# The amplitude of a sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal (t) = 5 cos 1600πt + 20 cos 1800πt +

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The amplitude of a sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal (t) = 5 cos 1600πt + 20 cos 1800πt + 5 cos 2000πt. The value of the modulation index is __________

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Concept: Modulation index is also called Depth of modulation.

Application: The general AM modulated signal expression is ${\rm{s}}\left( {\rm{t}} \right) = {{\rm{A}}_{\rm{c}}}\left( {1 + {\rm{\mu }}\cos {{\rm{\omega }}_{\rm{m}}}{\rm{t}}} \right)\cos {{\rm{\omega }}_{\rm{c}}}{\rm{t}}$

$\begin{array}{l} \Rightarrow {\rm{s}}\left( {\rm{t}} \right) = {{\rm{A}}_{\rm{c}}}{\rm{cos}}{{\rm{\omega }}_{\rm{c}}}{\rm{t}} + \frac{{{{\rm{A}}_{\rm{c}}}{\rm{\mu }}}}{2}\left[ {2\cos {{\rm{\omega }}_{\rm{m}}}{\rm{t}}\cos {{\rm{\omega }}_{\rm{c}}}{\rm{t}}} \right]\\ = {{\rm{A}}_{\rm{c}}}\cos {{\rm{\omega }}_{\rm{c}}}{\rm{t}} + \frac{{{{\rm{A}}_{\rm{c}}}{\rm{\mu }}}}{2}\left[ {\cos \left( {{{\rm{\omega }}_{\rm{n}}} + {{\rm{\omega }}_{\rm{c}}}} \right){\rm{t}}\cos \left( {{{\rm{\omega }}_{\rm{c}}} - {{\rm{\omega }}_{\rm{m}}}} \right){\rm{t}}} \right]\\ \Rightarrow {\rm{s}}\left( {\rm{t}} \right) = {{\rm{A}}_{\rm{c}}}{\rm{cos}}{{\rm{\omega }}_{\rm{c}}}{\rm{t}} + \frac{{{{\rm{A}}_{\rm{c}}}{\rm{\mu }}}}{2}\cos \left( {{{\rm{\omega }}_{\rm{m}}} + {{\rm{\omega }}_{\rm{c}}}} \right){\rm{t}} + \frac{{{{\rm{A}}_{\rm{c}}}{\rm{\mu }}}}{2}\cos \left( {{{\rm{\omega }}_{\rm{c}}} - {{\rm{\omega }}_{\rm{m}}}} \right){\rm{t}} \end{array}$

Thus, we see that two terms have same amplitudes. Comparing with

${\rm{s}}\left( {\rm{t}} \right) = 5\cos 1600{\rm{\pi t}} + 20\cos 1800{\rm{\pi t}} + 5\cos 2000{\rm{\pi t}}$

we have ${A_c} = 20$

and $\frac{{{{\rm{A}}_{\rm{c}}}{\rm{\mu }}}}{2} = 5 \Rightarrow {\rm{\mu }} = \frac{{2 \times 5}}{{20}} = \frac{1}{2} = 0.5$