信号与系统奥本海姆第二版答案

Systems(Second Edition) —Learning Instructions (Exercises Answers)

Department

ComputerEngineering

2005.12

ContentsChapter 17Chapter 35Chapter 62Chapter 83Chapter 109Chapter 119Chapter 132Chapter 140Chapter 10 160 Answers1.1 Converting from polar Cartesiancoordinates: 1.2converting from Cartesian polarcoordinates: limlim dtdt =cos(t).Therefore, limlim dtdt limlim limcos shiftedsignal shiftedsignal

flippedsignal

flippedsignal

newSignal

flippedsignal newsignal flippedsignal right.Therefore, weknow t>-2.Similarly,

x(2-t)

t>-1,Therefore,

(1-t)+x(2-t)

linearlycompression Therefore,x(3t) linearlycompression

Therefore,x(3t) periodicbecause fundamentalperiod FigureS1.6. Therefore, fundamentalperiod oddsignal, allvalues zerowhen zeroonly

when

periodiccomplex

multiplied

exponential.

10

10

complexexponential Therefore,

decayingexponential. complexexponential fundamentalperiod

periodicsignal.

periodicsignal.

fundamentalperiod

Weobtain fundamentalperiod complexexponential =3/5.We cannot find any integer integer.Therefore, periodic.1.10. x(t)=2cos(10t＋1)-sin(4t-1)

Period

firstterm

firstterm

overallsignal periodwhich leastcommon multiple secondterms. -3-1 -1-2 -3 -3-3 firstterm secondterm secondterm

overallsignal periodwhich leastcommon Multiple threeterms inn 35.1.12. figureS1.12. flippedsignal right.Therefore, no=-3.1.13 itsderivative FigureS1.14. Therefore [n-3]=2x [n-2]+4x [n-3]+4x

[n-4])=2x

[n-2]+5x

input-outputrelationship

y[n]=2x[n-2]+5x [n-3] 2x[n-4] input-outputrelationship does connectedseries reversed. Wecan easily prove [n-3])+4(x input-outputrelationship onceagain y[n]=2x[n-2]+ 5x [n-3] 2x[n-4] 1.16 memoryless because pastvalues wemay conclude systemoutput alwayszero causalbecause sometime may depend futurevalues Considertwo arbitrary inputs (sin(t))Let linearcombination

arbitraryscalars

givensystem

correspondingoutput linear.1.18.(a) Consider two arbitrary inputs

linearcombination

arbitraryscalars.

givensystem, arbitraryinput

.Consider

secondinput

correspondingoutput correspondingoutput

outputcorresponding Alsonote +1)B.Therefore +1)B.1.19 Considertwo arbitrary inputs (t-1)Let linearcombination arbitraryscalars. givensystem, correspondingoutput linear.(ii) Consider arbitraryinputs correspondingoutput .Consider secondinput outputcorresponding Alsonote Considertwo arbitrary inputs [n-2].Let linearcombination arbitraryscalars. givensystem,

correspondingoutput

linear.(ii)

Consider

arbitraryinput correspondingoutput .Consider secondinput outputcorresponding Alsonote Considertwo arbitrary inputs -1]Let

linearcombination

arbitraryscalars.

Consider

givensystem, arbitraryinput secondinput

correspondingoutput correspondingoutput

linear.(ii)

.Consider

outputcorresponding Alsonote Considertwo arbitrary inputs linearcombination correspondingoutput correspondingoutput outputcorresponding

arbitraryscalars. linear.(ii)

Consider

givensystem, arbitraryinputs secondinput Thereforex1

.Consider

Alsonote

systemliner

weknow x2(t)=cos(2(t-1/2))= linearityproperty, we may once again write x1 cos(3t-1)Therefore, x1 (t)=cos(2(t-1/2)) =cos(3t-1)1.21.The signals figureS1.21. Figure S1.21 1.22 figureS1.22 1.23 oddparts FigureS1.23 1/2-1/2 -1 0.50.5 3/2-3/2 x(2t+1)x(4-t/2) 1012 1/2-1/2 -1 1/2-1/2 1/2-1 -2 1/2-1 -2 1/2-1 -2 FigureS1.22 -4-1 -2 -7xo[n] 3t/2-3t/2 -1/2-7 FigureS1.24 -2 FigureS1.23 1/2 1/2-7 1/2-1 3/2-3/2 -1/2 101.24 oddparts FigureS1.24