Chapter 6 Part A Contents ●●● ●●● ●●● ●●● Part A:z-Transform z-Transform z-Transform Part B:Inverse z-Transform Part C:Transfer Function Part A:z-Transform 1.z-Transform 1.z-Transform The DTFT provides a frequency-domain In general,ZT can be thought of as a ◆z-Transform representation of discrete-time signals and LTI generalization of the DTFT.ZT is more Region of Convergence(ROC)of a discrete-time systems. complex than DTFT (both literally and Rational z-Transform Because of the convergence condition,in figuratively),but provides a great deal of many cases,the DTFT of a sequence may not insight into system design and behavior.For exist. discrete-time systems,ZT plays the similar role of Laplace-transform does in As a result,it is not possible to make use of such frequency-domain characterization in continuous-time systems.ZT characterizes these cases. signals or LTI systems in complex frequency domain
Chapter 6 z-Transform Part A z-Transform Contents Part A: z-Transform Part B: Inverse z-Transform Part C: Transfer Function 4 Part A: z-Transform z-Transform Region of Convergence (ROC) of a Region of Convergence (ROC) of a Rational z Rational z-Transform 5 1. z-Transform The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems. Because of the convergence condition, in Because of the convergence condition, in many cases, the DTFT of a sequence may not many cases, the DTFT of a sequence may not exist. As a result, it is not possible to make use of such frequency-domain characterization in these cases. 6 1. z-Transform In general, ZT can be thought of as a generalization of the DTFT. ZT is more complex than DTFT (both literally and figuratively), but provides a great deal of insight into system design and behavior. For discrete-time systems, ZT plays the similar role of Laplace Laplace-transform does in continuous-time systems. ZT characterizes signals or LTI systems in complex frequency complex frequency domain.
1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform ·d“is a one dimensional(single-variable) Recall that the definition of DTFT of a .A generalization of the DTFT defined by leads function which can be expressed in one- sequence g(n)can be expressed by dimensional plane.In order to extend the G(e)= g(n)e-jam G(e)=g(n)e DTFT to ZT.it is possible to replace the basic building block e"by a two dimensional(two- to the z-transform where G(")can be viewed as a Fourier variable)function.Hence,the new basic building block can described in a two- .z-transform may exist for many sequences for series and g(n)is the coefficients of this series. dimensional plane which the DTFT does not exist The basic building block in DTFT is Define a new two dimensional variable z= Moreover.use of z-transform techniques re",we obtain the expression of z-transform permits simple algebraic manipulations 1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform Consequently,z-transform has become an If we letz=rei then the z-transform reduces The contour=I is a circle in the z-plane of important tool in the analysis and design of to unity radius and is called the unit circle. digital filters G(re)=g(mr"em Like the DTFT.there are conditions on the For a given sequence g(n),its z-transform G(z) The above can be interpreted as the DTFT of convergence of the infinite series is defined as G)=g) the modified sequence (g(n)" .Forr=1 (i.e.,=1),z-transform reduces to For a given sequence,the set R of values ofz where z=Re(z)+jlm(z)is a complex variable its DTFT,provided the latter exists. for which its z-transform converges is called the region of convergence (ROC). 12
