Section G.4. MULTISECTION COMPLEX FSF FREQUENCY RESPONSE

G 4 MULTISECTION COMPLEX FSF FREQUENCY RESPONSE

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 One. Discrete Sequences and Systems
• DISCRETE SEQUENCES AND THEIR NOTATION
• SIGNAL AMPLITUDE, MAGNITUDE, POWER
• SIGNAL PROCESSING OPERATIONAL SYMBOLS
• INTRODUCTION TO DISCRETE LINEAR TIME-INVARIANT SYSTEMS
• DISCRETE LINEAR SYSTEMS
• TIME-INVARIANT SYSTEMS
• THE COMMUTATIVE PROPERTY OF LINEAR TIME-INVARIANT SYSTEMS
• ANALYZING LINEAR TIME-INVARIANT SYSTEMS
• REFERENCES

Chapter Two. Periodic Sampling

• Chapter Two. Periodic Sampling
• ALIASING: SIGNAL AMBIGUITY IN THE FREQUENCY DOMAIN
• SAMPLING LOW-PASS SIGNALS
• SAMPLING BANDPASS SIGNALS
• SPECTRAL INVERSION IN BANDPASS SAMPLING
• REFERENCES

Chapter Three. The Discrete Fourier Transform

• Chapter Three. The Discrete Fourier Transform
• UNDERSTANDING THE DFT EQUATION
• DFT SYMMETRY
• DFT LINEARITY
• DFT MAGNITUDES
• DFT FREQUENCY AXIS
• DFT SHIFTING THEOREM
• INVERSE DFT
• DFT LEAKAGE
• WINDOWS
• DFT SCALLOPING LOSS
• DFT RESOLUTION, ZERO PADDING, AND FREQUENCY-DOMAIN SAMPLING
• DFT PROCESSING GAIN
• THE DFT OF RECTANGULAR FUNCTIONS
• THE DFT FREQUENCY RESPONSE TO A COMPLEX INPUT
• THE DFT FREQUENCY RESPONSE TO A REAL COSINE INPUT
• THE DFT SINGLE-BIN FREQUENCY RESPONSE TO A REAL COSINE INPUT
• INTERPRETING THE DFT
• REFERENCES

Chapter Four. The Fast Fourier Transform

• Chapter Four. The Fast Fourier Transform
• RELATIONSHIP OF THE FFT TO THE DFT
• HINTS ON USING FFTS IN PRACTICE
• FFT SOFTWARE PROGRAMS
• DERIVATION OF THE RADIX-2 FFT ALGORITHM
• FFT INPUT/OUTPUT DATA INDEX BIT REVERSAL
• RADIX-2 FFT BUTTERFLY STRUCTURES
• REFERENCES

Chapter Five. Finite Impulse Response Filters

• Chapter Five. Finite Impulse Response Filters
• AN INTRODUCTION TO FINITE IMPULSE RESPONSE (FIR) FILTERS
• CONVOLUTION IN FIR FILTERS
• LOW-PASS FIR FILTER DESIGN
• BANDPASS FIR FILTER DESIGN
• HIGHPASS FIR FILTER DESIGN
• REMEZ EXCHANGE FIR FILTER DESIGN METHOD
• HALF-BAND FIR FILTERS
• PHASE RESPONSE OF FIR FILTERS
• A GENERIC DESCRIPTION OF DISCRETE CONVOLUTION
• REFERENCES

Chapter Six. Infinite Impulse Response Filters

• Chapter Six. Infinite Impulse Response Filters
• AN INTRODUCTION TO INFINITE IMPULSE RESPONSE FILTERS
• THE LAPLACE TRANSFORM
• THE Z-TRANSFORM
• IMPULSE INVARIANCE IIR FILTER DESIGN METHOD
• BILINEAR TRANSFORM IIR FILTER DESIGN METHOD
• OPTIMIZED IIR FILTER DESIGN METHOD
• PITFALLS IN BUILDING IIR FILTERS
• IMPROVING IIR FILTERS WITH CASCADED STRUCTURES
• A BRIEF COMPARISON OF IIR AND FIR FILTERS
• REFERENCES

Chapter Seven. Specialized Lowpass FIR Filters

• Chapter Seven. Specialized Lowpass FIR Filters
• FREQUENCY SAMPLING FILTERS: THE LOST ART
• INTERPOLATED LOWPASS FIR FILTERS
• REFERENCES

Chapter Eight. Quadrature Signals

• Chapter Eight. Quadrature Signals
• WHY CARE ABOUT QUADRATURE SIGNALS?
• THE NOTATION OF COMPLEX NUMBERS
• REPRESENTING REAL SIGNALS USING COMPLEX PHASORS
• A FEW THOUGHTS ON NEGATIVE FREQUENCY
• QUADRATURE SIGNALS IN THE FREQUENCY DOMAIN
• BANDPASS QUADRATURE SIGNALS IN THE FREQUENCY DOMAIN
• COMPLEX DOWN-CONVERSION
• A COMPLEX DOWN-CONVERSION EXAMPLE
• AN ALTERNATE DOWN-CONVERSION METHOD
• REFERENCES

Chapter Nine. The Discrete Hilbert Transform

• Chapter Nine. The Discrete Hilbert Transform
• HILBERT TRANSFORM DEFINITION
• WHY CARE ABOUT THE HILBERT TRANSFORM?
• IMPULSE RESPONSE OF A HILBERT TRANSFORMER
• DESIGNING A DISCRETE HILBERT TRANSFORMER
• TIME-DOMAIN ANALYTIC SIGNAL GENERATION
• COMPARING ANALYTIC SIGNAL GENERATION METHODS
• REFERENCES

