# Sidebands

#### Side-bands OR Side Frequencies:

Whenever a carrier is modulated by an information signal, new signals at different frequencies are generated as part of the process.  These new frequencies are called side frequencies or sidebands and occur in the frequency spectrum directly above and directly below the carrier frequency.

If the modulating signal is a single-frequency sine wave, the resulting new signals produced by modulation are called side frequencies.  If the modulating signal contains multiple frequencies such as voice, video, or digital signals, the result is a range of multiple side frequencies.  These are referred to as sidebands.

The sidebands occur at frequencies that are the sum and difference of the carrier and modulating frequencies. Assuming a carrier frequency of fc and a modulating frequency of fm, the upper sideband fUSB and lower sideband fLSB are computed as follows:

ƒUSB = ƒc + ƒm

ƒLSB = ƒc – ƒm

The existence of these additional new signals that result from the process of modulation can also be proven mathematically.  This can be done by starting with the equation for an AM signal v2.

V2 = vc sin 2πƒct + (Vm sin 2πƒmt)(sin 2 πƒct

There is a trigonometric identity that says that the product of two sine waves is

sin A sin B = cos(A – B)/2  -  cos(A+B)/2

By substituting this identity into the expression for our modulated wave, the instantaneous amplitude of the signal becomes

Carrier     +    LSB

e2 = vc sin 2πƒct  +   Vm/2  cos 2πt(ƒc – ƒm ) – Vm/2 cos 2πt(ƒc + ƒm)

As you can see, the second and third terms of this expression contain the sum ƒc + ƒm and difference ƒc – ƒm of the carrier and modulating signal frequencies.  The first element in the expression is simply the carrier wave to which is added the difference frequency and the sum frequency.

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