A simple procedure, applicable to any high order transfer function (n≥3), is described, which generates a family of transfer functions possessing identical phase/frequency responses but distinct gain/frequency responses. By application to a 10th-order all-pole Bessel function, it is shown that selected members of this particular family have significantly wider gain-bandwidths and correspondingly improved time domain behaviour.

Phase invariant delay approximations

CHAPTER 1 FILTERS IN ELECTRONICS 11 Types of Filters 12 Filter Applications 13 All-Pass Filters 14 Properties of Lattice Filters 15 Filter Building Blocks 16 Higher Order Filters 17 Coil-Saving Bandpass Filters 18 Frequency Range of Applications 19 Physical Elements of the Filter 110 Active Bandpass Filters 111 RC Passive and Active Filters 112 Microwave Filters 113 Parametric Filters CHAPTER 2 THEORY OF EFFECTIVE PARAMETERS 21 Power Balance 22 Types of General Network Equations 23 Effective Attenuation 24 Reflective (Echo) Attenuation 25 Transmission Function as a Function of Frequency Parameter, s 26 Polynomials of Transmission and Filtering Functions 27 Filter Networks 28 Voltage and Current Sources 29 The Function D (s) As An Approximation Function 210 Example of Transmission Function Approximation 211 Simplest Polynomial Filters in Algebraic Form 212 Introduction to Image=Parameter Theory 213 Bridge Networks 214 Examples of Realization in the Bridge Form 215 Hurwitz Polynomial 216 The Smallest Realizabel Networks, 217 Fourth-Order Networks 218 Fifth-Order Networks CHAPTER 3 FILTER CHARACTERISTICS IN THE FREQUENCY DOMAIN 31 Amplitude Responses 32 Phase-and Group-Delay Responses 33 Group Delay of an Idealized Filter 34 Group Delay-Attenuation Relationship 35 The Chebyshev Family of Response Characteristics 36 Gaussian Family of Response Characteristics 37 A Filter with Transitional Magnitude Characteristics 38 Legendre Filters 39 Minimum-Loss Characteristics 310 Synchronously Tuned Filters 311 Arithmetically Symmetrical Bandpass Filters 312 Attenuation Characteristics of Image Parameter Filters 313 Other Types of Filter Characteristics 314 Plots of the Attenuation and Group Delay Characteristics CHAPTER 4 ELLIPTIC FUNCTION AND ELEMENTS OF REALIZATION 41 Double Periodic Elliptic Functions 42 Mapping of s-Plane into u-Plane 43 First Basic Transformation of Elliptic Functions 44 Filtering Function in z-Plane 45 Graphical Representation of Parameters 46 Characteristic Value of D(s) 47 An Example of Filter Design 48 Consideration of Losses 49 Introduction of Losses by Frequency Transformation 410 Highpass Filters with Losses 411 Transmission Functions with Losses 412 Conclusions on Consideration of Losses 413 Realization Process 414 Bandpass Filter with a Minimum Number of Inductors 415 The Elements of a Coil-Saving Network 416 Consideration of Losses in Zig-Zag Filters 417 Realization Procedure 418 Numerical Example of Realization 419 Full and Partial Removal for a Fifth-Order Filter CHAPTER 5 THE CATALOG OF NORMALIZED LOWPASS FILTERS 51 Introduction to the Catalog 52 Real Part of the Driving Point Impedance 53 Lowpass Filter Design 54 Design of Highpass Filters 55 Design of LC Bandpass Filters 56 Design of Narrowband Crystal Filters 57 Design of Bandstop Filters 58 Catalog of Normalized Lowpass Models CHAPTER 6 DESIGN TECHNIQUES FOR POLYNOMIAL FILTERS 61 Introduction to Tables of Normalized Element Values 62 Lowpass Design Examples 63 Bandpass Filter Design 64 Concept of Coupling 65 Coupled Resonators 66 Second-Order Bandpass Filter 67 Design with Tables of Predistorted k and q Parameters 68 Design Examples using Tables of k and q Values 69 Tables of Lowpass Element Values 610 Tables of 3-dB Down k and q Values CHAPTER 7 FILTER CHARACTERISTICS IN THE TIME DOMAIN 71 Introduction to Transient Characteristics 72 Time and Frequency Domains 73 Information Contained in the Impulse Response 74 Step Response 75 Impulse Response of an Ideal Gaussian Filter 76 Residue Determination 77 Numerical Example 78 Practical Steps on the Inverse Transformation 79 Inverse Transform of Rational Spectral Functions 710 Numerical Example 711 Estimation Theory 712 Transient Response in Highpass and Bandpass Filters 713 The Exact Calculation of Transient Phenomena for Highpass Systems 714 Estimate of Transient Responses in Narrowband Filters 715 The Exact Transient Calculation in Narrowband Systems 716 Group Delay Versus Transient Response 717 Computer Determination of Filter Impulse Response 718 Transient Response Curves CHAPTER 8 CRYSTAL FILTERS 81 Introduction 82 Crystal Structure 83 Theory of Piezoelectricity 84 Properties of Piezoelectric Quartz Crystals 85 Classification of Crystal Filters 86 Bridge Filters 87 Limitation of Bridge Crystal Filters 88 Spurious Response 89 Circuit Analysis of a Simple Filter 810 Element Values in Image-Parameter Formulation 811 Ladder Filters 812 Effective Attenuation of Simple Filters 813 Effective Attenuation of Ladder Networks 814 Ladder Versus Bridge Filters 815 Practical Differential Transformer for Crystal Filters 816 Design of Narrowband Filters with the Aid or Lowpass Model 817 Synthesis of Ladder Single Sideband Filters 818 The Synthesis of Intermediate Bandpass Filters 819 Example of Band-Reject Filter 820 Ladder Filters with Large Bandwidth CHAPTER 9 HELICAL FILTERS 91 Introduction 92 Helical Resonators 93 Filter with Helical Resonators 94 Alignment of Helical Filters 95 Examples of Helical Filtering CHAPTER 10 NETWORK TRANSFORMATIONS 10 1 Two-Terminal Network Transformations 102 Delta-Star Transformation 103 Use of Transformer in Filter Realization 104 Norton's Transformation 105 Applications of Mutual Inductive Coupling 106 The Realization of LC Filters with Crystal Resonators 107 Negative and Positive Capacitor Tranformation 108 Bartlett's Bisection Theorem 109 Cauer's Equivalence 1010 Canonic Bandpass Structures 1011 Bandpass Ladder Filters Having a Cononical Number of Inductors without Mutual Coupling 1012 Impedance and Admittance Inverters 1013 Source and Load Transformation Bibliography Index

