版权说明：本文档由用户提供并上传，收益归属内容提供方，若内容存在侵权，请进行举报或认领
文档简介
§3.5-典型非周期信号的频谱单边指数信号双边指数信号矩形脉冲信号钟形脉冲信号符号函数升余弦脉冲信号信号-¥dt
<
¥TTTT
fi
¥2222f
(
t
)1功率：lim能量：
f
(
t
)
dt
<
¥-
¥从能量方面考虑:傅立叶变换的重要性质唯一性f1
(t
)
=
f
2
(t
)由F
[f1
(t
)]=F
[f
2
(t
)]=F
(jw
)则反之,由21-1-1[F
(
jw
)]
=
f
(t)F
[F
(
jw
)]
=
FF1
(w
)
=
F2
(w
)则给出简短证明如下：111\=¥2p
-¥¥2p
-¥2p
-¥¥
¥-¥1¥2p
-¥f1
(t)
=
e-
jw
(t
-t)
dw
=
d
(t
-t)
e
jwt
dw[
f2
(t)e
dt]e
dw-
jwt
jwt
F
(
jw
)e
jwt
dwd
(t)
=交换积分顺序1f1
(t)
=¥
¥2p
-¥
-¥[
e
jw
(t
-t)
dw
]
f
(t)dt2¥=
d(t
-t)
f
2(t)dt
=
f
2
(t)-¥f1
(t),f2
(t)在积分意义上相等。傅立叶变换的唯一性表明了信号及其频谱的唯一对应关系。证明p17
1-352tEt[
sa
(
wt
)];1
«
2pd
(w
)EG
(t
)
«tfi
¥以及lim
Gt
(t)=1和傅立叶变换的唯一性，有2tfi
¥lim
Et[sa(wt)]
=
2pE
d(w
)2sa(kt)]kpK
fi
¥
2pd(t)
=
lim[k
fi
¥d(t)
=
lim
K
[sa(
Kt
)]f1
(t)f2
(t)波形矩形t22t-tAf
(t
)E|
t
|£t/2|
t
|>t/2f
(t)
=0，E，f(t)不连续2p
tt2p-wtAF
(w
)E
t)wt2F
(jw
)
=
Et
Sa
(F
(w)与w
大致成反比形三角t
)t2
t
)E
(1
-f
(
t
)
=0(
t
<2t
‡
0dtdff
(t
)连续、 不连续F(
w
)
=
E
t
Sa
2
(
wt
)2
4F
(w)与w2大致成反比t22t–Et2p380,附录三4p
8pt
twtf(t)E升余弦2t–t2Etf(t)
=22
t[1
+
cos(
pt
)]t
<
tE220t
‡
t不连续连续f(t)
、dt
2d
2
fdtdf2pt2pt-wtAE
t2F
(w
)Ett4p118面2p2
1
-(wt)2wtF(w)
=
Et
2
Sa(
)F
(w)与w3大致成反比6pt信号表达式–幅频–相频f
(t
)
=
0
e
-at¥-¥(t
‡
0
)(t
<
0
)(a
>
0)F
(w
)
=f(t)e-
jw
t
dt
=
1
a2
+w
2a
+
jwF(w
)
=
1
)w
aarctg(j(w
)
=
-单边指数信号的频谱f(t)t0F
(w
)
1awj
(w
)p22-
pw2a13af
(t)
=
e-a
t
(-¥
<
t
<
+¥
)a
2
+
w
2F(w
)
=
2a
j
(w
)
=
0双边指数信号的频谱一.冲激函数的频谱¥-¥=
1F[d
(t)]
=
d
(t)e
-
jwt
dt=
e
-
jw
0¥coswtdw-¥2p1¥1.e
jwt
dw
=-¥2p1d
(t)
=?d
(t
)tjwF
(
jw
)1000d
(
t
-
t
)t
0F
(
j
w
)j
(
j
w
)0-
w
td
(t)
«
1e
-
jwt
0d(t
-
t0
)
«P80-81黎曼－勒贝格2-99和2－100limcosw
