DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

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A Signal Processing Perspective 115 Definition of the magnitude-squared Butterworth response: Note that in (3.6) N is the Butterworth filter order, 1 and the magnitude-squared response is 1/2 when independent of N. This is referred to as the 3 dB frequency. 2 Note, again, that by simple inspection, it is observed that (3.6) satisfies the Analog Filter Design Theorem. Therefore, since the Theorem is satisfied, there exists a transfer function that is time-invariant, causal, linear, rational with real coefficients, and is stable 3 with the magnitude-squared frequency response of (3.6). See Figure 3.1 for plots of (3.6) for several values of N. When plotted as shown, with the vertical scale in dB and a logarithmic frequency scale, the fall-off for is a straight line, and the slope is -N × 20 dB/decade. This is, of course, well known from a study of Bode plots (Siebert, 1986) 4 . Also see Figure 3.2 for detailed plots of (3.6) across the passband; note that larger values of N yield a closer approximation to the ideal lowpass response. 3.2 FILTER SELECTIVITY AND SHAPING FACTOR Applying (2.37), the definition of Filter Selectivity, to the square root of (3.6) results in Therefore, for a Butterworth filter, Filter Selectivity is directly related to filter order N, and inversely related to the 3 dB cutoff frequency 1 In Section 3.4 it is shown that N is the number of poles in H(s). 2 Actually, the attenuation is slightly greater than 3 dB. It is precisely -10 log(1/2). Throughout this book, the term “3 dB” precisely means -10 log(1/2). 3 See Section 2.6 for a discussion of imposed constraints: the constraints are summarized on page 57. Satisfaction of the Analog FilterDesign Theorem,summarized on page 61, implies the imposed constraints are met. 4 See Sections 2.8 and 2.9, especially page 67. Section 3.2 Filter Selectivity and Shaping Factor
116 DESIGN AND ANALYSIS OF ANALOG FILTERS: Let A be an arbitrary attenuation in dB relative to the DC value. From (3.6): For a given A, solving (3.8) for would be equivalent to solving for the bandwidth at that attenuation A: Using (3.9) and applying (2.38), the definition of Shaping Factor, the Butterworth filter Shaping Factor may be readily found: Chapter 3 Butterworth Filters
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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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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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CHAPTER 2 ANALOG FILTER DESIGN AND
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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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272 DESIGN AND ANALYSIS OF ANALOG F
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CHAPTER 10 PASSIVE FILTERS In this
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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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