What is the modulation index?

Hint: The modulation index is related to amplitude modulation as well as frequency modulation. After defining the AM and FM in mathematical expression we can get a modulation index for both of the cases. The equation of the carrier wave and the signal wave have to be known for defining the modulation index. This index is the identifier of the level of distortion or noise.

Complete step-by-step solution:
The modulation index is directly proportional to the ratio of the amplitude of the information signal and the amplitude of the carrier wave.
The equation of the carrier wave is represented by, $V = {V_0}\sin \Omega t$
${V_0}$ is the amplitude of the wave, $\Omega $ is the angular frequency.
The equation of the information signal wave is represented by, $v = {v_0}\sin \omega t$
${v_0}$ is the amplitude of the wave, $\omega $ is the angular frequency.
Here the changeable velocity $v$ is called the modulation voltage as this voltage is responsible for the modulation of the carrier wave.
Hence, the modulation index,
$\beta \propto \dfrac{{{v_0}}}{{{V_0}}}$
$ \Rightarrow \beta = k\dfrac{{{v_0}}}{{{V_0}}}$
$k$ is a dimensionless constant term.
The significance of the modulation index is to measure the level of noise or distortion. The higher value of the ratio denotes the high distortion where the lower value of the ratio denotes the low distortion or noise in a signal.


Note: The condition for the modulation is $\omega < < \Omega $ i.e. the frequency of the carrier is much much greater than that of the signal. In that case, the wavelength of the carrier is very low. In reality, the frequency of the carrier is almost $1000$ times or more than that of the signal wave.
The modulation voltage changes very slowly with the change of time. That’s why the determining wave is said to be a slow-wave.