Part A. z-transform. 1. z-transform

Transcription

1 Chapter 6 ztransform Part A ztransform Contents Part A: ztransform Part B: Inverse ztransform Part C: Transfer Function Part A: ztransform 1. ztransform 1. ztransform ztransform Reion of Converence (ROC) of a Rationa ztransformz 4 The DTFT provides a frequencydomain representation of discretetime sinas and LTI discretetime systems. Because of the converence condition, in many cases, the DTFT of a sequence may not exist. As a resut, it is not possibe to make use of such frequencydomain characterization in these cases. 5 In enera, ZT can be thouht of as a eneraization of the DTFT. ZT is more compex than DTFT (both iteray and fiurativey), but provides a reat dea of insiht into system desin and behavior. For discretetime systems, ZT pays the simiar roe of Lapacetransform does in continuoustime systems. ZT characterizes sinas or LTI systems in compex frequency domain. 6

2 Reca that the definition of DTFT of a sequence (n) can be expressed by j jn Ge ( ) ne ( ) n where G(e j ) can be viewed as a Fourier series and (n) is the coefficients of this series. The basic buidin bock in DTFT is e j. 7 e j is a one dimensiona (sinevariabe) function which can be expressed in onedimensiona pane. In order to extend the DTFT to ZT, it is possibe to repace the basic buidin bock e j by a two dimensiona (twovariabe) function. Hence, the new basic buidin bock can described in a twodimensiona pane Define a new two dimensiona variabe z= re j, we obtain the expression of ztransform 8 A eneraization of the DTFT defined by eads j jn Ge ( ) ne ( ) to the ztransform n ztransform may exist for many sequences for which the DTFT does not exist Moreover, use of ztransform techniques permits simpe aebraic manipuations 9 Consequenty, ztransform has become an important too in the anaysis and desin of diita fiters For a iven sequence (n), its ztransform G(z) is defined as n Gz ( ) nz ( ) n where z = Re(z) + jim(z) is a compex variabe. 10 j If we et z re, then the ztransform reduces to j n jn Gre ( ) nr ( ) e n The above can be interpreted as the DTFT of the modified sequence {(n)r n } For r = 1 (i.e., z = 1), ztransform reduces to its DTFT, provided the atter exists. 11 The contour z = 1 is a circe in the zpane of unity radius and is caed the unit circe. Like the DTFT, there are conditions on the converence of the infinite series n nz ( ) n For a iven sequence, the set of vaues of z for which its ztransform converes is caed the reion of converence (ROC). 12

3 From our earier discussion on the uniform converence of the DTFT, it foows that the series j n jn Gre ( ) nr ( ) e n converes if ( nz ) n is absoutey summabe, i.e., if n nr ( ) n 13 In enera, the ROC of a ztransform of a sequence (n) is an annuar reion of the z pane: R z R where 0 R R Note: The ztransform is a form of a Laurent series and is an anaytic function at every point in the ROC. 14 An exampe of ROC The ray reion represents the ROC 0 jim{z} R 1 Re{z} zpane unit circe 15 Comments The compex variabe z is caed the compex frequency iven by z= re j, where r is the attenuation and is the rea frequency. Since the ROC is defined in terms of the manitude r, the shape of the ROC is an annuus. Note that R may be equa to 0 and/or coud possiby be infinity. If R R, then the ROC is a nu space and the ZT does not exist. The function r=1 (or z= e j ) is a circe of unit radius in the zpane and is caed the unit circe. If the ROC contains the unit circe, then we can evauate G(z) on the unit circe. j Gz ( ) Ge ( ) ( ne ) j ze n jn Therefore the discretetime Fourier transform G(e j ) may be viewed as a specia case of the ztransform G(z). If (n)=h(n) is the impuse response of some system, its ztransform G(z)=H(z) is caed as System Function or Transfer Function of this system

4 Exampe 1: Cacuate the ZT of x( n) a n u( n) n n n 1 n X( z) x( n) z a z az n n0 n0 1 z 1 1az z a 1 Note that the above equation hods ony for az 1, i.e. z a Reion of converence 19 Exampe 2: n Cacuate the ZT of xn ( ) au( n1) 1 1 n n 1 n X( z) a z az n n 1 n 1 n z az a z n1 n1 z a 1 Note that the above equation hods ony for a z 1, i.e. z a Reion of converence 20 From the above two exampes, we find that Very different time functions can have the same ztransform. Because ROC pays an important roe in computin the ztransform or inverse ztransform. So we must specify not ony the ztransform correspondin to a time function, but its ROC as we. 21 In the case of LTI discretetime systems we are concerned with in this course, a invoved ztransforms are rationa functions of z 1 That is, they are ratios of two poynomias in z 1 : 1 ( M 1) Pz ( ) p0 pz 1 pm 1z pmz Gz ( ) 1 ( N1) Dz ( ) d dz d z d z M 0 1 N1 N1 N The deree of the numerator poynomia P(z) is M and the deree of the denominator poynomia D(z) is N An aternate representation of a rationa z transform is as a ratio of two poynomias in z: M M1 ( NM) p0z p1z pm 1z pm Gz ( ) z N N1 dz dz d zd 0 1 N1 N1 A rationa ztransform can be aternativey written in factored form as M 1 p 11 0 z Gz ( ) N 1 d 1 z z 0 1 M ( NM) p0 1 N d0 1 z z

