Technology & Engineering
Z Transform vs Laplace Transform
The Z-transform and Laplace transform are mathematical techniques used in signal processing and control systems. The Laplace transform is used for continuous-time signals and systems, while the Z-transform is used for discrete-time signals and systems. Both transforms are used to analyze and solve differential equations and transfer functions in engineering applications.
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12 Key excerpts on "Z Transform vs Laplace Transform"
eBook - PDF- Daniel Fleisch(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
5 The Z-Transform Another useful transform related to the Fourier and Laplace transforms is the Z-transform, which, like the Laplace transform, converts a time-domain function into a frequency-domain function of a generalized complex frequency parameter. But the Z-transform operates on sampled (or “discrete-time”) functions, often called “sequences” while the Laplace transform operates on continuous-time functions. Thus the relationship between the Z-transform and the Laplace transform parallels the relationship between the discrete- time Fourier transform and the continuous-time Fourier transform. Under- standing the concepts and mathematics of discrete-time transforms such as the Z-transform is especially important for solving problems and designing devices and systems using digital computers, in which differential equations become difference equations and signals are represented by sequences of data values. You can find comprehensive discussions of the Z-transform in several text- books and short introductions on many websites, so rather than duplicating that material, this chapter is designed to provide a bridge to the Z-transform from the discussion and examples of the Laplace transform and its characteristics in previous chapters. The foundations of that bridge are laid in Section 5.1 with the definition of the Z-transform and an explanation of its relationship to the Laplace transform, along with graphical depictions of the mapping of the Laplace s -plane to the z-plane and a description of the inverse Z-transform. Section 5.2 has examples of the Z-transform of several basic discrete-time functions, and the properties of the Z-transform are presented in Section 5.3. As in previous chapters, the final section of this chapter contains a set of problems which will allow you to check your understanding of the chapter’s material. 174
eBook - PDF- Alex Palamides, Anastasia Veloni(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
10 z-Transform In this chapter, we introduce the z -transform. z -transform is the counterpart of Laplace transform when dealing with discrete-time signals. It is employed to transform difference equations that describe the input = output relationships of discrete-time systems into algebraic equations and is a very useful tool for the analysis and design of discrete-time systems. 10.1 Mathematical De fi nition As Laplace transform is a more general transform compared to the Fourier transform for continuous-time signals, z -transform is a more general transform than discrete-time Fourier transform when dealing with discrete-time signals. A discrete-time signal is de fi ned in the discrete-time domain n ; that is, it is given by a function f [ n ], n 2 Z . z -transform is denoted by the symbol Z { } and expresses a signal in the z -domain, i.e., the signal is given by a function F ( z ). The mathematical expression is F ( z ) ¼ Z { f [ n ]} : (10 : 1) In other words, the z -transform of a function f [ n ] is a function F ( z ). The mathematical expression of the two-sided (or bilateral) z -transform is F ( z ) ¼ Z { f [ n ]} ¼ X 1 n ¼1 f [ n ] z n , (10 : 2) where z is a complex variable. Setting the lower limit of the sum from minus in fi nity to zero yields the one-sided (or unilateral) z -transform whose mathematical expression is F ( z ) ¼ Z { f [ n ]} ¼ X 1 n ¼ 0 f [ n ] z n : (10 : 3) In order to return from the z -domain back to the discrete-time domain, the inverse z -transform is applied. The inverse z -transform is denoted by the symbol Z 1 { }; that is, one can write f [ n ] ¼ Z 1 { F ( z )} : (10 : 4) 443 The mathematical expression of the inverse z -transform is x [ n ] ¼ 1 2 p j þ X ( z ) z n 1 dz : (10 : 5) The z -transform of a sequence is easily computed by Equations 10.2 or 10.3. Example Compute the z -transform of the sequence x [ n ] ¼ [3,5,4,3], 0 n 3. The z -transform of the sequence x [ n ] is directly computed from Equation 10.3.
