DOCSLIB.ORG

Teaching Digital Control and Filtering Using MATLAB and Vissim

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Teaching Digital Control and Filtering Using MATLAB and Vissim Teaching Digital Control and Filtering Using MATLAB andVisSim Raymond G. Jacquot and Jerry C.Hamann College of Engineering, University of Wyoming Laramie, WY 82071-3295 Abstract In the past, the authors have taught essentially the identical course material, but due to computational The authors report the revision for a course limitations the problem assignments have been rather entitled Discrete Data Systems which is a blend of simple, usually of the second-order variety. With the classical and modern digital control, and digital convenience offered by MATLAB for Windows [1] and filtering. This revision reflects computing resources now VisSim [2], the authors have made the assigned problems available and as a result no longer are low order system more realistic and can concentrate class time on problems assigned. The authors have incorporated algorithms for the design process rather than getting symbolic computing in several appropriate places where bogged down with detailed classroom calculations. In it can increase understanding and yield error-free addition, the Symbolic Toolbox of MATLAB allows algebraic manipulations. In most exercises, the some rather tedious, error-prone algebraic manipulations computing is of the numerical type using algorithms to be accomplished with zero errors. The authors and developed in the course. Simulation employing a others have reported previously the use of symbolic graphical-oriented simulation language lends reality as computation in a digital filter design course [3]. VisSim to how control laws and estimation algorithms are is a simulation environment which handles mixed implemented in practice and lets students explore the continuous- and discrete-time elements with a graphical effects of controller parameters and sampling interval on user interface. The learning curve for VisSim is responses. essentially negligible, and students progress rapidly in their ability to simulate complex systems. Simulink [4] is Introduction an alternative to the use of VisSim and runs seamlessly with MATLAB. A “theme problem” is employed for most of the At the University of Wyoming a lower level exercises in the course. This problem is one that is graduate course is offered to introduce graduate students, interesting from both a classical and a state space design in a variety of specialty areas, to discrete-time control point-of-view and one not amenable to hand calculations, and digital filtering. The students are typically from the thus motivating the students to develop an interest in the controls, power systems and signal processing areas with modern software tools. an occasional mechanical or chemical engineer thrown in for good measure. The course is a blend of classical discrete-data control topics, modern digital control topics Theme Problem Plant and digital filter synthesis techniques. A topical outline is given in Table 1. The plant chosen for the homework exercise is the thermal environmental chamber illustrated in Fig. 1. Table 1. Course Outline The outer chamber is heated by a flow of steam which is Review of z-transforms controlled by a valve for which the dynamics may not be The pulse transfer function neglected. Fig. 1 is labeled with the perturbed Single-loop controller design temperature variables (x1(t), x2(t), w(t)) and the perturbed PID and lead-lag controllers valve gate motion variable (x3(t)). This model is then Ragazzini controller design analyzed for small perturbations in the physical variables Digital filter synthesis about an equilibrium condition, with x1(t) and x2(t) being Discrete-time state variable representation the temperature perturbations in the inner and outer State feedback and associated algorithms chambers respectively. The variable x3(t) is the Prediction, current and reduced order observers perturbation in the valve opening while the current used Nonzero setpoints versus regulators to produce that perturbation is denoted as u(t). The L-Q optimal control perturbation in the environmental temperature is denoted Reciprocal root locus as w(t). w(t) b x2(t) u(t) x1(t) x3(t) N S S N K2/2 Coil K2/2 X From Steam Generator Valve Figure 1. Thermal Chamber System to be Controlled The governing differential equations for the perturbations Pulse Transfer Function are, assuming appropriate coefficients, dx1 = -x1 + x2 After a brief review of the z-transform it is dt appropriate to examine how a discrete-time dx2 representation of the plant and the associated data = x1 - 2x2 + 2x3 + w(t) (1) dt converters can be obtained. The A/D and D/A converters dx are assumed to operate synchronously at a rate specified 3 = -4x + u(t) dt 3 by the sampling interval T and to have infinitely precise If the temperature of the inner chamber is taken to be the representations of the digital data. The so-called pulse system output, the transfer function for the open-loop transfer function is then given by plant is z -1 ì -1 éG(s)ù ü G(z) = ZíL t=kT ý (5) X (s) z ëê s ûú G(s) = 1 = 2 (2) î þ U(s) s3 + 7s2 + 13s + 4 where Z{.