Chapter Ten. Sample Rate Conversion

• Chapter Ten. Sample Rate Conversion
• DECIMATION
• INTERPOLATION
• COMBINING DECIMATION AND INTERPOLATION
• POLYPHASE FILTERS
• CASCADED INTEGRATOR-COMB FILTERS
• REFERENCES

Chapter Eleven. Signal Averaging

• Chapter Eleven. Signal Averaging
• COHERENT AVERAGING
• INCOHERENT AVERAGING
• AVERAGING MULTIPLE FAST FOURIER TRANSFORMS
• FILTERING ASPECTS OF TIME-DOMAIN AVERAGING
• EXPONENTIAL AVERAGING
• REFERENCES

Chapter Twelve. Digital Data Formats and Their Effects

• Chapter Twelve. Digital Data Formats and Their Effects
• FIXED-POINT BINARY FORMATS
• BINARY NUMBER PRECISION AND DYNAMIC RANGE
• EFFECTS OF FINITE FIXED-POINT BINARY WORD LENGTH
• FLOATING-POINT BINARY FORMATS
• BLOCK FLOATING-POINT BINARY FORMAT
• REFERENCES

Chapter Thirteen. Digital Signal Processing Tricks

• Chapter Thirteen. Digital Signal Processing Tricks
• FREQUENCY TRANSLATION WITHOUT MULTIPLICATION
• HIGH-SPEED VECTOR MAGNITUDE APPROXIMATION
• FREQUENCY-DOMAIN WINDOWING
• FAST MULTIPLICATION OF COMPLEX NUMBERS
• EFFICIENTLY PERFORMING THE FFT OF REAL SEQUENCES
• COMPUTING THE INVERSE FFT USING THE FORWARD FFT
• SIMPLIFIED FIR FILTER STRUCTURE
• REDUCING A/D CONVERTER QUANTIZATION NOISE
• A/D CONVERTER TESTING TECHNIQUES
• FAST FIR FILTERING USING THE FFT
• GENERATING NORMALLY DISTRIBUTED RANDOM DATA
• ZERO-PHASE FILTERING
• SHARPENED FIR FILTERS
• INTERPOLATING A BANDPASS SIGNAL
• SPECTRAL PEAK LOCATION ALGORITHM
• COMPUTING FFT TWIDDLE FACTORS
• SINGLE TONE DETECTION
• THE SLIDING DFT
• THE ZOOM FFT
• A PRACTICAL SPECTRUM ANALYZER
• AN EFFICIENT ARCTANGENT APPROXIMATION
• FREQUENCY DEMODULATION ALGORITHMS
• DC REMOVAL
• IMPROVING TRADITIONAL CIC FILTERS
• SMOOTHING IMPULSIVE NOISE
• EFFICIENT POLYNOMIAL EVALUATION
• DESIGNING VERY HIGH-ORDER FIR FILTERS
• TIME-DOMAIN INTERPOLATION USING THE FFT
• FREQUENCY TRANSLATION USING DECIMATION
• AUTOMATIC GAIN CONTROL (AGC)
• APPROXIMATE ENVELOPE DETECTION
• A QUADRATURE OSCILLATOR
• DUAL-MODE AVERAGING
• REFERENCES

Appendix A. The Arithmetic of Complex Numbers

• Appendix A. The Arithmetic of Complex Numbers
• Section A.1. GRAPHICAL REPRESENTATION OF REAL AND COMPLEX NUMBERS
• Section A.2. ARITHMETIC REPRESENTATION OF COMPLEX NUMBERS
• Section A.3. ARITHMETIC OPERATIONS OF COMPLEX NUMBERS
• Section A.4. SOME PRACTICAL IMPLICATIONS OF USING COMPLEX NUMBERS
• REFERENCES

Appendix B. Closed Form of a Geometric Series

• Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

• Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

• Appendix D. Mean, Variance, and Standard Deviation
• Section D.1. STATISTICAL MEASURES
• Section D.2. STANDARD DEVIATION, OR RMS, OF A CONTINUOUS SINEWAVE
• Section D.3. THE MEAN AND VARIANCE OF RANDOM FUNCTIONS
• Section D.4. THE NORMAL PROBABILITY DENSITY FUNCTION
• REFERENCES

Appendix E. Decibels (dB and dBm)

• Appendix E. Decibels (dB and dBm)
• Section E.1. USING LOGARITHMS TO DETERMINE RELATIVE SIGNAL POWER
• Section E.2. SOME USEFUL DECIBEL NUMBERS
• Section E.3. ABSOLUTE POWER USING DECIBELS

Appendix F. Digital Filter Terminology

• Appendix F. Digital Filter Terminology
• REFERENCES

Appendix G. Frequency Sampling Filter Derivations

• Appendix G. Frequency Sampling Filter Derivations
• Section G.1. FREQUENCY RESPONSE OF A COMB FILTER
• Section G.2. SINGLE COMPLEX FSF FREQUENCY RESPONSE
• Section G.3. MULTISECTION COMPLEX FSF PHASE
• Section G.4. MULTISECTION COMPLEX FSF FREQUENCY RESPONSE
• Section G.5. REAL FSF TRANSFER FUNCTION
• Section G.6. TYPE-IV FSF FREQUENCY RESPONSE

Appendix H. Frequency Sampling Filter Design Tables

• Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing (2nd Edition)

ISBN: 0131089897
EAN: 2147483647

Year: 2004
Pages: 183

Authors: Richard G. Lyons

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