eBook - PDF- John Bird(Author)
- 2021(Publication Date)
- Routledge(Publisher)
Section L Z-Transforms Chapter 66 An introduction to z-transforms Why it is important to understand: An introduction to z-transforms In mathematics and signal processing, the z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It can be considered as a discrete-time equivalent of the Laplace transform. Laplace transform methods are widely used for analysis in linear systems and are used when a system is described by a linear differential equation, with constant coefficients. However, there are numerous systems that are described by difference equations - not differential equations - and these systems are common and different from those described by differential equations. Systems that satisfy difference equations include computer controlled systems - systems that take mea- surements with digital input/output boards or GPIB instruments (digital 8-bit parallel communications interface with data transfer rates up to 1 Mbyte/s), calculate an output voltage and output that voltage digitally. Frequently these systems run a program loop that executes in a fixed interval of time. Other systems that satisfy difference equations are those systems with digital filters - which are found anywhere digital signal processing/digital filtering is undertaken - that includes digital signal transmission systems like the telephone system or systems that process audio signals. A CD contains digital signal information, and when it is read off the CD, it is initially a digital signal that can be processed with a digital filter. There are an incredible number of systems used every day that have digital components which satisfy difference equations. In continuous systems Laplace transforms play a unique role. They allow system and circuit designers to analyse systems and predict performance, and to think in different terms - like frequency responses - to help understand linear continuous systems.
eBook - PDF- B. P. Lathi, Roger A. Green(Authors)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
Chapter 7 Discrete-Time System Analysis Using the z -Transform The counterpart of the Laplace transform for discrete-time systems is the z -transform. The Laplace transform converts integro-differential equations into algebraic equations. In the same way, the z -transform changes difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems. The z -transform method of analysis of discrete-time systems parallels the Laplace transform method of analysis of continuous-time systems, with some minor differences. In fact, we shall see in Sec. 7.8 that the z -transform is the Laplace transform in disguise . The behavior of discrete-time systems is, with some differences, similar to that of continuous-time systems. As shown in Sec. 5.5.4 , the frequency-domain analysis of discrete-time systems is based on the fact that the response of a linear time-invariant discrete-time (LTID) system to an everlasting exponential z n is the same exponential within a multiplicative constant, given by H ( z ) z n . More generally, when an input x [ n ] is a sum of (everlasting) exponentials of the form z n , the system response to x [ n ] is easily found as a sum of the system’s responses to all these exponential components. The tool that allows us to represent an arbitrary input x [ n ] as a sum of everlasting exponentials (complex frequencies) of the form z n is the z -transform. 7.1 The z -Transform Much like the Laplace transform, the z -transform comes in two flavors: bilateral and unilateral. We begin with the bilateral case and then treat the unilateral case second. 7.1.1 The Bilateral z -Transform Just as the Laplace transform can be derived from the Fourier transform by generalizing the fre-quency variable from jω to σ + jω , the z -transform can be obtained from the discrete-time Fourier transform by generalizing the frequency variable from j Ω to σ + j Ω.
eBook - PDF- Baolong Guo, Juanjuan Zhu, China Science Publishing & Media Ltd.(Authors)
- 2018(Publication Date)
- De Gruyter(Publisher)
6 The Z-transform and Z-domain analysis Please focus on the following key questions. 1. What is the mapping relation of the time-domain and Z-domain representations of discrete signals? 2. What is the method of applying the Z-transform for analyzing LTID systems? 6.0 Introduction As illustrated in Chapter 4, the continuous-time signal can be converted to discrete-time signals by sampling. In Chapter 5, the Laplace transform is used to analyze the LTIC systems by transforming the differential equations into algebraic equations. For LTID systems, the Z-transform is applied to solve the difference equations to deter-mine the output response. The Laplace transform uses e st as the basis function. The Z-transform expresses f ( k ) in terms of z k , where the independent variable z is given by z = e sT . In this chapter, we first introduce the bilateral Z-transform definition. For causal signals, the bilateral Z-transform reduces to the unilateral Z-transform. The properties of the Z-transform are given in Section 6.2, and the inverse Z-transform is proposed in Section 6.3. The relationship between the Laplace transform and the Z-transform is discussed in Section 6.4. Section 6.5 applies the Z-transform to solve the difference equation. The Z-transfer function and the stability analysis of the LTID system in the Z-domain is presented in Section 6.6. The signal flow graph and system simulation are introduced in Section 6.7. Section 6.8 defines the frequency response, including the magnitude and phase spectra for LTID systems. Finally, the chapter is concluded in Section 6.9. 6.1 Analytical development 6.1.1 From the Laplace transform to the Z-transform The continuous-time signal f ( t ) is sampled by the ideal impulse-train δ T ( t ) to obtain the discrete-time signal f s ( t ) .
