ch5-dt變換lec信號(hào)與系統(tǒng)

Autumn

2015信號(hào)與系統(tǒng)Chapter

5The

D-T

FourierTransform離散時(shí)間傅里葉變換Lecture

5-2于源，yu

回顧: DTFT

vs.

CTFT??[??]單元信號(hào)????????時(shí)域周期、頻域周期(????)分析公式∞??(??????)

=

??[??]???????????=?∞頻域連續(xù)、周期(????)??(??)????????時(shí)域周期、頻域非周期∞??(????)

=

??(??)??????????????∞頻域連續(xù)、非周期(一般)綜合公式??[??]

=????????(??????)??????????????????(??)

=??1信號(hào)與系統(tǒng)課程組?2015????∞??(????)?????????????∞2信號(hào)與系統(tǒng)課程組?2015離散時(shí)間非周期信號(hào)的傅里葉變換典型D-T非周期信號(hào)的傅里葉變換離散時(shí)間周期信號(hào)的傅里葉變換離散時(shí)間傅里葉變換的性質(zhì)離散時(shí)間傅里葉變換的卷積和相乘性質(zhì)對(duì)偶性線性差分方程描述D-T

LTI系統(tǒng)3.1

周期信號(hào)頻域描述:

從DTFS

到DTFT周期信號(hào)的D-T傅里葉級(jí)數(shù)周期信號(hào)的D-T傅里葉變換把周期信號(hào)與非周期信號(hào)的頻域分析方法

起來(lái)：用”幅度密度”和”相位”表征頻率分量信息??

??

=????

=????????????????????

=??????????(????

??)????=

????=

????

??

???????(????

??)????=

??3信號(hào)與系統(tǒng)課程組?2015回顧:

連續(xù)時(shí)間周期信號(hào)的CTFT∞??

????

??

=

????????

??(??

?

??????)??=?∞∞4信號(hào)與系統(tǒng)課程組?2015=??=?∞連續(xù)時(shí)間周期信號(hào)傅里葉變換是：一系列位于諧波頻率的離散沖激序列，沖激的“復(fù)振幅”包絡(luò)等于非周期中心脈沖傅里葉變換乘以頻譜間隔????(????????)????

??(??

?

??????)3.2

周期復(fù)指數(shù)信號(hào)??????????

的DTFT??????????

?

??????(??

?

????

?

??????)??=?∞= ??(??

?

????)??????????????

=

??????????2??∞???1∞??2??

?? ??

???

?2??????=?∞2??2????1= 2????(??

?

??

)????????

????信號(hào)與系統(tǒng)課程組?20155????[??]

=??????????????????=

??3.3

一般離散時(shí)間周期信號(hào)的DTFT??{????[??]}

=??????

??????????????=

??∞??=

??=??????????(??

?

??????

?

??????)??=?∞∞??=

??

??=?∞∞??????????(??

?

??????)??=?∞=??????????(??

?

(??

+

????)????)=??????

=

????????

=

????+????????????∞?

??????(??

?

????

?

??????)??=?∞6信號(hào)與系統(tǒng)課程組?2015CTFT

for

C-T

Periodic

Signals∞??

????

??

=

????????

??(??

?

??????)??=?∞D(zhuǎn)TFT

for

D-T

Periodic

Signals∞??

????[??]

=

????????

??(??

?

??????)??=?∞可以看出：周期信號(hào)的CTFT和DTFT具有相同的數(shù)學(xué)形式7信號(hào)與系統(tǒng)課程組?20153.3

一般離散時(shí)間周期信號(hào)的DTFT:

周期信號(hào)的幅度譜&

幅度密度譜傅里葉級(jí)數(shù)給出了C-T/D-T周期信號(hào)的諧波頻率分量的幅度，其數(shù)值通常為非零有限值幅度@

??????:

????

≡

????

or

????傅里葉變換給出的是各諧波分量“幅度密度”版本的幅度信息幅度密度

@

??????:??????????

???????→?????=

??????????(??

?

??????)8信號(hào)與系統(tǒng)課程組?2015CTFS:CTFT:????????????C-T:????(??)??(??)

=

??????????(??)????

=

????

????

=

??.

??,

??

=

??????:

周期信號(hào)的幅度譜&

幅度密度譜9信號(hào)與系統(tǒng)課程組?2015CTFS:CTFT:????????????????????????????????????????????????????????????????????????????????:

周期信號(hào)的幅度譜&

幅度密度譜10信號(hào)與系統(tǒng)課程組?2015DTFS:DTFT:????????(??????)D-T:????[??]????

