离散时间信号处理DSP习题.pptx

Chapter1Discrete-timesystemexercises

LinearityAdiscrete-timesystemislinearifandonlyifH{ax(n)}=aH{x(n)}andH{x1(n)+x2(n)}=H{x1(n)}+H{x2(n)}foranyconstanta,andanysequencesx(n),x1(n),andx2(n).2

TimeinvarianceAdiscrete-timesystemistimeinvariantifandonlyif,foranyinputsequencex(n)andintegern0,thenH{x(n-n0)}=y(n-n0)withy(n)=H{x(n)}.3

CausalityAdiscrete-timesystemiscausalifandonlyif,whenx1(n)=x2(n)fornn0,thenH{x1(n)}=H{x2(n)},fornn04

1.1(a)y(n)=(n+a)2x(n+4)1.1Characterizethesystemsbelowaslinear/nonlinear,causal/noncausalandtimeinvariant/timevarying.(a)y(n)=(n+a)2x(n+4)Linearity:H{ax(n)}=(n+a)2ax(n+4)=a(n+a)2x(n+4)=aH{x(n)}H{x1(n)+x2(n)}=(n+a)2[x1(n+4)+x2(n+4)]=(n+a)2x1(n+4)+(n+a)2x2(n+4)=H{x1(n)}+H{x2(n)}thereforey(n)islinear.5

1.1(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.6

1.1(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.7

1.1(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,supposingT0).Sothesystemisnoncausal.TimeinvarianceH{x(n-n0)}=ax[(n-n0)T+T]=y(n-n0)Thereforethesystemistimeinvariant.8

1.1(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)andthereforeth