eBook - ePub- Wim van Drongelen(Author)
- 2018(Publication Date)
- Academic Press(Publisher)
Chapter 12Laplace and z -Transform
Abstract
In this chapter we describe the use of Laplace- and z -transforms to solve differential and difference equations. First we introduce the formalisms for the Laplace transform of an expression and its derivatives. Then we apply this to the solution of a differential equation of a simplified ion channel model and the general expression for a time-invariant linear system. Next we introduce the effect of a time-delay to the Laplace transform and use this to introduce the z -transform. The z -transform is used to analyze the properties of the discrete-time differentiator. Throughout the text, the application of the transforms for characterizing systems by its transfer function is explained and illustrated. Appendices provide examples of Laplace and z -transform pairs, discuss the region of convergence, and provide examples of partial fraction expansion.Keywords
Lag operator; Laplace transform table; Partial fraction expansion; Region of convergence (ROC); Transfer function; z -Transform table12.1. Introduction
In this chapter we briefly summarize the use of the Laplace transform and the closely related z-transform . The former is used in the analysis of continuous time systems, while the latter is the equivalent for discrete time (sampled) data sets. Both transforms are related to the Fourier transform, therefore (for those not familiar with spectral analysis) it is recommended to review Chapters 5 –7 before proceeding with this chapter. The goal is to use the Laplace and z- transforms to analyze the input–output relationship of linear systems, which we will need specifically for the subsequent chapters which cover the application of analog and digital filters. The starting point for the mathematical description of these linear time-invariant (LTI) systems is their associated differential and difference equations that govern their input–output relationships (Chapters 11 and 13 , Eqs. 13.1a and 13.1.b ). It is strongly recommended to review Section 13.2
- Richard J. Tervo(Author)
- 2013(Publication Date)
- Wiley(Publisher)
CHAPTER 9 ¢ The z-Transform 9.1 Introduction In Chapter 8 the use of discrete time signals was described as an important consequence of using dig- ital computers in signal processing. The analysis of sampled signals revealed a number of special prop- erties of the Fourier transform with discrete time sig- nals (DTFT). From these observations, it emerges that the application of digital signal processing (DSP) may benefit from a different transform that simplifies both the mathematics and the computational require- ments by recognizing the special nature of discrete signals. The z -transform incorporates aspects of both the Fourier transform and the Laplace transform when dealing with discrete signals. 9.2 The z-Transform When dealing with discrete signals, the z -transform is generally preferred over the more general Fourier or Laplace transform techniques owing to the unique properties of discrete signals in both the time and frequency domains. The z -transform can be regarded as a generalization of the discrete time Fourier transform (DTFT) much as the Laplace transform is a generalization of the continuous Fourier transform. Consequently, the z -transform can also be described as a discrete time version of the Laplace transform. Like the Laplace transform, the z -transform incorporates the Fourier transform and adds another dimension to create a two-dimensional domain to be called the z -domain. The comparison to the Laplace transform is strengthened when dealing with z -transform poles and zeros, system stability, and regions of convergence.
eBook - PDFSignals and Systems
A Primer with MATLAB
- Matthew N. O. Sadiku, Warsame Hassan Ali(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
describing a broad variety of systems and their properties Just as Laplace transform is useful in handling signals that do not have Fourier transform, the z-transform enables us to treat discrete-time signals that do not have discrete-time Fourier transforms (DTFT) (For example, the unit sequence u [ n ] does not have a DTFT) Also, just as the Laplace transform converts integrodifferential equations into algebraic equations, the z-transform converts difference equations into algebraic equations that are easier to manipulate and solve Therefore, tech-niques in this chapter parallel the techniques used in Chapter 3 on Laplace trans-form Although the properties of the z-transform are similar to those of the Laplace transform, there are some differences Like Laplace transform, the z-transform is applicable to systems with initial conditions
eBook - PDFDSP for MATLAB™ and LabVIEW™ II
Discrete Frequency Transforms
- Forester W. Isen, Forester Isen(Authors)
- 2022(Publication Date)
- Springer(Publisher)
27 C H A P T E R 2 The z-Transform 2.1 OVERVIEW In the previous chapter, we took a brief look at the Fourier and Laplace families of transforms, and a more detailed look at the DTFT, which is a member of the Fourier family which receives a discrete time sequence as input and produces an expression for the continuous frequency response of the discrete time sequence. With this chapter, we take up the z-transform, which uses correlators having magnitudes which can grow, decay, or remain constant over time. It may be characterized as a discrete-time variant of the Laplace Transform. The z-transform can not only be used to determine the frequency response of an LTI system (i.e.,the LTI system’s response to unity-amplitude correlators), it reveals the locations of poles and zeros of the system’s transfer function, information which is essential to characterize and understand such systems. The z-transform is an indispensable transform in the discrete signal processing toolbox, and is virtually omnipresent in DSP literature. Thus, it is essential that the reader gain a good understanding of it. The z-transform mathematically characterizes the relationship between the input and output sequences of an LTI system using the generalized complex variable z, which, as we have already seen, can be used to represent signals in the form of complex exponentials. Many benefits accrue from this: • An LTI system is conveniently and compactly represented by an algebraic expression in the variable z; this expression, in general, takes the form of the ratio of two polynomials, the numerator representing the FIR portion of the LTI system, and the denominator representing the IIR portion. • Values of z having magnitude 1.0, which are said to “lie on the unit circle” can be used to evaluate the z-transform and provide a frequency response equivalent to the DTFT. • Useful information about a digital system can be deduced from its z-transform, such as location of system poles and zeros.
eBook - PDFIntroduction to Orthogonal Transforms
With Applications in Data Processing and Analysis
- Ruye Wang(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
The Laplace and z -transforms 325 where we have defined W (z) = X(z)/( ∑ N k =0 a k z −k ) as an intermediate variable, or in the time domain: N k =0 a k w[n − k] = x[n], or a N w[n − N ] = x[n] − N −1 k =0 a k w[n − k]. (6.262) Without loss of generality, we assume a N = 1, and the LTI system can now be represented as a block diagram, as shown in Fig. 6.15 (for M = 2 and N = 3). Figure 6.15 Block diagram of a discrete LTI system described by an LCCDE. To find the impulse response h[n] we first convert H(z) to a summation by partial fraction expansion: H(z) = M k =1 (z − z 0 k ) N k =1 (z − z 0 k ) = N k =1 c k 1 − p k z −1 , (6.263) (assume no repeated poles) and then carry out the inverse transform (the LTI system in Eq. (6.257) is causal) to get h[n] = Z −1 [H(z)] = N k =1 Z −1 c k 1 − p k z −1 = N k =1 c k p n k u[n]. (6.264) The output y[n] of the LTI system can be found by solving the difference equation in Eq. (6.257). Alternatively, it can also be found by the convolution y[n] = h[n] ∗ x[n], or the inverse z-transform: y[n] = Z −1 [Y (z)] = Z −1 [H(z)X(z)]. (6.265) As the LCCDE in Eq. (6.257) is an LTI system, it can also be solved in the following two steps. First, we assume the input on the right-hand side is simply x[n] and find the corresponding output y[n]. Then the response to the true input ∑ k b k x[n − k] can be found to be ∑ k b k y[n − k]. Note that the output y[n] obtained this way is only the particular solution due to input x[n], but the homogeneous solution due to any non-zero initial conditions is not represented by the bilateral Laplace transform. This problem will be addressed by the unilateral z-transform to be discussed later, which takes the initial conditions into consideration. 326 The Laplace and z -transforms Same as in the case of a continuous LTI system, here, the behavior of a discrete LTI system in terms of stability and oscillation is also dictated by the pole locations in the z-plane.