t
=
0wfi
¥二.冲激偶的傅立叶变换¥-¥e
jw
tdw2pd(t)
=
1
¥-
¥(
jw
)
e
jw
t
d
wd
[d
(
t
)
]=dt12
p
dtF
d
d(t)
=
jwnF
n
d(t)
=
(
jw)dt
dndwndnF(tn
)=
2p(j)n
[d(w
)]F
[d
(t)]
=
1三.sgn(t)的付立叶变换+1
t>0-1
t<0f
(t
)
=
sgn(
t
)
=e
-at
.....t
>
0f
2
(t)
=-
eat
.....t
<
0-¥¥002e
(
a
-
jw
)t
dt
+F
(
jw
)
=
-=a
2
+
w
2-
j2w2F
(
jw
)afi
0=
limF
(
jw
)
=jwa
2
+
w
2lim
-
j2w
2 =afi
02F
(
jw
)
=wf(
jw
)
=....w
>
0-
p22p
....w
<
0-1f
(t)1e
-(
a
+
jw
)t
dtF
(w)f2
(t)ttwwpjwdwjwsin
wt122p1\
sgn(
t)
«e
jwt
dw
=sgn(
t)
=¥-¥¥-¥210需从分配函数的观点解释。¥\
sin
wtdt
=
w¥
¥F
(
jw
)
=
-2
j
f
(t)
sin
wtdt
=
-2
j
sin
wtdt0
0四.常数的付立叶变换t2-t
2E2tE
tsa
(
wt
)EG
(
t
)
«2
t2psa
(
wt
)2F
[
E
]
=
lim
E
tSa
(
wt
)
=
2pE
limt
fi
¥
t
fi
¥P17.1-35[
k
sa (
kt
)]pk
fi
¥d
(
t
)
=
limF
[
E
]
=
2
p
E
d
(w
)F
[1]
=
2pd
(w
)f
(t)Etw2pEd(w)u(t)21F[u(t)]
=
[1
+
sgn(
t)]jw1F
(
jw
)
=
pd
(w
)
+wP168.3-30五.u(t)的付立叶变换方法一1
sgn(t)212tttu
(
t
)
=
lim
e
-
at
(
t
>
0
)a
fi
01
e
-
at
u
(
t
)
«1wa
+
jwae=
A(w
)
+
jB(w
)a
2
+
w
2-
ja
2
+
w
2=a
+
jwF
(
jw
)
=方法二:利用单边指数函数取极限(
w
„
0)(w
=
0)A(w)
=
lim
A(w)
=
¥A(w)
=
lim
A(w)
=
0afi
0afi
0=
¥
¥-¥¥－
¥arctg
w
=
pa
a
a
d
w
lima
fi
0
dw2a
fi
0
a
fi
0-¥
1
+
w
lim
A(w
)dw
=
lim
wjw=
pd
(w
)
+
1\
F
(w
)
=
A(w
)
+
jB(w
)B(w
)
=
lim
B(w
)
=
-
1afi
0§3.9
周期信号的傅立叶变换一般周期信号的傅立叶变换傅立叶级数FS与其单脉冲的傅立叶变换FT的关系正余弦信号的傅立叶变换FT复指数信号的傅立叶变换周期单位冲激序列的FS和FT周期矩形脉冲的FS和FT周期矩形脉冲与单矩形脉冲的关系1.
指数函数e
–jw
0t的傅立叶变换112p2pe
–
jw
0t-¥¥d
(w
w
0
)e
dw
=jwtF
-1
[d
(w
w
]
=0102pF[e
–
jw
0t
]FF
-1
[d
(w
w
)]
=0«
2pd
(w
w
)e
–
jw
0tRtjI00pd
(w
w
)-1pFF
[d
(w
w
)]
=
2\
F[e
]
=
2–
jwt用反证法:wF(jw)0
w
02
.
F
[cos
w
1
t
].
and
.