Handbook of Filter Synthesis

Approximation methods for electronic filter design

A realizable, low-pass, minimum-phase transfer function is constructed which produces "maximally flat" time delay as a function of frequency, and which is associated with an impulse response that resembles closely a Gaussian curve properly delayed with respect to t= 0. The delay and loss characteristics are particularly easy to evaluate, since this transfer function can be expressed in terms of conveniently tabulated functions. It is shown how to realize this transfer function as the transfer impedance of a ladder network containing only low-Q elements, and the response of this network is put into perspective to those of conventionally designed delay networks. Also, a class of nonminimum-phase transfer functions is generated from the above transfer functions in order to offer added flexibility. The delay and loss characteristics of the nonminimum-phase transfer functions are described, but their network realization is not considered in this paper.

Synthesis of Constant-Time-Delay Ladder Networks Using Bessel Polynomials

A lumped-constant equivalent of a transmission line can be obtained in general in the form of a symmetrical lattice, in which the series and lattice arms are inverse and approximate respectively to the short-circuit and open-circuit impedances of half the line. One such set of approximations can be derived from the infinite ladder networks (Cauer's canonical form) equivalent to these impedances. These approximations produce all-pass constant-impedance networks (dissipation being neglected) in which the delay is maximally flat in the sense that the first 2m ? 1 derivatives of the delay with respect to frequency are zero at the origin; m is an integer expressing the order of the approximation.

Phase invariant delay approximations

References

Handbook of Filter Synthesis

Approximation methods for electronic filter design

Synthesis of Constant-Time-Delay Ladder Networks Using Bessel Polynomials

Delay networks having maximally flat frequency characteristics

The analysis, design, and synthesis of electrical filters

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