Technology & Engineering
Inverse Z Transform
The inverse Z-transform is a mathematical operation that converts a function of a discrete complex variable, known as a Z-transform, back into a sequence of real or complex numbers. It is commonly used in digital signal processing and control system analysis to analyze and design discrete-time systems. The inverse Z-transform helps to understand the behavior of discrete systems in the time domain.
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10 Key excerpts on "Inverse Z Transform"
eBook - PDF- Tim Wescott(Author)
- 2011(Publication Date)
- Newnes(Publisher)
(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. Z Transforms 19 to investigate the general behavior of a system; in this case one picks signals that have easy closed-form transforms. Table 2.1 lists a number of time-domain signals and their z transforms. 2.4 The Inverse Z Transform Unlike the z transform given in (2.16), there is no useful, mathematically tidy and general expression that will get you from the z domain back to the time domain. There are, however, two very useful techniques that will always work if the signal in question is expressed as a ratio of polynomials in z . One uses the fact that control systems tend to generate responses which are easy to categorize and identify; since there is a one to one relationship between a time-domain signal and its z transform, the inverses can be found by inspection after a little work. The other method is a “brute force” one of synthetic polynomial division, which will always yield an answer when the z transform is in the form of a ratio of polynomials. By Identification When one solves problems in control systems using the z transform, one invari-ably gets results that are expressed as ratios of polynomials in z . Indeed, (2.18) is an example of a z transform that results in a small polynomial ratio. By using partial fraction expansion, these polynomial ratios can be broken down into terms that can be found in Table 2.1; because there is a one-to-one correspondence between a signal and its z transform, one can identify the time-domain signals from their z transforms to construct the time-domain signal.
eBook - PDFDSP for MATLAB™ and LabVIEW™ II
Discrete Frequency Transforms
- Forester W. Isen, Forester Isen(Authors)
- 2022(Publication Date)
- Springer(Publisher)
Additionally, an understanding will have been acquired of the inverse z-transform, and use of the z-transform to evaluate frequency response of various LTI systems such as the FIR and the IIR. 2.2 SOFTWARE FOR USE WITH THIS BOOK The software files needed for use with this book (consisting of m-code (.m) files, VI files (.vi), and related support files) are available for download from the following website: http://www.morganclaypool.com/page/isen The entire software package should be stored in a single folder on the user’s computer, and the full file name of the folder must be placed on the MATLAB or LabVIEW search path in accordance with the instructions provided by the respective software vendor (in case you have encountered this notice before, which is repeated for convenience in each chapter of the book, the software download only needs to be done once, as files for the entire series of four volumes are all contained in the one downloadable folder). See Appendix A for more information. 2.3 DEFINITION & PROPERTIES 2.3.1 THE Z-TRANSFORM The z-transform of a sequence x [n] is: X(z) = ∞ n=−∞ x [n]z −n where z represents a complex number. The transform does not converge for all values of z; the region of the complex plane in which the transform converges is called the Region of Convergence (ROC), and is discussed below in detail. The sequence z −n is a complex correlator generated as a power sequence of the complex number z and thus X(z) is the correlation (CZL) between the signal x [n] and a complex exponential the normalized frequency and magnitude variation over time of which are determined by the angle and magnitude of z. 2.3.2 THE INVERSE Z-TRANSFORM The formal definition of the inverse z-transform is x [n] = 1 2πj X(z)z n−1 dz (2.1) 2.3. DEFINITION & PROPERTIES 29 where the contour of integration is a closed counterclockwise path in the complex plane that sur- rounds the origin (z = 0) and lies in the ROC.
