transmitted. History The first U.S. patent application for SSB modulation was filed on December 1, 1915 by John Renshaw Carson. The U.S. Navy experimented with SSB over its radio circuits before World War I. SSB first entered commercial service on January 7, 1927, on the longwave transatlantic public radiotelephone circuit between New York and London. The high power SSB transmitters were located at Rocky Point, New York, and Rugby, England. The receivers were in very quiet locations in Houlton, Maine, and Cupar Scotland. SSB was also used over long-distance telephone lines, as part of a technique known as frequency-division multiplexing (FDM). FDM was pioneered by telephone companies in the 1930s. With this technology, many simultaneous voice channels could be transmitted on a single physical circuit, for example in L-carrier. With SSB, channels could be spaced (usually) only 4,000 Hz apart, while offering a speech bandwidth of nominally 300 Hz to 3,400 Hz. Amateur radio operators began serious experimentation with SSB after World War II. The Strategic Air Command established SSB as the radio standard for its aircraft in 1957. It has become a de facto standard for long-distance voice radio transmissions since then. Mathematical formulation Single-sideband has the mathematical form of quadrature amplitude modulation (QAM) in the special case where one of the baseband waveforms is derived from the other, instead of being independent messages: where is the message (real-valued), is its Hilbert transform, and is the radio carrier frequency. To understand this formula, we may express as the real part of a complex-valued function, with no loss of information: where represents the imaginary unit. is the analytic representation of which means that it comprises only the positive-frequency components of : where and are the respective Fourier transforms of and Therefore, the frequency-translated function contains only one side of Since it also has only positive-frequency components, its inverse Fourier transform is the analytic representation of and again the real part of this expression causes no loss of information. With Euler's formula to expand we obtain : Coherent demodulation of to recover is the same as AM: multiply by and lowpass to remove the "double-frequency" components around frequency . If the demodulating carrier is not in the correct phase (cosine phase here), then the demodulated signal will be some linear combination of and , which is usually acceptable in voice communications (if the demodulation carrier frequency is not quite right, the