B/� � T`QT`B2/�/2 /� +QMpQHmϽQ,
y(t) = h(t) ∗ x(t) F←−−−→ Y (jω) = H(jω)X(jω) UkX8eV
�bbBK- � T`QT`B2/�/2 /� +QMpQHmϽQ K�T2B� mK� +QMpQHmϽQ MQ
/QKőMBQ /Q i2KTQ TQ` mK T`Q/miQ MQ /QKőMBQ /� 7`2[māM+B�- Q [m2
û mK BKTQ`i�Mi2 `2bmHi�/Q M� �M€HBb2 /2 bBbi2K�b GAhX
kXjXN S`QT`B2/�/2 /� KmHiBTHB+�ϽQ
� T`QT`B2/�/2 /� KmHiBTHB+�ϽQ TQ/2 b2` BMi2`T`2i�/� +QKQ � T`Q@
T`B2/�/2 /m�H /� +QMpQHmϽQX lK� p2x [m2 � T`QT`B2/�/2 /� +QM@
pQHmϽQ 2bi�#2H2+2 [m2 mK� +QMpQHmϽQ MQ i2KTQ û K�T2�/� +QKQ
mK� KmHiBTHB+�ϽQ M� 7`2[māM+B�- � T`QT`B2/�/2 /� KmHiBTHB+�ϽQ
2bi�#2H2+2 2t�i�K2Mi2 Q QTQbiQ, mK� KmHiBTHB+�ϽQ MQ i2KTQ b2
K�T2B� +QKQ mK� +QMpQHmϽQ M� 7`2[māM+B�- �bbBK,
y(t) = x(t)c(t)
F←−−−→ Y (jω) = 1
2pi
X(jω) ∗ C(jω) UkX8dV
kXjX S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2` MQ i2KTQ +QMiőMmQ 9j
� T`QT`B2/�/2 /� KmHiBTHB+�ϽQ û #�bi�Mi2 ȹiBH MQ 2bim/Q /2 KQ@
/mH�ϽQ TQ` �KTHBim/2 U�JV- 2K i2H2+QKmMB+�ÏǤ2bX SQ` 2bi� `�x½Q-
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KQ/mH�ϽQX
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`B2` T�`� bBM�Bb +QMiőMmQb MQ i2KTQX
h�#2H� j Ĝ S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 6Qm`B2`X
S`QT`B2/�/2 aBM�H h`�Mb7Q`K�/� /2 6Qm`B2`
x(t) X(jω)
y(t) Y (jω)
GBM2�`B/�/2 ax(t) + by(t) aX(jω) + bY (jω)
.2bHQ+�K2MiQ
MQ i2KTQ x(t− t0) e
−jωt0X(jω)
.2bHQ+�K2MiQ
M� 7`2[māM+B� e
jω0tx(t) X (j(ω − ω0))
*QMDm;�ϽQ x∗(t) X∗(−jω)
_2~2t½Q MQ
i2KTQ x(−t) X(−jω)
Jm/�M� M�
2b+�H� /Q i2KTQ
2 M� 7`2[māM+B�
x(at)
1∣∣a∣∣X
(
jω
a
)
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JmHiBTHB+�ϽQ x(t)y(t) 1
2pi
X(jω) ∗ Y (jω)
.B72`2M+B�ϽQ MQ
i2KTQ
/x(t)
/t jωX(jω)
AMi2;`�ϽQ
∫ t
−∞
x(t) /t 1
jω
X(jω) + piX(0)δ(ω)
*QMiBMm� M� T`ƦtBK� T€;BM�
99 *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
S`QT`B2/�/2 aBM�H h`�Mb7Q`K�/� /2 6Qm`B2`
.B72`2M+B�ϽQ MQ
7`2[māM+B� tx(t) j
/X(jω)
/ω
aBK2i`B�
+QMDm;�/� T�`�
bBM�Bb `2�Bb
x(t) `2�H
X(jω) = X∗(−jω)
${X(jω)} = ${X(−jω)}
){X(jω)} = −){X(−jω)}∣∣X(jω)∣∣ = ∣∣X(−jω)∣∣
∠X(jω) = −∠X(−jω)
aBK2i`B� T�`�
bBM�Bb `2�Bb 2
T�`2b
x(t) `2�H 2 T�` X(jω) `2�H 2 T�`
aBK2i`B� T�`�
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őKT�`2b
x(t) `2�H 2 őKT�` X(jω) Tm`�K2Mi2 BK�;BM€@`BQ 2 őKT�`
.2+QKTQbBϽQ
T�`@őKT�` /2
bBM�Bb `2�BbR-k
xe(t) = E
{
x(t)
}
xo(t) = O
{
x(t)
} ${X(jω)}
j){X(jω)}
_2H�ϽQ /2
S�`b2p�H
∫ ∞
−∞
∣∣x(t)∣∣2 /t = 1
2pi
∫ ∞
−∞
∣∣X(jω)∣∣2 /ω
kX9 S�`2b i`�Mb7Q`K�/Qb /2 6Qm`B2` +QKmMb
� h�#2H� 9 +QMiûK Qb T�`2b i`�Mb7Q`K�/Qb K�Bb +QKmMb /� h`�Mb@
7Q`K�/� /2 6Qm`B2`X
R .2+QKTQbBϽQ S�`, E{x(t)} = x(t)+x(−t)2
k .2+QKTQbBϽQ ŐKT�`, O{x(t)} = x(t)−x(−t)2
kX9X S�`2b i`�Mb7Q`K�/Qb /2 6Qm`B2` +QKmMb 98
h�#2H� 9 Ĝ S�`2b i`�Mb7Q`K�/Qb /2 6Qm`B2`X
aBM�H h`�Mb7Q`K�/� /2 6Qm`B2`
∞∑
k=−∞
ake
jkω0t 2pi
∞∑
k=−∞
akδ(ω − kω0)
ejω0t 2piδ(ω − ω0)
+Qb(ω0t) pi [δ(ω + ω0) + δ(ω − ω0)]
bBM(ω0t)
pi
j
[δ(ω + ω0)− δ(ω − ω0)]
x(t) = 1 2piδ(ω)
x(t)
1,
∣∣t∣∣ ≤ T1
0, T1 <
∣∣t∣∣ ≤ T/2
2 x(t+ T ) = x(t)
∞∑
k=−∞
2 bBM(kω0T1)
k
δ(ω − kω0)
∞∑
n=−∞
δ(t− nT ) 2pi
T
∞∑
k=−∞
δ
(
ω − 2pik
T
)
x(t) =
1,
∣∣t∣∣ ≤ T1
0,
∣∣t∣∣ > T1 2 bBM(ωT1)ω
bBM(Wt)
pit
X(jω) =
1,
∣∣ω∣∣ ≤W
0,
∣∣ω∣∣ > W
δ(t) 1
u(t)
1
jω
+ piδ(ω)
δ(t− t0) e−jωt0
e−atu(t), ${a} > 0 1
a+ jω
te−atu(t), ${a} > 0 1
(a+ jω)2
*QMiBMm� M� T`ƦtBK� T€;BM�
9e *�TőimHQ kX h`�Mb7Q`K�/� /2 6Qm`B2`
aBM�H h`�Mb7Q`K�/� /2 6Qm`B2`
tne−atu(t), ${a} > 0 n!
(a+ jω)n+1
*�SŐhlGP j
h`�Mb7Q`K�/� /2
G�TH�+2
� �M€HBb2 /2 6Qm`B2`- �i`�pûb /� aû`B2 /2 6Qm`B2` Qm /� h`�Mb7Q`K�/�
/2 6Qm`B2`- û #�bi�Mi2 ȹiBH MQ 2bim/Q 2 M� �M€HBb2 /2 bBM�Bb 2 bBbi2K�b
GAhX 1bi� BKTQ`iM+B� û /2pB/� �Q 7�iQ /2 [m2 mK� �KTH� ;�K� /2
bBM�Bb TQ/2K b2` `2T`2b2Mi�/Qb �i`�pûb /2 +QK#BM�ÏǤ2b HBM2�`2b /2
2tTQM2M+B�Bb +QKTH2t�b- b2M/Q [m2 2bi�b 2tTQM2M+B�Bb +QKTH2t�b
b½Q �miQ7mMÏǤ2b /2 bBbi2K�b GAhX
*QM7Q`K2 pBbiQ M� a2ϽQ RXk- 2bi�b T`QT`B2/�/2b b½Q �i2M/B/�b
T�`� 7mMÏǤ2b 2tTQM2M+B�Bb +QKTH2t�b /� 7Q`K� est- +QK s mK� p�@
`B€p2H +QKTH2t� UBX2X s ∈ CVX � �M€HBb2 /2 6Qm`B2` û /2}MB/� bQ#`2
mK� p�`B€p2H s Tm`�K2Mi2 BK�;BM€`B�- Qm b2D�- T�`� s = jω UQm b2D�-
T�`� 2tTQM2M+B�Bb +QKTH2t�b /� 7Q`K� ejωVX Pb `2bmHi�/Qb /� a2ϽQ
RXk b½Q p€HB/Qb T�`� [m�H[m2` p�HQ` /2 s +QKTH2tQ- M½Q �T2M�b �
p�HQ`2b Tm`�K2Mi2 BK�;BM€`BQbX
� miBHBx�ϽQ /2 s +QKTH2tQ- /� 7Q`K� s = σ+jω U+QK σ = ${s}
2 ω = ){s}V- H2p� ¨ mK� ;2M2`�HBx�ϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`
T�`� Q i2KTQ +QMiőMmQ- +QM?2+B/� +QKQ h`�Mb7Q`K�/� /2 G�TH�+2X
93 *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
� ;`�M/2 p�Mi�;2K /� �M€HBb2/2 bBM�Bb 2 bBbi2K�b GAh �i`�pûb
/� h`�Mb7Q`K�/� /2 G�TH�+2 û [m2 � K2bK� û K�Bb �#`�M;2Mi2 [m2 �
h`�Mb7Q`K�/� /2 6Qm`B2`- TQ` 2t2KTHQ, � h`�Mb7Q`K�/� /2 6Qm`B2`
M½Q û +�T�x /2 `2T`2b2Mi�` bBbi2K�b BMbi€p2Bb- K�b � h`�Mb7Q`K�/�
/2 G�TH�+2 ûX
.� K2bK� 7Q`K� [m2 � h`�Mb7Q`K�/� /2 G�TH�+2 TQ/2 b2` pBbi�
+QKQ mK� ;2M2`�HBx�ϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2`- � h`�Mb7Q`@
K�/� /2 6Qm`B2` TQ/2 b2` +QMbB/2`�/� mK +�bQ 2bT2+B�H /� h`�Mb@
7Q`K�/� /2 G�TH�+2X � �T`2b2Mi�ϽQ /2bi� /2}MBϽQ û #�bi�Mi2 /2@
T2M/2Mi2 /� Q`/2K /2 �T`2M/Bx�/Q /2bi�b i`�Mb7Q`K�ÏǤ2bX
jXR � h`�Mb7Q`K�/� /2 G�TH�+2
� h`�Mb7Q`K�/� /2 G�TH�+2 /2 mK bBM�H [m�H[m2` x(t) û /2}MB/�
+QKQ,
X(s) =
∫ ∞
−∞
x(t)e−st /t UjXRV
2bi� /2}MBϽQ û +QKmK2Mi2 +?�K�/� /2 h`�Mb7Q`K�/� /2 G�TH�+2
"BH�i2`�H- /2pB/Q �Qb HBKBi2b /2 BMi2;`�ϽQ /2 −∞ � ∞- T�`� /B@
72`2M+B€@H� /� h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H U+QK HBKBi2b /2
BMi2;`�ϽQ /2 0 �∞VX 1bi2 i2tiQ QKBiB`€ � /2}MBϽQ #BH�i2`�H- b2M/Q
[m2- [m�M/Q M2+2bb€`BQ- 2bi�`€ 2tTHő+BiQ Q mbQ /� i`�Mb7Q`K�/� mMB@
H�i2`�H U� b2` 2bim/�/� M� a2ϽQ jXeVX
� /2}MBϽQ /� h`�Mb7Q`K�/� /2 G�TH�+2 M� 1[X jXR TQ/2 b2`
7�+BHK2Mi2 Q#iB/� T2H� 2tT�Mb½Q /� h`�Mb7Q`K�/� /2 6Qm`B2` U1[X
kXRjV- bm#biBimBM/Q � p�`B€p2H Tm`�K2Mi2 BK�;BM€`B� jω T2H� +QK@
TH2t� sX
� h`�Mb7Q`K�/� /2 G�TH�+2 TQ/2 b2` BM/B+�/�- /2 7Q`K� K�Bb
+QMp2MB2Mi2- �i`�pûb /Q QT2`�/Q` h`�Mb7Q`K�/� /2 G�TH�+2 L { · }-
Qm �i`�pûb /Q T�` i`�Mb7Q`K�/Q,
x(t)
L←−−−−→ X(s) UjXkV
jXRX � h`�Mb7Q`K�/� /2 G�TH�+2 9N
o�H2 MQi�`- /� T`ƦT`B� /2}MBϽQ M� 1[X jXR [m2- b2 � h`�Mb7Q`@
K�/� /2 G�TH�+2 7Q` �p�HB�/� T�`� s = jω- 2H� b2 `2/mxB`€ ¨ h`�Mb@
7Q`K�/� /2 6Qm`B2`X .2 mK� 7Q`K� K�Bb ;2`�H- � `2H�ϽQ 2Mi`2 �
h`�Mb7Q`K�/� /2 G�TH�+2 2 � /2 6Qm`B2` TQ/2 b2` 2tT`2bb� +QKQ,
X(σ + jω) =
∫ ∞
−∞
x(t)e−(σ+jω)t /t UjXjV
X(σ + jω) =
∫ ∞
−∞
[
x(t)e−σt
]
e−jωt /t UjX9V
QM/2 Q K2K#`Q /B`2BiQ /� 1[X jX9 TQ/2 b2` pBbiQ +QKQ � h`�Mb@
7Q`K�/� /2 6Qm`B2` /2 x(t)e−σtX �bbBK � h`�Mb7Q`K�/� /2 G�TH�+2
TQ/2 b2` pBbi� +QKQ � �THB+�ϽQ /� h`�Mb7Q`K�/� /2 6Qm`B2` �TƦb �
KmHiBTHB+�ϽQ T2H� 2tTQM2M+B�H `2�H e−σt- [m2 TQ/2 b2` +`2b+2Mi2 Qm
/2+`2b+2Mi2- +QM7Q`K2 Q bBM�H /2 σX
a�#2M/Q [m2 mK bBM�H- T�`� TQbbmB` h`�Mb7Q`K�/� /2 6Qm`B2`-
/2p2 b2` �#bQHmi�K2Mi2 BMi2;`€p2H U*QM/BÏǤ2b /2 .B`B+?H2iV,∫ ∞
−∞
∣∣x(t)e−σt∣∣ /t <∞ UjX8V
� 2tBbiāM+B� /� h`�Mb7Q`K�/� /2 G�TH�+2 TQ/2 2bi�` +QM/B+BQM�/� �
mK� 7�Bt� 2bT2+ő}+� /2 p�HQ`2b /2 σ- /2T2M/2M/Q /Q bBM�H x(t)X .2
mK� 7Q`K� ;2`�H- Q BMi2`p�HQ /2 p�HQ`2b /2 s T�`� Q [m�H � h`�Mb7Q`@
K�/� /2 G�TH�+2 +QMp2`;2 û +?�K�/� /2 _2;B½Q /2 *QMp2`;āM+B�-
Qm _P* U_2;BQM Q7 *QMp2`;2M+2VX
a2`€ pBbiQ M� b2[māM+B� [m2 � BM7Q`K�ϽQ /� _P* û /2 7mM@
/�K2Mi�H BKTQ`iM+B� MQb 2bim/Qb /� h`�Mb7Q`K�/� /2 G�TH�+2-
TQBb 2tBbi2K /B72`2Mi2b 7mMÏǤ2b MQ i2KTQ x(t) [m2 `2bmHi�K 2K mK�
K2bK� 2tT`2bb½Q X(s)- +mD� ȹMB+� /B72`2MÏ� û � _P*X
*QKQ s û mK� p�`B€p2H +QKTH2t�- � `2T`2b2Mi�ϽQ /� _P* TQ/2
b2` 72Bi�- ;`�}+�K2Mi2- �i`�pûb /Q TH�MQ sX � 6B;m`� 8 U�V BHmbi`�
mK� _P* T�`� Qb p�HQ`2b ${s} > σ0X
ú #�bi�Mi2 +QKmK [m2 �b 7mMÏǤ2b X(s) b2D�K `�+BQM�Bb- Qm b2D�,
X(s) =
N(s)
D(s)
UjXeV
8y *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
XX
!
X
j"
X
σ0
X
U�V
XX
!
X
j"
X
Ś
X
Ś
XŚ XÌX
´1 + j
X
´1´ j
X
´1, 5
X
0
X
U#V
6B;m`� 8 Ĝ U�V _2;B½Q /2 *QMp2`;āM+B� M� TH�MQ sc U#V .B�;`�K� /2
TQHQb 2 x2`Qb MQ TH�MQ sX
QM/2 N(s) û Q TQHBMƬKBQ 2K s /Q MmK2`�/Q` 2 D(s) û Q TQHBMƬKBQ
2K s /Q /2MQKBM�/Q`X �b `�őx2b /2 N(s) b½Q +?�K�/�b /2 x2`Qb /2
X(s)- 2M[m�MiQ �b `�őx2b /2 D(s) b½Q +?�K�/�b /2 TQHQb /2 X(s)X
ú +QKmK � `2T`2b2Mi�ϽQ /2 TQHQb 2 x2`Qb ;`�}+�K2Mi2- MQ TH�MQ sX
w2`Qb b½Q `2T`2b2Mi�/Qb TQ` dz◦Ǵ 2 TQHQb b½Q `2T`2b2Mi�/Qb TQ` dz×ǴX
� 6B;m`� 8 U#V- KQbi`� mK 2t2KTHQ /Q /B�;`�K� /2 TQHQb 2 x2`Qb
/� 7mMϽQ,
X(s) =
s
(s+ 1, 5)(s+ 1 + j)(s+ 1− j)
=
s
(s+ 1, 5)(s2 + 2s+ 2)
=
s
s3 + 3, 5s2 + 5s+ 3
UjXdV
+QK _P* ${s} > −1X
�bbBK- 2t+2iQ TQ` mK p�HQ` /2 2b+�H�- Q /B�;`�K� /2 TQHQb 2 x2`Qb
MQ TH�MQ s- DmMi�K2Mi2 +QK � _P* �bbQ+B�/� /2}M2 mMBpQ+�K2Mi2
� 7mMϽQ X(s)X
�HûK /BbbQ- �BM/� [m2 M½Q };m`2K 2tTHB+Bi�K2Mi2 M� 7Q`K� �H;û@
#`B+� /2 X(s)- û BMi2`2bb�Mi2 �M�HBb�` � T`2b2MÏ� /2 TQHQb 2fQm x2`Qb
MQ BM}MBiQX SQ` 2t2KTHQ- b2 X(s) TQbbmB Q MȹK2`Q /2 TQHQb K�BQ`
jXkX � _2;B½Q /2 *QMp2`;āM+B� /� h`�Mb7Q`K�/� /2 G�TH�+2 8R
[m2 Q MȹK2`Q /2 x2`Qb- ¨ K2/B/� [m2 Q p�HQ` /2 s �mK2Mi�- X(s)
i2M/2 � x2`Q- iőTB+Q +QKTQ`i�K2MiQ 2bT2`�/Q T�`� � T`2b2MÏ� /2
x2`Qb MQ BM}MBiQX a2 X(s) TQbbmB Q MȹK2`Q /2 x2`Qb K�BQ` [m2 Q /2
TQHQb- X(s) i2M/2 �Q BM}MBiQ- +QM7Q`K2 s +`2b+2 /2 7Q`K� BHBKBi�/�-
Q [m2 TQ/2 b2` 2Mi2M/B/Q +QKQ � T`2b2M� /2 TQHQb MQ BM}MBiQX
.2 mK� 7Q`K� ;2`�H- b2 n û � Q`/2K /Q /2MQKBM�/Q` UMȹK2`Q
/2 TQHQbV 2 m � Q`/2K /Q MmK2`�/Q` UMȹK2`Q /2 x2`QbV- 2Mi½Q b2
n > m- Q MȹK2`Q /2 x2`Qb MQ BM}MBiQ b2`€ n−mX �Hi2`M�iBp�K2Mi2-
b2 n < m- Q MȹK2`Q /2 TQHQb MQ BM}MBiQ b2`€ m− nX a2 n = m- M½Q
?�p2`€ M2K TQHQb M2K x2`Qb MQ BM}MBiQX
jXk � _2;B½Q /2 *QMp2`;āM+B� /� h`�Mb7Q`K�/� /2 G�TH�+2
*QM7Q`K2 �}`K�/Q M� a2ϽQ �Mi2`BQ`- � `2H�ϽQ x(t) L←−−−−→ X(s)
M½Q û ȹMB+�- b2M/Q M2+2bb€`B� � BM7Q`K�ϽQ /� _P* T�`� /2i2`@
KBM�`- b2K �K#B;mB/�/2- mK T�` i`�Mb7Q`K�/QX L2bi� b2ϽQ b2`½Q
2bim/�/�b �b T`QT`B2/�/2b /� _2;B½Q /2 *QMp2`;āM+B�- 2MmK2`�/�b
� b2;mB`,
RX � _P* /2 X(s) +QMbBbi2 /2 7�Bt�b p2`iB+�Bb- T�`�H2H�b �Q 2BtQ
jω- MQ TH�MQ sX
kX 1K h`�Mb7Q`K�/�b /2 G�TH�+2 `�+BQM�Bb- � _P* M½Q +QMiûK
M2M?mK TQHQX
jX a2 � 7mMϽQ x(t) i2K /m`�ϽQ }MBi� 2 û �#bQHmi�K2Mi2 BMi2@
;`€p2H- � _P* b2`€ iQ/Q Q TH�MQ sX
9X a2 x(t) 7Q` mK� 7mMϽQ H�i2`�H /B`2Bi�- 2 b2 � `2i� ${s} = σ0
T2`i2M+2` ¨ _P*- 2Mi½Q iQ/Qb Qb p�HQ`2b /2 s QM/2 ${s} > σ0
i�K#ûK 2bi�`½Q M� _P*X
8X a2 x(t) 7Q` mK� 7mMϽQ H�i2`�H 2b[m2`/�- 2 b2 � `2i� ${s} = σ0
T2`i2M+2` ¨ _P*- 2Mi½Q iQ/Qb Qb p�HQ`2b /2 s QM/2 ${s} < σ0
i�K#ûK 2bi�`½Q M� _P*X
8k *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
eX a2 x(t) 7Q` mK� 7mMϽQ #BH�i2`�H- 2 b2 � `2i� ${s} = σ0 T2`@
i2M+2` ¨ _P*- 2Mi½Q � _P* +QMbBbiB`€ /2 mK� 7�Bt� MQ TH�MQ
s [m2 +QMiûK � `2i� ${s} = σ0X
dX a2 � h`�Mb7Q`K�/� /2 G�TH�+2 X(s) /2 x(t) 7Q` `�+BQM�H- 2Mi½Q
bm� _P* û mK b2KBTH�MQ HBKBi�/Q TQ` TQHQb ¨ 2b[m2`/� 2 ¨
/B`2Bi�- Qm b2 2bi2M/2 �iû Q BM}MBiQ- ¨ /B`2Bi� Qm ¨ 2b[m2`/�X
�/B+BQM�HK2Mi2- M2M?mK TQHQ 2bi�`€ +QMiB/Q M� _P*X
3X a2 � h`�Mb7Q`K�/� /2 G�TH�+2 X(s) û `�+BQM�H 2 x(t) û H�i2`�H
/B`2Bi�- � _P* û Q b2KBTH�MQ MQ TH�MQ s ¨ /B`2Bi� /Q TQHQ K�Bb
¨ /B`2Bi�X �Hi2`M�iBp�K2Mi2- b2 x(t) û H�i2`�H 2b[m2`/�- � _P*
û Q b2KBTH�MQ MQ TH�MQ s ¨ 2b[m2`/� /Q TQHQ K�Bb ¨ 2b[m2`/�X
P +QM?2+BK2MiQ /�b T`QT`B2/�/2b /� _P* û ȹiBH MQ 2bim/Q /2
bBbi2K�b GAh- bBKTHB}+�M/Q � �M€HBb2 2K T`Q#H2K�b +QKTH2tQbX
jXj h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2
L� a2ϽQ jXR- 7QB KQbi`�/Q [m2 � h`�Mb7Q`K�/� /2 G�TH�+2 /2 x(t)
TQ/2 b2` BMi2`T`2i�/� +QKQ � h`�Mb7Q`K�/� /2 6Qm`B2` /2 x(t)e−σt-
mK� p2x [m2 s = σ + jωX 1b+`2p2M/Q 2bi2 `2bmHi�/Q M� 1[m�ϽQ /2
aőMi2b2 /� h`�Mb7Q`K�/� /2 6Qm`B2`,
x(t)e−σt = F−1
{
X(σ + jω)
}
=
1
2pi
∫ ∞
−∞
X(σ + jω)ejωt /ω
UjX3V
KmHiBTHB+�M/Q Qb /QBb H�/Qb /� B;m�H/�/2 TQ` eσt,
x(t) =
1
2pi
∫ ∞
−∞
X(σ + jω)e(σ+jω)t /ω UjXNV
lK� p2x [m2 s = σ + jω- TQ/2@b2 Q#i2` x(t) � T�`iB` /2 X(s) T�`�
mK p�HQ` }tQ /2 σ UT2`i2M+2Mi2 ¨ _P*V 2 p�`B�M/Q ω /2 −∞ � ∞X
�HûK /BbbQ- p2`B}+�M/Q [m2 /s = j/ω UT�`� σ +QMbi�Mi2V- 2Mi½Q �
jX9X S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 8j
/2}MBϽQ /� h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2 û,
x(t) =
1
j2pi
∫ σ+j∞
σ−j∞
X(s)est /s UjXRyV
P +€H+mHQ 7Q`K�H /� 1[X jXRy BMpQHp2 mK� BMi2;`�H /2 +QMiQ`MQ bQ@
#`2 Q TH�MQ +QKTH2tQ- b2M/Q � BMi2;`�ϽQ +QMp2`;2Mi2 T�`� [m�H[m2`
σ T2`i2M+2Mi2 ¨ _P*X 1K#Q`� TQbbőp2H- 2bi� QT2`�ϽQ û /2K�bB�/�@
K2Mi2 i`�#�H?Qb�- b2M/Q +QKmK2Mi2 miBHBx�/�b iû+MB+�b �Hi2`M�iBp�b
T�`� � Q#i2MϽQ /� h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2X
jX9 S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2
aBKBH�`K2Mi2 ¨ h`�Mb7Q`K�/� /2 6Qm`B2`- � h`�Mb7Q`K�/� /2 G�@
TH�+2 �T`2b2Mi� mK +QMDmMiQ/2 T`QT`B2/�/2b ȹi2Bb M� �M€HBb2 /2
T`Q#H2K�b +QKTH2tQb 2 MQ 2bim/Q �T`Q7mM/�/Q /� h`�Mb7Q`K�/� /2
G�TH�+2X 1bi�b T`QT`B2/�/2b b½Q /Bb+miB/�b 2K /2i�H?2 M� b2[māM+B�X
jX9XR GBM2�`B/�/2
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXRRV
2 [m2,
y(t)
L←−−−−→ Y (s), +QK _P* = Ry UjXRkV
2Mi½Q,
ax(t) + by(t)
L←−−−−→ aX(s) + bY (s), +QK _P* ⊃ Rx ∩Ry
UjXRjV
*QM7Q`K2 BM/B+�/Q- � _P* `2bmHi�Mi2 +QMiûK � BMi2`b2ϽQ /�b
_P*b +QMbiBimBMi2bX 1K +�bQb 2ti`2KQb- � +QK#BM�ϽQ HBM2�` `2bmH@
i�Mi2 TQ/2 TQbbmB` mK +QMDmMiQ p�xBQ +QKQ _P*- /2 7Q`K� [m2 �
h`�Mb7Q`K�/� /2 G�TH�+2 M½Q û /2}MB/�X
89 *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
jX9Xk .2bHQ+�K2MiQ MQ i2KTQ
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXR9V
2Mi½Q,
x(t− t0) L←−−−−→ e−st0X(s), +QK _P* = Rx UjXR8V
1bi� T`QT`B2/�/2 TQ/2 b2` Q#iB/� T2H� bBKTH2b bm#biBimBϽQ /2 p�@
`B€p2Bb M� /2}MBϽQ /� h`�Mb7Q`K�/� /2 G�TH�+2X
jX9Xj .2bHQ+�K2MiQ MQ /QKőMBQ s
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXReV
2Mi½Q,
es0tx(t)
L←−−−−→ X(s− s0), +QK _P* = Rx + $
{
s0
} UjXRdV
Pm b2D�- � _P* û /2bHQ+�/� /2 ${s0}X 1t2KTHB}+�M/Q- b2 X(s)
TQbbmB mK TQHQ 2K s = a- X(s−s0) TQbbmB`€ mK TQHQ 2K s = a+s0X
lK +�bQ 2bT2+B�H BKTQ`i�Mi2 /2 MQi� û s0 = jω0- QM/2,
ejω0tx(t)
L←−−−−→ X(s− jω0), +QK _P* = Rx UjXR3V
L2bi2 +�bQ- Q bBM�H x(t) KQ/mH� � 2tTQM2M+B�H +QKTH2t�- 2 � 7mMϽQ
X(s−jω0) TQ/2 b2` BMi2`T`2i�/� +QKQ mK /2bHQ+�K2MiQ MQ b2MiB/Q
T�`�H2HQ �Q 2BtQ jωX
jX9X9 Jm/�M� M� 2b+�H� /Q i2KTQ
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXRNV
jX9X S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 88
2Mi½Q,
x(at)
L←−−−−→ 1∣∣a∣∣X
(
s
a
)
, +QK _P* = aRx UjXkyV
.2p2 b2` Q#b2`p�/Q [m2 ?€ mK 2b+�H�K2MiQ M� _P*- /2T2M/2M/Q
/Q p�HQ` /2 sX a2 0 < a < 1- ?€ mK� +QKT`2bb½Q MQ p�HQ` /� _P*-
b2 a > 1- ?€ mK� 2tT�Mb½Q /� _P*X a2 a < 0- ?€ � `2p2`b½Q /�
_P*X
� `2p2`b½Q MQ i2KTQ û mK +�bQ 2bT2+B�H /� Km/�MÏ� /2 2b+�H�-
/2}MB/� +QKQ,
x(−t) L←−−−−→ X(−s), +QK _P* = −Rx UjXkRV
jX9X8 *QMDm;�ϽQ
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXkkV
2Mi½Q,
x∗(t) L←−−−−→ X∗(s∗), +QK _P* = Rx UjXkjV
�bbBK- b2 x(t) û `2�H- 2Mi½Q X(s) = X∗(s∗)X
jX9Xe S`QT`B2/�/2 /� *QMpQHmϽQ
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXk9V
2 [m2,
y(t)
L←−−−−→ Y (s), +QK _P* = Ry UjXk8V
2Mi½Q,
x(t) ∗ y(t) L←−−−−→ X(s)Y (s), +QK _P* ⊃ Rx ∩Ry UjXkeV
*QKQ M� T`QT`B2/�/2 /2 HBM2�`B/�/2- � _P* `2bmHi�Mi2 TQ/2 b2`
K�BQ` [m2 � BMi2`b2ϽQ /2pB/Q �Q TQbbőp2H +�M+2H�K2MiQ /2 TQHQb 2
x2`Qb 2Mi`2 �b 7mMÏǤ2bX
8e *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
jX9Xd .B72`2M+B�ϽQ i2KTQ`�H
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXkdV
2Mi½Q,
/x(t)
/t
L←−−−−→ sX(s), +QK _P* ⊃ Rx UjXk3V
1bi� T`QT`B2/�/2 TQ/2 b2` T`Qp�/� �THB+�M/Q � /2`Bp�/� M� /2}MBϽQ
/� h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2,
x(t) =
1
j2pi
∫ σ+j∞
σ−j∞
X(s)est /s UjXkNV
`2bmHi�M/Q,
/x(t)
/t =
1
j2pi
∫ σ+j∞
σ−j∞
sX(s)est /s UjXjyV
*QKQ � KmHiBTHB+�ϽQ TQ` s TQ/2 +�M+2H�` mK TQHQ M� Q`B;2K-
?€ � TQbbB#BHB/�/2 /2 mK� _P* K�Bb �#`�M;2Mi2 [m2 � Q`B;BM�HX
jX9X3 .B72`2M+B�ϽQ MQ /QKőMBQ s
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXjRV
2Mi½Q,
−tx(t) L←−−−−→ /X(s)/s , +QK _P* = Rx UjXjkV
1bi� T`QT`B2/�/2 TQ/2 b2` T`Qp�/� �THB+�M/Q � /2`Bp�/� M� /2}MBϽQ
/� h`�Mb7Q`K�/� /2 G�TH�+2,
X(s) =
∫ ∞
−∞
x(t)e−st /t UjXjjV
`2bmHi�M/Q 2K,
/X(s)
/s =
∫ ∞
−∞
−tx(t)e−st /t UjXj9V
jX9X S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 8d
jX9XN AMi2;`�ϽQ i2KTQ`�H
�bbmKBM/Q [m2,
x(t)
L←−−−−→ X(s), +QK _P* = Rx UjXj8V
2Mi½Q,∫ t
−∞
x(τ) /τ L←−−−−→ 1
s
X(s), +QK _P* ⊃ Rx ∩
{${s} > 0}
UjXjeV
1bi� T`QT`B2/�/2 TQ/2 b2` 7�+BHK2Mi2 T`Qp�/� � T�`iB` /� T`Q@
T`B2/�/2 /� +QMpQHmϽQ- TQBb � +QMpQHmϽQ /2 mK� 7mMϽQ x(t) +QK
Q /2;`�m mMBi€`BQ u(t) û MmK2`B+�K2Mi2 2[mBp�H2Mi2 ¨ BMi2;`�ϽQX
jX9XRy h2Q`2K�b /2 o�HQ` 6BM�H 2 AMB+B�H
a2 x(t) = 0 T�`� t < 0 2 b2 HBK
t→∞x(t) <∞- Q h2Q`2K� /Q o�HQ` 6BM�H
�}`K� [m2,
HBK
t→∞x(t) = HBKs→0 sX(s) UjXjdV
aQ# �b +QM/BÏǤ2b [m2 x(t) = 0 T�i� t < 0- 2 [m2 x(t) M½Q TQbbmB
BKTmHbQb Qm Qmi`�b bBM;mH�`B/�/2b /2 Q`/2K K�Bb �Hi� M� Q`B;2K-
