The Mathematical Radio: Inside the Magic of AM, FM, and Single-Sideband

How a modern radio works, told through mathematics, history, and selected puzzles The modern radio is a wonder, and behind that magic is mathematics. In The Mathematical Radio , Paul Nahin explains how radios work, deploying mathematics and historical discussion, accompanied by a steady stream of intriguing puzzles for math buffs to ponder. Beginning with oscillators and circuits, then moving on to AM, FM, and single-sideband radio, Nahin focuses on the elegant mathematics underlying radio technology rather than the engineering. He explores and explains more than a century of key developments, placing them in historical and technological context. Nahin, a prolific author of books on math for the general reader, describes in fascinating detail the mathematical underpinnings of a technology we use daily. He explains and solves, for example, Maxwell’s equations for the electromagnetic field. Readers need only a familarity with advanced high school–level math to follow Nahin’s mathematical discussions. Writing with the nonengineer in mind, Nahin examines topics including impulses in time and frequency, spectrum shifting at the transmitter, the superheterodyne, the physics of single-sideband radio, and FM sidebands. Chapters end with “challenge problems” and an appendix offers solutions, partial answers, and hints. Readers will come away with a new appreciation for the beauty of even the most useful mathematics. Cover Contents Foreword by Andrew J. Simoson Preface Chapter 1. Radio Mathematics, Oscillators, and Transmitters 1.1 Kirchhoff’s Laws and FitzGerald’s Oscillating Circuit 1.2 Laplace Transforms, AC Impedance, and Transfer Functions 1.3 Van der Pol’s Negative Resistance Oscillator Equation 1.4 Filters and the Wiener-Paley Theorem 1.5 An Electronic Phase-Shift Oscillator 1.6 Fourier Series and Transforms, and Signal Spectrums 1.7 Impulses in Time and Frequency Challenge Problems Chapter 2. More Radio Mathematics: Circuits That Multiply 2.1 Spectrum Shifting at the Transmitter 2.2 Large-Carrier AM 2.3 Convolution, and Multiplying by Squaring and Filtering 2.4 Fleming’s Vacuum Tube Diode and the Detection of Radio-Frequency Waves 2.5 Multiplying by Sampling and Filtering 2.6 AM Envelope Detection, Tuning, and Impedance Matching Challenge Problems Chapter 3. The AM Radio Receiver 3.1 Prelude to the Superheterodyne 3.2 The Superheterodyne 3.3 Fancy Heterodyning Challenge Problems Chapter 4. SSB Radio 4.1 The Origin of SSB and Carson’s SSB Transmitter 4.2 Hilbert Transforms and Hartley’s SSB Transmitter 4.3 The Mathematics of the Hilbert Transformer 4.4 The Physics of the Hilbert Transformer 4.5 Receiving an SSB Signal 4.6 Weaver’s SSB Transmitter Challenge Problems Chapter 5. FM Radio 5.1 Why FM? 5.2 Instantaneous Frequency 5.3 FM Sidebands 5.4 The FM Receiver 5.5 Bilotti’s Phase Quadrature Detector Challenge Problems 6. American AM Broadcast Radio: A Historical Postscript A Final Author’s Note to the Reader Appendix—Maxwell’s Theory, the Poynting Vector, and a Simple Radio Transmitting Antenna Solutions, Partial Answers, and More Hints to Most of the End-of-Chapter Challenge Problems Acknowledgments Illustration Credits Index