# Sum and Difference Frequencies

When you superimpose two sine waves of different frequencies, you get components at the sum and difference of the two frequencies. This can be shown by using a sum rule from trigonometry. For equal amplitude sine waves

The first term gives the phenomenon of beats with a beat frequency equal to the difference between the frequencies mixed. The beat frequency is given by

since the first term above drives the output to zero (or a minimum for unequal amplitudes) at this beat frequency. Both the sum and difference frequencies are exploited in radio communication, forming the upper and lower sidebands and determining the transmitted bandwidth.

When you say that the beat frequency is f1-f2 rather than (f1-f2)/2, that requires some explanation. For the difference frequency you can just say that you get a minumum when the modulating term reaches zero, which it does twice per cycle, so that the number of minima per second is f1-f2.

 Wave envelope for two tones
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# Carriers and Sidebands

Radio transmission involves putting audio frequency information on a much higher frequency electromagnetic wave called a carrier wave. The process of superimposing the "electrical image" of the sound information on the carrier wave is called modulation, and there are two commonly used schemes: amplitude modulation (AM) and frequency modulation (FM). Either form of modulation produces frequencies which are the sum and the difference of the carrier and modulation frequencies - these frequencies are sometimes called sidebands.

Because of the existence of the sidebands, the frequency range or bandwidth necessary for radio transmission depends on the range of modulating frequencies.

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# Bandwidth for Communication

 When you quote a frequency for a radio station, you generally quote the frequency of the carrier. But when you superimpose a signal on the carrier by AM or FM, you produce sidebands at the sum and difference of the carrier and modulation frequencies. This means that the transmitted signal is spread out in frequency over a bandwidth which is twice the highest frequency in the signal.

Bandwidths are assigned for all types of broadcast communication and this imposes a maximum signal frequency which may be transmitted. The bandwidths assigned to AM and FM radio are such as to limit the fidelity of music broadcasts in AM, but permit the luxury of stereo high-fidelity broadcasts by FM. The high signal frequencies associated with video broadcasting require higher bandwidths for channels assigned to television.

 Sum and difference frequencies Sidebands
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