F
[sin
w
1
t
]21+
e
-
jw
1t
)cos
w
t
=
1
(
e
jw
1tcosw1t
«
p[d(w
+w1
)
+d(w
-w1
)]112
j(
e
jw
t1
-
e
-
jw
1
t
)sin
w
t
=jp[d
(w
+
w
1
)
-
d
(w
-
w
1
)]sin
w
1t
«25f
(t)F
(w)pwwjp-
w0w0w0-
w0
w0njn
w
1
t3.一般周期信号的傅立叶变换¥n
=
-¥f
(
t
)
=
F
e12pd
(w
-
w
)11=
2
p¥¥F
d(w
-
n
w
)F F
[e
]
e
jw
1t
«F
[
f
(t
)]
=n
=
-¥njn
w
tn
=
-¥n01nTw
=nw
1F
=
1
F
(
jw
)P147.例3－10周期单位冲激序列的FS¥
¥n
=-¥nn
=-¥T1F
.e1jnw
td
(t
-
nT
)
=d
(t
)
=11211T1T12TTdt
=Fn
=
T--
jn
w
1td
(t
).e¥TeTjnwt1
n=-¥d
(t)
=
1周期单位冲激序列的FT¥¥¥=n
=
-¥nn
=
-¥Te
jnw
tT1
n
=
-¥11F
.e
jnw
td
(t
-
nT
)
=d
(t
)
=1Td
(w
-
nw1
)F
[
f
(t
)]
=
2p
1
¥1
n
=-¥¥F
(w
)
=
F[dT
(t)]
=
w1
d(w
-
nw1
)n=-¥d(t)0tF0
(w
)1(1)0wTd
(t)1TFn00ww1-w12w1
w2w1w1-w1-
2w
1tF
(w
)w111TFSFT(p148,例题3-11)20F
(
jw)
=
EtSa(
wt
fi)1101T
2E
t
Sa
(
n
w
1t
)F
(
jw
)Tn
w
1n=F
=T1
2n
=-¥f
(t)
=
Et
Sa
(
nw
1t
)e
jw
t¥¥1
1
21¥)d(w
-
nw
)nw
tSa(F(
jw)
=2p
Fnd(w
-nw1)
=
Etwn=-¥n=-¥令:秒201t
=w1=
8p=
2p
=
2pT1
0.251T
=
s4单个矩形脉冲的变换nw
08p-
40p
40p周期信号的频谱密度F
(
jw
)15F
(
jw
)4.周期矩形脉冲的FS和FTE0f
(t)22-
t
0
ttT1-T1Ef
(t)F(w
)FntFTFSFT周期重复2p
t-
2ptwtAF
(w
)E
tE
t2pt2pt-Fnnw2p
t-
2ptwtAF
(w
)E
t5.周期矩形脉冲与单矩形脉冲的关系-11211T12Tf
(t
).e
dtTFn
=-
jn
w
t-
1
20T12Tf
0
(t
).e
dt-
jw
tF
(w
)
=01nTw
=
nw
1F
=
1
F
(w
)
2
0F
(w
)
=
EtSa
wt1T1
2Tn
0=
EtSa(
nw1t)F
=
1
F
(w
)w
=nw111jnw
t.eSaT1Et
¥
nwt2n
=-¥f
(t
)
=2
d(w
-nw1)1Sa
nw1t
F(w
)
=
Etw1¥n=-¥由单脉冲联想FS的FnFSFT小结:单脉冲和周期信号的傅立叶变换的比较单脉冲的频谱F0(w)是连续谱，它的大小是有限值；周期信号的谱F(w)是离散谱，含谱密度概念，它的大小用冲激表示；F0(w)
是F(w)
的包络的
1w
。1.物理意义不同,Fn是单个复简谐波成份的复振幅,F
(jw)是单位带宽内所有复简谐波成分的和的复振幅值。.单位不同,Fn的单位是伏特或安培,而F
(jw)
的单位则是(伏特/赫,安培/赫).Fn代表的是信号的功率分配,而F
(jw
)代表了信号的能量分布.*.FS和FT表示举例f(t)级数系数频谱密度函数F(w)直流EF0
=
E2pEd(w)Ecosw0tF
=
F
=
E1
-1
2pE[d(w
+
w0
)
+
d(w
-
w0
)]Esinw0tF
=
F
=
E1
-1
2jpE[d(w
+
w0
)
-
d(w
-
w0
)]*四种时频对应关系基本性质。与抽样定理有关的性质。周期信号的频谱:周期性-抽样性抽样信号的频谱:抽样性-周期性与单边特性有关的性质。
(希尔泊特交换）解析信号-单频谱与功率谱有关的性质。相关函数-功率谱§3.7傅立叶变换的基本性质对称性和叠加性奇偶虚实性尺度变换特性时移特性和频移特性微分和积分特性卷积定理(§3.8)Paseval定理F
(w
)1:对称性若f
(t
)«2pf
(-w
)则F
(t)«2pf
(w
)若
f
(t)为偶函数，
则
F
(t)
«或12pf
(w
)F
(t
)
«证明见p123¥-¥
F
(w)e
jw
t
dw2p
-¥1f
(t)
=¥F
(w)
=
f
(t)e
-
jw
t
dttwtwF
(t
)2pd
(w
)若f(t)为偶函数，则时域和频域完全对称直流和冲激函数的频谱的对称性是一例子F
(w
)d
(
t
)0000f
(t)F
(w
)2
p
tt
22
p
t-tw100f
(t)F
(w
)cwcw-2cw2ctw2pw
c12
p
0
2
p-
w
0-
t
2w
c
.Sa
(
w
c
t
)2p
2wt
2t
Sa+11
t
2例题一:求:Fa
2
+w
22ae
«-
a
t\«=
pe-
w2«
2p
1
e-
w112\
1
e-
tt
2
+11
+w
2解：p114.双边指数信号f
(t
)
=
e
-
atFTa
+
jwF
(w
)=
1
11=
?