7 1. 1 Definition of z-Transform Recall that the definition of DTFT of a sequence g(n) can be expressed by where G(ej¹) can be viewed as a Fourier series and g(n) is the coefficients of this series. The basic building block in DTFT is ej¹ . - ( ) () j j n n Ge gne 8 1. 1 Definition of z-Transform ej¹ is a one dimensional (single-variable) function which can be expressed in onedimensional plane. In order to extend the DTFT to ZT, it is possible to replace the basic building block ej¹ by a two dimensional (twovariable) function. Hence, the new basic building block can described in a twodimensional plane Define a new two dimensional variable z= rej¹ , we obtain the expression of z-transform 9 1. 1 Definition of z-Transform A generalization of the DTFT defined by leads to the z-transform z-transform may exist for many sequences for which the DTFT does not exist Moreover, use of z-transform techniques permits simple algebraic manipulations - ( ) () j j n n Ge gne 10 1. 1 Definition of z-Transform Consequently, z-transform has become an important tool in the analysis and design of digital filters For a given sequence g(n), its z-transform G(z) is defined as where z = Re(z) + jIm(z) is a complex variable. - () () n n Gz gnz 11 1. 1 Definition of z-Transform If we let , then the z-transform reduces to The above can be interpreted as the DTFT of the modified sequence modified sequence {g(n)r-n} For r = 1 (i.e., |z| = 1), z-transform reduces to its DTFT, provided the latter exists. - ( ) () j n jn n G re g n r e j z re 12 1. 1 Definition of z-Transform The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle unit circle. Like the DTFT, there are conditions on the convergence of the infinite series For a given sequence, the set of values of z for which its z-transform converges is called the region of convergence region of convergence (ROC). ( ) n n gnz
1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform From our earlier discussion on the uniform In general,the ROC R of a z-transform of a An example of ROC ,m母 convergence of the DTFT,it follows that the sequence g(n)is an annular region of the z- Z-plane series Gre")=g(nr"e plane: R.<R converges if g(n)"is absolutely summable, where0≤R-<R,≤o ·ReH i.e.,if Elscrk Note:The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC. The gray region unit cirele represents the ROC 1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform Comments 3 If R.<R-,then the ROC is a null space 1 The complex variable z is called the complex and the ZT does not exist. Therefore the discrete-time Fourier transform frequency given by z=re",where r is the G(")may be viewed as a special case of the ④The function=l(or=ed“)is a circle of unit attenuation and is the real frequency. z-transform G(z). radius in the z-plane and is called the unit 2 Since the ROC is defined in terms of the circle.If the ROC contains the unit circle. 5 If g(n)=h(n)is the impulse response of some magnitude r.the shape of the ROC is an then we can evaluate G(z)on the unit circle. system,its z-transform G()=H(z)is called as annulus.Note that R.may be equal to 0 System Function or Transfer Function of this and/or Rcould possibly be infinity. G()l=G(e)=g(me system. 1
13 1. 1 Definition of z-Transform From our earlier discussion on the uniform convergence of the DTFT, it follows that the series converges if is absolutely summable, i.e., if ( ) n g n z ( ) () j n jn n G re g n r e - ( ) n n gnr 14 1. 1 Definition of z-Transform In general, the ROC of a z-transform of a sequence g(n) is an annular region of the zplane: where Note: The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC. g g R zR 0 g g R R 15 g R g R 1. 1 Definition of z-Transform An example of ROC unit circle 1 z-plane Re{z} jIm{z} 0 The gray region represents the ROC 16 1. 1 Definition of z-Transform Comments ķ The complex variable z is called the complex frequency given by z= rej¹, where r is the attenuation and ¹ is the real frequency. real frequency. ĸ Since the ROC is defined in terms of the magnitude r, the shape of the ROC is an annulus. Note that may be equal to 0 and/or could possibly be infinity. g R g R 17 1. 1 Definition of z-Transform Ĺ If , then the ROC is a null space and the ZT does not exist. ĺ The function r=1 (or z= ej¹) is a circle of unit radius in the z-plane and is called the unit circle. If the ROC contains the unit circle contains the unit circle, then we can evaluate G(z) on the unit circle. g g R R - - ( )| ( ) ( ) j j jn z e n Gz Ge g n e 18 1. 1 Definition of z-Transform Therefore the discrete-time Fourier transform G(ej¹) may be viewed as a special case of the z-transform G(z). Ļ If g(n)=h(n) is the impulse response of some system, its z-transform G(z)=H(z) is called as System Function or Transfer Function of this system.