} represents the taking of the z-transform and -1 From relations (1) the continuous-time state description L [.] represents the taking of the inverse Laplace matrices are transform. MATLAB is particularly useful utilizing the é-1 1 0 ù é0ù ‘residue’ function for evaluation of the partial fraction coefficients for G(s)/s. The ‘deconv’ function is F = ê 1 -2 2 ú , g = ê0ú (3) ê ú ê ú particularly attractive for the reassembly of the z-domain ëê 0 0 -4ûú ëê1ûú fractions into a single polynomial ratio for G(z). The and pulse transfer function for this plant for a sampling cT = [1 0 0] (4) interval of T = 0.2 s is It is in this setting that the problem exercises are 0.001899(z + 2.6629)(z + 0.1865) constructed. Of course, the problems could be of higher G(z) = (6) (z - 0.9625)(z - 0.5924)(z-.4493) order, but in the authors’ opinions this system is of sufficient complexity to be instructive while illustrating the power of contemporary computing tools. Cascade Controller Design State Variable Representation This content focuses on the synthesis of a The first step in the discrete-time state variable discrete-time controller with transfer function D(z) which design process is to find a discrete-time state variable will reshape either the root locus diagram or the open- representation of the plant and the associated data loop frequency response to yield performance which is converters. The plant and data converters can be improved. Here the polynomial handling facilities of represented by the matrix-vector difference equation MATLAB are invaluable particularly in the synthesis of x(k +1) = Ax(k) + bu(k) (7) finite settling time and ripple-free controllers. Easy The process of evaluation of the A and b matrices can be calculation and graphical display are highly desirable for carried out either analytically or numerically. The these trial-and-error designs. The ease of mixed discrete- analytical process involves the inversion of Laplace time and continuous-time system simulation make transforms for which the ‘residue’ function is invaluable. VisSim attractive for design evaluation and for The numeric process involves the calculation of the understanding of control law implementation. matrix exponential, for which MATLAB is particularly well-suited. The discrete-time state variable matrices Digital Filter Synthesis computed for a sampling interval of T = 0.2s are é0.8341 0.1494 0.0254ù é0.0019ù In this course there is in-depth discussion of ê A = 0.1494 0.6847 0.2227ú , b = ê0.0273ú (8) several approaches to digital filter synthesis. The reason ê ú ê ú for examination of this topic is that often controllers are ëê 0 0 0.4493ûú ëê0.1377ûú synthesized in the s-domain and are to then be At this point it is also advantageous to explore system implemented in discrete-time form. The two methods controllability and observability employing the ‘rank’ addressed are the bilinear transform method (Tustin’s function in MATLAB. method) and the matched-pole-zero technique. The authors have previously discussed the use of State and Estimated State Feedback symbolic computing for the bilinear-transform synthesis of digital filters [3]. The approach taken is that the The design process begins with the designer starts with a unit frequency lowpass prototype of consideration of pole placement using full state feedback. Butterworth, Chebyshev, inverse Chebyshev or elliptic This involves the expansion of a symbolic determinant type. The first step is to perform either a lowpass, and equating it to the desired closed-loop characteristic bandpass, bandstop or highpass transformation to yield equation ac(z) an s-domain filter of the correct type. The next step is to det[zI - A + bf T ] = a (z) (9) prewarp the poles of this filter so they will have the c This nasty algebraic expansion involves the symbol z and correct critical frequencies after the bilinear transform to the symbolic elements of the vector fT and as such is a the z-domain has been performed. Depending upon the process that is quite prone to human error in attempted extent to which one desires to involve the student in the manipulation and simplification. This is an ideal process, the MATLAB Symbolic Toolbox can be used to application of the MATLAB Symbolic Toolbox which is advantage for both of these processes while demanding a also incorporated into the Student Edition of MATLAB substantial degree of mastery of the process on the part of [5]. In the absence of a complete set of state the student. Alternatively, purely numerical alternatives measurements, the appropriate control law is to feed back which demand very little process expertise can also be estimates of the states obtained from an asymptotic employed, for example the functions 'lp2bp' and observer either of the prediction or current variety.
Load more

Recommended publications
  • Moving Average Filters
    CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiple- pass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In M &1 this equation, x[ ] is the input signal, y[ ] is ' 1 % y[i] j x [i j ] the output signal, and M is the number of M j'0 points used in the moving average. This equation only uses points on one side of the output sample being calculated. Where x[ ] is the input signal, y[ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: x [80] % x [81] % x [82] % x [83] % x [84] y [80] ' 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: x[78] % x[79] % x[80] % x[81] % x[82] y[80] ' 5 This corresponds to changing the summation in Eq.
