离散时间信号处理DSP习题.ppt

Chapter 1 Discrete-time system exercises,2,Linearity,A discrete-time system is linear if and only if H a x(n) = a H x(n) and H x1(n) + x2(n) = H x1(n) + H x2(n) for any constant a, and any sequences x(n), x1(n), and x2(n).,3,Time invariance,A discrete-time system is time invariant if and only if, for any input sequence x (n) and integer n0, then H x(n-n0)=y(n-n0) with y (n)= H x(n).,4,Causality,A discrete-time system is causal if and only if, when x1(n) = x2(n) for n n0, then H x1(n) = H x2(n), for n n0,5,1.1 (a) y(n)=(n+a)2x(n+4),1.1 Characterize the systems below as linear/nonlinear, causal/noncausal and time invariant/time varying. (a) y(n)=(n+a)2x(n+4) Linearity: Hax(n)=(n+a)2ax(n+4)=a(n+a)2x(n+4)=aHx(n) Hx1(n)+x2(n)=(n+a)2x1(n+4)+x2(n+4) = (n+a)2x1(n+4)+(n+a)2x2(n+4) =Hx1(n)+Hx2(n) therefore y(n) is linear.,6,1.1 (a) y(n)=(n+a)2x(n+4),Causality Because y(n)=(n+a)2x(n+4) i.e. the output for a certain time t = n of this system depends on the time after n (i.e. t = n+4). So the system is noncausal.,7,1.1 (a) y(n)=(n+a)2x(n+4),Time invariance Hx(n-n0)=(n+a)2x (n-n0+4) y(n-n0)=(n-n0+a)2x(n-n0+4) If y(n) = Hx(n), then y(n-n0)Hx(n-n0). Therefore the system is time varying.,8,1.1 (b) y(n)=ax(nT+T),(b) y(n)=ax(nT+T) Linearity Hbx(n)=abx(nT+T)=bHx(n) Hx1(n)+x2(n)=ax1(nT+T)+x2(nT+T)=Hx1(n)+Hx2(n) Therefore y(n) is linear. Causality The output for a certain time t = n of this system depends on the time after n (i.e. t = nT+T, supposing T0). So the system is noncausal. Time invariance Hx(n-n0)=ax(n-n0)T+T=y(n-n0) Therefore the system is time invariant.,9,1.1 (f) y(n)=x(n)/x(n+3),(f) y(n)=x(n)/x(n+3) Linearity Hax(n)=ax(n)/ax(n+3)=x(n)/x(n+3)ay(n) Hx1(n)+x2(n)=x1(n)+x2(n)/x1(n+3)+x2(n+3)y1(n)+y2(n) and therefore the system is nonlinear. Causality The output for a certain time t = n of this system depends on the time after n (i.e. t = n+3). So the system is noncausal. Time invariance Hx(n-n0)=x(n-n0)/x(n-n0+3)=y(n-n0) so the system is time invariant.,10,Periodic sequence,A sequence x(n) is defined to be periodic if and only if there is an integer N0 such that x(n) = x(n + N) for all n. In such a case, N is called the period of the sequence.,11,1.2,1.2 For each of the discrete signals below, determine whether they are period or not. Calculate the periods of those that are periodic. (a) (c) Solution (a) N is an integer and N 0.,12,1.2,if x(n) = x(n+N), that is x(n) is periodic, and the period is N = 15. (c) if x(n) = x(n+N), that is x(n) is periodic, and the period is N = 54.,13,1.4 (b),Solution:,* If x(i), ;y(j), , then,16,1.17 (a),1.17 Discuss the stability of the systems described by the impulse responses below: h(n)=2-nu