DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

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A Signal Processing Perspective 305 where the zeros, and the poles, may be, and in general are, complex, then consider the results of the lowpass-to-bandstop transformation. Case I Consider the case where is real. To determine the corresponding bandstop filter poles, let It follows that or the two poles corresponding to the one lowpass prototype pole, are Theoretically, could be real however such will not be the case for practical bandstop filters. Therefore, consider Also, note that since the numerator as well as the denominator of the transfer function is multiplied by in order to form the above quadratic equation, it follows that two transfer function zeros are introduced at Therefore, each real pole in will yield a pair of complex conjugate poles in . as developed above, and two zeros at Note that if > > then the imaginary part of those complex conjugate poles will be approximately Example 9.17 Suppose and It follows that will be the resultant poles, and there will two zeros at Section 9.4 Lowpass-to-Bandstop Transformation .
306 DESIGN AND ANALYSIS OF ANALOG FILTERS: Case II Consider the case where is imaginary, that is, where there are zeros on the axis, such as is the case for a Chebyshev Type II or elliptic lowpass prototype. To make the imaginary property of explicit, let where is real. To determine the corresponding bandstop filter zeros, let It follows that or the two zeros corresponding to the one lowpass prototype zero, are where it is noted that both zeros are imaginary, but not the same magnitude (not conjugates). However, if the lowpass prototype has a zero at there will also be a zero at Therefore, the four bandstop zeros corresponding to the two lowpass prototype zeros at are as follows: Also, note that since the denominator as well as the numerator is multiplied by for each zero in the lowpass prototype transfer function in order to form the above quadratic equations, it follows that four transfer function poles are introduced at However, it should be noted that these imaginary axis poles will cancel imaginary axis zeros at introduced by real poles (or complexconjugate poles as discussed below) in the lowpass prototype transfer function. Therefore, each pair of conjugate imaginary zeros in will yield two pair of conjugate imaginary zeros in as developed above, and two pair of conjugate imaginary poles at Example 9.18 Suppose and there are two lowpass prototype zeros located at ±j 3. It follows that the corresponding bandstop filter zeros will be as follows: ± j 4.68, and ± j 5.34, and there will also be two conjugate imaginary poles introduced at s = ± j 5. Chapter 9 Frequency Transformations
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DESIGN AND ANALYSIS OF ANALOG FILTE
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Design and Analysis of A Signal Pro
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PREFACE nalog filters, that is cont
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The term “approximation” is use
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PREFACE Chapter 1. 2. INTRODUCTION
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6.7 6.8 6.9 6.10 6.11 6.12 TABLE OF
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CHAPTER 1 INTRODUCTION Iimportance
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114 DESIGN AND ANALYSIS OF ANALOG F
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130 (f) (g) (h) DESIGN AND ANALYSIS
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CHAPTER 5 CHEBYSHEV TYPE II FILTERS
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6. 1 INTRODUCTION CHAPTER 6 ELLIPTI
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CHAPTER 7 BESSEL FILTERS Although a
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CHAPTER 8 OTHER FILTERS In this cha
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APPENDIX B CONTENTS OF THE ACCOMPAN
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APPENDIX D THE MATLAB m-FILE EXAMP6
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A Active filters, 359-390 advantage
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