    [Show full text]
  • Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks
    NATIONAL Application Note 097 INSTRUMENTS® The Software is the Instrument ® Designing Filters Using the Digital Filter Design Toolkit Rahman Jamal, Mike Cerna, John Hanks Introduction The importance of digital filters is well established. Digital filters, and more generally digital signal processing algorithms, are classified as discrete-time systems. They are commonly implemented on a general purpose computer or on a dedicated digital signal processing (DSP) chip. Due to their well-known advantages, digital filters are often replacing classical analog filters. In this application note, we introduce a new digital filter design and analysis tool implemented in LabVIEW with which developers can graphically design classical IIR and FIR filters, interactively review filter responses, and save filter coefficients. In addition, real-world filter testing can be performed within the digital filter design application using a plug-in data acquisition board. Digital Filter Design Process Digital filters are used in a wide variety of signal processing applications, such as spectrum analysis, digital image processing, and pattern recognition. Digital filters eliminate a number of problems associated with their classical analog counterparts and thus are preferably used in place of analog filters. Digital filters belong to the class of discrete-time LTI (linear time invariant) systems, which are characterized by the properties of causality, recursibility, and stability. They can be characterized in the time domain by their unit-impulse response, and in the transform domain by their transfer function. Obviously, the unit-impulse response sequence of a causal LTI system could be of either finite or infinite duration and this property determines their classification into either finite impulse response (FIR) or infinite impulse response (IIR) system.
    [Show full text]
  • Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli
    Finite Impulse Response (FIR) Digital Filters (II) Ideal Impulse Response Design Examples Yogananda Isukapalli 1 • FIR Filter Design Problem Given H(z) or H(ejw), find filter coefficients {b0, b1, b2, ….. bN-1} which are equal to {h0, h1, h2, ….hN-1} in the case of FIR filters. 1 z-1 z-1 z-1 z-1 x[n] h0 h1 h2 h3 hN-2 hN-1 1 1 1 1 1 y[n] Consider a general (infinite impulse response) definition: ¥ H (z) = å h[n] z-n n=-¥ 2 From complex variable theory, the inverse transform is: 1 n -1 h[n] = ò H (z)z dz 2pj C Where C is a counterclockwise closed contour in the region of convergence of H(z) and encircling the origin of the z-plane • Evaluating H(z) on the unit circle ( z = ejw ) : ¥ H (e jw ) = åh[n]e- jnw n=-¥ 1 p h[n] = ò H (e jw )e jnwdw where dz = jejw dw 2p -p 3 • Design of an ideal low pass FIR digital filter H(ejw) K -2p -p -wc 0 wc p 2p w Find ideal low pass impulse response {h[n]} 1 p h [n] = H (e jw )e jnwdw LP ò 2p -p 1 wc = Ke jnwdw 2p ò -wc Hence K h [n] = sin(nw ) n = 0, ±1, ±2, …. ±¥ LP np c 4 Let K = 1, wc = p/4, n = 0, ±1, …, ±10 The impulse response coefficients are n = 0, h[n] = 0.25 n = ±4, h[n] = 0 = ±1, = 0.225 = ±5, = -0.043 = ±2, = 0.159 = ±6, = -0.053 = ±3, = 0.075 = ±7, = -0.032 n = ±8, h[n] = 0 = ±9, = 0.025 = ±10, = 0.032 5 Non Causal FIR Impulse Response We can make it causal if we shift hLP[n] by 10 units to the right: K h [n] = sin((n -10)w ) LP (n -10)p c n = 0, 1, 2, ….
