Digital Signal Processing Module 3 Z-Transforms Objective: 1. To have a review of z-transforms. 2. Solving LCCDE using Z-transforms. Introduction: The z-transform is a useful tool in the analysis of discrete-time signals and systems and is the discrete-time counterpart of the Laplace transform for continuous-time signals and systems. The z-transform may be used to solve constant coefficient difference equations, evaluate the response of a linear time-invariant system to a given input, and design linear filters. Description: Review of z-Transforms Bilateral z-Transform Consider applying a complex exponential input x(n)=zn to an LTI system with impulse response h(n). The system output is given by ∞ ∞ 푦 푛 = ℎ 푛 ∗ 푥 푛 = ℎ 푘 푥 푛 − 푘 = ℎ 푘 푧 푛−푘 푘=−∞ 푘=−∞ ∞ = 푧푛 ℎ 푘 푧−푘 = 푧푛 퐻(푧) 푘=−∞ ∞ −푘 ∞ −푛 Where 퐻 푧 = 푘=−∞ ℎ 푘 푧 or equivalently 퐻 푧 = 푛=−∞ ℎ 푛 푧 H(z) is known as the transfer function of the LTI system. We know that a signal for which the system output is a constant times, the input is referred to as an eigen function of the system and the amplitude factor is referred to as the system’s eigen value. Hence, we identify zn as an eigen function of the LTI system and H(z) is referred to as the Bilateral z-transform or simply z-transform of the impulse response h(n). 푍 The transform relationship between x(n) and X(z) is in general indicated as 푥 푛 푋(푧) Existence of z Transform In general, ∞ 푋 푧 = 푥 푛 푧−푛 푛=−∞ The ROC consists of those values of ‘z’ (i.e., those points in the z-plane) for which X(z) converges i.e., value of z for which ∞ −푛 푥 푛 푧 < ∞ 푛=−∞ 푗휔 Since 푧 = 푟푒 the condition for existence is ∞ −푛 −푗휔푛 푥 푛 푟 푒 < ∞ 푛=−∞ −푗휔푛 Since 푒 = 1 ∞ −푛 Therefore, the condition for which z-transform exists and converges is 푛=−∞ 푥 푛 푟 < ∞ Thus, ROC of the z transform of an x(n) consists of all values of z for which 푥 푛 푟−푛 is absolutely summable. Relation between Z and Discrete Time Fourier transform 푗휔 푗휔 ∞ −푗휔푛 When 푧 = 푒 , 푋 푒 = 푛=−∞ 푥 푛 푒 corresponds to the Discrete Time Fourier transform (DTFT) of x(n), i.e.,푋 푧 푧 = 푒푗휔 = ℱ{푥 푛 }. Thez transform also bears a straight forward relationship to the DTFT when the complex variable 푧 = 푟푒푗휔 . To see this relationship, consider X(z) with 푧 = 푟푒푗휔 . ∞ 푋 푧 = 푟푒푗휔 = 푥 푛 푟−푛 푒−푗휔푛 푛=−∞ or ∞ 푋 푧 = 푟푒푗휔 = [푥 푛 푟−푛 ]푒−푗휔푛 = ℱ{푥 푛 푟−푛 } 푛=−∞ Unilateral z-Transform have considerable value in analyzing causal systems and particularly, systems specified by linear constant coefficient difference equations with non- zero initial conditions( i.e., systems that are not initially at rest). The Unilateral z- transform of a discrete time signal x(n) is defined as ∞ 푋 푧 = 푥 푛 푧−푛 푛=0 Properties of ROC: The Z-transform has two parts which are, the expression and Region of Convergence respectively. Whether the Z-transform X(z) of a signal x(n) exists or not depends on the complex variable ‘z’ as well as the signal itself. All complex values of ‘z=rejω’ for which the summation in the definition converges form a region of convergence (ROC) in the z-plane. A circle with r=1 is called unit circle and the complex variable in z-plane is represented as shown in the Figure 3.1. Figure 3.1 Unit circle in z-plane Property 1:The ROC of X(z) consists of a ring in the z-plane centered about the origin. Property 2:If the z-transform X(z) of x(n) is rational, then the ROC does not contain any poles but is bounded by poles or extend to infinity. Property 3: If x(n) is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and / or z=∞ Property 4: If x(nt) is a right sided sequence, and if the circle |z|=ro is in the ROC, then all finite values of z for which |z|>ro will also be in the ROC. Property 5: If x(n) is a left sided sequence, and if the circle|z|=ro is in the ROC, then all values of z for which 0<|z|<ro will also be in the ROC. Property 6: If x(n) is two sided, and if the circle |z|=ro is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle |z|=ro. Property 7: Ifthe z-transform X(z) of x(n) is rational, and if x(n) is right sided, then the ROC is the region in the z-plane outside the outermost pole i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). Property 8:If the z-transform X(z) of x(n) is rational, and if x(n) is left sided, then the ROC is the region in the z-plane inside the innermost pole i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0. Properties of Z-Transform The Properties of z-transform simplifies the work of finding the z-domain equivalent of a time domain function when different operations are performed on discrete signal like time shifting, time scaling, time reversal etc. These properties also signify the change in ROC because of these operations. These properties are also used in applying z- transform to the analysis and characterization of Discrete Time LTI systems. 1. Linearity 푍 If 푥1(푛) 푋1(푧) with ROC = R1 푍 and푥2(푛) 푋2(푧) with ROC = R2 푍 then 푎푥1 푛 + 푏푥2 푛 푎푋1 푧 + 푏푋2(푧), with ROC containing 푅1 ∩ 푅2 2. Time Shifting 푍 If 푥(푛) 푋(푧)with ROC= R ℒ then 푥 푛 − 푚 푧−푚 푋 푧 with ROC= R, except for the possible addition or deletion of the origin or infinity 3. Scaling in the z-Domain 푍 If 푥(푛) 푋(푧) with ROC= R 푍 푛 푧 then 푧표 푥 푛 푋 with ROC= |zo|R where, |zo|R is the scaled version of R. 푧표 4. Time Reversal 푍 If 푥(푛) 푋(푧) with ROC= R 푍 1 1 then 푥 −푛 푋 with ROC= 푧 푅 5. Conjugation 푍 If 푥(푛) 푋(푧) with ROC= R ℒ then 푥∗ 푛 푋∗ 푧∗ with ROC= R 6. The Convolution Property 푍 If 푥1(푛) 푋1(푧) with ROC = R1 푍 and푥2(푛) 푋2(푧) with ROC = R2 ℒ then 푥1 푛 ∗ 푥2 푛 푋1 푧 . 푋2(푧), with ROC containing 푅1 ∩ 푅2 7. Accumulation 푍 If 푥(푛) 푋(푧) with ROC = R then ℒ 1 푛 푥(푘) 푋 푧 . , with ROC containing 푅 ∩ { 푧 > 1} 푘=−∞ 1−푧 −1 8. Differentiation in the z-Domain 푍 If 푥(푛) 푋(푧) with ROC= R 푍 푑푋(푧) then 푛푥(푛) − 푧 with ROC = R 푑푧 9. The Initial Value Theorems If x(n)=0, for n < 0 then initial value of x(n) i.e., 푥 0 = lim푧→∞ 푋(푧) 10. The Final Value Theorem If x(n) is causal and X(z) is the z-transform of x(n) and if all the poles of X(z) lie strictly inside the unit circle except possibly for a first order pole at z=1 then lim 푥 푛 = lim(1 − 푧−1)푋(푧) 푁→∞ 푧→1 Inverse z-transform Inverse z-transform maps a function in z-domain back to the time domain. Since discrete system analysis is usually easier in z-domain, the process is to convert the discrete system time domain representation to z-domain (both system and inputs),perform system analysis in z-domain and then convert back to the time domain representation for the response. The reason to do this process in this convoluted way is that due to its properties, the z-transform converts the Linear Constant Coefficient Difference Equations (LCCDE) that describe system behaviour to a polynomial. Also the convolution operation which describes the system action on the input signals is converted to a multiplication operation. These two properties make it much easier to do systems analysis in the z-domain. Inverse z-transform is performed using Long Division Method (Power Series Expansion method), Partial Fraction Expansion and Residue method (Contour Integral Method). Partial Fraction Expansion Method As we know that the rational form of X(z) can be expanded into partial fractions, Inverse z-transform can be taken according to location of poles and ROC of X(z). Following steps are to be performed for partial fraction expansions: Step 1: Arrange the given X(z) as, 푋(푧) 푛푢푚푒푟푎푡표푟 푝표푙푦푛표푚푖푎푙 = 푧 푧 − 푝1 푧 − 푝2 … (푧 − 푝푁) Step 2: 푋(푧) 퐴 퐴 퐴 퐴 = 1 + 2 + 3 + ⋯ + 푁 푧 푧 − 푝1 푧 − 푝2 푧 − 푝3 푧 − 푝푁 Where Ak for k=1, 2,…N are the constants to be found in partial fractions. Poles may be of multiple order. The coefficients will be calculated accordingly. Step 3: Above equation can be written as 퐴 푧 퐴 푧 퐴 푧 퐴 푧 푋 푧 = 1 + 2 + 3 + ⋯ + 푁 푧 − 푝1 푧 − 푝2 푧 − 푝3 푧 − 푝푁 퐴1 퐴2 퐴3 퐴푁 = −1 + −1 + −1 + ⋯ + −1 1 − 푝1푧 1 − 푝2푧 1 − 푝3푧 1 − 푝푁푧 퐴푘 Step 4:All the terms in above step are of the form −1.