On December 24, 1906, fishermen and ship radio operators along the Atlantic coast heard something impossible crackle through their headsets: a human voice reciting from the Gospel of Luke, followed by Handel's "Largo" played on a phonograph, and a violin solo of "O Holy Night." Their equipment was supposed to receive only Morse code dots and dashes. The voice belonged to Reginald Fessenden, a Canadian-born inventor broadcasting from Brant Rock, Massachusetts — and he had just performed the world's first audio radio broadcast.

The patent that made it possible was US Patent 706,737, filed September 28, 1901, and granted August 12, 1902. Titled "Wireless Signaling," it described the heterodyne principle — the foundational technique that still underpins essentially every radio, television, cell phone, and Wi-Fi receiver built in the last 120 years.

What it actually did: Marconi's contemporary wireless systems worked by generating crude bursts of electromagnetic noise — a spark gap fired, broadcast a damped wave, then went silent. You could send dots and dashes, but you couldn't carry a voice on something so jagged. Fessenden's insight was to use a continuous wave at a high, steady frequency, then modulate its amplitude with the much slower waveform of a human voice. At the receiver, he mixed the incoming carrier wave with a locally-generated wave of slightly different frequency. The interference between them produced a third, lower "beat" frequency — the heterodyne — which carried the audio in a form the listener could actually hear.

This is, line for line, how AM radio works. It is also how your phone's RF front-end works. Every time your iPhone tunes a 2.4 GHz Wi-Fi signal down to a baseband signal a processor can decode, it is performing a heterodyne mix that Fessenden patented when Theodore Roosevelt was president.

The supporting cast: To make this practical, Fessenden also needed a continuous-wave transmitter that didn't exist. He commissioned Ernst Alexanderson at General Electric to build a high-frequency alternator — essentially a giant electrical generator spinning fast enough (at 50,000+ RPM) to produce 50 kHz directly. The Alexanderson alternator (US Patent 1,008,577, 1911) was a freakish piece of machinery: two-meter rotors spinning at the edge of mechanical failure, generating radio waves the way a power plant generates 60 Hz mains.

The rediscovery: Edwin Armstrong refined the heterodyne into the superheterodyne receiver in 1918 (US Patent 1,342,885), which became the dominant radio architecture by the 1930s and remains so today. Software-defined radios, the basis of modern 5G base stations and the SDRs hobbyists buy for $30 on Amazon, are essentially digital heterodynes — they sample a wide chunk of spectrum, then perform the frequency mixing in software instead of with analog circuits. The math is identical to what Fessenden wrote down in 1901.

The tragedy: Fessenden was a difficult man, prone to lawsuits, and his backers at the National Electric Signaling Company forced him out in 1911. Marconi got the Nobel Prize. Armstrong got the radio fortune. Fessenden died in 1932 having sued his way to a modest settlement, his name largely erased from the history of an industry he had invented. Every smartphone in your pocket performs his mathematics millions of times per second.