5 At a root z of the numerator poynomia, G(, and as a resut, these vaues of z ) 0 are known as the zeros of G(z) At a root z of the denominator poynomia, Gz ( ) 0, and as a resut, these vaues of z are known as the poes of G(z) 25 Consider: Gz ( ) z M ( NM) p0 1 N d0 1 Note G(z) has M finite zeros and N finite poes If N > M there are additiona NM zeros at z = 0 (the oriin in the zpane) If N < M there are additiona MN poes at z = 0 z z 26 A physica interpretation of the concepts of poes and zeros can be iven by pottin the omanitude 20o 10 G(z) as shown on next side for z 2.88z Gz ( ) z 0.64z Reion of Converence of a Rationa ztransform 2. Reion of Converence of a Rationa ztransform Two zeros at z=1.2j1.2 Two poes at z=0.4j ROC of a ztransform is an important concept. Without the knowede of the ROC, there is no unique reationship between a sequence and its ztransform. Hence, the ztransform must aways be specified with its ROC. 29 Moreover, if the ROC of a ztransform incudes the unit circe, the DTFT of the sequence is obtained by simpy evauatin the ztransform on the unit circe. There is a reationship between the ROC of the ztransform of the impuse response of a causa LTI discretetime system and its BIBO stabiity. 30

6 2. Reion of Converence of a Rationa ztransform The ROC of a rationa ztransform is bounded by the ocations of its poes To understand the reationship between the poes and the ROC, it is instructive to examine the poezero pot of a ztransform In enera, there are four types of ROCs for z transforms, and they depend on the type of the correspondin time functions. Finiteenth sequence Rihtsided sequence Leftsided sequence Twosided (infinite duration) sequence Finiteenth Sequence A finiteenth sequence (n) is defined for MnN with M and N positive, and (n) <. In enera, its ROC incudes the entire zpane except possibe z=0 or/and z= For finite duration sequences, the condition of converence is that every term in the ZT is converent. Except the z=0 and z=, the ZT of a finite sequence is converent in the entire zpane Rihtsided Sequence A rihtsided sequence u(n) with nonzero sampe vaues ony for nm jim{z} If M0, If M <0, z z Leftsided Sequence A eftsided sequence v(n) with nonzero sampe vaues ony for nn jim{z} If M=0, u(n) is caed a causa sequence ROC 0 R 1 Re{z} Comment A causa sequences (or the impuse responses of LTI systems) are rihtsided, whie not a rihtsided sequences correspond to causa systems. unit circe ROC 0 1 Re{z} unit circe 36

7 If N >0, 0 z If N 0, R 0 z R 0 R 0 If N=0, v(n) is caed a anticausa sequence Twosided Sequence The ztransform of a twosided sequence w(n) can be expressed as 1 n n n n n0 n W( z) w( n) z w( n) z w( n) z A rihtsided sequence + A eftsided sequence Obviousy, the ROC of W(z) is the intersection of z and z. If, its ROC has the foowin form jim{z} 0 R 1 Re{z} But, if, its ROC is a nu space, i.e., the transform does not exist z R z Determine the ROC by MATLAB An exampe Consider the twosided sequence x(n)=a n, where a can be either compex or rea. Its ztransform is iven by 1 n n n n n0 n X( z) az az z a z a There is no overap between these two reions. Hence, its z transform does not exist 40 Summary In enera, if the rationa ztransform has N poes with R distinct manitudes, then it has R+1 ROCs Thus, there are R+1 distinct sequences with the same ztransform Hence, a rationa ztransform with a specified ROC has a unique sequence as its inverse ztransform. 41 The ROC of a rationa ztransform can be easiy determined usin MATLAB [z,p,k] = tf2zp(num,den) determines the zeros, poes, and the ain constant of a rationa ztransform with the numerator coefficients specified by the vector num and the denominator coefficients specified by the vector den. 42

8 2.2 Determine the ROC by MATLAB 2.2 Determine the ROC by MATLAB 2.2 Determine the ROC by MATLAB [num,den] = zp2tf(z,p,k) impements the reverse process The factored form of the ztransform can be obtained usin sos = zp2sos(z,p,k) where sos stands for secondorder order section The above statement computes the coefficients of each secondorder factor iven as an L6 matrix sos b01 b11 b21 a01 a11 a12 b02 b12 b22 a02 a12 a 22 sos b0l b1l b2l a0l a1l a2l where L 1 2 b0k b1 kz b2kz Gz ( ) 1 2 a a z a z k 1 0k 1k 2k The poezero pot is determined usin the function zpane The ztransform can be either described in terms of its zeros and poes: zpane(zeros,poes) or, it can be described in terms of its numerator and denominator coefficients: zpane(num,den)