Chapter1Discrete-timesystemexercisesLinearityAdiscrete-timesystemislinearifandonlyifH{ax(n)}=aH{x(n)}andH{x1(n)+x2(n)}=H{x1(n)}+H{x2(n)}foranyconstanta,andanysequencesx(n),x1(n),andx2(n).2TimeinvarianceAdiscrete-timesystemistimeinvariantifandonlyif,foranyinputsequencex

(n)andintegern0,thenH{

x(n-n0)}=y(n-n0)withy

(n)=H{

x(n)}.3CausalityAdiscrete-timesystemiscausalifandonlyif,whenx1(n)=x2(n)forn<n0,thenH{x1(n)}=H{x2(n)},forn<n041.1(a)y(n)=(n+a)2x(n+4)CausalityBecausey(n)=(n+a)2x(n+4)i.e.theoutputforacertaintimet=nofthissystemdependsonthetimeaftern(i.e.t=n+4).Sothesystemisnoncausal.61.1(a)y(n)=(n+a)2x(n+4)TimeinvarianceH{x(n-n0)}=(n+a)2x(n-n0+4)y(n-n0)=(n-n0+a)2x(n-n0+4)Ify(n)=H{x(n)},theny(n-n0)≠H{x(n-n0)}.Thereforethesystemistimevarying.71.1(b)y(n)=ax(nT+T)(b)y(n)=ax(nT+T)LinearityH{bx(n)}=abx(nT+T)=bH{x(n)}H{x1(n)+x2(n)}=a[x1(nT+T)+x2(nT+T)]=H{x1(n)}+H{x2(n)}Thereforey(n)islinear.CausalityTheoutputforacertaintimet=nofthissystemdependsonthetimeaftern(i.e.t=nT+T,supposingT>0).Sothesystemisnoncausal.TimeinvarianceH{x(n-n0)}=ax[(n-n0)T+T]=y(n-n0)Thereforethesystemistimeinvariant.81.1(f)y(n)=x(n)/x(n+3)(f)y(n)=x(n)/x(n+3)LinearityH{ax(n)}=ax(n)/ax(n+3)=x(n)/x(n+3)≠ay(n)H{x1(n)+x2(n)}=[x1(n)+x2(n)]/[x1(n+3)+x2(n+3)]≠y1(n)+y2(n)andthereforethesystemisnonlinear.CausalityTheoutputforacertaintimet=nofthissystemdependsonthetimeaftern(i.e.t=n+3).Sothesystemisnoncausal.TimeinvarianceH{x(n-n0)}=x(n-n0)/x(n-n0+3)=y(n-n0)sothesystemistimeinvariant.9PeriodicsequenceAsequencex(n)isdefinedtobeperiodicifandonlyifthereisaninteger

N≠0suchthatx(n)=x(n

+

N)foralln.Insuchacase,Niscalledtheperiodofthesequence.101.21.2Foreachofthediscretesignalsbelow,determinewhethertheyareperiodornot.Calculatetheperiodsofthosethatareperiodic.(a)(c)Solution(a)

NisanintegerandN≠0.111.2ifx(n)=x(n+N),thatis

x(n)isperiodic,andtheperiodisN=15.(c)ifx(n)=x(n+N),thatisx(n)isperiodic,andtheperiodisN=54.121.4(b)Solution:13*****Ifx(i),;y(j),,then1.17(a)1.17Discussthestabilit