UNIVERSITI MALAYSIA PERLIS

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1 UNIVERSITI MALAYSIA PERLIS SCHOOL OF COMPUTER & COMMUNICATIONS ENGINEERING EKT 230 SIGNALS AND SYSTEMS LABORATORY MODULE LAB 5 : LAPLACE TRANSFORM & Z-TRANSFORM 1

2 LABORATORY OUTCOME Ability to describe the concept of zeros and poles of the LTI object system Ability to determine the transfer function representation of the system and plot the locations of the poles and zeros on s-domain. Ability to demonstrate the desired simulation waveform for a system. Ability to analyze the system created on the EDA/CAD tool. EQUIPMENTS/COMPONENTS Computer Unit with Microsoft Windows XP Operating System Scilab v4.0 or later software INTRODUCTION Laplace transform The Laplace transform represents continuous-time signals as weighted superposition of complex exponentials, which are more general signals than complex sinusoids. Correspondingly, the Laplace transform represents a more general class of signals than does the Fourier transform, including signals that are not absolutely integrable. Hence, we may use the Laplace transform to analyze signals and LTI systems that are not stable. The transfer function is the Laplace transform of the impulse response and offers another description for the input-output characteristics of an LTI system. Figure 5.1: Basic Laplace Transform 2

3 The Laplace transform converts the convolution of time signals to multiplication of Laplace transforms, so the Laplace transform of an LTI system output is the product of the Laplace transform of the input and the transfer function. The locations of the poles and zeros of a transfer function in the s-plane offer yet another characterization of an LTI system, providing information regarding the system s stability, causality, invertibility and the frequency response. Complex exponentials are parameterized by a complex variable- s. The Laplace transform is a function of s and is represented in a complex plane termed the s-plane. The Fourier transform is obtained by evaluating the Laplace transform on the j -axis that is by setting s = j. The properties of the Laplace transform are analogous to those of the Fourier transform. The frequency response of an LTI system is obtained from the transfer function by setting s = j. The Bode diagram uses the poles and zeros of an LTI system to depict the magnitude response in dbs and phase response as logarithmic function of frequency. z-transform The z-transform represents the discrete-time signals as a weighted superposition of complex exponential, a more general signal class than complex sinusoid, so the z- Transform can represent a broader class of discrete-time signals than the DTFT, including signals that are not absolutely summable. Thus, we may use the z-transform to analyze discrete-time signals and LTI systems that are not stable. The transfer function of the discrete-time LTI system is the z-transform of its impulse response. The transfer function offers another description of the input-output characteristic of an LTI system the z-transform converts convolution of time signals into multiplication of z-transform, so the z-transform of a system s output is the product of the z-transform of the input and the system s transfer function. A complex exponential is described by a complex numbers. Hence, the z-transform is a function of complex variable z represented in the complex plane. The DTFT is obtained by evaluating the z-transform of the unit circle, z = 1, by setting z = e j. The properties of the z-transform are analogous to those of the DTFT. The ROC define the value of z for which the z-transform converges. The ROC must be specified in order to have a unique relationship between the time signals and its z-transform. The relative location of the ROC and z-transform poles determine whether the corresponding time signal is right-sided, left-sided or both. The location of z-transform s poles and zeros offer another representation of the input-output characteristic of an LTI system providing information regarding the system s stability, causality, invertibility and frequency response. 3

4 Figure 5.2: Basic z-transform 4

5 PROCEDURE Laplace transform and z-transform only work in linear systems as described earlier. In order to transform f(t) F(s) for Laplace and x(n) X(z) for z-transforms, understanding and memorizing the Laplace transforms and z-transforms are very useful to convert from the time-domain into frequency-domain. But when it comes to inverting the system back to its time-domain, these transforms however when can be applied via computational methods and when applied in Scilab are very closely related to polynomials and partial fractions. New Scilab call functions introduced in this lab will be : Scilab syntax csim(a,t,sysh) horner(sysh,x) roots(p) pfss(sysh) Description Simulation time response of linear system after a syslin is declared a = t = vector sysh = linear system of H polynomials Polynomial / rational evaluation sysh = linear system of H polynomials or rational matrix x = array of numbers@ polynomials@rationals Calculates the roots of a given polynomial p = polynomial with real or complex coefficients or vector Partial fraction decomposition sysh = linear system of H polynomials [n d k]=factors(sysh) Returns the factors of linear system polynomial H together with gain n = numerator d = denominator k = gain derivat(p) plzr(sysh) Computes the derivative of a polynomial, p Pole-zero plot of linear system H Some basic examples of the new call functions. Try out this Scilab simple program clear; clc; xdel( ); s = poly(0, s ); num = 2 + s; den = s^ *s ; sysh = syslin( c,num,den); dden = derivat(den); rd = roots(den); [n d k] = pfss(sysh); --> continued x = 2 : 25; y = horner(sysh,x); t = 0:50; simh = csim( step,t,sysh); subplot(211); plot2d1(x,y); subplot(212); plot2d1(t,simh); scf; plzr(sysh); 5

$Partial Fractions Expansions Partial fractions involve the process of splitting up of a given fraction into two or more fractions with only one type of factor in the denominator.$

6 Partial Fractions Expansions Partial fractions involve the process of splitting up of a given fraction into two or more fractions with only one type of factor in the denominator. They are used to express a ratio of polynomials as a sum of order of lower order polynomials. One particular method of expanding partial fractions is by using the method of residues, which is based on manipulating the partial fraction expansion so as to isolate each residue. The following equations can be used to apply this method : eq. 5.1 and if there exists repeated roots we use eq. 5.2 Using Scilab, the trick to determine whether the partial fraction expansion requires the usage of eq5.1 or eq5.2 or both depends on the call function pfss. If there exists multiple roots then other call functions are needed as demonstrated in Example 5.1 and 5.2. Example 5.1 a simple partial fraction expansion Expand the following fraction in Scilab Solution : F(s) = 6

7 Example 5.2 partial fraction expansions with multiple roots Evaluate the partial fraction decomposition of the following linear system Solution : Drill Question 5.1 : Manually show the partial fraction technique to obtain the partial fraction decomposition for example 5.1 and 5.2 above. Relating to System Descriptions A system may be described in terms of differential equation, transfer function, poles and zeros, frequency response or state variables. Relating the polynomials and partial fractions in Scilab to a certain domain (either Fourier, Laplace or Z-transforms) can link up the LTI system and the characteristics. Example 5.3 With reference to the Laplace transform table, transform the solution in example5.1 to time-domain. Solution : Drill Question 5.2 : Obtain the inverse Laplace transform in example

8 Example 5.4 Consider a system containing 1. zeros at s = 0 and s = j10 and 2. poles at s = j 5, s = -3, and s = - 4 and 3. with gain equals 2. Determine the transfer function representation of this system, plot the locations of the poles & zeros in the s-plane, and plot the system s magnitude response using the Scilab commands. Solution : 8

9 EXERCISE QUESTIONS 1. Obtain the poles and zeros of the following transfer function and plot the system using poles-zero plot : a. b. c. 2. Using the Scilab repfreq call function, plot the magnitude and phase response for the LTI System having the transfer function : 3. Obtain the inverse transform for the following Laplace transform. Show the partial fraction expansion both in Scilab commands and manual calculations before changing to time-domain a. b. c. d. e. 4. Calculate the Laplace transform for the following functions. Plot and compare the f(t) and F(s) signals obtained : a. b. c. 9

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand