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

by Paul J. Nahin, Andrew Simoson (Foreword)

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.

Genres: Mathematics, Nonfiction, Science

376 pages, Hardcover

Published January 16, 2024

About the author

Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire and the author of many best-selling popular math books, including The Logician and the Engineer and Will You Be Alive 10 Years from Now? (both Princeton).

Reviews

[Steve Donoghue]

This book is full of equations, but it's also full of odd, dorky passion. It's the author's spirited defense of all the complicated math that goes into the technology behind the humble FM/AM radio, and from that description, you'd hardly think it had any chance of keeping you reading. But it kept me reading, and my understanding of math is exactly equal to that of an 8-year-old beagle. And it goes without saying that this book will be required reading for actual radio enthusiasts. My full review is here: https://openlettersreview.com/posts/t...

[Tim Robinson]

Nahin has gone too far this time. He thinks he can teach electrical engineering without teaching electrical engineering, which is of course impossible. He assumes his audience knows complex numbers and some fairly advanced calculus, but nothing at all about electronics. I don't think this audience exists in the 21st century.

Anyway, the maths is difficult, and perhaps more difficult than it needs to be. Nahin lets the equations do the talking. It is his explicit intention to teach radio from a purely mathematical standpoint, but I don't think he succeeds. He uses the Laplace Transform, which is an engineer's weapon.

[Kam Yung Soh]

A fascinating book that looks at the mathematics behind the operation of AM, FM and Single-Sideband radios to show how they work as transmitters and receivers. The book also goes through the history of such radios, from curiosities to hobbyists, to commercialisation and public broadcasting of radio channels and the creation of commercial radio shows.

Starting with the mathematics behind electronic components like resistors, inductors and capacitors, diodes, triodes, oscillators, amplifiers and so on, the author then uses mathematical identities and techniques (like the Fourier transform) to show how combinations of electronic components can be used to modulate an oscillating signal in various ways to produce an AM or FM signal. This modulated signal can then be received by a demodulator to extract back a representation of the information on the modulated signal.

The explanation for Single-Sideband Radio was interesting, as it is a topic mentioned during my engineering course but never fully explained. Here, the author first shows a brute-force method for a Single-Sideband Radio (using filters to remove a sideband) before showing other, more elegant techniques (using an 'all-pass' filter) for generating and receiving a Single-Sideband Radio signal. One interesting aspect here, that pops out of the mathematics, is that the receiver actually receives a 'scrambled' signal, where the phase components of the signal are shifted by different amounts depending on the original frequency. In practice, this phase shifting does not matter, as the human ear is able to compensate for it and still able to hear the signal.

While looking at the mathematics behind the various radios, the author also goes through the history of AM and FM radio. It starts with 'spark-gap' radios and crystal radios, which were sensitive and hard to operate and only appealed to hobbyist. The vacuum tube would herald the age of commercial radio, making radio receivers much easier to operate.

While the appeal of radio may now be less, due to the internet, it was an important technology that helped spread news and entertainment to the world, and it is good to have a book that shows how radios work mathematically, even if the actual components used to implement a radio now would be different from those used in the past.

[Stephen]

Good book. I enjoyed the math but the deep dives into the weeds of signal processing using limiters, shifters, LPFs, HPFs, etc. were not as enjoyable. Still, Nahin is Nahin which is to say an excellent author. Recommended book especially if you're keen on EE more than on the physics of this topic.

[Todd N]

Perfect book for someone who is nostalgic about radio and wants to brush up on their vector calculus

[Liquidlasagna]

the exposition could be a lot better

but the book has some clarity to it before dumping equations on people, which is something in which most books either do way too little, or way too much, with little more than cryptic haze given to the reader