1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform Example上 Example2生 From the above two examples,we find that Calculate the ZT of x(n)=a"u(n) Calculate the ZT of x(n)=-a"u(-n-1) Very different time functions can have the Xe)=2x0=2a2=2(e xe=-42”-2( same z-transform.Because ROC plays an 0 important role in computing the z-transform 1-, =2(m=(a2= or inverse z-transform. 2-a Note that the above equation holds only for .So we must specify not only the z-transform Note that the above equation holds only for az i<a corresponding to a time function,but its ROC as well. Region of convergence Region of convergence 20 1.2 Rational z-Transform 1.2 Rational z-Transform 1.2 Rational z-Transform In the case of LTI discrete-time systems we .The degree of the numerator polynomial P(z) A rational z-transform can be alternatively are concerned with in this course,all involved is M and the degree of the denominator written in factored form as z-transforms are rational functions of polynomial D(z)is N That is,they are ratios of two polynomials in An alternate representation of a rational z- Ge=凸Π1-5) transform is as a ratio of two polynomials in z: dΠ-4z G()=P()=+p+py Ge=nP“+n++p2tu do2+dza+…dw-2+dw- =zw凸Π(2-) D() d。+d2+-dw-2a0+dw-2 d。Πa(2-】
19 1. 1 Definition of z-Transform Calculate the ZT of Note that the above equation holds only for , i.e. 1 az 1 z a Region of convergence 1 0 0 1 () () 1 1 n n nn n nn X z x n z a z az z az z a Example 1: Example 1: () () n x n aun 20 1. 1 Definition of z-Transform Calculate the ZT of Note that the above equation holds only for , i.e. -1 a z 1 z a Region of convergence 1 1 1 1 1 1 1 ( ) n n n n n n n n n X z a z az z az a z z a Example 2: Example 2: ( ) ( 1) n xn au n 21 1. 1 Definition of z-Transform From the above two examples, we find that Very different time functions can have the same z-transform. Because ROC plays an important role in computing the z-transform or inverse z-transform. So we must specify not only the z-transform corresponding to a time function, but its ROC as well. 22 1. 2 Rational z-Transform In the case of LTI discrete-time systems we are concerned with in this course, all involved z-transforms are rational functions of zˉ1 That is, they are ratios of two polynomials in zˉ1: 1 ( 1) 01 1 1 ( 1) 01 1 1 ( ) ( ) ( ) M M M M N N N N P z p pz p z p z G z Dz d dz d z d z 23 1. 2 Rational z-Transform The degree of the numerator polynomial P(z) is M and the degree of the denominator polynomial D(z) is N An alternate representation of a rational ztransform is as a ratio of two polynomials in z: 1 ( ) 01 1 1 01 1 1 ( ) M M N M M M N N N N p z pz p z p Gz z dz dz d z d 24 1. 2 Rational z-Transform A rational z-transform can be alternatively written in factored form as 1 1 0 1 0 1 ( ) 0 1 0 1 1 ( ) 1 M l l N l l M N M l l N l l p z G z d z p z z d z
1.2 Rational z-Transform 1.2 Rational z-Transform 1.2 Rational z-Transform At a root z=5 of the numerator polynomial, Consider:G()=zwn凸ΠH(e-s) .A physical interpretation of the concepts of G()=0,and as a result,these values of z poles and zeros can be given by plotting the are known as the zeros of G(z) .Note G(z)has Mfinite zeros and N finite log-magnitude 20logG()as shown on next slide for .At a root 2=乙，of the denominator poles polynomial,G(z)=0,and as a result,these .If N>Mthere are additional N-Mzeros at z G()= 1-2.4z+2.88z-2 values of z are known as the poles of G(z) =0 (the origin in the z-plane) 1-0.8z+0.64z2 If N<Mthere are additional M-N poles atz =0 2.Region of Convergence of a 2.Region of Convergence of a 1.2 Rational z-Transform Rational z-Transform Rational z-Transform Two poles at 30.4±j0.6928 ROC of a z-transform is an important concept. Moreover.if the ROC of a z-transform includes the unit circle,the DTFT of the Without the knowledge of the ROC.there is no sequence is obtained by simply evaluating the unique relationship between a sequence and its z-transform on the unit circle. z-transform. .There is a relationship between the ROC of the Hence.the z-transform must always be z-transform of the impulse response of a causal specified with its ROC. LTI discrete-time system and its BIBO Two zeros at stability. 2-1.2±12 2 10
