Figure P.7.21 shows a carrier of frequency vc being amplitude modulated by a sine wave of frequency ω_m, that is,

E = E₀(1 + α cos ω_mt) cos ω_ct

Show that this is equivalent to the superposition of three waves of frequencies ω_c, ω_c + ω_m, and ω_c - ω_m. When a number of modulating frequencies are present, we write E as a Fourier series and sum over all values of ω_m. The terms ω_c + ω_m constitute what is called the upper sideband, and all the ω_c - ω_m terms form the lower sideband. What bandwidth would you need in order to transmit the complete audible range?

Figure P.7.21

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E