離散2015數(shù)字信號(hào)處理第3章

1、13.2 Properties of Region of ConvergenceFor any given sequence, the set of values of z, for which the z transform converges, is call the ROCz transform convergence condition: nX ( z) xn z (3.8)n In general, the ROC of a ZT is a ringhe z-plane, centeredat the origin: R z RggThe ROC cannot contain any

2、 poles of X(z)R gR g冪級(jí)數(shù)是否收斂The region of convergence唯一取決于|z| nX ( z) xn z (3.8)n In general, the ROC of a ZT is a ringhe z-plane, centered at the origin:DTFT convergence condition:R ROC of X(z) includes the unitgRg circle (|z|= 1)The ROC cannot contain any poles of X(z)R gz RgRational z-transformX(z

3、) is a rational functionX (z) P(z) Q(z) The values of z for X(z)=0 are called the zeros of X(z).The values of z for X(z)= are called the poles of X(z)Poles are the root of Q(z)Poles may occur at z=0 or z=relationship betn ZT and DTFT:z-plane and unit circleX ( z ) xnz nz ejn z e j X (z) |ze j xnejn

4、n X (e j )|z|= 1, ZT is just DTFTunit circle: |z|= 1ZT evaluated on the unit circle corresponds to DTFT3.1 z-transformdefinition of z-transform:X ( z ) Zxn xnz n(3.2)n where z Re(z) j Im(z) is a complex variableone-side z-transform & two-side z-transformX (z) xnznX ( z ) xnz nn0n E.g.xn nX (z) nzn 1

5、nCh3. The z-transform2E.g. , determine the z-transform, including the ROC, for each of the following sequen. xn RN n xn anun xn bnunN 11 z NX (z) xnzn zn 0 z nn01 z1從上式的分母可知在z=1處有一個(gè)極點(diǎn)，但是從分子處看出z=1處有一個(gè)零點(diǎn)，零極點(diǎn)剛好對(duì)消。X (z) anzn (az1)n 1 z | z | a |n0n01 az1z a01bX (z) bn zn bn zn | z | b |nn01 b1z b zE.g.

6、Determine the z-transform, including the ROC, for the following sequen.x1n unx2n un 11不同收斂域的組合X (z) unzn zn同一封閉表達(dá)式與nn0表示完全不同的序X (z) 1z 1列11 z 11X 2 (z) un 1z (z ) n nnnz1 (zm ) z 1m11 z1 z1ROC of Two-sided sequenTwo-sided sequence: an infinite-duration sequence extending from n = to n = + Left-sided

7、 sequence + Right-sided sequence = Two-sided sequence0 z min pLk pjpi max pRk z ROC :pizp jROC of Left-sided and Right-sided sequenLeft-sided: Right-sided:xn 0, n N 2 0, n N1 0, n Nxn 0, n N210 z min pk max pk z ROCROCN2 0, 0 z min pk N1 0, max pk z ROC of Finite-duration sequenxn 0, N1 n N2 0, othe

8、rwiseNNn12N 2X ( z ) xnz n xnz n n n N1east 0 z ROC N1 0, 0 z N2 0, 0 z ROC of different sequenFinite-duration sequenceLeft-sided sequenceRight-sided sequenceTwo-sided sequenceX (z) P(z) N(z)MQ(z) (z p )kk 1pk poles of X(z)3Inspection methodanun z 1,z a1 az 1 1 nz xn un 2 The z-transform pairsable 3

9、.1 (p104) are invaluable in applying the inspection methodX (z) 1,z 1 1 1 z1223.3 The inverse z-transformDefine: Given X(z) and its ROC, compute xnxn Z-1X (z)Methods of determining the inverse z-transformInspection MethodPartial fraction expaner series expanE.g. Determine the region of c and the z-t

10、ransform, including the ROC, for the following sequence.xn c n , c is real1X (z) c|n|zn cn zn cn|zn cn zn cn znnnn0n1n0 cz 1c 1c 2 1cz11 c21X (z) c z 1 cz 1 cz1(1 cz)(1 cz1)ccc 1czcz1 1z 1cTable 3.1 some common z-transform pairsp104, table 3.1Stability, Causality and the ROC Let H ( z) Zhn Stability