eBook - PDF- Tim Wescott(Author)
- 2011(Publication Date)
- Newnes(Publisher)
(2.14) Inserting (2.14) into (2.11) and simplifying we get 0.005 0.005 0 a = , (2.15) from which we can easily conclude that a 0 = 1. 18 Chapter 2 2.3 The Z Transform One can use the Laplace transform to solve linear time invariant differential equations, and to deal with many common feedback control problems using continuous-time control. With a sampled-time system one deals with linear shift invariant difference equations, and the tool for analysis is the z transform. By definition, the z transform takes an expression for a signal x k which is dependent on the time variable k and transforms it into an expression X ( z ) that depends only on the variable z . Note that the signal hasn’t changed; its description will look totally different, but the z transform preserves all information about the signal. That’s the transform part: the problem is transformed from one in the sampled time domain k , and put it into the z domain. 3 The z transform of x is denoted as Z{x}. In this book we’ll use the single-ended z transform which is defined as: X z = x = x z k k= k ( ) { } ∞ − ∑ Z 0 . (2.16) Because we have defined the transform as starting at k = 0 the signal must be restricted to x k = 0 for all k < 0. In practical control-systems problems this restriction causes no problems, and saves us from having to worry about what’s been happening to our signals since before the beginning of time! As an example of the z transform, if you have a signal x = k < a k k k 0 0 0 ≥ (2.17) you can plug it into (2.16) and get: X z = a z = z z a k k k= ( ) − − ∞ ∑ 0 . (2.18) Obviously (2.16) can be used to find the z transform for any arbitrary sig-nal—you just may not be able to get it into a nice closed form as was done in (2.18), which is tedious, however. In practice one is interested in using signals 3 Later in this chapter we will see how the z domain can be viewed in terms of frequency. For now just take it as a mathematical convenience.
- Alexander D. Poularikas(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
8 The z -Transform The z -transform method provides a powerful tool for solving difference equations of any order and, hence, plays a very important role in digital systems analysis. This chapter includes a study of the z -transform, its properties, and its applications. The z -transform method provides a technique for transforming a difference equation into an algebraic equation. Speci fi cally, the z -transform converts a sequence of numbers { y ( n )} into a function of complex variable Y ( z ), thereby allowing algebraic process and well-de fi ned mathematical procedures to be applied in the solution process. In this sense, the z -transform plays the same general role in the solution of difference equations that the Laplace transform (LT) plays in the solution of differential equations and in a roughly parallel way. Inversion procedures that parallel one another also exist. 8.1 The z -Transform To understand the essential features of the z -transform, consider a one-sided sequence of numbers { y ( n )} taken at uniform time intervals. This sequence might be the values of a continuous function that has been sampled at uniform time intervals; it could, of course, be a number sequence, for example, the values of the amount that are present in a bank account at the beginning of each month that includes the interest. This number sequence is written as { y ( n )} ¼ { y (0), y (1), y (2), . . . , y ( n ), . . . } (8 : 1) We now create the series Y ( z ) ¼ y (0) z 0 þ y (1) z þ y (2) z 2 þ ¼ y (0) þ y (1) z 1 þ y (2) z 2 þ (8 : 2) In this expression, z denotes the general complex variable and Y ( z ) denotes the z -transform of the sequence { y ( n )}. In this more general form, the one-sided z -transform of a sequence { y ( n )} is written as 8 -1 Y ( z ) ¼ D Z { y ( n )} ¼ X 1 n ¼ 0 y ( n ) z n (8 : 3) This expression can be taken as the de fi nition of the one-sided z -transform.
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