    [Show full text]
  • ELEG 5173L Digital Signal Processing Ch. 5 Digital Filters
    Department of Electrical Engineering University of Arkansas ELEG 5173L Digital Signal Processing Ch. 5 Digital Filters Dr. Jingxian Wu [email protected] 2 OUTLINE • FIR and IIR Filters • Filter Structures • Analog Filters • FIR Filter Design • IIR Filter Design 3 FIR V.S. IIR • LTI discrete-time system – Difference equation in time domain N M y(n) ak y(n k) bk x(n k) k 1 k 0 – Transfer function in z-domain N M k k Y (z) akY (z)z bk X (z)z k 1 k 0 M k bk z Y (z) k 0 H (z) N X (z) k 1 ak z k 1 4 FIR V.S. IIR • Finite impulse response (FIR) – difference equation in the time domain M y(n) bk x(n k) k 0 – Transfer function in the Z-domain M Y (z) k H (z) bk z X (z) k 0 – Impulse response h(n) [b ,b ,,b ] 0 1 M • The impulse response is of finite length finite impulse response 5 FIR V.S. IIR • Infinite impulse response (IIR) – Difference equation in the time domain N M y(n) ak y(n k) bk x(n k) k1 k0 – Transfer function in the z-domain M k bk z Y(z) k0 H (z) N X (z) k 1 ak z k1 – Impulse response can be obtained through inverse-z transform, and it has infinite length 6 FIR V.S. IIR • Example – Find the impulse response of the following system. Is it a FIR or IIR filter? Is it stable? 1 y(n) y(n 2) x(n) 4 7 FIR V.S.
    [Show full text]
  • Digital Signal Processing Filter Design
    2065-27 Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis 26 October - 20 November, 2009 Digital Signal Processing Filter Design Massimiliano Nolich DEEI Facolta' di Ingegneria Universita' degli Studi di Trieste via Valerio, 10, 34127 Trieste Italy Filter design 1 Design considerations: a framework |H(f)| 1 C ıp 1 ıp ıs f 0 fp fs Passband Transition Stopband band The design of a digital filter involves five steps: Specification: The characteristics of the filter often have to be specified in the frequency domain. For example, for frequency selective filters (lowpass, highpass, bandpass, etc.) the specification usually involves tolerance limits as shown above. Coefficient calculation: Approximation methods have to be used to calculate the values hŒk for a FIR implementation, or ak, bk for an IIR implementation. Equivalently, this involves finding a filter which has H.z/ satisfying the requirements. Realisation: This involves converting H.z/ into a suitable filter structure. Block or flow diagrams are often used to depict filter structures, and show the computational procedure for implementing the digital filter. 1 Analysis of finite wordlength effects: In practice one should check that the quantisation used in the implementation does not degrade the performance of the filter to a point where it is unusable. Implementation: The filter is implemented in software or hardware. The criteria for selecting the implementation method involve issues such as real-time performance, complexity, processing requirements, and availability of equipment. 2 Finite impulse response (FIR) filter design A FIR filter is characterised by the equations N 1 yŒn D hŒkxŒn k kXD0 N 1 H.z/ D hŒkzk: kXD0 The following are useful properties of FIR filters: They are always stable — the system function contains no poles.
    [Show full text]
  • Lecture 2 - Digital Filters
    Lecture 2 - Digital Filters 2.1 Digital filtering Digital filtering can be implemented either in hardware or software; in the first case, the numerical processor is either a special-purpose chip or it is assembled out of a set of digital integrated circuits which provide the essential building blocks of a digital filtering operation – storage, delay, addition/subtraction and multiplication by constants. On the other hand, a general-purpose mini-or micro-computer can also be programmed as a digital filter, in which case the numerical processor is the computer’s CPU, GPU and memory. Filtering may be applied to signals which are then transformed back to the analogue domain using a digital to analogue convertor or worked with entirely in the digital domain. 33 2.1.1 Reasons for using a digital rather than an analogue filter The numerical processor can easily be (re-)programmed to implement a num- • ber of different filters. The accuracy of a digital filter is dependent only on the round-off error in • the arithmetic. This has two advantages: – the accuracy is predictable and hence the performance of the digital signal processing algorithm is known apriori. – The round-off error can be minimized with appropriate design techniques and hence digital filters can meet very tight specifications on magnitude and phase characteristics (which would be almost impossible to achieve with analogue filters because of component tolerances and circuit noise). The widespread use of mini- and micro-computers in engineering has greatly • increased the number of digital signals recorded and processed. Power sup- ply and temperature variations have no effect on a programme stored in a computer.
