The golden age of Hollywood was a time of classic movies and classic movie stars. A time of ‘frankly my dear’, ‘play it again’ and ‘whoops Mr Parson’ (I made the last one up). Yet only one star was billed as ‘The most beautiful woman in the world’. This was Hedy Lamarr: an Austrian-born actress, former wife of an arms dealer, international movie star, and occasional inventor. Her most celebrated invention was something without which today’s mobile phone and Wi-Fi technology would not be possible: frequency-hopping. Frequency hopping is the idea of transmitting radio signals while rapidly changing frequencies. This makes any transmission impossible to intercept, jam, or accidentally cross with your neighbour’s order to the Chinese takeaway. It’s also a technology currently estimated to be worth over $100bn a year. Yet Hedy never earned a penny from her idea, giving the patent to the US Navy, who allowed it to expire while Hedy lived a modest Hollywood retirement.
Frequency-hopping spread spectrum
Modern frequency-hopping involves multiplying a data signal by a code sequence. The code sequence occurs at
a much faster rate than the data signal, which causes the latter to be spread among a far greater range of frequencies.
Although spreading increases the range of frequencies a message will use, not all of that range will be in use at any one time, allowing many people to use the same range of frequencies simultaneously. It is this property of spread-spectrum that makes mobile phone technology possible.
The combined signal is then used to modulate a carrier signal and transmitted via radio. The original data signal is recovered by re-multiplying the non-modulated signal by a locally generated copy of the code sequence, causing the code sequence to cancel itself out.
Without knowledge of the code sequence, the transmission should be indistinguishable from white noise. Ideally, code sequences must appear as noise-like as possible, yet be easily reproducible at the receiving end. Such sequences are called pseudorandom sequences.
For example, an m-sequence (or maximum length sequence) starts with a binary string of length $n$ and iteratively generates a sequence of strings. The next string is the sum of the last two digits (using modulo-2 addition of the previous string), followed by the first $n-1$, which remain unchanged.
For example, when $n=4$,
Continuing in this way, the procedure eventually generates all binary strings of length $n$ (except for the zero string), before returning to the start.
However, Hedy’s original idea was pre-digital, when streaming was just something rivers did, and Spotify was a natural part of adolescence. Instead, Hedy had a much more militaristic application in mind: as a method to control submarine torpedoes remotely. So how did a Hollywood star know so much about munitions?
The arms dealer
Hedy’s love of inventing came from her father, a bank director in Vienna, yet Hedy still dropped out of school at sixteen in order to become an actress or, in her own words, “I wanted to be famous”. This is a sensible ambition and nothing bad will come of it.
Hedy landed her first starring role at the age of eighteen in the film Ecstasy. This was a sensitive look at a young woman trapped in a loveless marriage to an older man, but who eventually finds new love, new freedom and finally gains her independence. It is mostly remembered because Hedy gets her nips out and pretends to have some sex.
As the first non-pornographic movie to include a sex scene, Ecstasy was ahead of its time, so its time decided to condemn it. Instead of fame, Hedy found infamy—which is like fame but with creepier fans. One such fan was Fritz Mandl, arms dealer to the Nazis and massive jerk.
I don’t know what first attracted Hedy to third-richest-man-in-Austria Fritz Mandl, but once they married, Hedy found herself living a life of luxury. These luxuries included three hunting lodges, cars, planes, a yacht, servants and eating from gold plates. Something even Donald Trump might think was a little gauche.
Yet it was a gilded cage, with a possessive and jealous husband. Hedy was no longer allowed to act, while Fritz supposedly spent a fortune trying to find and destroy every copy of Ecstasy. The stories of Hedy’s escape—drugging a maid and escaping in her uniform—may have been an invention of MGM Studios, but by the time she arrived in America, Hedy Mandl was now Hedy Lamarr—a fully minted movie star.
Within a couple of decades, Hedy had starred in over twenty movies and divorced six times. Men wanted Hedy Lamarr, women wanted to be Hedy Lamarr. Dogs wanted to be petted by Hedy Lamarr, cats were not fussed.
Yet when Hedy was not busy making movies or divorcing people, she found time to invent.
Name that tune
It is said that Hedy came up with frequency-hopping while playing a game at the piano with her composer friend George Antheil.
The game involved one of them playing a tune on the piano for the other to guess and then jump in on duet. To make this game more difficult, George would keep changing the key, forcing Hedy to follow. This game of two piano players remaining synchronised while constantly changing keys gave Hedy her idea. If a transmitter and receiver could remain synchronised while hopping from frequency to frequency, then without knowledge of the pattern of hops, the signal could not be blocked (also known as jammed).
