大二下課件信號(hào)與系統(tǒng)

第3章傅里葉變 StateStateKeyLabofASIC&Systems,FudanUniversity,Jinmeift 級(jí)數(shù)Fn 離散1ft 變換F 連續(xù)

統(tǒng)一的分析方法 變非周期StateKeyLabofASIC&Systems,FudanUniversity,Jinmei一．正弦信號(hào) 由

cos0tsint

21ej0tej0t 2

1ej01ej0

20

StateKeyLabofASIC&Systems,FudanUniversity,JinmeiFFsin0t頻譜圖

F002StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 變?cè)O(shè)信號(hào)周期: 1T TFTFfT

ft

F

2πFn1n11F 2πFn1n111 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei FT2πFn1n1f 位置: 諧波頻率強(qiáng)度:2πFn與F(n)成正比 離散 (2)譜線的幅值不是有限頻率范圍無(wú)限小,幅度為 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei

求Fn1即單個(gè)脈沖的F0與周期信號(hào)f0

Tt的譜系數(shù)Fn1的關(guān)fT

T設(shè)ftF

F ftejtd 2 jnfT

Fn1 nFFn11

d StateKeyLabofASIC&Systems,FudanUniversity,JinmeiT F fT

d

fT

Fn1nTTFn

d

在TT內(nèi)ft與ft相 2

所以Fn1F111 111可由F求周期函數(shù)ft的譜Fn StateKeyLabofASIC&Systems,FudanUniversity,Jinmei 變

t

tnT1

T T因 tT

n

1

1

所 Tt的傅氏級(jí)數(shù)譜系

Fn 1 所以Fn1F1T 11

T所 T

Fn1 e 1StateKeyLabofASIC&Systems,FudanUniversity,Jinmei頻 jn1t1F

2πn1 1

F T1

T1n

n1

nT1n

1Fn 11T

Fn1

F

2 o Tt的頻譜密度函數(shù)仍是沖激序列,強(qiáng)度和間隔都是StateKeyLabofASIC&Systems,FudanUniversity,Jinmei五．周期矩形脈沖序列的傅

2

ff0fE2

F0()F(n1)F(F()ESa

所以Fn1F11 211

1 1

ESanF()

Fn1n12π

n

1n

StateKeyLabofASIC&Systems,FudanUniversity,Jinmei方法 f(t)

(t)

TttnT1nF()F0()1(n1 2 n 1(n1

Sa

21 21 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3

變換 StateKeyLabofASIC&Systems,FudanUniversity,JinmeiffF(F(m0

HH(m0

f11 Sm0m

H( H(mC(Smfst)Fs,與F的關(guān) StateKeyLabofASIC&Systems,FudanUniversity,Jinmeif

fsotF1EEotsoStateKeyLabofASIC&Systems,FudanUniversity,Jinmei二．理想抽樣（周期單位沖激抽樣f

fs

ftFT

ftF 周期單位沖激抽樣信號(hào) p(t)T(t)(tnTs)s(ns fs(t)f(t)T(tsFs

1F

T

FnssnStateKeyLabofASIC&Systems,FudanUniversity,JinmeiotF1ssosEotEotot s

sStateKeyLabofASIC&Systems,FudanUniversity,Jinmeiss

Fs1s含原信號(hào)的全部信息,幅 差Ts 2F以為周期的連續(xù)譜 新的頻率成分,即F的周

s為Ts截止頻率mcsm濾除高頻成分，即可重現(xiàn)原信號(hào)。

smsStateKeyLabofASIC&Systems,FudanUniversity,Jinmei otff fs

連續(xù)信

f

oot抽樣脈沖序列抽樣信號(hào)

fstft

StateKeyLabofASIC&Systems,FudanUniversity,Jinmei關(guān)關(guān)連續(xù)信號(hào) f抽樣連續(xù)信號(hào) f抽樣脈沖序 ：抽樣信號(hào) fsfst

(mmffstftP()2Pn(nsnPn12p(tjnsdts2F1FPsPnF(ns StateKeyLabofASIC&Systems,FudanUniversity,JinmeiPPn12jnsdt12jnsTsTs222Fs2)F(ssStateKeyLabofASIC&Systems,FudanUniversity,JinmeiotF1 理想抽otF122TsEoEotsofsfsot

Fs

StateKeyLabofASIC&Systems,FudanUniversity,Jinmei舉例說(shuō)明抽樣信號(hào)與原信號(hào)頻譜的設(shè)設(shè):1T2πss2T,ss因?yàn)镕sTnsFnn ns22s2偶nnnnn

1F2F 1SaπF1F n 2 F 12F1F n

StateKeyLabofASIC&Systems,FudanUniversity,Jinmein2

Fs nF

12F31F3 n 2 3π

F()1F(3)1F()1F(

1F()1F(3 StateKeyLabofASIC&Systems,FudanUniversity,JinmeiffF(F(m0

HH(m0

f11 Sm0m

H( H(mC(Smfst)Fs,與F的關(guān) StateKeyLabofASIC&Systems,FudanUniversity,Jinmei第3

變換 StateKeyLabofASIC&Systems,FudanUniversity,Jinmei抽樣otF1oot1FsoStateKeyLabofASIC&Systems,FudanUniversity,Jinmei奈奎斯特(Nyquist抽樣率和抽樣22πf sTsmms即抽樣頻率

2

是必要條件，或抽樣間隔Ts

2fmStateKeyLabofASIC&Systems,FudanUniversity,Jinmei抽樣一個(gè)頻帶受限的信號(hào)f(t)

~的范圍，則信號(hào)ft可用等間隔的抽樣值來(lái)維一

2

，即Ts

2

2π otFotF1oo