Z Transform

8 Key excerpts on "Z Transform"

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  eBook - PDF

  Bird's Higher Engineering Mathematics

  • 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.

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  eBook - ePub

  Mathematics for Circuits and Filters

  • Wai-Kai Chen(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  z transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the z-transform can be seen as a generalization of the discrete-time Fourier transform (FT)

  X

  e

  j ω

  =

  ∑

  n = − ∞

  + ∞

  x

  [ n ]

  e

  − j ω n

  (5.1)

  The above expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the z-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.

  The z-transform, as pointed out by Jury in his classical text [3 ], is not new. It can be traced back to the early 18th century and the times of DeMoivre, who introduced the notion of the generating function, extensively used in probability theory

  Γ ( z ) =

  ∑

  n = − ∞

  + ∞

  p

  [ n ]

  z n

  (5.2)

  where p[n] is the probability that the discrete random variable n will take a value n [8 ]. By comparing (5.2) and (5.3) below, we can see that the generating function Γ(1/z) is the z-transform of the sequence p[n] = p{n = n}. After these initial efforts, and due to the fast development of digital computers, a renewed interest in the z-transform occurred in the early 1950s, and the z-transform has been used for analyzing discrete-time systems ever since.

  This section is intended as a brief introduction to the theory and application of the z-transform. For a rigorous mathematical treatment of the transform itself, the reader is referred to the book by one of the pioneers in the development of analysis of sampled data systems, Jury [3 ], and the references therein. For a more succinct account of the z-transform, its properties and use in discrete-time systems, consult, for example, [7 ]. A few other texts which contain parts on the z-transform include [1 ,2 ,5 ,6 ,10

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  Signals and Systems

  • Simon Haykin(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  Thus, we may use the z-transform to analyze discrete-time signals and LTI systems that are not stable. The transfer function of a discrete-time LTI system is the z-transform of its impulse response. The transfer function offers another description of the input-output characteristics of an LTI system. The z-transform converts convolution of time signals into multiplication of z-transforms, 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 number. Hence, the z-transform is a function of a complex variable z represented in the complex plane. The DTFT is ob- tained by evaluating the z-transform on the unit circle, lzl = 1, by setting z = ei 0 . The properties of the z-transform are analogous to those of the DTFT. The ROC defines the val- ues of z for which the z-transform converges. The ROC must be specified in order to have a unique relationship between the time signal and its z-transform. The relative locations of the ROC and z-transform poles determine whether the corresponding time signal is right sided, left sided, or both. The locations of z-transform's poles and zeros offer another representation of the input-output characteristics of an LTI system, providing information regarding the system's stability, causality, invertibility, and frequency response. The z-transform and DTFT have many common features . However, they have distinct roles in signal and system analysis. 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.

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  Practical Signals Theory with MATLAB Applications

  • 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.