25 1. 2 Rational z-Transform At a root of the numerator polynomial, , and as a result, these values of z are known as the zeros of G(z) At a root of the denominator polynomial, , and as a result, these values of z are known as the poles of G(z) l z () 0 G l () 0 G zl l z 26 1. 2 Rational z-Transform Consider: Note G(z) has M finite zeros and N finite poles If N > M there are additional NˉM zeros at z = 0 (the origin in the z-plane) If N < M there are additional MˉN poles at z = 0 ( ) 0 1 0 1 ( ) M N M l l N l l p z Gz z d z 27 1. 2 Rational z-Transform A physical interpretation of the concepts of poles and zeros can be given by plotting the log-magnitude 20log10|G(z)| as shown on next slide for 1 2 1 2 1 2.4 2.88 ( ) 1 0.8 0.64 z z G z z z 28 1. 2 Rational z-Transform Two poles at z=0.4fj0.6928 Two zeros at z=1.2fj1.2 29 2. Region of Convergence of a Rational z-Transform ROC of a z-transform is an important concept. Without the knowledge of the ROC, there is no unique relationship between a sequence and its z-transform. Hence, the z-transform must always be specified with its ROC. 30 2. Region of Convergence of a Rational z-Transform Moreover, if the ROC of a z-transform includes the unit circle, the DTFT of the sequence is obtained by simply evaluating the z-transform on the unit circle. There is a relationship between the ROC of the z-transform of the impulse response of a causal LTI discrete-time system and its BIBO stability.
2.Region of Convergence of a Rational z-Transform 2.1 General Form of ROC 2.1 General Form of ROC -Finite-length Sequence The ROC of a rational z-transform is bounded .In general,there are four types of ROCs forz- A finite-length sequence g(n)is defined for-MnN by the locations of its poles transforms,and they depend on the type of the corresponding time functions. with M and N positive,and To understand the relationship between the In general,its ROC includes the entire z-plane except possible poles and the ROC,it is instructive to examine -Finite-length sequence 0oand ee the pole-zero plot of a z-transform -Right-sided sequence -Left-sided sequence For finite duration sequences,the condition of convergence is that every term in the ZT is convergent.Except the ==0 and -Two-sided (infinite duration)sequence z=,the ZT of a finite sequence is convergent in the entire -plane. 2.1 General Form of ROC 2.1 General Form of ROC 2.1 General Form of ROC -Right-sided Sequence -Left-sided Sequence A right-sided sequenee u(n)with nonzero sample values only IfM20.RR= A left-sided sequence vn)with nonzero sample values only for oH≥M fM0,R<水oR<因 nN tim树 If M-0.m(n)is called a causal sequence Comment All causal sequenees (or the impulse responses of LTI Re systems)are right-sided,while not all right-sided sequences correspond to causal systems. ROC unit cirele 35 unit circle
31 2. Region of Convergence of a Rational z-Transform The ROC of a rational z-transform is bounded by the locations of its poles by the locations of its poles To understand the relationship between the poles and the ROC, it is instructive to examine the pole-zero plot of a z-transform 32 2.1 General Form of ROC In general, there are four types of ROCs for ztransforms, and they depend on the type of the corresponding time functions. – – Finite-length sequence length sequence – – Right-sided sequence sided sequence – – Left-sided sequence sided sequence – – Two-sided (infinite duration) sequence sided (infinite duration) sequence 33 2.1 General Form of ROC – Finite-length Sequence A finite-length sequence g(n) is defined for ˉMİnİN with M and N positive, and |g(n)|<Ğ. In general, its ROC includes the entire z-plane except possible z=0 or/and z=Ğ For finite duration sequences, the condition of convergence is that every term in the ZT is convergent. Except the z=0 and z=Ğ, the ZT of a finite sequence is convergent in the entire z-plane. 34 2.1 General Form of ROC – Right-sided Sequence A right-sided sequence u(n) with nonzero sample values only for nıM g R 1 Re{z} jIm{z} 0 ROC unit circle 35 2.1 General Form of ROC Comment All causal sequences (or the impulse responses of LTI systems) are right-sided, while not all right-sided sequences correspond to causal systems. g If Mı0, R z g R g If M <0, R z g R If M=0, u(n) is called a causal sequence 36 2.1 General Form of ROC – Left-sided Sequence A left-sided sequence v(n) with nonzero sample values only for nİN g R unit circle 1 Re{z} jIm{z} 0 ROC