    [Show full text]
  • Digital Signal Processing Tom Davinson
    First SPES School on Experimental Techniques with Radioactive Beams INFN LNS Catania – November 2011 Experimental Challenges Lecture 4: Digital Signal Processing Tom Davinson School of Physics & Astronomy N I VE R U S I E T H Y T O H F G E D R I N B U Objectives & Outline Practical introduction to DSP concepts and techniques Emphasis on nuclear physics applications I intend to keep it simple … … even if it’s not … … I don’t intend to teach you VHDL! • Sampling Theorem • Aliasing • Filtering? Shaping? What’s the difference? … and why do we do it? • Digital signal processing • Digital filters semi-gaussian, moving window deconvolution • Hardware • To DSP or not to DSP? • Summary • Further reading Sampling Sampling Periodic measurement of analogue input signal by ADC Sampling Theorem An analogue input signal limited to a bandwidth fBW can be reproduced from its samples with no loss of information if it is regularly sampled at a frequency fs 2fBW The sampling frequency fs= 2fBW is called the Nyquist frequency (rate) Note: in practice the sampling frequency is usually >5x the signal bandwidth Aliasing: the problem Continuous, sinusoidal signal frequency f sampled at frequency fs (fs < f) Aliasing misrepresents the frequency as a lower frequency f < 0.5fs Aliasing: the solution Use low-pass filter to restrict bandwidth of input signal to satisfy Nyquist criterion fs 2fBW Digital Signal Processing D…igi wtahl aSt ignneaxtl? Processing Digital signal processing is the software controlled processing of sequential data derived from a digitised analogue signal. Some of the advantages of digital signal processing are: • functionality possible to implement functions which are difficult, impractical or impossible to achieve using hardware, e.g.
    [Show full text]
  • Lecture 5 - Digital Filters
    Lecture 5 - Digital Filters 5.1 Digital filtering Digital filtering can be implemented either in hardware or software; in the first case, the numerical processor is either a special-purpose chip or it is assembled out of a set of digital integrated circuits which provide the essential building blocks of a digital filtering operation – storage, delay, addition/subtraction and multiplica- tion by constants. On the other hand, a general-purpose mini-or micro-computer can also be programmed as a digital filter, in which case the numerical processor is the computer’s CPU and memory. Filtering may be applied to signals which are then transformed back to the analogue domain using a digital to analogue convertor or worked with entirely in the digital domain (e.g. biomedical signals are digitised, filtered and analysed to detect some abnormal activity). 5.1.1 Reasons for using a digital rather than an analogue filter The numerical processor can easily be (re-)programmed to implement a number of different filters. The accuracy of a digital filter is dependent only on the round-off error in the arithmetic. This has two advantages: – the accuracy is predictable and hence the performance of the digital signal processing algorithm is known a priori. – The round-off error can be minimized with appropriate design tech- niques and hence digital filters can meet very tight specifications on magnitude and phase characteristics (which would be almost impossi- ble to achieve with analogue filters because of component tolerances and circuit noise). 59 The widespread use of mini- and micro-computers in engineering has greatly increased the number of digital signals recorded and processed.
    [Show full text]
  • Oversampling with Noise Shaping • Place the Quantizer in a Feedback Loop Xn() Yn() Un() Hz() – Quantizer
    Oversampling Converters David Johns and Ken Martin University of Toronto ([email protected]) ([email protected]) University of Toronto slide 1 of 57 © D.A. Johns, K. Martin, 1997 Motivation • Popular approach for medium-to-low speed A/D and D/A applications requiring high resolution Easier Analog • reduced matching tolerances • relaxed anti-aliasing specs • relaxed smoothing filters More Digital Signal Processing • Needs to perform strict anti-aliasing or smoothing filtering • Also removes shaped quantization noise and decimation (or interpolation) University of Toronto slide 2 of 57 © D.A. Johns, K. Martin, 1997 Quantization Noise en() xn() yn() xn() yn() en()= yn()– xn() Quantizer Model • Above model is exact — approx made when assumptions made about en() • Often assume en() is white, uniformily distributed number between ±∆ ⁄ 2 •∆ is difference between two quantization levels University of Toronto slide 3 of 57 © D.A. Johns, K. Martin, 1997 Quantization Noise T ∆ time • White noise assumption reasonable when: — fine quantization levels — signal crosses through many levels between samples — sampling rate not synchronized to signal frequency • Sample lands somewhere in quantization interval leading to random error of ±∆ ⁄ 2 University of Toronto slide 4 of 57 © D.A. Johns, K. Martin, 1997 Quantization Noise • Quantization noise power shown to be ∆2 ⁄ 12 and is independent of sampling frequency • If white, then spectral density of noise, Se()f , is constant. S f e() ∆ 1 Height k = ---------- ---- x f 12 s f f f s 0 s –---- ---- 2 2 University of Toronto slide 5 of 57 © D.A. Johns, K. Martin, 1997 Oversampling Advantage • Oversampling occurs when signal of interest is bandlimited to f0 but we sample higher than 2f0 • Define oversampling-rate OSR = fs ⁄ ()2f0 (1) • After quantizing input signal, pass it through a brickwall digital filter with passband up to f0 y ()n 1 un() Hf() y2()n N-bit quantizer Hf() 1 f f fs –f 0 f s –---- 0 0 ---- 2 2 University of Toronto slide 6 of 57 © D.A.