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 - PDFApplied Digital Signal Processing
Theory and Practice
- Dimitris G. Manolakis, Vinay K. Ingle(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
3 The z -transform We have seen that discrete-time signals or sequences are defined, generated, and processed by systems in the n -domain or time-domain , that is, as functions of the discrete index n . In this sense, we also say that the implementation of discrete-time systems takes places in the time-domain. The purpose of this chapter is threefold. First, we introduce a new representation of sequences, known as the z -transform. Second, we study how the properties of a sequence are related to the properties of its z -transform. Finally, we use the z -transform to study LTI systems described by a convolution sum or a linear constant-coefficient difference equation. Study objectives After studying this chapter you should be able to: • Understand how to represent a sequence of numbers with a function of a complex variable called the z -transform. • Change a sequence by manipulating its z -transform and vice versa. • Possess a basic understanding of the concept of system function and use it to investigate the properties of discrete-time LTI systems. • Determine the output of systems described by linear constant-coefficient difference equations using the z -transform. 90 The z -transform 3.1 Motivation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • In Section 2.4, we exploited the decomposition of an arbitrary sequence into a linear combination of scaled and shifted impulses, x [ n ] = ∞ k =−∞ x [ k ] δ [ n − k ] , (3.1) to show that every LTI system can be represented by the convolution sum y [ n ] = ∞ k =−∞ x [ k ] h [ n − k ] = ∞ k =−∞ h [ k ] x [ n − k ] . (3.2) The impulse response sequence h [ n ] specifies completely the behavior and the properties of the associated LTI system. In general, any sequence that passes through a LTI system changes shape.
eBook - PDF- Simon Haykin(Author)
- 2017(Publication Date)
- Wiley(Publisher)
The z-transform is generally used to study LTI system characteristics such as stability and causality, to develop computational structures for implementing discrete-time systems, and in the design of digital filters, the subject of Chapter 8. The z-transform is also used for transient and stability analysis of sampled-data control systems, a topic we visit in Chapter 9. The unilateral z-transform applies to causal signals and offers a convenient tool for solving problems associated with LTI systems defined by difference equations with nonzero initial conditions. None of these problems are addressable with the DTFT. Instead, the DTFT is usually used as a tool for representing signals and to study the steady-state characteristics of LTI systems, as we illustrated in Chapters 3 and 4. In these problems, the DTFT is easier to visualize than the z-transform, since it is a function of the real-valued frequency n, while the z-transform is a function of a complex number z = rei 0 . I FURTHER READING 1. The following text is devoted entirely to z-transforms: "' Vich, R., Z Transform Theory and Applications (D. Reidel Publishing, 1987) 2. The z-transform is also discussed in most texts on signal processing, including the following: ,.. Proakis, J. G., and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, 3rd ed. (Prentice Hall, 1995) 11> Oppenheim, A. V., R. W. Schafer, and J. R. Buck, Discrete Time Signal Processing, 2nd ed . (Prentice Hall, 1999) The book by Oppenheim et a!. discusses the relationship between the magnitude and phase responses of minimum-phase discrete-time systems. 3. Evaluation of the inverse z-transform using Eq. (7.5) is discussed in : Additional Problems 607 .,. Oppenheim, A. V., R. W. Schafer, and J. R. Buck, op. cit. and an introductory treatment of the techniques involved in contour integration is given in: ,.. Brown,]., and R. Churchill, Complex Variables and Applications, (McGraw-Hill, 1996) 4.
eBook - PDF- F Gentili, L Menini;A Tornamb??;L Zaccarian;(Authors)
- 1998(Publication Date)
- WSPC(Publisher)
CHAPTER 7 THE ^-TRANSFORM System and Control Theory benefits extensively from the use of z-trans-forms. Their application often simplifies the computations and gives a deeper insight into the structure of the solution of some particular problems. For instance, by means of z-transforms, it is easier to solve difference equations (through the solution of algebraic equations, in the field of rational functions), to compute the sum of series, and to characterize completely discrete random variables (through probability generating functions). In this chapter, the z-transform is formally defined and some basic results are derived, which allow direct and inverse z-transforms to be computed in many cases of practical in-terest. A final section of applications gives a short overview of some problems that can be solved by means of z-transforms. Additional applications can be found in Chapters 1, 6, and 9. 7.1 Definitions and Properties In this section, the unilateral ^-transform, which is most widely used in discrete-time system theory, is formally defined and some of its fundamental properties are stated. For the sake of completeness, the following definition of bilateral ^-transform is included. Let X(X, Y) denote the set of all the functions /(•) : X -4 Y having domain X and co-domain Y (where X and Y are two general spaces), and define #(Z,Q := {*(-): Z-> Q , *(C,C) := {X(>):C->Q. 321 322 7 - The z-Transform Definition 7.1 (Bilateral z-transform). The (direct) bilateral z-transform is an operator, Z 6 { . } : # ( Z , Q -» * ( C , C ) , which transforms a function x(-) of the integer variable t, into a complex-valued function X b (-) of the complex variable z, X h (-)€X(C,C), which is defined for z € C by means of the following formal series: 1 X b (z):= E x{t)z-*. t=—oo In the following, a simplified notation, which is not formally correct, will be used to refer to the bilateral z-transform.
