刘丹6离散系统Z分析.ppt

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1、第六章第六章 z z变换与离散系统的频域分析变换与离散系统的频域分析 6.1 6.1 Z Z变换的定义变换的定义vX(zX(z)=)=Z Zx(nx(n)=)=(-n )(-n|a|-|a|-在极点在极点外侧外侧默认默认 非因果信号非因果信号,|z|,|z|小有利收敛小有利收敛 例例Z-aZ-an nu(-n-1)=-au(-n-1)=-a-1-1z z1 1-a-a-2-2z z2 2-a-a-3-3z z3 3-=-a =-a-1-1z/(1-az/(1-a-1-1z)z)=1/(1-az =1/(1-az-1-1)C:|zC:|z|a|z|0 C:|z|0 4.2 Z的性质定理 线性线性

2、反转反转 x(-n)x(-n)X(zX(z-1-1)算子作用律算子作用律 ZH(ZH(q q)x(n)x(n)=)=H(z)Zx(nH(z)Zx(n)线性调制线性调制nx(nnx(n)z z-1-1(d/dz(d/dz-1-1)X(z)X(z)时移时移Z Zq qm mx(nx(n)=z zm m Z Zx(nx(n)复共轭复共轭 x x*(n)n)X X*(z(z*)时域卷积时域卷积x x1 1(n)*x(n)*x2 2(n)(n)X X1 1(z)X(z)X2 2(z)(z)指数调制指数调制 a an nx(n)x(n)X(zX(z/a)/a)初值初值-x(nx(n)因果因果 x(0)=x

3、(0)=X(z)|zX(z)|z终值终值-x(nx(n)右向右向,减或等幅减或等幅 x(+)=(1-zx(+)=(1-z-1-1)X(z)|z)X(z)|z1 1例例(1)0.8(1)0.8 n nu(n)u(n)1/(1-0.8z1/(1-0.8z-1-1)|z|0.8)|z|0.8(2)q(2)q-2-21,4,31,4,3z z-2-2(1+4z(1+4z-1-1+3z+3z-2-2)(8)0.5(8)0.5 n nu(n)-u(n-Nu(n)-u(n-N)=(1-0.5 =(1-0.5 N Nq q-N-N)0.5)0.5 n nu(nu(n)(1-0.5(1-0.5 N Nz z-N

4、-N)/(1-0.5z)/(1-0.5z-1-1),|z|0.5),|z|0.5 6.4 逆逆 Z变变 换换 算子法算子法1 1 右向或有限长右向或有限长x(nx(n)-|z|R-)-|z|R-x(nx(n)=)=X(q)X(q)(n(n)=)=X(z)|X(z)|z zq q (n(n)例例:z:z2 2(2-z(2-z-2-2+4z+4z-3-3)q q2 2(2-q(2-q-2-2+4q+4q-3-3)(n)(n)=q =q+2+22,0,-1,42,0,-1,4例例:X(zX(z)=z)=z-1-1/(1-0.5z/(1-0.5z-1-1),|z|0.5),|z|0.5 x(nx(n)

5、=q)=q-1-1/(1-0.5q/(1-0.5q-1-1)(n),)(n),=q =q-1-1 0.50.5n n u(nu(n)幂级数展开幂级数展开将将X(zX(z)展开为展开为z,zz,z-1-1级数级数,z z-n-n系数为系数为x(nx(n)X(zX(z)=e)=ez z-1-1 =1+z =1+z-1-1+(1/2)z+(1/2)z-2-2+(1/n!)z+(1/n!)z-n-n+x(nx(n)=(1/n!)u(n)=(1/n!)u(n)6.4.2 部分分式法部分分式法 b b0 0+b+b1 1z z-1-1+b+b2 2z z-2-2+b bM Mz z-M-M X X(z z

6、)=-)=-1+a 1+a1 1z z-1-1+a+a2 2z z-2-2+a aN Nz z-N-N =K(zK(z)+)+r ri i/(1-p/(1-pi i z z-1-1)p pi i单级点时单级点时:r ri i=(1-p=(1-pi iz z-1-1)X(z)|z=p)X(z)|z=pi i 例例:X(zX(z)=)=z z-1-1/(1-/(1-z z-1-1)(1-0.5)(1-0.5z z-1-1)=r1/(1-=r1/(1-z z-1-1)+r2/(1-0.5)+r2/(1-0.5z z-1-1)r1=(1-r1=(1-z z-1-1)X(z)|)X(z)|z=1z=1=

7、1/(1-0.5)=2=1/(1-0.5)=2 r2=(1-0.5 r2=(1-0.5z z-1-1)X(z)|)X(z)|z=0.5z=0.5=2/(1-2)=-2 =2/(1-2)=-2 X(zX(z)=2/(1-)=2/(1-z z-1-1)-2/(1-0.5)-2/(1-0.5z z-1-1)x(nx(n)=2u(n)-2(0.5)=2u(n)-2(0.5)n nu(n)u(n)M语言X(zX(z)=)=z z-3-3/(1-/(1-z z-1-1)(1-0.5)(1-0.5z z-1-1)|z|1)|z|1 r,p,kr,p,k=residuez(0,0,0,1,1,-1.5,0.5

