数字信号处理(双语)习题Ch8-CH8习题

8.1

Considerthesequencex\n]givenbyx[n]=anu\n\.Assume\a\<1.Aperiodic

sequencex\n]isconstructedfromx\n\inthefollowingway:

00

x[n]={x[n+rN]

r=-8

(a)DeterminetheFouriertransformX(e^)ofx\ri\.

(b)DeterminethediscreteFourierseriesofX[k]ofx\n\.

(c)HowisX[k]relatedtoX(e；£0).

8.2

ComputetheDFTofeachofthefollowingfinite-lengthsequencesconsideredtobeof

lengthN(whereNiseven).

(a)x[n]=S[n],

(b)x[n]=6[n-nQ],0<n0<N-1,

1,neven,0<n<N—1,

(c)x[n]=

0,nodd,0<n<N-1,

N

0<n<2_t

(d)x[n]=N

-<n<N-1,

0<n<N—1,

(e)x[n]=L

otherwise.

8.3

Considerthefinite-lengthsequencesx[n]inFigure8-1.LetX(z)bethez-transform

ofx[n].IfwesampleX(z)atz=/。万//匕fc=o,l,2,3,weobtain

Xjk]=X(^)|z=e，(2〃/4)的k=0,l,2,3.

Sketchthesequencexx[n]obtainedastheinverseDFTofXx[k].

x[n]

1

।>

----------------------------------------------->

-2-101234567n

Figure8-1

8.4

Supposewehavetwofour-pointsequencesx\rt\andh\n\asfollows:

Tin

x\n\=cos—,n=0,1,23

h[n]=2n,n=0,1,2,3.

(a)Calculatethefour-pointDFTX[k].

(b)Calculatethefour-pointDFTH[k].

(c)Calculatey[n]=x[n]@/i[n]bydoingthecircularconvolutiondirectly.

(d)Calculatey[n]ofPart(c)bymultiplyingtheDFTsofx[n]andh[n]and

performinganinverseDFT.

8.5

Figure8-2(a)and(b)showstwosequences[n]andx2[n].Thevalueofx2[n]at

timen=3isnotknown,butisshownasavariablea.Figure8-2(c)showsy[n],

thefour-pointcircularconvolutionofxr[n]andx2[n].Basedonthegraphofy[n],

canyouspecifyauniquely?Ifso,whatisa?Ifnot,givetwopossiblevaluesa

thatwouldyieldthesequencey[n]asshown.

11

11()

-2-101234567n

(a)

x2[n]a

1

(

-2-101234567n

(b)

y[n]JJ

>

(,।»

-1-1

(C)

Figure8-2

8.6

Figure8-3showstwofinite-lengthsequences.Sketchtheir/V-pointcircular

convolutionforN=6andforN=IO.

xAn]

6

's

3

2

-2-101234567n

X2[n]

11

__I__________I________>

-2-101234567n

Figure8-3

8.7

Twofinite-lengthsequencesx1[n]andx2[n],whicharezerooutsidetheinterval

0<n<99,arecircularlyconvolvedtoformanewsequencey[n];i.e.,

99

y[n]=XiM]?产M=2久i同小[（（九一①）ioo，0<n<99.

k=o

If,infact,[n]isnonzeroonlyfor10<n<73,determinethesetofvaluesofn

forwhichy[n]isguaranteedtobeidenticaltothelinearconvolutionofx1\n]and

%2[叫

8.8

Letx[n]beareal-valued5-pointsequencewhose7-pointDFTisdenotedbyX[k].

IfReal{X[k]}isthe7-pointDFTofg[n],showthatg[0]=x[0],anddetermine

therelationshipbetweeng[l]andx[l].Justifyyouranswer.

8.9

Supposexr[n]isaninfinite-length,stable(i.e.,absolutelysummable)sequencewith

z-transformgivenby

1

Xi(z)=z