a
+
jt
F
(w
)
=
FT
对称性+
a
wF1
(w
)
=
2
p
f
(
-
w
)
=
2
p
e例题二:
a
>
1, t
>
0(对称性部分成立的例子
）t换成wf
换成换成tF
1w例题三：试求函数tsin
t的傅立叶变换解：若直接用t¥-¥¥-¥=sin
te-
jwt
dtf
(t)e-
jwt
dtF
(w
)
=来求出sin
t的傅立叶变换将是不t容易的。这里可用对称性来求解.分析:2wtsin
wtEt
2
f
(t)
«f
(t)
={10t
<
1t
>
1«
F
(w
)
=
F[
f
(t)]
=
2
sin
wwt2
2t-根据偶函数对称性可得
0w
<
1w
>
1F
[
F
(t
)]
=
F
[
2
sin
t
]
=
2pf
(w
)
=
2pt上式两端同乘以1/2得
0w
<
1w
>
1=
p
[u
(w
+
1)
-
u
(w
-
1)]tF
[
sin
t
]
=
pww21F(t)f
(t)f
(w)F(w)我们也可以用此来求dtt+¥-¥
-¥+¥
sin
atSa
=
f
(t
)dt
=
att+¥
1
2a
sin
at-¥
2+¥+¥
sin
at-¥
-¥=
e-
jwt
dt
=
p[u(w
+
a)
-
u(w
-
a)]F
(
jw
)=
f(t)e-
jwt
dt
=
e-
jwt
dt当w
=0
时F
(
jw
)
w
=
0-
¥\
S
a
=
F
(
jw
)
w
=
0
=
p+¥=
f
(t
)
dt
=
S
a-
awap2、线性（叠加性）若：FT
[fi
(t)]=Fi
(w
)
n
ni
=1
i
=1则：FT
ai
fi
(t)
=
ai
Fi
(w
)例题四:求f(t)的傅立叶变换f
(t)-1-t
2
t
22tf
(t
)
=
[u
(t
+
t
)
-
u
(t
-
t
)]
+
[u
(t
+
t)
-
u
(t
-
t)]2
2F
(w
)
=
t[
Sa
(wt
/
2
)
+
2
Sa
(wt
)]3、
奇偶虚实性无论f(t)是实函数还是复函数，下面两式均成立if
F[
f
(t)]
=
F
(w
)then
F[
f
*
(t)]
=
F
*
(-w
)if
F[
f
(t)]
=
F
(w
)then
F
[f
(-t)]
=
F
(-w
)时域反摺频域也反摺时域共轭频域共轭并且反摺更广泛地讲,函数f(t)是t的复数;令f
1(t)
和
f
2(t)
分别代表它们的实部和虚部.f
(
t
)
+
jf
(
t
)1
2f
(
t
)
==cos
wt
-j
sin
wt带入把尤拉公式：e-jwt上式整理得出：F
(
jw
)
=
R
(w
)
+
jX
(w
)¥F
(
jw
)
=
[
f1
(t
)
+
jf
2
(t
)]e
dt-
jwt-¥+¥\
R
(w
)
=
[
f1
(t
)
cos
w
t
+
f
2
(t
)
sin
w
t
]dt-
¥+¥X
(w
)
=
-
[
f1
(t
)
sin
w
t
-
f
2
(t
)
cos
w
t
]dt-
¥e
jw
t+
¥-
¥
1
2pd
w
......
