数字信号处理基础答案第三章sol

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1、Chapter3Solutions3.1(a)(i)x[0]=3(ii)x[3]=5(iii)x[–1]=2x[n-2](b)(i)n(ii)Thesketchforx[n+1]doesnotshowthevalueofthesamplex[5],sincethisinformationisnotprovidedinthequestion.x[n+1]n3.2Theimpulsefunctioniszeroeverywhereexceptn=0.(a)d[–4]=0(b)d[0]=1(c)d[2]=03.3(a)Thisfunctionisamirrorimageoftheim

2、pulsefunctionacrosstheverticalaxis,whichmeansnochangeoccurs.x[n]n(b)Thisfunctionshiftstheimpulsetwostepstotherightandincreasesitsamplitudeto2.x[n]n(c)Thisfunctionisthesumoftwoimpulsefunctions.x[n]n3.4Thestepfunctioniszeroforn<0andoneeverywhereelse.(a)u[–3]=0(b)u[0]=1(c)u[2]=13.5(a)x[n]=4u[n–

3、1]x[n]n(b)x[n]=–2u[n]x[n]n(c)x[n]=2u[–n]x[n]n(d)x[n]=u[n–2]x[n]n(e)x[n]=u[2–n]x[n]nx[n]3.6(a)n(b)x[n]n3.7(a)Thissignalisasumofshiftedstepfunctions,eachwithamplitudeone.x[n]=0.1u[n–1]+0.1u[n–2]+0.1u[n–3]+…x[n]n(b)Thissignalisasumofimpulsefunctionswithincreasingamplitude.x[n]=0.1d[n–1]+0.2d[n–

4、2]+0.3d[n–3]+…x[n]n3.8x[n]=(u[n]–u[n–2])+(u[n–5]–u[n–7])+(u[n–10]–u[n–12])x[n]n3.9x[n]=2d[n]–3d[n–1]+d[n–2]–d[n–3]+3d[n–4]3.10(a)x[n]=d[n–3]+d[n–4]+d[n–5]+d[n–6]–d[n–7]–d[n–8]–d[n–9]–d[n–10]–d[n–11](b)x[n]=u[n–3]–2u[n–7]+u[n–12]3.11x[n]=u[n]+2u[n–4]3.12x[n]n3.13Thesignalhasvalues1,0.5,0.25,0

5、.125,etc.Thesevaluescanbegeneratedfromthefunction0.5n,whereeachvalueistheamplitudeofanimpulsefunction.Thesignalmaybeexpressedas=d[n]+0.5d[n–1]+0.25d[n–2]+0.125d[n–3]+…3.14(a)ByEuler’sidentity,(b)From(a),.Therefore,.Thedigitalperiodis8samples.3.15(a)n01.00001.0010.5000–j0.86601.002–0.5000–j0.

6、86601.003–1.00001.004–0.5000+j0.86601.0050.5000+j0.86601.00(b)Themagnitudeofanycomplexnumberisthesquarerootofthesumofthesquaresoftherealandimaginaryparts.Themagnitudesinthelastcolumnin(a)showthat=1.Thisequationistrueforalln.3.16Thefrequencyoftheanalogsignalisf=w/2p=200/2pHz.Thesamplingfreque

7、ncyfS=1/TS=1/(25x10–3)=40Hz.ThedigitalfrequencyisW=2pf/fS=200/40=5rads.Thesampledsignalisx[n]=5sin(nW)=5sin(5n).3.17Check2p/Wforeachfunction.Thefunctionisperiodicifthisratioisrational.(a)2p/W=2p/(4/5)=10p/4=5p/2Thisratioisnotrational,sothesinusoidi