eBook - PDF- Alexander D. Poularikas(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Therefore, the inverse Z -transform is given by f ( nT ) ¼ d ( n ) n ¼ 0 4(1) n 1 3 2 1 2 n 1 1 2 n 1 2 n 1 n 1 8 < : Example Now let us assume the same example but with j z j < 1 = 2. This indicates that the output signal is anticausal. Hence, from F ( z ) ¼ 4 z z 1 2 z z 1 2 z 2 z 1 2 2 and Table A.6.3, we obtain f ( nT ) ¼ 4(1) n þ 2 1 2 n þ ( n þ 1) 1 2 n n 1 Similarly from F ( z ) ¼ 1 þ 4 1 z 1 3 2 1 z 1 2 1 2 z z 1 2 2 and Table A.6.4, we obtain f ( nT ) ¼ d ( n ) n ¼ 0 4(1) n 1 þ 3 2 1 2 n 1 þ 1 2 n 1 2 n 1 n 1 8 < : Z-Transform 6 -23 6.1.2.3.3 Integral Inversion Formula THEOREM 6.1 If F ( z ) ¼ X 1 m ¼1 f ( mT ) z m (6 : 96) converges to an analytic function in the annular domain R þ < j z j < R , then f ( nT ) ¼ 1 2 p j þ C F ( z ) z n dz z (6 : 97) where C is any simple closed curve separating j z j ¼ R þ from j z j ¼ R and it is traced in the counterclockwise direction. Proof Multiply Equation 6.96 by z n 1 and integrate around C . Then 1 2 p j þ C F ( z ) z n dz z ¼ X 1 m ¼1 f ( mT ) 1 2 p j þ C z n m dz z (6 : 98) Set z ¼ Re j u with R þ < R < R to obtain 1 2 p j þ C z n m dz z ¼ 1 2 p j ð 2 p 0 R n m 1 e j u ( n m 1) Rje j u d u ¼ 1 2 p R k ð 2 p 0 e j u k d u ¼ 1 k ¼ 0 0 elsewhere & (6 : 99) Hence, the summation on the right-hand side of Equation 6.98 reduces to f ( nT ). Let { a k } be the set of poles of F ( z ) z n 1 inside the contour C and { b k } be the set of poles of F ( z ) z n 1 outside C in a fi nite region of the z -plane. By Cauchy ’ s residue theorem f ( nT ) ¼ X k Res F ( z ) z n 1 , a k n 0 (6 : 100) f ( nT ) ¼ X k Res F ( z ) z n 1 , b k n < 0 (6 : 101) Example Let F ( z ) ¼ 1 (1 z 1 )(1 a T z 1 ) a < 1, j z j > 1 The function F ( z )z n 1 ¼ z n þ 1 = ( z 1)( z a T ) has two poles enclosed by C for n 0.
