soph2信号与系统-同步来金梅课件

来金复旦大学微电子系楼229 第3章傅里叶变 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei频域分三角函数形式的傅里叶级数StateKeyLabofASIC&Systems,FudanUniversity,Jinmei发展历 第3章傅里叶变 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei且常kiStateKeyLabofASIC&Systems,FudanUniversity,JinmeiStateKeyLabofASIC&Systems,FudanUniversity,Jinmei三角函数集0 0t0

2 mncos

1tcos

1tdt

mt0

sinntsinmtdt

mn0 0

mStateKeyLabofASIC&Systems,FudanUniversity,JinmeiT ft,周期为T,基波角频率为T 1f(t)a0ancosn1tbnsinn1t

t0f(t)d00 0

t0

f(t)cosntd

T 0 0

t0

f(t)sinntd

T StateKeyLabofASIC&Systems,FudanUniversity,Jinmei StateKeyLabofASIC&Systems,FudanUniversity,Jinmeia

2f(t)dStateKeyStateKeyLab

ASIC&

T22an 22

eim m2其它形f(f(t)c0cncosn1tna2bnn

bn

narctan f(t)f(t)d0dnsinn1tn

cos

bncnsin

arctanan

a2na2nnbandnsin bndnStateKeyLabofASIC&Systems,FudanUniversity,Jinmei幅度频率特性和相位频率周期信号可分解为直流，基波（1）cn~dcn~dn~ ~ StateKeyLabofASIC&Systems,FudanUniversity,Jinmei二．指数函数形式的傅里叶StateKeyLabofASIC&Systems,FudanUniversity,Jinmei二．指数函数形式的傅里叶1复指数正交函数集n1tn0,1,2 1

f(t)F(n)ejn1t 11T1F(n)1

f(t)ejn1td0T1ejn1tejn1td0也可写

f(t)ejn1td0

StateKeyLabofASIC&Systems,FudanUniversity,Jinmei1T1T

1T1f(t)ejnt1

11f(t)F(n)1

(4StateKeyLabofASIC&Systems,FudanUniversity,Jinmeinnf(t) nnejnStateKeyLabofASIC&Systems,FudanUniversity,Jinmei111TTF(n)1TT

f(t)ejntd f(t)cosntdtj T T 1ajb2F(n)

Tf(t)cosntdtj

Tf(t)sinntd T T 1ajb FF(nF(n)11FnF(n)ejn11StateKeyLabofASIC&Systems,FudanUniversity,Jinmei复数频复数复数

F(n1

aa2nnbn

1 narctan 关于的偶函数（实n取正值F(n1关于的奇函数（实关于的偶函n取正值n1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei频谱cn~ Fn~ n~ StateKeyLabofASIC&Systems,FudanUniversity,Jinmeif(t)的平均功率P与傅立叶系数的f(t)a0

f(t)

n

F(n1)

jn1t

f

2

(a2b0T01

0nn0nnc2 2 2时域和时域和频域的能量守恒（帕塞瓦尔Parseval定理周期信号f(t)

2=c0cncos(n1tn2

t0f(t)d0

an

t0

f(t)cosn1td n

t0nFnFn 11T0f(tejn1d511

f(t)sinn1td

f(t)

F(n)ejn1teState ie四．总

cn~，n~

~，n~

F(n)1cn

Fc

F(n1)F(n1相位频谱为奇(n1)(n1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei四．归纳唯一性：f(t)的谱线唯一引入负对于双边频谱，负频率(n1)，只有数学意义，而无物理意义。为什么引入负频率?

数，必须有共轭ejn1和ejn1，才能保证f(t)的实函数的性质不变StateKeyLabofASIC&Systems,FudanUniversity,JinmeiStateKeyLabofASIC&Systems,FudanUniversity,Jinmei偶函EfEf(t f(t)f(tbn

t0

f(t)sinn1td

f(t)cosntdt

t0

f(t)cosntd0T T 0T

T FF(n)1

jb1

Fn为实函数StateKeyLabofASIC&Systems,FudanUniversity,Jinmei奇函

对称的：f(t)f(t1a012

f(t1tTOf(t1tTO2an

2

Tf(t)sinntd

T2f(t)

tdt T T FF(n)1ajb1 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei奇谐函

f(t 2a

f(t) TT T2

1

ff0 0T T2

f(t T

2f(t)0 0T 0T

2f(t)00TT 0TT

2f(t)0n1,3,5

Tf(t)cosntd4

f(t)0

ntd1

mnn2,4,6时 mns,FudanUniversity,Jinmei偶谐函f(tf(tOt2 T ftft 1

1 2 当n1,3,5

anbn44当n2,4,6 an

f(t)cosntd1

f(t)sinntdnT nT1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 奇谐函数：只有奇次谐波分量；奇谐函数：只有奇次谐波分量 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei fta acosntbsinnt 取前(2N取前(2N1)项 近f(tSNa0

