The frequency response of a guaranteed-stable complex N-section FSF, when r < 1, is Hgs,cplx(z) with the z variable in Eq. (7-18) replaced by ejw, giving

To temporarily simplify our expressions, we let q = w – 2pk/N, giving

Equation G-23

Factoring out the half-angled exponentials, and accounting for the r factors, we have

Equation G-24

Converting all the terms inside parentheses to exponentials (we'll see why in a moment), we have

Equation G-25

The algebra gets a little messy here because our exponents have both real and imaginary parts. However, hyperbolic functions to the rescue. Recalling when a is a complex number, sinh(a) = (ea – e–a)/2, we have

Equation G-26

Replacing angle q with w – 2pk/N, canceling the –2 factors, we have

Equation G-27

Rearranging and combining terms we conclude with

Equation G-28

(Whew! Now we see why this frequency response expression is not usually found in the literature.)


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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

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

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