Editing Sideband (section)

== Sideband creation ==

We can illustrate the creation of sidebands with one trigonometric identity''':'''

:<math>\cos(A)\cdot \cos(B) \equiv \tfrac{1}{2}\cos(A+B) + \tfrac{1}{2}\cos(A-B)</math>

Adding <math>\cos(A)</math> to both sides''':'''

:<math>\cos(A)\cdot [1+\cos(B)] = \tfrac{1}{2}\cos(A+B) + \cos(A) + \tfrac{1}{2}\cos(A-B)</math>

Substituting (for instance) &nbsp;<math>A \triangleq 1000\cdot t</math>&nbsp; and &nbsp;<math>B \triangleq 100\cdot t,</math>&nbsp; where <math>t</math> represents time''':'''

:<math>\underbrace{\cos(1000\ t)}_{\text{carrier wave}}\cdot \underbrace{[1+\cos(100\ t)]}_{\text{amplitude modulation}}
= \underbrace{\tfrac{1}{2}\cos(1100\ t)}_{\text{upper sideband}}
+ \underbrace{\cos(1000\ t)}_{\text{carrier wave}}
+ \underbrace{\tfrac{1}{2}\cos(900\ t)}_{\text{lower sideband}}.</math>

Adding more complexity and time-variation to the amplitude modulation also adds it to the sidebands, causing them to widen in bandwidth and change with time.  In effect, the sidebands "carry" the information content of the signal.<ref>Tony Dorbuck (ed.), ''The Radio Amateur's Handbook, Fifty-Fifth Edition'', American Radio Relay League, 1977, p. 368</ref>

===Sideband Characterization===

In the example above, a [[cross-correlation]] of the modulated signal with a pure sinusoid, <math>\cos(\omega t),</math> is zero at all values of <math>\omega</math> except 1100, 1000, and 900.  And the non-zero values reflect the relative strengths of the three components.  A graph of that concept, called a [[Fourier transform]] (or ''spectrum''), is the customary way of visualizing sidebands and defining their parameters.

[[File:Modulated radio signal frequency spectrum.svg|thumb|upright=1.4|[[Frequency]] spectrum of a typical modulated AM or FM radio signal.]]