So Hedy decided to invent a radio-controlled torpedo—something she knew a bit about thanks to the many dinner parties she endured entertaining the Nazi generals who were guests of Fritz Mandl. Radio control and frequency-hopping would allow a ship to correct the direction of a torpedo for a more accurate hit. Hedy’s idea was entirely original but, according to patent law, Hedy would need to demonstrate some form in which the idea could be put into practice. This would become George Antheil’s biggest contribution, thanks to his experience with self-playing pianos.
Patents and the player piano
Self-playing pianos were extremely popular in the days before radio. A tune was represented by holes on a roll of paper which was then loaded into the player piano. The holes in the paper passed over a vacuum that would pull down the keys and cause the piano to play. Player pianos are very cool—if slightly ghostly.
As a young composer, Antheil had written Ballet Mécanique, a musical experience involving several synchronised self-playing pianos, as well as saws, hammers, electric bells and three aeroplane propellers. This was early experimental electronic music, a Skrillex of its time and about as well received. The 1926 debut in Paris ended in boos and a riot.
So when Hedy needed a practical demonstration of her idea, George thought of the self-playing piano. Hedy and George’s patent described a ship and torpedo that would rapidly, and synchronously, change frequency in a pattern controlled by ribbons—perforated paper like that of a player piano—which would determine both frequency and duration.
With hundreds of hops per minute, switching unpredictably between frequencies, the signal would be impossible for the enemy to jam. Whimsically, George chose to use 88 different frequencies, one for every key on a piano.
In August 1942, Hedy and George’s patent for a ‘Secret Communication System’ was approved. However, the US Navy ultimately rejected the idea, citing that the mechanism would be too heavy to fit in the torpedoes. This is where Hedy and George had made their mistake. In an attempt to better explain their idea, they had mentioned the player piano. Although their system could have been reduced to the size of a dollar, it appears that the navy imagined torpedoes containing parts from player pianos themselves.
Mysteriously, the navy acquired the patent for frequency-hopping, which was then filed away and left to expire. Instead, Hedy was told that she could serve the nation better by using her celebrity status to sell war bonds.
Meanwhile, far from the glamour of Hollywood, the mathematical foundation of spread-spectrum technology was being established by a man called Claude Shannon, who proved that communication could be made more reliable, and with less chance of error, by increasing the range of frequencies.
The father of information theory
Shannon was a mathematician and electrical engineer working for Bell Labs in New York. His work was finally made public in his 1948 paper A Mathematical Theory of Communication. This was a landmark paper that became the foundation of information theory, and even coined the term ‘bit’ as a unit of information.
The capacity, $C$, of an analogue channel, subject to additive white Gaussian noise, can be calculated from the physical properties of that channel using the Shannon–Hartley theorem:
$$C = B \log_2(1 + S/N),$$
where $B$ is the range of frequencies, or bandwidth, of the channel in hertz; $S$ is the average received signal power over the bandwidth, measured in watts; $N$ is the average power of the interfering Gaussian noise over the bandwidth, also measured in watts; and $S/N$ is known as the signal-to-noise ratio.
The Shannon–Hartley theorem shows that a channel with little noise can transmit high amounts of data nearly error-free, while to do the same on a noisy channel you need only increase the bandwidth. This is the principle on which spread spectrum works today.
A channel’s capacity is a theoretical limit only and represents the best that can be achieved with any code and modulation method. Modern techniques can now achieve data rates close to that limit.
Tragically, as Hedy’s beauty faded so did her movie career. Similarly, her contribution to the growing technology of frequency-hopping and spread-spectrum went unacknowledged for decades. Finally, a campaign in the 1990s resulted in Hedy Lamarr and George Antheil receiving the 1997 Pioneer Award from the Electronic Frontier Foundation for their contributions to the technology. On hearing she was to receive the award, Hedy remarked to her son “it’s about time”.
Hedy died in 2000, aged 85. Some of her ashes were placed in the Vienna Central Cemetery. Recently, a memorial was added there that, when viewed correctly, creates an image of Hedy. The memorial is made from 88 stainless steel rods—to represent the 88 frequencies in Hedy’s original patent—and features a last few words from Hedy: “Films have a certain place and a certain time period. Technology is forever.”
[ Pictures: Banner, Frequency hopping, George Antheil, Piano, Patent: Public Domain. Poster: Image used with permission from fffmovieposters.com. Shannon: Tekniska Museet, CC BY-SA 2.0. Grave: Papergirl, CC BY-SA 4.0 ]
More from Chalkdust
- Never be stumped by a maths problem again, with this crash course from the ever-competent Stephen Muirhead
- Colin Wright juggles Euler, doodling and Millennium problems
- Make your own treasures, guaranteed to be priceless on a future episode of Antiques Roadshow
- Pythagoras gave us so much more than a² + b² = c²
- Exploring the connection between maths and art.
- Sit in your favourite chair and do away with those tedious algebraic proofs