TQ/2@b2 miBHBx�` � h`�Mb7Q`K�/� /2 G�TH�+2 T�`� +�H+mH�` /B`2i�@
K2Mi2 b2m p�HQ` BMB+B�H x(0+) UBX2X b2m HBKBi2 b2 �T`QtBK�M/Q T2H�
/B`2Bi�VX �bbBK- Q h2Q`2K� /Q o�HQ` AMB+B�H 2bi�#2H2+2 [m2,
x(0+) = HBK
s→∞ sX(s) UjXj3V
� h�#2H� 8 bmK�`Bx� �b T`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�@
TH�+2X
83 *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
h�#2H� 8 Ĝ S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2X
S`QT`B2/�/2 aBM�H h`�Mb7Q`K�/�/2 G�TH�+2 _P*
x(t) X(s) Rx
y(t) Y (s) Ry
GBM2�`B/�/2 ax(t) + by(t) aX(s) + bY (s) ⊃ Rx ∩Ry
.2bHQ+�K2MiQ
MQ i2KTQ x(t− t0) e
−st0X(s) = Rx
.2bHQ+�K2MiQ
2K s e
s0tx(t) X(s− s0) = Rx + $
{
s0
}
Jm/�M� M�
2b+�H� /Q
i2KTQ
x(at)
1∣∣a∣∣X
(
s
a
)
= aRx
*QMDm;�ϽQ x∗(t) X∗(s∗) = Rx
*QMpQHmϽQ x(t) ∗ y(t) X(s)Y (s) ⊃ Rx ∩Ry
.B72`2M+B�ϽQ
MQ i2KTQ
/x(t)
/t sX(s) ⊃ Rx
.B72`2M+B�ϽQ
2K s −tx(t)
/X(s)
/s = Rx
AMi2;`�ϽQ
∫ t
−∞
x(τ) /τ 1
s
X(s) ⊃ Rx∩
{${s} > 0}
h2Q`2K� /Q
o�HQ` 6BM�H
a2 x(t) = 0 T�`� t < 0 2 HBK
t→∞x(t) <∞- 2Mi½Q,
HBK
t→∞x(t) = HBKs→0 sX(s)
*QMiBMm� M� T`ƦtBK� T€;BM�
jX8X S�`2b +QKmMb /� h`�Mb7Q`K�/� /2 G�TH�+2 8N
S`QT`B2/�/2 aBM�H h`�Mb7Q`K�/�/2 G�TH�+2 _P*
h2Q`2K� /Q
o�HQ` AMB+B�H
a2 x(t) = 0 T�`� t < 0 2 x(t) M½Q +QMiûK BK@
TmHbQb Qm bBM;mH�`B/�/2b /2 Q`/2K K�Bb �Hi� M�
Q`B;2K- 2Mi½Q,
x(0+) = HBK
s→∞ sX(s)
jX8 S�`2b +QKmMb /� h`�Mb7Q`K�/� /2 G�TH�+2
*QM7Q`K2 /BiQ M� a2ϽQ jXj- � Q#i2MϽQ /� h`�Mb7Q`K�/� AMp2`b� /2
G�TH�+2 � T�`iB` /� /2}MBϽQ û mK T`Q+2bbQ K�i2K€iB+Q /2K�bB�/�@
K2Mi2 +QKTH2tQX lK� �Hi2`M�iBp� +QKmK û � miBHBx�ϽQ /Qb T�`2b
i`�Mb7Q`K�/Qb +QKmMb- K�Bb �b T`QT`B2/�/2b /� h`�Mb7Q`K�/� /2
G�TH�+2- T�`� � Q#i2MϽQ /� h`�Mb7Q`K/� AMp2`b� /2 G�TH�+2X
� bBKTHB}+�ϽQ /2 mK� 7mMϽQ X(s) �`#Bi`€`B� T�`� Qb +QKTQ@
M2Mi2b K�Bb bBKTH2b [m2 };m`�K M� h�#2H� e û +?�K�/� /2 2tT�Mb½Q
2K 7`�ÏǤ2b T�`+B�BbX 1bi2 T`Q+2/BK2MiQ û 2tTHB+�/Q 2K /2i�H?2 MQ
�TāM/B+2 �X � h�#2H� e bmK�`Bx� Qb T�`2b i`�Mb7Q`K�/Qb /2 G�TH�+2
K�Bb +QKmMbX
h�#2H� e Ĝ S�`2b +QKmMb /� h`�Mb7Q`K�/� /2 G�TH�+2X
aBM�H h`�Mb7Q`K�/�/2 G�TH�+2 _P*
δ(t) 1 iQ/Q s
u(t)
1
s
${s} > 0
−u(−t) 1
s
${s} < 0
*QMiBMm� M� T`ƦtBK� T€;BM�
ey *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
aBM�H h`�Mb7Q`K�/�/2 G�TH�+2 _P*
tnu(t)
n!
sn+1
${s} > 0
−tnu(−t) n!
sn+1
${s} < 0
e−atu(t)
1
s+ a
${s} > −a
−e−atu(−t) 1
s+ a
${s} < −a
tne−atu(t)
n!
(s+ a)n+1
${s} > −a
−tne−atu(−t) n!
(s+ a)n+1
${s} < −a
δ(t− t0) est0 iQ/Q s
+Qb(ω0t)u(t)
s
s2 + ω20
${s} > 0
bBM(ω0t)u(t)
ω0
s2 + ω20
${s} > 0
e−at +Qb(ω0t)u(t)
s+ a
(s+ a)2 + ω20
${s} > −a
e−at bBM(ω0t)u(t)
ω0
(s+ a)2 + ω20
${s} > −a
δn
′
(t) =
/nδ(t)
/tn s
n iQ/Q s
un(t) = u(t) ∗ . . . ∗ u(t)︸ ︷︷ ︸
n p2x2b
1
sn
${s} > 0
jXe h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H
�iû �;Q`�- 7QB �M�HBb�/� � h`�Mb7Q`K�/� /2 G�TH�+2 "BH�i2`�HX L2bi�
a2ϽQ b2`€ 2bim/�/� � h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H- [m2 TQb@
bmB ;`�M/2 p�HQ` M� �M€HBb2 /2 bBbi2K�b +�mb�Bb UbBbi2K�b +mD� b�ő/�
jXeX h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H eR
� mK /�/Q i2KTQ û /2T2M/2Mi2 �T2M�b /� 2Mi`�/� �Q K2bKQ i2KTQ-
Qm mK i2KTQ �Mi2`BQ`V 2- T�`iB+mH�`K2Mi2- bBbi2K�b /2i2`KBM�/Qb
TQ` 2[m�ÏǤ2b /B72`2M+B�Bb HBM2�`2b +QK +Q2}+B2Mi2b +QMbi�Mi2b 2 +QK
+QM/BÏǤ2b BMB+B�Bb M½Q@MmH�b UQm b2D�- bBbi2K�b [m2 M½Q 2bi½Q BMB+B�H@
K2Mi2 2K `2TQmbQVX
� h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H û /2}MB/� TQ`,
X (s) =
∫ ∞
0−
x(s)e−st /t UjXjNV
M2bi2 i2tiQ- b2`€ miBHBx�/� X (s) T�`� `2T`2b2Mi�` � h`�Mb7Q`K�/�
/2 G�TH�+2 lMBH�i2`�H- T�`� 2pBi�` +QM7mb½Q +QK � h`�Mb7Q`K�/� /2
G�TH�+2 "BH�i2`�H X(s)X
P HBKBi2 BM72`BQ` /2 BMi2;`�ϽQ 0−- bB;MB}+� [m2 b½Q BM+Hmő/Qb MQ
BMi2`p�HQ /2 BMi2;`�ϽQ [m�Bb[m2` BKTmHbQb Qm bBM;mH�`B/�/2b /2 �Hi�Q`/2K [m2 TQbb�K �T�`2+2` M� Q`B;2KX �M�HQ;�K2Mi2 ¨ h`�Mb7Q`@
K�/� /2 G�TH�+2 "BH�i2`�H- � h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H
i�K#ûK TQbbmB mK� MQi�ϽQ �#`2pB�/�,
x(t)
UL←−−−−−→ X (s) X (s) = U L {x(t)} UjX9yV
aBM�Bb [m2 /B72`2K �T2M�b T�`� t < 0 i2`½Q h`�Mb7Q`K�/�b /2
G�TH�+2 lMBH�i2`�H B;m�Bb- K�b h`�Mb7Q`K�/�b /2 G�TH�+2 "BH�i2`�Bb
/B72`2Mi2bX aBM�Bb [m2 b2D�K MmHQb T�`� t < 0 TQbbm2K h`�Mb7Q`K�@
/�b /2 G�TH�+2 lMBH�i2`�H 2 "BH�i2`�H B/āMiB+�bX
.2 7Q`K� �Hi2`M�iBp�- � h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H /2
x(t) TQ/2 b2` �#Q`/�/� +QKQ � h`�Mb7Q`K�/� /2 G�TH�+2 "BH�i2`�H
/2 x(t)u(t)- Qm b2D� X (s) = L {x(t)u(t)}X
jXeXR S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H
�M�HQ;�K2Mi2 ¨ h`�Mb7Q`K�/� /2 G�TH�+2 "BH�i2`�H- � h`�Mb7Q`@
K�/� /2 G�TH�+2 lMBH�i2`�H i�K#ûK �T`2b2Mi� T`QT`B2/�/2b ;2`�Bb-
KmBi�b /2H�b B/āMiB+�b ¨b /� h`�Mb7Q`K�/� /2 G�TH�+2 "BH�i2`�H- K�b
�H;mK�b bm#bi�M+B�HK2Mi2 /B72`2Mi2bX
ek *�TőimHQ jX h`�Mb7Q`K�/� /2 G�TH�+2
lK� /�b /B72`2M�b K�Bb K�`+�Mi2b /� h`�Mb7Q`K�/� /2 G�TH�+2
lMBH�i2`�H û [m2 bm� _P* �bbQ+B�/� û b2KT`2 Q b2KBTH�MQ ¨ /B`2Bi�
/Q TQHQ K�Bb ¨ /B`2Bi�- /�ő � `�x½Q /2 M½Q ?�p2` `272`āM+B�b 2tTHő+Bi�b
¨ _P* M�b T`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H-
bmK�`Bx�/�b M� h�#2H� dX
h�#2H� d Ĝ S`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�HX
S`QT`B2/�/2 aBM�H
h`�Mb7Q`K�/� /2
G�TH�+2 lMBH�i2@
`�H
x(t) X (s)
y(t) Y(s)
GBM2�`B/�/2 ax(t) + by(t) aX (s) + bY (s)
.2bHQ+�K2MiQ 2K s es0tx(t) X (s− s0)
Jm/�M� M� 2b+�H�
/Q i2KTQ x(at)- a > 0
1
a
X
(
s
a
)
*QMDm;�ϽQ x∗(t) X ∗(s)
*QMpQHmϽQR x(t) ∗ y(t) X (s)Y(s)
.B72`2M+B�ϽQ MQ
i2KTQ
/x(t)
/t sX (s)− x(0
−)
.B72`2M+B�ϽQ 2K s −tx(t) /X (s)/s
AMi2;`�ϽQ
∫ t
0−
x(τ) /τ 1
s
X (s)
a2 x(t) M½Q +QMiûK BKTmHbQb Qm bBM;mH�`B/�/2b /2 Q`/2K K�Bb
�Hi� 2K t = 0- 2Mi½Q,
h2Q`2K� /Q o�HQ`
6BM�H
HBK
t→∞x(t) = HBKs→0 sX (s)
*QMiBMm� M� T`ƦtBK� T€;BM�
R amTQM/Q x(t) 2 y(t) MmHQb T�`� t < 0X
jXeX h`�Mb7Q`K�/� /2 G�TH�+2 lMBH�i2`�H ej
S`QT`B2/�/2 aBM�H
h`�Mb7Q`K�/� /2
G�TH�+2 lMBH�i2@
`�H
h2Q`2K� /Q o�HQ`
AMB+B�H x(0
+) = HBK
s→∞ sX (s)
�H;mK�b /�b Km/�MÏ�b BKTQ`i�Mi2b /2 MQi� b½Q,
Ç � T`QT`B2/�/2 /� +QMpQHmϽQ û /2}MB/� �T2M�b T�`� bBM�Bb
BMB+B�HK2Mi2 2K `2TQmbQ UMmHQb T�`� t < 0VX
Ç Pb HBKBi2b /� T`QT`B2/�/2 /2 BMi2;`�ϽQ b½Q �Hi2`�/Qb T�`� Q
MQpQ BMi2`p�HQ /2 BMi2`2bb2 /� h`�Mb7Q`K�/� /2 G�TH�+2 lMB@
H�i2`�H UBX2X [0−,∞)VX
Ç � �Hi2`�ϽQ K�Bb K�`+�Mi2- TQ`ûK- û M� T`QT`B2/�/2 /� /B@
72`2M+B�ϽQ MQ i2KTQ- QM/� ?€ � �/BϽQ /2 mK i2`KQ BM/2@
T2M/2Mi2 −x(0−) ¨ T`QT`B2/�/2 �M€HQ;� M� h`�Mb7Q`K�/� /2
G�TH�+2 "BH�i2`�HX L� b2[māM+B� û /2b2MpQHpB/� � T`Qp� /2bi�
T`QT`B2/�/2,∫ ∞
0−
/x(t)
/t e
−st /t = x(t)e−st
∣∣∣∣∣
∞
0−
+ s
∫ ∞
0−
x(t)e−st /t
= sX (s)− x(0−)
UjX9RV
�TāM/B+2b
�SĀL.A*1�
1tT�Mb½Q 2K 6`�ÏǤ2b
S�`+B�Bb
� Q#i2MϽQ /� h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2 � T�`iB` /� bm�
/2}MBϽQ U1[X jXRyV- û mK T`Q+2/BK2MiQ �M�HőiB+Q `�xQ�p2HK2Mi2 2M@
pQHpB/Q- b2M/Q /2b2D€p2H mK� bQHmϽQ �Hi2`M�iBp�X JmBiQ 2K#Q`� b2D�
TQbbőp2H Q#i2`- 2K +�bQb K�Bb bBKTH2b- � h`�Mb7Q`K�/� AMp2`b� /2
G�TH�+2 �i`�pûb /� bBKTH2b BMbT2ϽQ `2p2`b� /2 mK� i�#2H� /2 T�@
`2b i`�Mb7Q`K�/Qb Uh�#2H� eV- 2 +QKTH2K2Mi�`K2Mi2- mK� i�#2H� /2
T`QT`B2/�/2b /� h`�Mb7Q`K�/� /2 G�TH�+2 Uh�#2H� 8V- ?€ +�bQb QM/2
� 2tT`2bb½Q X(s) bBKTH2bK2Mi2 M½Q TQbbmB M2K `2KQi� bBKBH�`B/�/2
� M2M?mK T�` i`�Mb7Q`K�/Q +QM?2+B/QX
L2bi2b +�bQb- Q T`Q+2/BK2MiQ b2;mB/Q û dz[m2#`�`Ǵ � 7mMϽQ X(s)
2K b2mb +QKTQM2Mi2b K�Bb bBKTH2b- QM/2 2bT2`�@b2 [m2 2bi2b };m`2K
M� i�#2H� /2 T�`2b i`�Mb7Q`K�/QbX h�H T`Q+2/BK2MiQ /2 dz[m2#`�Ǵ û
+?�K�/Q /2 2tT�Mb½Q 2K 7`�ÏǤ2b T�`+B�Bb 2- �T2b�` /Q MQK2- M�/�
K�Bb û [m2 Q +�KBM?Q BMp2`bQ /2 KőMBKQ KȹHiBTHQ +QKmKX
e3 �SĀL.A*1 �X 1tT�Mb½Q 2K 6`�ÏǤ2b S�`+B�Bb
�bbmKBM/Q mK� 2tT`2bb½Q X(s)- `�+BQM�H- /� 7Q`K�,
X(s) = A
sm + bm−1sm−1 + bm−2sm−2 + . . .+ b1s+ b0
sn + an−1sn−1 + an−2sn−2 + . . .+ a1s+ a0
U�XRV
b2M/Q A mK� +QMbi�Mi2 [m�H[m2`- Qm- /2 K�M2B`� K�Bb 7Q`K�H,
X(s) = A
sm +
m−1∑
k=0
bks
k
sn +
n−1∑
k=0
aks
k
U�XkV
/Bi� 2bi�` M� 7Q`K� +�MƬMB+�- TQBb Qb +Q2}+B2Mi2b /2 K�Bb �Hi� Q`/2K
MQ MmK2`�/Q` 2 MQ /2MQKBM�/Q` b½Q mMBi€`BQbX P MmK2`�/Q` û mK
TQHBMƬKBQ 2K s +QK m `�őx2b 2 Q /2MQKBM�/Q` û mK TQHBMƬKBQ 2K
s +QK n `�őx2bX �bbBK- X(s) û mK� 7mMϽQ +QK n TQHQb 2 m x2`QbX
�bbmKBM/Q iQ/Qb Qb TQHQb 2 x2`Qb /2 X(s) `2�Bb- TQ/2@b2 `2/mxB`
Qb TQHBMƬKBQb ¨ bm�b `�őx2b +QMbiBimBMi2b /� 7Q`K�R,
X(s) = A
(s− z1)(s− z2) . . . (s− zm−1)(s− zm)
(s− p1)(s− p2) . . . (s− pn−1)(s− pn) U�XjV
Qm- MQp�K2Mi2- /2 K�M2B`� K�Bb 7Q`K�H,
X(s) = A
m∏
k=1
(
s− zk
)
n∏
k=1
(
s− pk
) U�X9V
b2M/Q zk Q k@ûbBKQ x2`Q /2 X(s) 2 pk Q k@ûbBKQ TQHQ /2 X(s)X
a2 ?Qmp2` `�őx2b +QKTH2t�b +QMDm;�/�b- � T`€iB+� +QKmK û /2B@
t�` M� 7Q`K� [m�/`€iB+� B``2/miőp2H 2K MȹK2`Qb `2�Bb- /2 7Q`K�,
X(s) = A
m1∏
k=1
(
s− zk
) m2∏
k=1
(
s− zk
)(
s− z∗k
)
n1∏
k=1
(
s− pk
) n2∏
k=1
(
s− pk
)(
s− p∗k
) U�X8V
R SQ` bBKTHB+B/�/2- � �M€HBb2 �;Q`� b2`€ BHmbi`�/� �T2M�b T�`� TQHQb 2 x2`Qb
`2�Bb- K�Bb �/B�Mi2 b2`€ KQbi`�/Q Q K2+�MBbKQ T�`� TQHQb 2 x2`Qb +QKTH2tQb
+QMDm;�/QbX
eN
Qm- �Hi2`M�iBp�K2Mi2,
X(s) = A
m1∏
k=1
(
s− zk
) m2∏
k=1
(
s2 − 2s${zk}+ ∣∣zk∣∣2)
