來金復旦大學微電子系樓229 第3章傅里葉變 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei頻域分三角函數形式的傅里葉級數StateKeyLabofASIC&Systems,FudanUniversity,Jinmei發展歷 第3章傅里葉變 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei且常kiStateKeyLabofASIC&Systems,FudanUniversity,JinmeiStateKeyLabofASIC&Systems,FudanUniversity,Jinmei三角函數集0 0t0

2 mncos

1tcos

1tdt

mt0

sinntsinmtdt

mn0 0

mStateKeyLabofASIC&Systems,FudanUniversity,JinmeiT ft,周期為T,基波角頻率為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&

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eim m2其它形f(f(t)c0cncosn1tna2bnn

bn

narctan f(t)f(t)d0dnsinn1tn

cos

bncnsin

arctanan

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

f(t)ejn1td0T1ejn1tejn1td0也可寫

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復數頻復數復數

F(n1

aa2nnbn

1 narctan 關于的偶函數（實n取正值F(n1關于的奇函數（實關于的偶函n取正值n1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei頻譜cn~ Fn~ n~ StateKeyLabofASIC&Systems,FudanUniversity,Jinmeif(t)的平均功率P與傅立葉系數的f(t)a0

f(t)

n

F(n1)

jn1t

f

2

(a2b0T01

0nn0nnc2 2 2時域和時域和頻域的能量守恒（帕塞瓦爾Parseval定理周期信號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．總

cn~，n~

~，n~

F(n)1cn

Fc

F(n1)F(n1相位頻譜為奇(n1)(n1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei四．歸納唯一性：f(t)的譜線唯一引入負對于雙邊頻譜，負頻率(n1)，只有數學意義，而無物理意義。為什么引入負頻率?

數，必須有共軛ejn1和ejn1，才能保證f(t)的實函數的性質不變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為實函數StateKeyLabofASIC&Systems,FudanUniversity,Jinmei奇函

對稱的：f(t)f(t1a012

f(t1tTOf(t1tTO2an

2

Tf(t)sinntd

T2f(t)

tdt T T FF(n)1ajb1 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei奇諧函

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時 mns,FudanUniversity,Jinmei偶諧函f(tf(tOt2 T ftft 1

1 2 當n1,3,5

anbn44當n2,4,6 an

f(t)cosntd1

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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周期信號傅里葉級數分析 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3章傅里葉變 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei典型周期信號的傅StateKeyLabofASIC&Systems,FudanUniversity,Jinmei一．頻譜脈寬為脈寬為 周期為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頻譜及其取T1 E

F(n1其最大值在n0E4離散譜（諧波性

1 1

2

當n時取

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為無限小 ft由周期信號非周期信號 n而衰減到零StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 1.問題提出EEF(n1O1StateKeyLabofASIC&Systems,FudanUniversity,JinmeiP

周期矩形脈沖信號T1f2(t)dtF周期矩形脈沖信號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周期矩形脈沖信號的功率：頻帶寬

2π或 1，帶寬與脈寬成反 對于一般周期信號，將幅度下降為

Fn1max3．系統的通頻帶>信號的帶寬，“滿足一定失真條件

有效帶寬約 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3章傅里葉變 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei T1f(t)：周期信 1譜系數F(n1)11

22

f(t)ejn1td

0譜線間隔

再用Fn1表示頻譜就不合適StateKeyLabofASIC&Systems,FudanUniversity,Jinmei

F(n1)

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

limT1Fn1

lim

f(t)ejn1td

T1

2XX

f(t t

StateKeyLabofASIC&Systems,FudanUniversity,頻譜密度函數的 dtFfj由由f(t)求F稱為傅里葉變換F一般為復信號故可表示F()|F()|F~幅度頻~相位頻StateKeyLabofASIC&Systems,FudanUniversity,Jinmei反變

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傅里葉變 dtFff(t)

d ftFF(n1)

1f1

f(t)

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

實 虛F

f(t)fetfo實信號偶分量奇分量f(t)ejtd

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關于X2f(t)sintd

關于 arctanXR

關于的偶函數關于ft偶

ft奇

F為虛奇函數，只有X，相位 StateKe