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Contetlts
xiii
4.3.4. Transmission Line Topology Codes, 83
4.3.5. Transmissjon Line Summary, 85
4.4. Nonadjacent Node Bridging, 86
4.4.1. Derivation of Bridged-T Chain Parameters, 86
4.4.2. A Group Delay Equalizer, 88
4.4.3. Interpolation of Nonadjacent Node-Bridging Current, 89
4.4.4. Summary of Nonadjacent Node Bridging, 91
4.5. fnput and Transfer Network Responses, 91
4.5.]. Jmpedance and Power Response Functions, 91
4.5.2. Scattering Parameters, 92
4.5.3. Wave Response Functions, 94
4.5.4. Conclusion to Network Responses, 96
4.6. Time Response From Frequency Response, 97
4.6.1. Real, Causal Fourier Integrals, 98
4.6.2. Numerical Convolution of Time Functions, 98
4.6.3. Time Response Summary, 100
4.7. Sensitivities, 101
4.7.1. Sensitivity Relationships, 101
4.7.2. Approximate Sensitivity, 103
4,7.3. Exact Partial Derivatives by Tellegen's Theorem, lO4
4.7.4. Summary of Sensitivities, 109
Problems, 109
5, Gradient Optimization
5.1. Quadratic Forms and Ellipsoids, 116
5.1.1. Quadratic Functions, 118
5.1.2. Gradients and Minima, 119
5.1.3. Quadratic Forms and Graphics, J2J
5.1.4. Taylor Series, 123
5.1.5. Newton's Method, 125
5.1.6. Summary of Quadratic Forms and Ellipsoids, 125
5.2. Conjugate Gradient Search, 126
5.2.1. Linear Search, 127
5.2.2. Elementary Search Schemes, 129
5.2.3. More Quadratic Function Properties, )J I
5.2.4. Fletcher-Reeves Conjugate Gradient Search Directions, 134
5.2.5. Summary of Conjugate Gradient Search, 136
5.3. Linear Search, 137
5.3.1. Slope in the Linear Search Direction, J38
5.3.2. Finding the Order of Magnitude of the First Step, 138
5.3.3. Extrapolation, Bounding, and Interpolation, 139
5.3.4. Fletcher's Linear Search Strategy, 141
5.3.5. Summary of Linear Searches, 143
5.4. The Fletcher-Reeves Optimizer, 143
5.4.1. Summary of Fletcher-Reeves Strategy, 144
5.4.2. The BASIC Language Computer Program, 145
5.4.3. The Rosenbrock 'Example, 145
113