N1N1

cosn1t

sinnt

Ntf(t)SNT 2(t)1 t0T1 2(t)dT 010EN

2(t)

21a0n2a0n

2 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei StateKeyLabofASIC&Systems,FudanUniversity,JinmeiP

2(t)

t0

f2(t)dt

E t

T2

versity,Jinmei StateKeyLabofASIC&Systems,FudanUniversity,Jinmei nudann周期信号傅里叶级数分析 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3章傅里叶变 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei典型周期信号的傅StateKeyLabofASIC&Systems,FudanUniversity,Jinmei一．频谱脉宽为脉宽为 周期为ff(tE22tStateKeyLabofASIC&Systems,FudanUniversity,Jinmei1f(tE

ftbn0,只有a0StateKey

f(t)a0anaa

y,Jinmei2f2f(t)nFjn1)121f(t)ejn1td12T11F(n)1jn1T

jn T 2

1tdt

ejn12jn1T1

sin 12 n

sin

12 ESanESanT112121FnF(n1)

2StateKeyLabofASIC&Systems,FudanUniversity,Jinmei频谱及其取T1 E

F(n1其最大值在n0E4离散谱（谐波性

1 1

2

当n时取

函数），幅度/相Fn0，相位 0，Fn0,相位为πStateKeyLabofASIC&Systems,FudanUniversity,Jinmei 幅度 谱线间 1 1当T，时

为无限小 ft由周期信号非周期信号 n而衰减到零StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 1.问题提出EEF(n1O1StateKeyLabofASIC&Systems,FudanUniversity,JinmeiP

周期矩形脉冲信号T1f2(t)dtF周期矩形脉冲信号T1n以以111s4

2

F2

F22

F32

1111F211111

F221

F321

F4210.181E21 0.181E21

2F

Sa

(t)dtT1T

11

12

f(t)

n

F(n)ejn1tStateKeyLabofASIC&Systems,FudanUniversity,Jinmei周期矩形脉冲信号的功率：频带宽

2π或 1，带宽与脉宽成反 对于一般周期信号，将幅度下降为

Fn1max3．系统的通频带>信号的带宽，“满足一定失真条件

有效带宽约 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3章傅里叶变 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei T1f(t)：周期信 1谱系数F(n1)11

22

f(t)ejn1td

0谱线间隔

再用Fn1表示频谱就不合适StateKeyLabofASIC&Systems,FudanUniversity,Jinmei

F(n1)

f(t)ejn1td 2 Fn111f当T1f10,111f当T1f10,F(n)Fn有界函nT111f111F

limT1Fn1

lim

f(t)ejn1td

T1

2XX

f(t t

StateKeyLabofASIC&Systems,FudanUniversity,频谱密度函数的 dtFfj由由f(t)求F称为傅里叶变换F一般为复信号故可表示F()|F()|F~幅度频~相位频StateKeyLabofASIC&Systems,FudanUniversity,Jinmei反变

1f(t)F(n1除以除以1，再乘以

F

limT1F(n1F(n1f(t)

n

F(n1

1

ejn

d,n

F(nlim

F

f

12

StateKeyLabofASIC&Systems,FudanUniversity,Jinmei傅里叶变 dtFff(t)

d ftFF(n1)

1f1

f(t)

F1)e1)eT11StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 相

实 虚F

f(t)fetfo实信号偶分量奇分量f(t)ejtd

f(t)f(t)costjsintd fe(t)costdt fo(t)sintd实 虚StateKeyLabofASIC&Systems,FudanUniversity,JinmeiR2f(t)costd

关于X2f(t)sintd

关于 arctanXR

关于的偶函数关于ft偶

ft奇

F为虚奇函数，只有X，相位 StateKe