n1∏
k=1
(
s− pk
) n2∏
k=1
(
s2 − 2s${pk}+ ∣∣pk∣∣2) U�XeV
QM/2 m1 û Q MȹK2`Q /2 x2`Qb `2�Bb- m2 û Q MȹK2`Q /2 T�`2b /2 x2`Qb
+QKTH2tQb +QMDm;�/Qb- /2 7Q`K� [m2 m = m1 + 2m2X .� K2bK�
7Q`K� n1 û Q MȹK2`Q /2 TQHQb `2�Bb- n2 û Q MȹK2`Q /2 T�`2b /2 TQHQb
+QKTH2tQb +QMDm;�/Qb- /2 7Q`K� [m2 n = n1 + 2n2X
P T`Q+2/BK2MiQ +QKTH2iQ T�`� `2T`2b2Mi�` mK� 7mMϽQ `�+BQM�H
X(s) �i`�pûb /2 bm�b 7`�ÏǤ2b T�`+B�Bb û /�/Q TQ`,
RX o2`B}+�K Q MȹK2`Q /2 x2`Qb m 2 Q MȹK2`Q /2 TQHQb nX
Ç a2 n > m TQ/2@b2 BMB+B�` /2 7Q`K� /B`2i� � 2tT�Mb½Q 2K
7`�ÏǤ2b T�`+B�BbX
Ç a2 n ≤ m- /2p2@b2 `2�HBx�` � /BpBb½Q HQM;� /2 TQHBMƬKBQb
�iû [m2- M� 7`�ϽQ `2bi�Mi2- � Q`/2K /Q /2MQKBM�/Q` b2D�
K�BQ` [m2 � Q`/2K /Q MmK2`�/Q`X
kX .�/� mK� 2tT`2bb½Q `�+BQM�H- +QK � Q`/2K /Q MmK2`�/Q`
K2MQ` [m2 � Q`/2K /Q /2MQKBM�/Q`- � 2tT�Mb½Q 2K 7`�ÏǤ2b
T�`+B�Bb TQ/2 b2` mK- Qm mK� +QK#BM�ϽQ /2 i`āb +�bQbX
Ç P /2MQKBM�/Q` û mK T`Q/miƦ`BQ /2 n `�őx2b /BbiBMi�bX
Ç P /2MQKBM�/Q` +QMiûK `�őx2b +QK KmHiBTHB+B/�/2X
Ç P /2MQKBM�/Q` +QMiûK `�őx2b +QKTH2t�b +QMDm;�/�bX
lK� /�/� 7mMϽQ X(s) TQ/2 +QMi2` mK� +QK#BM�ϽQ BM/2T2M@
/2Mi2 /2 +�/� mK� /2bi�b +QM/BÏǤ2b- [m2 /2p2K b2` �#Q`/�/�b b2@
T�`�/�K2Mi2X L� b2[māM+B�- Q K2+�MBbKQ /2 bQHmϽQ /2 +�/� +�bQ
û BHmbi`�/Q 2K mK 2t2KTHQ
dy �SĀL.A*1 �X 1tT�Mb½Q 2K 6`�ÏǤ2b S�`+B�Bb
�XR .BpBb½Q HQM;� /2 TQHBMƬKBQb
hQK�@b2 +QKQ 2t2KTHQ � 7mMϽQ X(s),
X(s) =
s3 + 3s2 + 3s+ 1
s2 + 5s+ 6
U�XdV
MQi�@b2 [m2 � Q`/2K /Q MmK2`�/Q` û K�BQ` [m2 � Q`/2K /Q /2MQ@
KBM�/Q`- b2M/Q M2+2bb€`B� � /BpBb½Q HQM;�- /� 7Q`K�,
s3 + 3s2 + 3s+ 1 s2 + 5s+ 6
−(s3 + 5s2 + 6s) s− 2
−2s2 − 3s+ 1
−(−2s2 − 10s− 12)
7s+ 13
�bbBK- TQ/2@b2 `22b+`2p2` X(s) +QKQ,
X(s) = s− 2 + 7s+ 13
s2 + 5s+ 6
U�X3V
M2bi� 2[m�ϽQ- � 7`�ϽQ `2bi�Mi2 TQ/2 b2` �THB+�/� 2tT�M/B/� 2K
7`�ÏǤ2b T�`+B�BbX
�Xk n TQHQb /BbiBMiQb
hQK�@b2 +QKQ 2t2KTHQ � 7mMϽQ X(s),
X(s) =
1
s3 + 3s2 + 2s
=
1
s(s+ 1)(s+ 2)
U�XNV
û TQbbőp2H `22b+`2p2` X(s) /� b2;mBMi2 K�M2B`�,
X(s) =
1
s(s+ 1)(s+ 2)
=
A
s
+
B
s+ 1
+
C
s+ 2
U�XRyV
iQK�M/Q � B;m�H/�/2 /2 BMi2`2bb2,
1
s(s+ 1)(s+ 2)
=
A
s
+
B
s+ 1
+
C
s+ 2
U�XRRV
�XjX SQHQb +QK KmHiBTHB+B/�/2 dR
TQ/2@b2 2M+QMi`�` A- B 2 C � T�`iB` /Q b2;mBMi2 /2b2MpQHpBK2MiQ,
1 = A(s+ 1)(s+ 2) +Bs(s+ 2) + Cs(s+ 1)
1 = A(s2 + 3s+ 2) +B(s2 + 2s) + C(s2 + s)
1 = (A+B + C)s2 + (3A+ 2B + C)s+ 2A
U�XRkV
/2 QM/2- TQ` BMbT2ϽQ- TQ/2@b2 Q#i2` Q b2;mBMi2 bBbi2K�,
A+B + C = 0
3A+ 2B + C = 0
2A = 1
U�XRjV
+mD� bQHmϽQ û,
A =
1
2
B = −1 C = 1
2
U�XR9V
�bbBK- X(s) TQ/2 b2` 2b+`Bi� +QKQk,X(s) =
1
s3 + 3s2 + 2s
=
1
2
1
s
− 1
s+ 1
+
1
2
1
s+ 2
U�XR8V
SQ` BMbT2ϽQ M� i�#2H� /2 T�`2b i`�Mb7Q`K�/Qb /2 G�TH�+2- 2
miBHBx�M/Q � T`QT`B2/�/2 /� HBM2�`B/�/2 U2 �bbmKBM/Q- TQ` +QMp2MB@
āM+B�- mK� _P* ${s} > 0V- û TQbbőp2H Q#i2`- � T�`iB` /� 2tT�Mb½Q
2K 7`�ÏǤ2b T�`+B�Bb- [m2 � h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2 /2 X(s)
û,
x(t) =
1
2
u(t)− e−tu(t) + 1
2
e−2tu(t) U�XReV
P T`Q+2/BK2MiQ BHmbi`�/Q û T`QMi�K2Mi2 2tT�Mbőp2H T�`� mK Mȹ@
K2`Q �`#Bi`€`BQ /2 TQHQb /2 X(s)X
�Xj SQHQb +QK KmHiBTHB+B/�/2
hQK�@b2 +QKQ 2t2KTHQ � 7mMϽQ X(s),
X(s) =
1
s4 + 3s3 + 3s2 + s
=
1
s(s+ 1)3
U�XRdV
k ú TQbbőp2H `2�HBx�` Q T`Q+2bbQ BMp2`bQ- �i`�pûb /2 KőMBKQ KȹHiBTHQ +QKmK-
T�`� p2`B}+�` � 2tT�Mb½Q 2K 7`�ÏǤ2b T�`+B�BbX
dk �SĀL.A*1 �X 1tT�Mb½Q 2K 6`�ÏǤ2b S�`+B�Bb
MQi�@b2 [m2 X(s) TQbbmB mK TQHQ M� Q`B;2K Us = 0V 2 i`āb TQHQb
2K s = −1- Qm b2D�- ?€ [m�i`Q TQHQb MQ iQi�H- b2M/Q [m2 mK /2H2b-
�T`2b2Mi� KmHiBTHB+B/�/2 i`BTH�X
L2bi2 +�bQ � 2tT�Mb½Q 2K 7`�ÏǤ2b T�`+B�Bb iQK� � 7Q`K�,
X(s) =
1
s(s+ 1)3
=
A
s
+
B
(s+ 1)3
+
C
(s+ 1)2
+
D
s+ 1
U�XR3V
MQi�@b2 [m2 MQb TQHQb +QK KmHiBTHB+B/�/2- � 2tT�Mb½Q 2K 7`�ÏǤ2b
T�`+B�Bb H2p� 2K +QMi� iQ/�b �b TQiāM+B�b BMi2B`�b TQbbőp2Bb- �iû �
mMB/�/2X
�THB+�M/Q Q K2bKQ KûiQ/Q /2 BMbT2ϽQ miBHBx�/Q M� a2ϽQ �M@
i2`BQ`,
A = 1 B = −1 C = −1 D = −1 U�XRNV
/2 7Q`K� [m2 X(s) TQ/2 b2` `22b+`Bi� +QKQ,
X(s) =
1
s4 + 3s3 + 3s2 + s
=
1
s
− 1
(s+ 1)3
− 1
(s+ 1)2
− 1
s+ 1
U�XkyV
/2 QM/2- MQp�K2Mi2- �i`�pûb /� i�#2H� /2 T�`2b i`�Mb7Q`K�/Qb- /�
T`QT`B2/�/2 /� HBM2�`B/�/2- 2 �bbmKBM/Q mK� _P* ${s} > 0- TQ/2@
b2 Q#i2` � h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2,
x(t) = u(t)− 1
2
t2e−tu(t)− te−tu(t)− e−tu(t) U�XkRV
J�Bb mK� p2x- Q K2+�MBbKQ BHmbi`�/Q û 2tT�Mbőp2H T�`� � T`2@
b2M� /2 /B72`2Mi2b TQHQb +QK /B72`2Mi2b KmHiBTHB+B/�/2bX
�X9 SQHQb +QKTH2tQb +QMDm;�/Qb
hQK�@b2 +QKQ 2t2KTHQ � 7mMϽQ X(s),
X(s) =
s2 − 4
s3 + s2 + s+ 1
=
(s+ 2)(s− 2)
(s+ 1)(s2 + 1)
U�XkkV