2.1 General Form of ROC 2.1 General Form of ROC 2.1 General Form of ROC -Two-sided Sequence fN0,00 The z-Transform of a two-sided sequence w(n)can be Obviously,the ROC of W()is the intersection of >R and R.If R>R,its ROC has the following form IfN0,0≤HR. 国4<回 numerator coefficients specified by the Hence,a rational z-transform with a vector num and the denominator There is no overlap between these two regions.Hence,its a- specified ROC has a unique sequence as its coefficients specified by the vector den. transform does not exist inverse z-transform. 42
37 2.1 General Form of ROC 0 If N >0, 0 z Rg Rg 0 If N İ 0, Rg If N=0, v(n) is called a anticausal anticausal sequence 0 g z R 38 2.1 General Form of ROC – Two-sided Sequence The z-Transform of a two-sided sequence w(n) can be expressed as 1 0 () () () () nn n n nn W z wnz wnz wnz A right-sided sequence A left-sided + sequence g- z R g z R 39 2.1 General Form of ROC Obviously, the ROC of W(z) is the intersection of and . If , its ROC has the following form g z R g z R Rg g R g R g R 1 Re{z} jIm{z} 0 But, if , its ROC is a null space, i.e., the transform does not exist R R g g 40 2.1 General Form of ROC Consider the two-sided sequence x(n)=an, where a can be either complex or real. Its z-Transform is given by An example 1 0 ( ) nn nn n n X z az az z a z a There is no overlap between these two regions. Hence, its ztransform does not exist 41 2.1 General Form of ROC Summary In general, if the rational z-transform has N poles with R distinct magnitudes, then it has R+1 ROCs Thus, there are R+1 distinct sequences with the same z-transform Hence, a rational z-transform with a specified ROC has a unique sequence as its inverse z-transform. 42 2.2 Determine the ROC by MATLAB The ROC of a rational z-transform can be easily determined using MATLAB [z,p,k] = tf2zp(num,den) determines the zeros, poles, and the gain constant of a rational z-transform with the numerator coefficients specified by the vector num and the denominator coefficients specified by the vector den.
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)implements the The pole-zero plot is determined using the reverse process function zplane a2a12 The factored form of the z-transform can be 505= The z-transform can be either described in obtained using sos=zp2sos(z,p,k)where terms of its zeros and poles: sos stands for second-order section boi. zplane(zeros.poles) The above statement computes the where or.it can be described in terms of its coefficients of each second-order factor c)= +42+b22 numerator and denominator coefficients: given as an LX6 matrix sos dos +duz+a zplane(num,den) 44
43 2.2 Determine the ROC by MATLAB [num,den] = zp2tf(z,p,k) implements the reverse process The factored form of the factored form z-transform can be obtained using sos = zp2sos(z,p,k) where sos stands for second-order section The above statement computes the coefficients of each second-order factor given as an L h6 matrix sos 44 2.2 Determine the ROC by MATLAB 01 11 21 01 11 12 02 12 22 02 12 22 012 0 1 2 LL L L L L bbba aa bbba aa bbb a aa sos 1 2 01 2 1 2 1 01 2 ( ) L kk k k kk k b bz bz G z a az az where 45 2.2 Determine the ROC by MATLAB The pole-zero plot zero plot is determined using the function zplane The z-transform can be either described in terms of its zeros and poles: zeros and poles: zplane(zeros,poles) or, it can be described in terms of its numerator and denominator coefficients: numerator and denominator coefficients: zplane(num,den)
