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Digital Filters1Four signals separated by four digital filters0.8Magnitude0.60.40.200 2 4 6 8Hz10 12 14 16x 10 4Figure 11.43. Frequency separations for the signals of Example 11.21In the following example, we will demonstrate the MATLAB filter function that is being used toremove unwanted frequency components from a function. But before we use the filter function,we must design a filter that is capable of removing those unwanted components.Example 11.22In Chapter 7, Subsection 7.4.4, Page 7−20, we found that the half−wave rectifier with no symmetrycan be represented by the trigonometric Fourier seriesft ()=A---π+A--- sint2–A---πcos2t-------------3cos4t cos6t cos8t+ ------------- + ------------- + ------------- + …15 35 63In this example, we want to filter out just the first 2 terms, in other words, to remove all cosineterms. To simplify this expression, we let A = 3π and we truncate it by eliminating all cosineterms except the cos2t term. Then,gt () = 3+1.5sint–cos2t(11.99)The problem now reduces to design a low−pass digital filter, and use the filter command toremove the cosine term in (11.99).Solution:We will use a 6–pole digital low−pass Butterworth filter because we must have a sharp transitionbetween the 1 and 2 rad⁄s frequency range. Also, since the highest frequency componentis 2 rad ⁄ s, to avoid aliasing, we must specify a sampling frequency of ω S = 4 rad ⁄ s. Thus, theSignals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications11−63
Digital FiltersThe analog and digital inputs and outputs are shown in Figures 11.45 and 11.46 respectively.6Function g(t)Filtered Output y(t)4200 2 4 6 8 10 12Continuous Time t64200 2 4 6 8 10 12Continuous Time tFigure 11.45. Continuous time input and output waveforms for Example 11.22Discrete Function g(n*T)Filtered Output y(n*T)64200 2 4 6 8 10 12Discrete Time nT64200 2 4 6 8 10 12Discrete Time nTFigure 11.46. Discrete time input and output waveforms for Example 11.22We conclude this section with one more example to illustrate the use of the MATLAB findfunction. This function displays the subscripts of an array of numbers where a relational expressionis true. For example,x=−2:5; % Display the integers in the range -2 0.8); % Find the subscripts of the numbers for which x > 0.8Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications11−67
Page 7: 4.2 Complex Impedance Z(s).........
Page 10 and 11: 8.6.4 The Transform of ft () = Acos
Page 12 and 13: 11.2.1 RC Low−Pass Filter .......
Page 14 and 15: MATLAB ComputingPages D−3, D−4,
Page 16 and 17: The Unit Step FunctionThus, the pul
Page 18 and 19: Chapter 1 Elementary Signals6. To d
Page 20 and 21: Chapter 2 The Laplace Transformatio
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Page 24 and 25: Chapter 3 The Inverse Laplace Trans
Page 26 and 27: Chapter 4Circuit Analysis with Lapl
Page 28 and 29: Solutions to End−of−Chapter Exe
Page 30 and 31: Chapter 5 State Variables and State
Page 32 and 33: A: [−1 −1; 3/4 0]B: [1 0]’C:
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Page 40 and 41: Chapter 8 The Fourier Transform8.2
Page 42 and 43: Chapter 8 The Fourier TransformExam
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Page 46 and 47: num=[0 1 0 0];% The coefficients of
Page 48 and 49: Chapter 10 The DFT and the FFT Algo
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Page 52 and 53: Basic Analog Filters11.2.3 RLC Band
Page 54 and 55: Chapter 11 Analog and Digital Filte
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Page 60: Appendix E Window Functionsh[n]A(f)
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Page 65 and 66: IndexSymbols Cauchy’s residue the
Page 67 and 68: Fourier transform - properties of f
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