=

??,

??

=

??????????:

周期信號(hào)的幅度譜&

幅度密度譜11信號(hào)與系統(tǒng)課程組?2015????DTFS:DTFT:????????(??????)???????????????????????????????????????

?????????????????????????:

周期信號(hào)的幅度譜&

幅度密度譜12信號(hào)與系統(tǒng)課程組?201513信號(hào)與系統(tǒng)課程組?2015離散時(shí)間非周期信號(hào)的傅里葉變換典型D-T非周期信號(hào)的傅里葉變換離散時(shí)間周期信號(hào)的傅里葉變換離散時(shí)間傅里葉變換的性質(zhì)離散時(shí)間傅里葉變換的卷積和相乘性質(zhì)對(duì)偶性線性差分方程描述D-T

LTI系統(tǒng)4.1

Periodicity

&

LinearityLinearity

(線性)????????????[??]

+

????[??]

????

??????+

????

????????

????(??+????)=

??

??????Periodicity

(周期性)14信號(hào)與系統(tǒng)課程組?2015??

??

?

?????????????????????????

????????

??

????????????????????

????(???????)4.2 Time

&

Frequency

Shifting

(時(shí)域/頻域平移)??(??

?

??

)?????????????????????

????????????????(??)???????????? ??(??

?

????)一致CT

vs.

DT

:15信號(hào)與系統(tǒng)課程組?20154.3

共軛與共軛對(duì)稱性??

?? =

???

?when

??

is

realIf

??

is

real,

then

??(?)

is

conjugate

symmetric.i.e.??(?)here

is

a

Hermitian

function(厄米函數(shù))???

?????????????

??????????????????

(??)????????一致CT

vs.

DT

:16信號(hào)與系統(tǒng)課程組?2015??????????(??)Real

-

EvenImaginary

-

EvenReal

-

OddImaginary

-

Odd??

????實(shí)偶

實(shí)偶虛偶

虛偶實(shí)奇

虛奇虛奇

實(shí)奇Real

-

EvenImaginary

-

EvenImaginary

-

OddReal

-

Odd4.5

Odd-Even

Imaginary-Real

Symmetry(奇偶虛實(shí)性)??[??]??

??????CT

vs.

DT

????????一致17信號(hào)與系統(tǒng)課程組?20154.4 Time

Differencing

(時(shí)域差分)??

?? ??? ??

?

??????????(??

????????)??

??????????????

????????????????

????????:??

????????18信號(hào)與系統(tǒng)課程組?2015?

????:????4.5

Accumulation

(累加)????[??]??=?∞??????????

????????

?

???????

+

????

??????

??(??

?

??????)∞??=?∞Corresponding

to

DC

oraverage

value????

??

?????∞????????

??

????????+

????(??)??(??)????:??

?????????

????:????DTFT

的周期性19信號(hào)與系統(tǒng)課程組?20154.6

Differentiation

in

Frequency(頻域微分)???????

??????????????????

?????????????(??)????????????????

????CT

vs.DT:

一致20信號(hào)與系統(tǒng)課程組?20154.7 Time

Reversal(時(shí)域反褶)??

?????????????

?????????(???)??????????

?????CT

vs.DT:

一致21信號(hào)與系統(tǒng)課程組?20154.8 Time

Expansion(時(shí)域擴(kuò)展)??(??)

??=??

??/??

,??,22信號(hào)與系統(tǒng)課程組?2015??/??

∈

?others??:擴(kuò)展系數(shù)，??=??,??,

??,…??[??]??(??)[??]??[??]??(??)[??]Padding

zeros

(補(bǔ)零)4.8 Time

Expansion(時(shí)域擴(kuò)展)23信號(hào)與系統(tǒng)課程組?2015??(??)??

=??

??/??

,??,??/??

∈

?others(??)??

????????????

????????4.8 Time

Expansion(時(shí)域擴(kuò)展)??-fold

time

expansion,

??-fold

frequency

compression(

水平方向：時(shí)域擴(kuò)展

??

倍，頻域壓縮

??

倍

)CT

vs.DT:

一致24信號(hào)與系統(tǒng)課程組?2015??

=

????

=

??4.8 Time

E低頻:

數(shù)值為偶數(shù)倍

??

附近的頻率高頻:

數(shù)值為奇數(shù)倍

??

附近的頻率25信號(hào)與系統(tǒng)課程組?20154.9

Parseval’s

relation(帕斯瓦爾定理)????????????[??]