11、:ROC of H(z) includes the unit circle (|z|= 1)Causality:ROC of H(z) : max pk z Stable and Causal:ROC of H(z) : max pk z , max pk 1Importance of ROC序列z變換的封閉表達(dá)式與其收斂域密不可分。z變換的收斂域一定是為收斂半徑 Rg 和Rg 所包含的環(huán)狀區(qū)域。右邊序列的z變換收斂域?yàn)?z Rg ，左邊序列的z變換的收斂域?yàn)?z Rg，雙邊序列z變換的收斂域?yàn)镽g z Rg 。4Exle 3.10X (z) z2 0.5z 1 0.5z1 1, n 20.5

12、, n 1xn 1, n 0 0.5, n 10, otherwiseor, xn n 2 0.5 n 1 n 0.5 n 13.3.3er series expanX (z) xn z n n x2z2 x1z x0 x1z1 x2z 2 If X(z) is given as aer series of z, the coefficient of the term involving zn simply corresponds the sequence xn.For X(z) = P(z) / Q(z), we can perform dividing P(z) by Q(z) and t

13、he quotient is aer series of z. Long diviExercise. Determine the inverse z-transform ofX (z) 2z 2z A1 A22 2 2 j 2 2 j 2 2 j 1 2 2 j 1 z z 12z12z22A 1 2 2 j z1 X (z) 1X (z) 111 21 e j 4 z1 1 e j 4 z1z 2 2 j2 2 2 j 1 xn (1)n e jn 4 e jn 4 unA2 1z X (z) 12z 2 2 j3n2 2cosun 42z2 2zX (z) z 1z2 2z 1Exle 3

14、.8X(z) 1, z 1(1 1 z1)(1 1 z1)2 M=0, N=2 42X(z) A1 A2 1 1 z1 1 1 z142X (z) 21 ,z 1 1 1 z1 1 1 z1224xn 2( 1 ) n un ( 1 )n un24(1)X(z)1,(11)X(z)23.3.2 Partial fraction expanM (1 c z 1 )bkX ( z) 0 k 1(3 39)a N10(1 dk z )k 1If MN, and the poles are allorder,X ( z) N Ak (3 40)1A (1 d z 1 ) X ( z)(3 41)k 1

15、 1 dk zkkz dkIf MN, and the poles are allorder,M NN AX (z) B zr k (3 43)r1 d z1r 0k 1kIf MN and if X(z) has a pole of order s at z di, and all otholes are-order,M NN As CX (z) B zr k m (3 44)r1 d z1(1 d z1)m r 0k 1,k ikm1i3.3.2 Partial fraction expanM b z kP ( z )kX ( z ) k 0(3.37)NQ ( z ) a z kkk 0

16、For rational X(z), a partial-fraction expanis carried ou, and then the inverse z- transforms of the simple terms are identified, and summarized to get the desirable result xn.5SummaryDefinition of Z-transfromROCInverse z-transfrom3.4 z-transform properties注意收斂域的變化Conjugation of a complex sequencex*n

17、z X (z*), ROC RxTime reversalx*nz X *(1/ z*), ROC 1/ RxConvolution of sequenx2Initial-value theoremIf xn is causal, then x0 lim X (z)zE.g. 雙邊序列，移位后收斂域不會(huì)發(fā)生變化；單邊序列在z=0或z=處收斂域可能有變化。Let xn n, determine the z-transform, including the ROC, for xn 1 and xn 1 n 1在z平面處處收斂 n 1 1在z=0處不收斂 n 1 1在z=處不收斂3.4 z-transform properties (p126, table3.2)Linearityax n bx n z aX ( z) bX (z), ROC R R1212x1x 2Time shiftingxn n z zn0 X (z), ROC R0 x(may include or exclude z = 0 or |z| = )Multiplication by an exponential sequencez n xnz X