Q#b2`p�@b2 [m2 Q i2`KQ MQ /2MQKBM�/Q` (s2 + 1) û B``2/miőp2H 2K
`2�Bb- Qm b2D�- HBKBi�/Q � +Q2}+B2Mi2b `2�Bb- M½Q TQ/2 b2` 2tT�M/B/Q
T�`� /m�b `�őx2b /BbiBMi�bX
�X9X SQHQb +QKTH2tQb +QMDm;�/Qb dj
L2bi2b +�bQb- � 2tT�Mb½Q 2K 7`�ÏǤ2b T�`+B�Bb iQK� 7Q`K�,
X(s) =
s2 − 4
(s+ 1)(s2 + 1)
=
A
s+ 1
+
Bs+ C
s2 + 1
U�XkjV
Qm b2D�- [m�M/Q Q i2`KQ MQ /2MQKBM�/Q` û [m�/`€iB+Q 2 B``2/miőp2H
2K `2�Bb- Q i2`KQ MQ MmK2`�/Q }+� /2 Q`/2K mMBi€`B�X
a2;mBM/Q Q K2bKQ T`Q+2/BK2MiQ /�b a2ÏǤ2b �Mi2`BQ`2b- 2M+QMi`�@
b2 [m2,
A = −3
2
B =
5
2
C = −5
2
U�Xk9V
_22b+`2p2M/Q X(s),
X(s) = −3
2
1
s+ 1
+
5
2
s
s2 + 1
− 5
2
1
s2 + 1
U�Xk8V
/2 QM/2- TQ` BMbT2ϽQ M� i�#2H� /2 T�`2b i`�Mb7Q`K�/Qb- �THB+�M/Q
� T`QT`B2/�/2 /� HBM2�`B/�/2- 2 �bbmKBM/Q mK� _P* ${s} > 0,
x(t) = −3
2
e−tu(t) +
5
2
+Qb(t)u(t)− 5
2
bBM(t)u(t) U�XkeV
�X9XR *QKTH2i�` Q [m�/`�/Q
�H;mMb T`Q#H2K�b /2 h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2 2MpQHp2M/Q
TQHQb +QKTH2tQb +QMDm;�/Qb TQ/2K b2 iQ`M�` bm#bi�M+B�HK2Mi2 K�Bb
2MpQHpB/Qb [m2 Q +�bQ bBKTH2b /Q ȹHiBKQ 2t2KTHQX S�`� +QMi2KTH�`
2bi2b +�bQb- QM/2 7�x@b2 M2+2bb€`B� � miBHBx�ϽQ /� iû+MB+� /2 dz+QK@
TH2i�` Q [m�/`�/QǴ- [m2 Q b2;mBMi2 2t2KTHQ û �T`2b2Mi�/QX
X(s) =
2s2 + 4s+ 2
s3 + 2s2 + 2s
=
2s2 + 4s+ 2
s(s2 + 2s+ 2)
U�XkdV
b2M/Q Q 7�iQ` s2 + 2s+ 2 B``2/miőp2H 2K `2�BbX
1tT�M/BM/Q 2K 7`�ÏǤ2b T�`+B�Bb,
X(s) =
2s2 + 4s+ 2
s(s2 + 2s+ 2)
=
A
s
+
Bs+ C
s2 + 2s+ 2
U�Xk3V
b2M/Q TQbbőp2H Q#i2`,
A = 1 B = 1 C = 2 U�XkNV
d9 �SĀL.A*1 �X 1tT�Mb½Q 2K 6`�ÏǤ2b S�`+B�Bb
_22b+`2p2M/Q X(s),
X(s) =
1
s
+
s+ 2
s2 + 2s+ 2
U�XjyV
AM72HBxK2Mi2- Q b2;mM/Q i2`KQ `�+BQM�H M½Q 2bi€ 2tTHő+BiQ 2K i�@
#2H�b /2 T�`2b i`�Mb7Q`K�/Qb /2 G�TH�+2- Q [m2 2tB;B`€ mK i`�#�H?Q
�H;û#`B+Q �/B+BQM�HX
1K#Q`� b2D� TQbbőp2H /2b2MpQHp2` Q T`Q#H2K� miBHBx�M/Q �b `�ő@
x2b +QKTH2t�b +QMDm;�/�b- mK� bQHmϽQ K�Bb bBKTH2b û b�#2` [m2 Q
TQHBMƬKBQ s2 + 2s + 2 TQbbmB � 7Q`K� ;2`�H (s + a)2 + b2 U[m2 2bi€
T`2b2Mi2 M�b i�#2H�b /2 T�`2b i`�Mb7Q`K�/Qb /2 G�TH�+2V- QM/2 a 2
b b½Q +QMbi�Mi2b `2�BbX 1M+QMi`�` Qb p�HQ`2b a 2 b [m2 dz+QKTH2i�K Q
[m�/`�/QǴ û #�bi�Mi2 bBKTH2b,
s2 + 2s+ 2 = (s+ a)2 + b2
s2 + 2s+ 2 = s2 + 2as+ a2 + b2
U�XjRV
Q#i2M/Q a = 1 2 b = 1X
�bbBK- X(s) û `22b+`Bi� MQp�K2Mi2 +QKQ,
X(s) =
1
s
+
s+ 2
(s+ 1)2 + 1
U�XjkV
/2Bt�M/Q Q MmK2`�/Q 2K +QM7Q`KB/�/2 +QK Qb T�`2b i`�Mb7Q`K�/Qb,
X(s) =
1
s
+
s+ 2
(s+ 1)2 + 1
=
1
s
+
s+ 1 + 1
(s+ 1)2 + 1
=
1
s
+
s+ 1
(s+ 1)2 + 1
+
1
(s+ 1)2 + 1
U�XjjV
/2 QM/2 û TQbbőp2H Q#i2` � h`�Mb7Q`K�/� AMp2`b� /2 G�TH�+2 UMQp�@
K2Mi2- BMbT2+BQM�M/Q � i�#2H� /2 T�`2b i`�Mb7Q`K�/Qb- miBHBx�M/Q �
T`QT`B2/�/2 /2 HBM2�`B/�/2 2 +QMbB/2`�M/Q mK� _P* ${s} > 0V,
x(t) = u(t) + e−t +Qb(t)u(t) + e−t bBM(t)u(t) U�Xj9V
	Lista de Figuras
	Lista de Tabelas
	Sumário
	Introdução
	Séries de Fourier
	Perspectiva histórica
	Resposta de sistemas LIT às exponenciais complexas
	Representação de sinais periódicos através de Séries de Fourier
	Convergência da Série de Fourier
	Propriedades da Série de Fourier
	Linearidade
	Deslocamento no tempo
	Reflexão no tempo
	Mudança na escala de tempo
	Multiplicação
	Conjugação e simetria conjugada
	Relação de Parseval
	Séries de Fourier para sinais periódicos comuns.
	Transformada de Fourier
	Representação de sinais através da Transformada de Fourier
	Definição da Transformada de Fourier
	Convergência da Transformada de Fourier
	Transformada de Fourier para sinais periódicos
	Propriedades da Transformada de Fourier no tempo contínuo
	Linearidade
	Deslocamento no tempo
	Conjugação e simetria conjugada
	Diferenciação e integração
	Mudança na escala de tempo e na frequência
	Dualidade
	Relação de Parseval
	Propriedade da convolução
	Propriedade da multiplicação
	Pares transformados de Fourier comuns
	Transformada de Laplace
	A Transformada de Laplace
	A Região de Convergência da Transformada de Laplace
	Transformada Inversa de Laplace
	Propriedades da Transformada de Laplace
	Linearidade
	Deslocamento no tempo
	Deslocamento no domínio s
	Mudança na escala do tempo
	Conjugação
	Propriedade da Convolução
	Diferenciação temporal
	Diferenciação no domínio s
	Integração temporal
	Teoremas de Valor Final e Inicial
	Pares comuns da Transformada de Laplace
	Transformada de Laplace Unilateral
	Propriedades da Transformada de Laplace Unilateral
	Apêndices
	Expansão em Frações Parciais
	Divisão longa de polinômios
	n polos distintos
	Polos com multiplicidade
	Polos complexos conjugados
	Completar o quadrado