=??(??????)????????????=??=

??????????(??????)

????

????????頻率分量幅度??(??????)????????對(duì)頻率分量??????

??????

????

?

????????平均功率：??(??????)=??

????????????????(通常趨零)??

????????→∞= lim

????

??????= lim

????→∞??

????????????????=??

????????

????26信號(hào)與系統(tǒng)課程組?2015????能量：4.9

Parseval’s

relation(帕斯瓦爾定理)∞??

????=?∞????????????

=

??

????????????把??=????

范圍內(nèi)所有頻率分量的能量相加，即為能量信號(hào)的總能量27信號(hào)與系統(tǒng)課程組?2015Check

Yourself??[??]??(??)[??]????(??)[??

?

??]例

5.9:

試求

??{??[??]}??

?? =

rect4[??

?

2]??[??]

?

?????2??

sin(5??/2)sin(??/2)??(??)[??]

?

?????4??

sin(5??)sin(??)????(??)[??

?

??]

?

2????????????4??

sin(5??)sin(??)??[??]??

??=

??(??)

?? +

????(??)[??

???]??

??28信號(hào)與系統(tǒng)課程組?2015?

(1

+

2???????)?????4??

sin(5??)sin(??)29信號(hào)與系統(tǒng)課程組?2015離散時(shí)間非周期信號(hào)的傅里葉變換典型D-T非周期信號(hào)的傅里葉變換離散時(shí)間周期信號(hào)的傅里葉變換離散時(shí)間傅里葉變換的性質(zhì)離散時(shí)間傅里葉變換的卷積和相乘性質(zhì)對(duì)偶性線性差分方程描述D-T

LTI系統(tǒng)5.1 DTFT

時(shí)域卷積性質(zhì)??

??????????=

??[??]

?

??[??]

??

??????=

??(??????)??

??????∞??

??

??[??

?

??]??=?∞??[??]??(??????)??(??????)30信號(hào)與系統(tǒng)課程組?2015??[??]

??[??]??(??????)時(shí)域卷積，頻域乘積例

5.11:

時(shí)移系統(tǒng)輸入輸出關(guān)系：??[??]

?

??[??]

???[??]??[??]??[??]??

?? =

??[??

?

????]系統(tǒng)頻響特性：??

??????

=

?? ??

??

?

????=

???????????=

??

????????

??????

??

??????=

?????????????

????????

??=

??????????????????

??????=

??[??

?

????]??5.1 DTFT

時(shí)域卷積性質(zhì)??[??]

?

??31信號(hào)與系統(tǒng)課程組?2015??[??]??????(??????)??[??]??????(??????)例

5.14: Find

the

frequency

response

??

??????(???)??

(???)????

???

??

???????????where

??????(??????)

is

a

ideal

loss

filter

with

????

=

??/????????(??????)??5.1 DTFT

時(shí)域卷積性質(zhì)32信號(hào)與系統(tǒng)課程組?2015??[??]??????(??????)??[??]??????(??????)(???)??????[??](???)??????[??]

????[??]????[??]??????????(??????)

=

??????

??(??????)????

?? =

(???)????[??]

=

??????????[??]????(??????)

=

??(????(?????))????(??????)

=

??????(??????)????(??????)????(??????)

=

????(????(?????))

=

??????=

??????????

?????????

???????(????(???????))??(??????)(???)??=

????????Frequency

shift33信號(hào)與系統(tǒng)課程組?2015??[??]??????(??????)??[??]??????(??????)(???)??????[??](???)??????[??]

????[??]????[??]??

??????=

??????????+

??????????=

??????????

???????

??????=??????????

?????+

??????+

??????????????????

??

????????

????????

??????=??

????????

??????=

??????????

?????+

????????????34信號(hào)與系統(tǒng)課程組?2015?

??

????

?????????????????(??????)????

??

????

?????????????(??????)??????????

????????????=

??????

+

??????????

?????Lo ss

+Highpass

=Bandstop(帶阻)?35信號(hào)與系統(tǒng)課程組?20155.2 DTFT

時(shí)域相乘性質(zhì)??

??=

??

??

??

????????????

????????????=??????

????????

????(?????)????Periodic

convolution(周期卷積)36信號(hào)與系統(tǒng)課程組?2015例

5.15: Find

??

??[??]??

?? =

????

??

????

????????

??????=

??

?

??

??

??

????

??????

????????

????????????=

??

???????????

