信号与系统第二版课后习题解答(34)奥本海姆

SignalandSystem

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Chap3

3.1Acontinuous-timeperiodicsignalx(t)isrealvalueandhasafundamentalperiodT=8.ThenonzeroFourierseriescoefficientsforx(t)are

.

Expressx(t)intheform

Solution:

Fundamentalperiod.

3.2Adiscrete-timeperiodicsignalx[n]isrealvaluedandhasafundamentalperiodN=5.ThenonzeroFourierseriescoefficientsforx[n]are

,,,

Expressx[n]intheform

Solution:

for,, , ,,

for

也就是说,

即12k<100，k<=8，故当k>8时，ak＝0。

3.35.Consideracontinuous-timeLTIsystemSwhosefrequencyresponseis

Whentheinputtothissystemisasignalx(t)withfundamentalperiodandFourierseriescoefficients,itisfoundthattheoutputy(t)isidenticaltox(t).

Forwhatvaluesofkisitguaranteedthat?

Solution:

T=,.

for ,

thatis , andisinteger,so.

Let,,itneeds,for.

Chap4

4.1UsetheFouriertransformanalysisequation(4.9)tocalculatetheFouriertransformsof;

(a)(b)

SketchandlabelthemagnitudeofeachFouriertransform.

Solution:

(a).

(b).

4.2UsetheFouriertransformanalysisequation(4.9)tocalculatetheFouriertransformsof:

(a)(b)

SketchandlabelthemagnitudeofeachFouriertransform.

Solution:

(a).

(b).

4.5UsetheFouriertransformsynthesisequation(4.8)todeterminetheinverseFouriertransformof,where

Useyouranswertodeterminethevaluesoftforwhichx(t)=0.

Solution:

Ifx(t)=0then

Thatis

4.6Giventhatx(t)hastheFouriertransform,expresstheFouriertransformsofthesignalslistedbelowinthetermsof.YoumayfindusefultheFouriertransformpropertieslistedinTable4.1.

(a)

(b)

(c)

Solution:

AccorrdingtothepropertiesoftheFouriertransform,we’llget:

(a).

then

(b).

(c).

4.11Giventherelationships

，And

Andgiventhatx(t)hasFouriertransformandh(t)hasFouriertransform,useFouriertransformpropertiestoshowthatg(t)hastheform

DeterminethevaluesofAandB.

Solution:

for

and

then

4.14Considerasignalx(t)withFouriertransform.Supposewearegiventhefollowingfacts:

1.x(t)isrealandnonnegative.

2.whereAisindependentoft.

3..

Determineaclosed-formexpressionforx(t).

Solution:

From(1),weknowisrealand;

From(2),weknow: