The frequency response of a single-section complex FSF is Hss(z) evaluated on the unit circle. We start by substituting ejw for z in Hss(z), because z = ejw defines the unit circle. Given an Hss(z) of

we replace the z terms with ejw, giving

Equation G-7

Factoring out the half-angled exponentials e–jwN/2 and e–j(w/2 – pk/N), we have

Equation G-8

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

Equation G-9

Canceling common factors and rearranging terms in preparation for our final form, we have the desired frequency response of a single-section complex FSF:

Equation G-10

Next we derive the maximum amplitude response of a single-section FSF when its pole is on the unit circle and H(k) = 1. Ignoring those phase shift factors (complex exponentials) in Eq. (G-10), the amplitude response of a single-section FSF is

Equation G-11

We want to know the value of Eq. (G-11) when w = 2pk/N, because that's the value of w at the pole locations, but |Hss(e jw)|w=2pk/N is indeterminate as

Equation G-12

Applying the Marquis de L'Hopital's Rule to Eq. (G-11) yields

Equation G-13

The phase factors in Eq. (G-10), when w = 2pk/N, are

Equation G-14

Combining the result of Eqs. (G-13) and (G-14) with Eq. (G-10), we have

Equation G-15

So the maximum magnitude response of a single-section complex FSF at resonance is |H(k)|N, independent of k.

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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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