离散时间信号处理DSP习题

1、chapter 1 discrete-time systemexercises2linearity 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).3time invariance a discrete-time system is time invariant if and only if, for any

2、 input sequence x (n) and integer n0, thenh x(n-n0)=y(n-n0) with y (n)= h x(n).4causality 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 tim

3、e invariant.91.1 (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). s

4、o the system is noncausal. time invariancehx(n-n0)=x(n-n0)/x(n-n0+3)=y(n-n0)so the system is time invariant.10periodic 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.

5、111.2 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 0.22( )cos ()15x nn( )cos(31)27x nnnnnx154cos2121152cos)(2nnnnx154154cos2121)(121.2 if x(n) = x(n+n), that is x(n)

6、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.kknknnnn152152154154154cos2121154cos2121( )cos(31)27x nnknknnnn54227312727cos3127cos131.4 (b)( )* ( )( ) ()(0) ( )(1) (1)(2) (2)(7) (7)(8) (8)( )(1)(2)(7)(8)kx nh nx k h nkxh nxh nxh nxh nx

7、h nh nh nh nh nh 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;

8、12: ( )* ( )(12)(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 bd161.17 (a)1.17 discuss the stability of the systems described by the impulse responses below:(a)h(n)=2-nu(n)(f) h(n)=0.5nu(n)-0.5nu(4-n)solution (a) the system is stable