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e st x(t) =12j C

X(s)estds;

X(s) =1

1 x(t)estdt:

X(s) =1

0 x(t)estdt:

X(z) =1X

n=0x[n]zn?????

X(z) =1X

n=02 nzn;

X(z) =1X

n=0 2z n S=1X n=0a n R;?? ????? ??? ??? ??? ?? ??? ?????? ??z???? ????jzj> R:

S= limN!1SN= limN!1N

X n=0a n: S

N= (1 +a+a2+:::+aN):

aS

N= (a+a2+:::+aN+aN+1):

S

NaSN= (1aN+1)

S

N(1a) = (1aN1):

???? ??a= 1;?? ???? ????SN=N+ 1:????a6= 1;??? ?????? ???? ????? ??(1a)?? ?????? S

N=1aN+11a

S= limN!1SN= limN!11aN+11a;

????? ???? ???? ?? ????? ????jaj<1;??? ????? ?? ????

S=11a:

S=N 2X n=N1a n= (aN1+aN1+1+:::+aN2) aS= (aN1+1+aN1+2+:::+aN2+aN2+1);

S(1a) = (aN1aN2+1)

S=aN1aN2+11a

N lim N

2!1S= limN

2!1a

N1aN2+11a=aN11a;

2X n=N1a n=aN1aN2+11a;???a6= 1????? ???1 X n=N1a n=aN11a;???jaj<1:?????

X(1) =1X

n=02 n1n 1X n=02 n;

Z(3) =1X

n=02 n3n 1X n=0 23
n 1123
= 3: n=0 2z n

X(z) =1X

n=0 2z n 112z
;2z <1 zz2;jzj>2;

X(z) =1X

n=0a nzn 1X n=0 az n;

X(z) =zza:

az <1,jajjzj<1, jajjaj:

X(z) =zza;jzj>jaj:

X aX X

3(z) =1X

n=0(ax1[n] +bx2[n])zn 1X n=0ax

1[n]zn+1X

n=0bx

2[n]zn

=a1X n=0x

1[n]zn+b1X

n=0x

2[n]zn

=aX1(z) +bX2(z):

X(z) =1X

n=0a nzn=zza;jzj>jaj ddz

X(z) =ddz

1X n=0a nzn! ddz zza ;jzj>jaj =1X n=0na nzn1=a(za)2;jzj>jaj: zddz

X(z) =1X

n=0na nzn=az(za)2;jzj>jaj: 1X n=0na nzn=az(za)2;jzj>jaj: nx[n], zddz X(z) 12 n(n1)x[n],a2z(za)3;jzj>jaj;

1m!n(n1)(nm+ 1)an,amz(za)m+1;jzj>jaj:

X(z) =1X

n=0cos(!n)zn 1X n=012 (ej!n+ej!n)zn 12 1 X n=0(ej!z1)n+12 1 X n=0(ej!z1)n 12

11ej!z1+12

11ej!z1;jzj>jej!j= 1

12 zzej!+12 zzej!;jzj>1 12 z(zej!)z

2z(ej!+ej!) + 1+z(zej!)z

2z(ej!+ej!) + 1

;jzj>1 z2zcos(!)z

22zcos(!) + 1;jzj>1:

x[n] =12jej!nej!n

12jej!n12jej!n

X(z) =12jzzej!12jzzej!;jzj>1

12jz(zej!)z(zej!)z

22zcos(!) + 1;jzj>1

zsin(!)z

22zcos(!) + 1;jzj>1:

[n],1: [nk],zk; u[n],zz1;jzj>1:

Z(ay1[n] +by2[n]) =aY1(z) +bY2(z):

1 X n=0(ay1[n] +by2[n])zn=a1X n=0y

1[n] +b1X

n=0y 2[n] =aY1(z) +bY2(z):

Z(y[nk]u[nk]) =zkY(z):

Z(y[nk]u[nk]) =1X

n=0y[nk]u[nk]zn 1X n=ky[nk]zn 1X m=0y[m]z(m+k) =zkY(z); ????? ??? ?????? ???? ??????? ????u[nk]????? ???? ???n < k??? ??? ????? ???? ??????? ???? ?????? ???

Z(y[nk]) =zk"

Y(z) +kX

m=1y[m]zm# y[nk] =y[nk]u[nk] +kX m=1y[m][nk+m]

Z(y[nk]) =zkY(z) +kX

m=1y[m]z(km) =zk"

Y(z) +kX

m=1y[m]zm#

Z(y[n+k]u[n]) =zk"

Y(z)k1X

m=0y[m]zm#

Z(y[n+k]u[n]) =1X

n=0y[n+k]zn =zk1X n=0y[n+k]z(n+k) =zk1X m=ky[m]zm =zk 1X m=0y[m]zmk1X m=0y[m]zm! =zk

Y(z)k1X

m=0y[m]zm! y[n] =1X m=1h[m]x[nm]:

Y(z) =1X

n=0y[n]zn 1X n=01 X m=1h[m]x[nm]zn 1X m=1h[m]1X n=0x[nm]zn 1X m=1h[m]X(z)zm =X(z)1X m=1h[m]zm =X(z)H(z); 1 X k=1x[nk]y[nk]h[k]u[k] =nX k=0x[nk]h[k] !X(z)H(z); y[n] =12j

Y(z)zn1dz

Y(z) =b0+b1z+:::+bMzMa

0+a1z+:::+aNzN:

Y(z) =zza

zza=1 + az +a2z 2 za)z za0 +a+ 0

0 +aa2z

0 + 0 +

a2z

0 + 0 +a2z

a3z 2

Y(z) = 1 +az1+a2z2+a3z3+

=y[0] +y[1]z1+y[2]z2+y[3]z3+; ???? ????? ?? ??? ????? ????y[n] =an;n0: ???????x[n+ 1]u[n],z(X(z)x[0]) nX k=0x[k]y[nk],X(z)Y(z) k=1x[k]y[nk],X(z)Y(z) ??????? ??k y[n+k]u[n],zk"

Y(z)k1X

m=0y[m]zm#

Y(z) +kX

x[n],X(z) =1X n=0x[n]znROCX [n] =(

1; n= 0

0; n6= 0,1???z

[nk],( zk; k0

0; k <0z6= 1

a n,zzajzj>jaj na n,az(za)2jzj>jaj a nsin(!n),azsin(!)z

22azcos(!) +a2jzj>jaj

a ncos(!n),1azcos(!)z

22azcos(!) +a2jzj>jaj

u[n] =(

1; n= 0

0; n6= 0,zz1jzj>1

1,zz1jzj>1

X(z) =b0+b1z+bMzMa

0+a1z++aNzN=b0+b1z+bMzM(zr1)(zr2)(zrN);

X(z) =NX

k=1A kzzk; A k= (zrk)X(z)jz=rk;

X(z) =NX

k=1A kzzk=NX k=1z

1Akzzrk,x[n] =NX

k=1A krn1 ku[n1];

Y(z) =z1(z2)(z3):

Y(z) =z1(z2)(z3)=A1z2+A2z3;

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