Modulation by Several Sine Waves

When a signal is modulated by several sine waves, the resulting modulated signal is a combination of the individual modulated signals. The modulation process can be carried out for each individual sine wave, and the modulated signals can be combined using the principle of superposition.

Let's consider an example where a carrier wave with a frequency of fc is modulated by two sine waves with frequencies f1 and f2. The modulated signal can be expressed as:

Modulation by Several Sine Waves

s(t) = Ac [1 + ka1 sin(2πf1t) + ka2 sin(2πf2t)] cos(2πfct)

where Ac is the amplitude of the carrier wave, ka1 and ka2 are the modulation indices for the two modulating sine waves, and cos(2πfct) is the carrier wave.

The modulated signal can be expanded using the trigonometric identity for the product of two sinusoids:

s(t) = Ac cos(2πfct) + ka1 Ac/2 [cos(2π(f1+fc)t) - cos(2π(f1-fc)t)] + ka2 Ac/2 [cos(2π(f2+fc)t) - cos(2π(f2-fc)t)]

From the above equation, we can see that the modulated signal consists of the carrier wave and four sidebands, two for each modulating sine wave. The sidebands are located at frequencies f1+fc, f1-fc, f2+fc, and f2-fc.

The power of each sideband is proportional to the square of the corresponding modulation index. Therefore, the power of the sidebands can be controlled by adjusting the modulation indices of the modulating sine waves.

In summary, when a signal is modulated by several sine waves, the resulting modulated signal is a combination of the individual modulated signals. The modulated signal contains the carrier wave and sidebands at frequencies that are the sum and difference of the carrier and modulating frequencies. The power of the sidebands can be controlled by adjusting the modulation indices of the modulating sine waves.