(1)F
(
jw
)
ejw
t
f
(
t
)
=F
(
jw
)
=
R
(
jw
)
+
jX
(w
).....
(2
)把(2)，(3)带入(1)式整理得=
cos
wt
+
j
sin
wt.....(3)\
f
(
t
)
=
11
2
p+¥
R
(w
)
cos
w
t
-
X
(w
)
sin(
w
t
]
d
w-
¥+¥
R
(w
)
sin
w
t
+
X
(w
)
cos
w
t
]
d
w-
¥1f
2
(
t
)
=
2
p+¥R
(w
)
=
f
(t
)
cos
w
tdt-
¥X
(w
)
=
-
f
(t)
sin
wtdt-¥特殊情况的讨论：a.实数函数设f(t)是t的实函数,则
F
(w
)
的实部与虚部将分别等于
(f2(t)=0,f(t)=f1(t))+¥从上式可以得出结论:X(-w
)
=-X
(w
)R(-w
)
=
R(w
)F(w
)
=
R(w
)
+
jX(w
)F(-w
)
=
R(w
)-
jX(w
)\
F(-w
)
=
F*(w
)实信号的频谱具有很重要的特点,正负频率部分的频谱是相互共轭的.¥-¥、f(t)是实函数¥-¥f
(t)sinw
tdtf
(t)
cosw
tdt
-
jF(w
)
=
X
(w
)R(w)
=
R(-w)F(-w
)
=
F*
(w
)偶函数奇函数FT
[
f
(-t)]
=
F
(-w
)FT
[
f
(-t)]
=
F
*
(w
)R
(w
)X(w)
=-X(-w)f1(t)
=0则-
¥b
.虚函数设f(t)是纯虚函数
f
(t)
=
jf2
(t)+¥R
(w
)
=
f
2
(t
)
sin
w
tdt-
¥+
¥X
(w
)
=
f
2
(t
)
cos
w
tdt因而
R(w)是
w
的奇函数,而X(w)是w
的偶函数.
\
F(-w
)
=
-F*
(w
)反之也正确.
-¥+¥
+¥+¥-¥
-¥=-
jwtf
(t)
coswtdt
-
j f
(t)sinwtdtf
(t)e dt
=c.偶实函数：f
(t)=f
(-t)F(w
)+¥
+¥
R
(w
)
=
f
(t
)
cos
w
tdt
=
2
f
(t
)
cos
w
tdt-
¥
0X
(w
)
=
0反之,若一实函数f(t)的付立叶积分也是实函数,则f(t)必是偶函数.实偶函数的傅立叶变换仍为实偶函数f
(t)
=
e-a
t
(-¥
<
t
<
+¥
)22a2a
+
wF
(w
)
=j
(w
)
=
0d奇实函数设f(-t)=-f(t)
则+¥R(w
)
=
f
(t)
温馨提示
- 1. 本站所有资源如无特殊说明，都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等，如果需要附件，请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸，网页内容里面会有图纸预览，若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对用户上传分享的文档内容本身不做任何修改或编辑，并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容，请与我们联系，我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 土地居间合同协议书样板
- 自愿提前解除合同协议书
- 采购合同与结算协议书
- 家庭施工雇佣合同协议书
- 补偿合同补充协议书模板
- 2025房屋买卖中介服务合同范本
- 计算机三级嵌入式各类试题及答案
- 平台化下的软件设计师考试试题及答案
- 2025年机电工程课程考核内容及试题及答案
- 店铺股份赠与合同协议书
- 驾驶员劳务派遣投标方案
- 稀土生产工艺流程图矿的开采技术
- 教育学 综合实践活动
- 地基钎探技术交底
- PE管道热熔施工技术交底
- 经济合同审批会签单
- 人教版四年级数学下册：第八单元：统计 单元测试卷（含答案）
- 电商售后客服试用期考核表范本
- 2023年四川省水电投资经营集团普格电力有限公司招聘笔试题库含答案解析
- JGJ106-2014 建筑基桩检测技术规范
- 2023届湖北省武汉市东湖高新区数学六年级第二学期期末综合测试试题含解析
评论
0/150
提交评论