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  eBook - ePub

  Advanced Engineering Mathematics

  A Second Course with MatLab

  • Dean G. Duffy(Author)
  • 2022(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 The Z-Transform Since the Second World War, the rise of digital technology has resulted in a corresponding demand for designing and understanding discrete-time (data sampled) systems. These systems are governed by difference equations in which members of the sequence y n are coupled to each other. One source of difference equations is the numerical evaluation of integrals on a digital computer. Because we can only have values at discrete time points t k = k T for k = 0, 1, 2, …, the value of the. integral y (t) = ∫ 0 t f (τ) d τ is y (k T) = ∫ 0 k T f (τ) d τ = ∫ 0 (k − 1) T f (τ) d τ + ∫ (k − 1) T k T f (τ) d τ (3.0.1) = y [ (k − 1) T ] + ∫ (k − 1)[. --=PLGO-SEPARATOR=--]T k T f (τ) d τ = y [ (k − 1) T ] + T f (k T), (3.0.2) because ∫ (k − 1) T k T f (τ) d t ≈ T f (k T). The right side of Equation 3.0.2 is an example of a first-order difference equation because the numerical scheme couples the sequence value y (kT) directly to the previous sequence value y [(k − 1) T ]. If Equation 3.0.2 had contained y [(k − 2) T ], then it would have been a second-order difference equation, and so forth. Although we could use the conventional Laplace transform to solve these difference equations, the use of z-transforms can greatly facilitate the analysis, especially when we only desire responses at the sampling instants. Often the entire analysis can be done using only the transforms and the analyst does not actually find the sequence y (kT). In this chapter we will first define the z-transform and discuss its properties. Then we will show how to find its inverse. Finally, we shall use them to solve difference equations. Figure 3.1.1 : Schematic of how a continuous function f (t) is sampled by a narrow-width pulse sampler f * S (t) and an ideal sampler f S (t). 3.1 THE RELATIONSHIP OF THE Z-TRANSFORM TO THE LAPLACE TRANSFORM 1 1 Gera (Gera, A. E., 1999: The relationship between the z-transform and the discrete-time Fourier transform. IEEE Trans. Auto

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  Introductory Signal Processing

  • Roland Priemer(Author)
  • 1990(Publication Date)
  • WSPC

    (Publisher)

  CHAPTER NINE 477 Z-TRANSFORM AND DISCRETE TIME SYSTEMS It has become commonplace to employ discrete time methods, i.e., discrete time systems, for continuous time signal processing in fields such as telecommunications, process control, medical instrumentation, and other fields too numerous to list here. This has occurred because of the advent in recent years of economical digital hardware for implementing discrete time methods. We shall see in this and the next chapter that in addition to economy there are other advantages that can be realized by employing digital hardware instead of analog hardware for signal processing. We have come to view the Fourier and Laplace transforms as methods for representing continuous time signals in terms of exponential functions. Properties of the exponential function and the superposition principle permitted our extensive application of these transforms. As for continuous time signals and LTI (linear and time invariant) systems, discrete time signals and LH systems can also be analyzed by a transform method, which is also referred to as a frequency domain technique. The transform that is used is called the Z-transform.

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  Signals and Systems

  A Primer with MATLAB

  • Matthew N. O. Sadiku, Warsame Hassan Ali(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  311 z-Transform We begin by defining the z -properties and studying its important properties The z-transform of some common discrete-time signals is derived We present methods for finding the inverse z-transform Two important applications of the z-transform are discussed Finally, we consider using MATLAB ® to find z-transform and its inverse 7.2 DEFINITION OF THE z-TRANSFORM The z-transform is the generalization of the discrete-time Fourier transform (DTFT) We recall that from Chapter 6, the DTFT of a signal x [ n ] is given by X x n e j n n ( ) [ ] W = -W =-¥ ¥ å (71) Inserting a factor ρ − n in Equation 71 leads to X x n e x n e n j n n j n n ( ) [ ] [ ] ( ) W = = --W =-¥ ¥ W -=-¥ ¥ å å r r (72) If we let z = ρ e j Ω Then, Equation 72 becomes X z x n z n n ( ) [ ] = -=-¥ ¥ å (73) This is the two-sided (or bilateral) z-transform The one-sided (or unilateral) z-transform is X z x n x n z n n ( ) { [ ]} [ ] = = -= ¥ å Z 0 (74) Only the one-sided z-transform will be discussed in this chapter X ( z ) and x [ n ] con-stitute a z-transform pair x n X z Z [ ] ( ) ¾ ® ¾ (75) Note the following: 1 From Equation 74, we notice that the z-transform of x [ n ] is a power series of z −1 whose coefficients are x [ n ], that is, X z x n z x x z x z x z n n ( ) [ ] [ ] [ ] [ ] [ ] = = + + + + -= ¥ ---å 0 1 2 3 0 1 2 3 midhorizellipsis (76) In this power series, z − n can be interpreted as indicating the n th sampling instant

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  DSP 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.

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