离散时间信号处理DSP习题ppt课件

1、Chapter 1 Discrete-time systemexercisesLinearity 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).Time invariance A discrete-time system is time invariant if and only if, for any i

2、nput sequence x (n) and integer n0, then H x(n-n0)=y(n-n0) with y (n)= H x(n).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 0). So the system is noncausal.Time invarianceHx(n-n0)=ax(n-n0)T+T=y(n-n0)Therefore the system is time i

3、nvariant.1.1 (f) y(n)=x(n)/x(n+3)(f) y(n)=x(n)/x(n+3)LinearityHax(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

4、system is noncausal.Time invarianceHx(n-n0)=x(n-n0)/x(n-n0+3)=y(n-n0)so the system is time invariant.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.1.2 1.2 F

5、or 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 0.22( )cos ()15x nn( )cos(31)27x nnnnnx154cos2121152cos)(2NnNnx154154cos2121)(1.2 if x(n) = x(n+N), that is x(n) is periodic,

6、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.kkNkNNnn152152154154154cos2121154cos2121( )cos(31)27x nnkNkNNnn54227312727cos3127cos1.4 (b)( )* ( )( ) ()(0) ( )(1) (1)(2) (2)(7) (7)(8) (8)( )(1)(2)(7)(8)kx nh nx k h nkxh nxh nxh nxh nxh nh nh nh nh n

7、h ntherwise , 041 , )( and therwise , 087 , 163 , 020 , 1)(onnnhonnnnxSolution:6: ( )* ( )(6)(5)(4)( 1)( 2)4;8: ( )* ( )(8)(7)(6)(1)(0)1;9: ( )* ( )(9)(8)(7)(2)(1)3;10: ( )* ( )(10)(9)(8)(3)(2)5;11: ( )* ( )(11)(10)(9)nx nh nhhhhhnx nh nhhhhhnx nh nhhhhhnx nh nhhhhhnx nh nhhh (4)(3)7;12: ( )* ( )(12

8、)(11)(10)(5)(4)4;: ( )* ( )0.hhnx nh nhhhhhnotherwise x nh n1: ( )* ( )(1)(0)( 1)( 6)( 7)1;2: ( )* ( )(2)(1)(0)( 5)( 6)3;3: ( )* ( )(3)(2)(1)( 4)( 5)6;4: ( )* ( )(4)(3)(2)( 3)( 4)9;5: ( )* ( )(5)(4)(3nx nh nhhhhhnx nh nhhhhhnx nh nhhhhhnx nh nhhhhhnx nh nhhh)( 2)( 3)7;hh)8()7()2() 1()()()(nhnhnhnhnhnhnx*If x(i), ;y(j), , then , ia b , jc d( )* ( ),x iy jac bd1.17 (a)1.17 Discuss the stability of the systems described by the impulse responses below:h(n)=2-nu(n)(f) h(n)=0.5nu(n)-0.5nu(4-n)Solution (a) the system is st