The frequency response of a single-section even-N Type-IV FSF is its transfer function evaluated on the unit circle. To begin that evaluation, we set Eq. (7-23)'s |H(k)| = 1, and denote a Type-IV FSF's single-section transfer function as

where the "ss" subscript means single-section. Under the assumption that the damping factor r is so close to unity that it can be replaced with 1, we have the simplified FSF transfer function

Equation G-39

Letting wr = 2pk/N to simplify the notation, and factoring HType-IV,ss(z)'s denominator gives

Equation G-40

in which we replace each z term with ejw, as

Equation G-41

Factoring out the half-angled exponentials, we have

Equation G-42

Using Euler's identity, 2jsin(a) = eja – e–ja, we obtain

Equation G-43

Canceling common factors, and adding like terms, we have

Equation G-44

Plugging 2pk/N back in for wr, the single-section frequency response is

Equation G-45

Based on Eq. (G-45), the frequency response of a multisection even-N Type-IV FSF is

Equation G-46

To determine the amplitude response of a single section, we ignore the phase shift terms (complex exponentials) in Eq. (G-45) to yield

Equation G-47

To find the maximum amplitude response at resonance we evaluate Eq. (G-47) when w = 2pk/N, because that's the value of w at the FSF's pole locations. However, that w causes the denominator to go to zero causing the ratio to go to infinity. We move on with one application of L'Hôpital's Rule to Eq. (G-47) to obtain

Equation G-48

Eliminating the pk terms by using trigonometric reduction formulae sin(pk–a) = (–1)k[-sin(a)] and sin(pk+a) = (–1)k[sin(a)], we have a maximum amplitude response of

Equation G-49

Equation (G-49) is only valid for 1 k (N/2)–1. Disregarding the (–1)k factors, we have a magnitude response at resonance, as a function of k, of

Equation G-50

To find the resonant gain at 0 Hz (DC) we set k = 0 in Eq. (G-47), apply L'Hôpital's Rule (the derivative with respect to w) twice, and set w = 0, giving

Equation G-51

To obtain the resonant gain at s/2 Hz we set k = N/2 in Eq. (G-47), again apply L'Hôpital's Rule twice, and set w = p, yielding

Equation G-52


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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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