    [Show full text]
  • Oversampling Techniques Using the Tms320c24x Family
    Oversampling Techniques using the TMS320C24x Family Literature Number: SPRA461 Texas Instruments Europe June 1998 IMPORTANT NOTICE Texas Instruments and its subsidiaries (TI) reserve the right to make changes to their products or to discontinue any product or service without notice, and advise customers to obtain the latest version of relevant information to verify, before placing orders, that in- formation being relied on is current and complete. All products are sold subject to the terms and conditions of sale supplied at the time of order acknowledgement, including those pertaining to warranty, patent infringement, and limitation of liability. TI warrants performance of its semiconductor products to the specifications applicable at the time of sale in accordance with TI’s standard warranty. Testing and other quality control techniques are utilized to the extent TI deems necessary to support this warranty. Specific testing of all parameters of each device is not necessarily performed, except those mandated by government requirements. CERTAIN APPLICATIONS USING SEMICONDUCTOR PRODUCTS MAY INVOLVE POTENTIAL RISKS OF DEATH, PERSONAL INJURY, OR SEVERE PROPERTY OR ENVIRONMENTAL DAMAGE (“CRITICAL APPLICATIONS”). TI SEMICONDUCTOR PRODUCTS ARE NOT DESIGNED, AUTHORIZED, OR WARRANTED TO BE SUITA- BLE FOR USE IN LIFE-SUPPORT DEVICES OR SYSTEMS OR OTHER CRITICAL AP- PLICATIONS. INCLUSION OF TI PRODUCTS IN SUCH APPLICATIONS IS UNDERS- TOOD TO BE FULLY AT THE CUSTOMER’S RISK. In order to minimize risks associated with the customer’s applications, adequate design and operating safeguards must be provided by the customer to minimize inherent or pro- cedural hazards. TI assumes no liability for applications assistance or customer product design.
    [Show full text]
  • The Z-Transform
    CHAPTER 33 The z-Transform Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the z-transform. The overall strategy of these two transforms is the same: probe the impulse response with sinusoids and exponentials to find the system's poles and zeros. The Laplace transform deals with differential equations, the s-domain, and the s-plane. Correspondingly, the z-transform deals with difference equations, the z-domain, and the z-plane. However, the two techniques are not a mirror image of each other; the s-plane is arranged in a rectangular coordinate system, while the z-plane uses a polar format. Recursive digital filters are often designed by starting with one of the classic analog filters, such as the Butterworth, Chebyshev, or elliptic. A series of mathematical conversions are then used to obtain the desired digital filter. The z-transform provides the framework for this mathematics. The Chebyshev filter design program presented in Chapter 20 uses this approach, and is discussed in detail in this chapter. The Nature of the z-Domain To reinforce that the Laplace and z-transforms are parallel techniques, we will start with the Laplace transform and show how it can be changed into the z- transform. From the last chapter, the Laplace transform is defined by the relationship between the time domain and s-domain signals: 4 X(s) ' x(t) e &st dt m t '&4 where x(t) and X(s) are the time domain and s-domain representation of the signal, respectively.
    [Show full text]
  • Eindhoven University of Technology MASTER Deconvolution Of
    Eindhoven University of Technology MASTER Deconvolution of radiometric imaging data by means of spatial filter design Lennartz, C.H.F. Award date: 1995 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain EINDHOVEN UNIVERSITY OF TECHNOLOGY FACULTY OF ELECTRICAL ENGINEERING TELECOMMUNICATIONS DIVISION Deconvolution ofradiometric imaging data by means ofspatial filter design by C.H.F. Lennartz Report of graduation work, performed from November 1992 to August 1993. Supervisors: profdr. ir. G. Brussaard dr.ir. M.H.A.J. Herben dr. ir. P.J.I. de Maagt The Faculty ofElectrical Engineering ofthe Eindhoven University of Technology disclaims all responsibility for the contents of training and graduation reports.
    [Show full text]