????=

????????(??????)

with

????

=????????

??????????????=

??

???????????

????=

????????(??????)

with

????

=??????????????????∞=??=?∞??????????????

(??

???????)

?37信號(hào)與系統(tǒng)課程組?2015??????(??????)????理想低通濾波器5.2 DTFT

時(shí)域相乘性質(zhì)??

????????????????=??????????

????????(?????)????????????????????????????

?

???????????????????

??????????????????????????????????如果能把

??

式的周期卷積轉(zhuǎn)化為求常規(guī)卷積(積分上下限為±∞)，可以簡(jiǎn)化計(jì)算。38信號(hào)與系統(tǒng)課程組?2015(??)周期卷積轉(zhuǎn)換為

常規(guī)卷積39信號(hào)與系統(tǒng)課程組?2015令????????

??

=????,

???

<

??

<

??其他??

??????=????????????????????????(?????)?????????????????????

???????

?

??????????????????????????????,?????????∞?????∞=??????????????????

(?????)????±∞區(qū)間積分常規(guī)卷積???????????

?????0.50.25??

??????40信號(hào)與系統(tǒng)課程組?2015??????????????????????

?

???????????????????

?????????????????????????????????????????????

?????0.50.25??

??????41?????????

??

?

????

?????????

?????????????????

??????=??????∞?????∞??????????????

(?????)??????

?????????????

????

?

????

??????5.2 DTFT

時(shí)域相乘性質(zhì)421.離散時(shí)間非周期信號(hào)的傅里葉變換典型D-T非周期信號(hào)的傅里葉變換離散時(shí)間周期信號(hào)的傅里葉變換離散時(shí)間傅里葉變換的性質(zhì)離散時(shí)間傅里葉變換的卷積和相乘性質(zhì)對(duì)偶性線性差分方程描述D-T

LTI系統(tǒng)信號(hào)與系統(tǒng)課程組?20156.1

對(duì)偶性情況CTFT連續(xù)時(shí)間傅里葉變換DTFS離散時(shí)間傅里葉級(jí)數(shù)分析公式∞??(????)

=??(??)??????????????∞??????

=

????=

????

??

?????????????綜合公式??(??)

=???????∞∞??(????)????????

??????[??]

=??=

??????????????????對(duì)偶性??

?? ?

??

??????

?? ?

??????

?????[??]

?

?????????? ?

??

??

???利用分析/綜合公式的數(shù)學(xué)對(duì)偶性，通過(guò)變量代換即可獲得時(shí)頻域波形的對(duì)偶性6.1

對(duì)偶性情況DTFT離散時(shí)間傅里葉變換CTFS連續(xù)時(shí)間傅里葉級(jí)數(shù)分析公式∞??(??????)

=??=?∞??[??]?????????????????

=

????

??

?????????????????綜合公式????????????[??]

=??(??????)????????????∞??

??

=??=?∞????????????????兩者各自的分析/綜合公式間沒(méi)有數(shù)學(xué)對(duì)偶性，時(shí)頻域波形間不再有對(duì)偶性對(duì)偶性但是：DTFT

和CTFS

之間存在某種對(duì)偶性456.2 DTFT

與CTFS

間的對(duì)偶性∞??(??????)

=

??[??]???????????=?∞是頻域內(nèi)以

????

為周期的連續(xù)函數(shù)????

=??????????

?????????????????????????

????????=?∞=??構(gòu)造一個(gè)時(shí)域內(nèi)以

????

為周期的連續(xù)函數(shù)

??(??????)其傅里葉級(jí)數(shù)(??=????,????

=??)：∞??????????????[??]

=????????????(??????)????????????由DTFT可知：??(??????)的時(shí)域波形為：????

=

??[???]466.2 DTFT

與CTFS

間的對(duì)偶性上述的分析表明：??????????(??????)若

??

??則??

????????????????[???]利用這一對(duì)偶關(guān)系，可以將DTFT

的若干特性對(duì)偶到CFS

中去；或者反之。信號(hào)與系統(tǒng)課程組?201547小結(jié)：對(duì)偶性CTFT??

??

?

??

??????

????

?

??????

???連續(xù)非周期連續(xù)非周期CTFS??

??

?

????連續(xù)周期離散非周期DTFT??[??]

?

??(??????)離散非周期連續(xù)周期DTFS??[??]

?

????????

?

??

??

?????離散周期離散周期??

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LTI系統(tǒng)

（

）49信號(hào)與系統(tǒng)課程組?20157.1 A

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