Question

The carrier wave is represented by C (t) = 5 Sin (10πt) volt. A modulating signal is a square wave as shown in the figure. Determine the modulation index.

Answer 1

- Hint: To find the modulation index, we apply the formula of it using the required variables given in the equation of the carrier wave and the figure. We look at the general form of the carrier wave to derive the amplitude from the given equation.

Complete step-by-step solution -
Given data,
Carrier wave, C (t) = 5 Sin (10πt)
A carrier wave is a wave of constant frequency. It has the properties of a sine wave. We know the general equation of a sine wave is C = A sin (ωt), where A is the amplitude of the wave.
Comparing this general equation to the given carrier wave equation, C (t) = 5 Sin (10πt)
We can say that the amplitude of the carrier wave is ${{\text{A}}_{\text{c}}} = 5$.

Now let us look at the modulating wave given in the figure,
We know the amplitude of a wave is the maximum value a particle in a wave reaches from its equilibrium position.
Therefore from the figure we could say that the amplitude of the modulating wave is${{\text{A}}_{\text{m}}} = 2$.
Modulating Index is defined as the range of the modulated variable of a carrier signal varies around its unmodulated level.
It is given by the formula
${\text{M}}{\text{.I = }}{\mu _{\text{a}}} = \dfrac{{{{\text{A}}_{\text{m}}}}}{{{{\text{A}}_{\text{c}}}}}$
$ \Rightarrow {\text{M}}{\text{.I = }}\dfrac{2}{5} = 0.4$

The modulation index of the wave is 0.4

Note – In order to answer this type of question the key is to know the general form of a carrier wave is a sine wave, using this we find its amplitude. Also looking at the graph we infer the amplitude of the modulating wave. We do not consider the sign while counting the amplitude, we only consider its magnitude. There are different modulation indexes for amplitude modulation and frequency modulation respectively.