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- Roland Priemer(Author)
- 1990(Publication Date)
- WSPC(Publisher)
Since this x(t) and consequently x(kT) has a finite duration, eq. 9.7 contains a finite number of terms, and it was easy to express X(z) as a ratio of polynomials inz. To take the inverse Z-transform means that the number sequence (discrete time function) x(kT) must be retrieved from X(z). The inverse Z-transform operation is denoted by Z' l Q[(z))=x(kT). From the form of X(z) given in eq. 9.8 it is not obvious that Z~*(X(z)) is the number sequence of eq. 9.6. However, we can recognize x(kT) if we convert X(z) as given in eq. 9.8 to the form of eq. 9.5. This can be done by a long division of P(z) by Q(z) to write P(z)IQ{z) as ^=... + jc(-2r)z 2 +x(-r)z 1 +^(0T)z 0 +x(r)z-1 +-and here we get 2z 1 2 Z + 1 + b-' | ** + 2z + 1 z 2 2z + 1 2z 1 0 The coefficients of the different powers of z 1 are x(kT). Since there is no z 2 or higher order term, we know that x(-2T) = 0, x(-3T) = 0, etc. The coefficient of z +1 gives x(-lT) = , and likewise for z° and z 1 . Since there is no z~ 2 term, we know that x(2T) = 0, and x(kT) = 0 for k £ 2. For anyX(z)=P(z)/j2(z),along division, although inconvenient, can be regarded as a method for takingtheinverseZ-transform.X(z)is also referred to as the generating function for the number sequence x(kT). This means that X(z) determines x(kT). This is similar to using the function, for example, l/(/co+2) to determine e^u^it), and the inverse Fourier transform integral is the oparation with which e'^u^it) can be retrieved (generated) with l/(/<0+2). We will shortly see another interpretation forX(z). Z-TRANSFORM 481 Example 9.2. Suppose that for some number sequence x(kT) 9 Z(x(kT)) =X(z) = zl(z -1/2). To determine x(kT% we perform a long division, i.e. 1 2 -2 1 -1 1 -2 1 + -z 1 + -, * + 1 2 — -2 1 2 Here, the long division continues indefinitely, while in Example 9.1 it terminated. However, we obtained enough terms to recognize a general form for x(kT). First, we see that x(kT) = 0 for k < 0. And, for k > 0, we have that x(kT) = Q]*.
- B. A. Shenoi(Author)
- 2005(Publication Date)
- Wiley-Interscience(Publisher)
Property 2.3: Time Reversal If X(z) is the z transform of a causal sequence x(n), n ≥ 0, then the z transform of the sequence x(−n) is X(z −1 ). The sequence x(−n) is obtained by reversing the sequence of time, which can be done only by storing the samples of x(n) and generating the sequence x(−n) by reversing the order of the sequence. If a discrete-time sequence or data x(n) is recorded on an audiocassette or a magnetic tape, it has to be played in reverse to generate x(−n). The sequence x(−n) and its z transform X(z −1 ) are extensively used in the simulation and analysis of digital signal processing for the purpose of designing digital signal processors, although the sequence x(−n) cannot be generated in real time, by actual electronic signal processors. Let X(z) = ∑ ∞ n=0 x(n)z −n . Then, ∑ 0 n=−∞ x(−n)z n = ∑ ∞ n=0 x(n)z n = X(z −1 ). Example 2.18 Z[(0.5) n u(n)] = ∞ n=0 (0.5) n z −n = 1 1 − 0.5z −1 = z z − 0.5 = X(z) Then Z[(0.5) −n u(−n)] = 0 n=−∞ (0.5) −n z n = X(z −1 ) = z −1 z −1 − 0.5 = 1 1 − 0.5z If F (z) = 0.1 + 0.25z −1 + 0.6z −2 1.0 + 0.4z −1 + 0.5z −2 + 0.3z −3 + 0.08z −4 (2.61) 74 TIME-DOMAIN ANALYSIS AND z TRANSFORM is the z transform of f (n)u(n), multiplying both the numerator and denominator polynomial by z 4 , we can express F (z) in the form F (z) = 0.1z 4 + 0.25z 3 + 0.6z 2 z 4 + 0.4z 3 + 0.5z 2 + 0.3z + 0.08 (2.62) The z transform of f (−n)u(−n) is obtained by replacing z by z −1 in (2.62) for F (z), and therefore the z transform of f (−n)u(−n) is F (z −1 ) = 0.1z −4 + 0.25z −3 + 0.6z −2 z −4 + 0.4z −3 + 0.5z −2 + 0.3z −1 + 0.08 (2.63) = 0.1 + 0.25z + 0.6z 2 1.0 + 0.4z + 0.5z 2 + 0.3z 3 + 0.08z 4 (2.64) which is different from either (2.61) or (2.62) for F (z) given above.
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