Question

A carrier is angle modulated by two sinusoidal modulating waveforms simultaneously to give a signal v(t) = cos(ωt + β₁cosω₁t + β₂cosω₂t). Show that this waveform has sidebands separated from the carrier, not only at multiples of ω₁ and ω₂ but also has sidebands at separations of multiples of ω₁ + ω₂ and ω₁ - ω₂

Answer 1

A carrier wave angle-modulated by two sinusoidal signals simultaneously creates sidebands not just at multiples of the individual signals' frequencies, but also at the sums and differences of their frequencies due to trigonometric identities applied to the modulation process.

Step-by-step explanation:

When a carrier is angle modulated by two sinusoidal modulating waveforms simultaneously, the result is a more complex spectrum than when modulated by a single waveform. The sidebands are produced at frequencies that are the sum and difference of the modulating signals' frequencies in addition to the multiples of the individual frequencies. This phenomenon can be explained by using trigonometric identities which, when applied to angle modulation, reveal that the spectrum consists not only of frequencies at ω₁ and ω₂ but also at (ω₁ + ω₂) and (ω₁ - ω₂). This effect creates additional sidebands spaced at integer multiples away from the carrier. To visualize this, consider the expansion of β₁cosω₁t and β₂cosω₂t using the sum and difference frequencies, then apply the trigonometric identity for the product of two cosines, which results in the appearance of these additional sidebands in the frequency spectrum.