8、)=residuez(0,0,0,1,1,-1.5,0.5)r=2 -8 r=2 -8 留数留数 p=1.0 0.5 p=1.0 0.5 极点极点 k=6 2 k=6 2 多项式多项式 X(zX(z)=(6-2)=(6-2z z-1-1)+2/(1-)+2/(1-z z-1-1)-8/(1-0.5)-8/(1-0.5z z-1-1)x(nx(n)=(6-2)=(6-2q q-1-1)(n(n)+2u(n)-8(0.5)+2u(n)-8(0.5)n nu(n)u(n)=6,-2 +2u(n)-8(0.5)=6,-2 +2u(n)-8(0.5)n nu(n)u(n)多重极点展开vp p1 1为为s

9、 s重极点重极点 r1 r2 r1 r2 rsrs-+-+-+-+-(1-p(1-p1 1z z-1-1)(1-p)(1-p1 1z z-1-1)2 2 (1-p (1-p1 1z z-1-1)s srsrs=(1-p(1-p1 1z z-1-1)s sX(z)|z=p1X(z)|z=p1r rs-1s-1=(-p=(-p1 1)-1-1(d/dz(d/dz-1-1)(1-p(1-p1 1 z z-1-1)s sX(z)|X(z)|z=pz=p1 1r rs-ms-m=(-p=(-p1 1)-m-m(1/m!)(d/dz(1/m!)(d/dz-1-1)m m1-p1-p1 1z z-1-1 s

10、 sX(z)|X(z)|z=pz=p1 1例例X(zX(z)=)=z z-1-1/(1-/(1-z z-1-1)(1-0.5)(1-0.5z z-1-1)2 2 r,p,k=residuez(0,1,1,-2,1.25,-0.25)r,p,k=residuez(0,1,1,-2,1.25,-0.25)r=4,-2,-2 r=4,-2,-2 p=1,0.5 p=1,0.5 0.50.5 k=k=X(zX(z)=)=4 4/(1-/(1-z z-1-1)-2/(1-0.5)-2/(1-0.5z z-1-1)-2/(1-0.5)-2/(1-0.5z z-1-1)2 2 x(nx(n)=4-2+2(n

11、+1)0.5)=4-2+2(n+1)0.5n nu(n)u(n)例例:因果与非因果信号因果与非因果信号因果因果:a an nu(nu(n)=1/(1-az)=1/(1-az-1-1)|a|z|)|a|z|非因果非因果:-a:-an nu(-n-1)=1/(1-azu(-n-1)=1/(1-az-1-1)|z|a|)|z|a|例例:X(zX(z)=1/(1-0.8)=1/(1-0.8z z-1-1)-1/(1-0.5)-1/(1-0.5z z-1-1)(1)C:0.8|(1)C:0.8|z|2z|2极点均因果极点均因果 x(nx(n)=0.8)=0.8n nu(n)-0.5u(n)-0.5n n

12、u(n)u(n)(2)C:0.5|z|0.8(2)C:0.5|z|0),n0全响应y(ny(n)=)=yzi(n)+yzs(nyzi(n)+yzs(n)例例:H(zH(z)=1/(1-0.5z)=1/(1-0.5z-1-1)(1-z)(1-z-1-1)=1/(1-1.5z)=1/(1-1.5z-1-1+0.5z+0.5z-2-2)y(-2)=1,y(-1)=0.5,y(-2)=1,y(-1)=0.5,x(nx(n)=0.5)=0.5n nu(n),u(n),X(zX(z)=1/(1-0.5z)=1/(1-0.5z-1-1),),yzs(n):r,p,kyzs(n):r,p,k=residuez

13、(1,conv(1,-1.5,0.5,1,-0.5)=residuez(1,conv(1,-1.5,0.5,1,-0.5)r=4,-2,-1;p=1,0.5,0.5:r=4,-2,-1;p=1,0.5,0.5:yzs(nyzs(n)=4)=4 u(n)+-2-(n+1)0.5u(n)+-2-(n+1)0.5n nu(n)u(n)yzi(n):r,p,kyzi(n):r,p,k=residuez(-1.5,0.5,0*0.5+0.5,0,0*1=residuez(-1.5,0.5,0*0.5+0.5,0,0*1 ,1-1.5,0.5),1-1.5,0.5)r=0,0.25;p=0,0.5:r=0,0.25;p=0,0.5:yzi(nyzi(n)=0.25*0.5)=0.25*0.5n nu(n)u(n)y(ny(n)=)=yzs(n)+yzi(nyzs(n)+yzi(n)=4u(n)-(n+2.75)0.5)=4u(n)-(n+2.75)0.5n nu(n)u(n)自然响应自然响应 自然和受迫响应自然和受迫响应 稳态响应稳态响应 暂态响应暂态响应