二、z變換定F(二、z變換定F(z)f(k)zkF(z)f(k)zk雙邊z變F(z)=Z[f(k)] (2)f2(k)={1,2,3,F(z)(k)zk(k)k k其收斂域為整個z平面F2(zz2+2z32z-1+z-2收斂域為0<z<f2(k)的單邊z2收斂域為z例3的z變bk(解F(z) (bz) (b 1kNb1z(b1z)N1b可見，b-1z<1,即z<b時，其zF(z)fzzo變換到zfS(t)f(t)T(t)f(kT)(tkTFSb(s)f(kT)kf(kT)→f(k)三、三、收斂f(k)zkk例例2的z變F(z) akzk (az1)klim1(az1)N1可見，僅當az-1<1F(z)yzz例f(k)=2k例f(k)=2k(k)←→Ff2(k)=–2k(–k,,z常用序列的z變換：(k1–(–k ，z>1z1一、線性 f1(k)←→F1(z)1<z<f2(k)←→F2(z)2<z<a1f1(k)+a2f2(k)←→例：2(k3(k←→2+z1f(k)←→F(z),|z|，且有整數(shù)m>0f(k-1)←→z-1F(z)+f(-f(k-2)←→z-2F(z)+f(-2)+f(-1)z-f(km)zmF(z)f(kkf(km)zmF(z)f(kf(k+1)←→zF(z)–f(k+2)←→z2F(z)–f(0)z2–mk例雙邊序列yfak0的z F(z)Fy(z)F(z)1234 z變換的性 二、二、移位特若kf(km)zk f(n)znzmzmFnkm((kmN解:(kmNzmN1z zN解:f(k+1k+1)ε(k+1)k+1)ε(kf(kzF(z)–zf(0)=F(z)+zF(z)=(zf1(k)←→F1(z)f2(k)←→F2(z)f(k﹡f(kF(z)F12 解：f(kzzz=ε(k)﹡ε(k-1)z1z (zf(k←→F(z<z<，設(shè)有整數(shù)m，且k+m>0，kf(k)zmF(),F)kk(k) 1z(1)dz z1zln( zln(z zZZ[f(k–f(km)zkf(km)zkf(kmkkk上式第二項令k三三、序列乘ak(z域尺度變換若f(kF(z),<z<，且有常數(shù) akf(k)←→F(z/a),Z[af(k)]=kaf(k)z kkkzaF()zz)=0.5(ejk+e-jk)ε(k)zejzej 若f(k←→F(z),則kf(k)zdF ,d解：(kk(k)ddzz1 (z1)z(z (zf(kf(k←→F(z),<z<f(i)F(z),z明例：求序列(a為實數(shù)ai(k≥0)的zkai ai(i)k z1z證zMF(z)=f(M)+f(M+1)z-1+f(M+2)z-f(M)limzmF 逆z變fff(k)=f2(k)+f1(k)=f(k)(–k–1)+f(k)F(z)=F2(z)+F1(z)，<|z|F Ff(k)f七、七、k域反轉(zhuǎn)(僅 若f(k←→F(z),fkF(z-1例：akkz，|z|zkz1，|z|ak1(k1)，|z|<乘aa(k1)z1 ，|z|<1/az1)f(M)limzFmf(0)limFf()limf(k)lim F(z)lim(z1)F(z)z2z2zz2/(z2-z-2)=1+z-1+3z-2+5z-3+ z:z/(–2–z–z22z2z3z4 z5 1 5kB(z)bzmbzm1 bz z az zn m F(z)可展開為：F(z)K0 z....zF(z)K0Kii1zak(k)z(z34z29z)F(z) 解F(z)f(k)(1)(k)2(k)(2)(k1)(3)(kkkk2F(z)=Z[f(k) f(k)zkF(z)=Z[f(k)(–k f(k，|z|>，|z|<例：已知象函數(shù)F(z)zz（1）|z|>(2)|z|<1(3)1<|z|<1z 2zF(z)3z1zF(z) ，z1F(z)3，z<2zzF(z)z1z2z3 F(z)· zz311 1 11122F(z) (z1)(z其收斂域分別為：（1）z>2(2)z<1(3)F(z)z (z1)(z2)z1z3(1)當f(k)[(1)(2)](k1k2kf(k)[(1)k(2)k](k12 (3)當f(k)1(1)k(k)2(2)k(k 若z若z>(za)rk(k (kr(rakr1(k1kz兩邊對aZ[0.5k(k-1)ak-2(k)]=(z zzaniy(ki)bmjf(k ani[zY(z) y(k]bm[[y(ki)zk]( zj)F差分方程為y(ky(k12y(k2f(k)+2f(k解方程取單邊z得(12z1)y(1)2y(2) 12z221z12z1z12zFz2z2z2z2zY(z) (z2)(z z z yzi(k)[2(2)(1) Y(z) z3y(k)[2k1(1)z 2z12z(2)(2)F(z)F(z) zc zcF(z)Ke K 若z>,f(k)K1k若z,f(kK1kk+)(–k–例：已知象函數(shù)例：已知象函數(shù)F(z(z (z (zz3，z>1(z zK11(z3F11d(z1)3F(z)2dz2F(z)(z (z zf(k)=[k(k-1)+3k+1]YY(z)M(z) (z)Y(z)Y令H Fy(k)–y(k–1)–2y(k–2)=f(k)+2f(k–已知y(–1)=2，y21/2，f(k(k)。求系統(tǒng)fff(k-1FD1H(z)1z213z1 z23z z1z h(k)=[2–)=Yzs(z)z23z2z1(z1)2(zz2 3zz zyzs(k)=[2k+3–21*[2-k例：求2k(–k*[2-kz0.5,|z|2k(k)z1 z4 4(z0.5)(z z0.5z 原式=3(0.5)k(k) 4yy(k)[()3k21k 9)]1k(kH(z)Yzs(z)z23zz21z z z h(k)=[3(1/2)k–2(–y(k) y(k1)y(k2)f(k)2f(k11z- z-2X(z)解:(1)畫z域框圖設(shè)中間變量X(z)=3z-1X(z)–2z-2X(z)X(z)13z1Yzs(z)=X(z)–3z-1X(z)