# Effective Modulation index of a Multi-tone AM signal

In a single-tone AM, message signal is a single-tone $i.e,&space;m(t)&space;=&space;A_{m}cos&space;2\pi&space;f_{m}t$ being modulated by a carrier signal and generates a single-tone modulated signal, where as in Multi-tone environment  message signal is a composite signal formed by number of frequencies f1,f2,f3 …..fn … being modulated by a carrier signal to generate an Amplitude  Modulated signal.

i.e, Multi-tone message signal is

$\therefore&space;m(t)&space;=&space;A_{1}cos&space;2\pi&space;f_{1}t&space;+A_{2}cos&space;2\pi&space;f_{2}t+A_{3}cos&space;2\pi&space;f_{3}t+....+A_{n}cos&space;2\pi&space;f_{n}t+....$

Now from the equation of General AM signal $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos&space;2\pi&space;f_{c}t$

the Multi-tone modulated signal can be obtained as

$S_{AM}(t)=A_{c}(1+k_{a}(A_{1}cos&space;2\pi&space;f_{1}t&space;+A_{2}cos&space;2\pi&space;f_{2}t+A_{3}cos&space;2\pi&space;f_{3}t+....+A_{n}cos&space;2\pi&space;f_{n}t+....))cos&space;2\pi&space;f_{c}t$

$S_{AM}(t)=A_{c}(1+k_{a}A_{1}cos&space;2\pi&space;f_{1}t&space;+k_{a}A_{2}cos&space;2\pi&space;f_{2}t+k_{a}A_{3}cos&space;2\pi&space;f_{3}t+....+k_{a}A_{n}cos&space;2\pi&space;f_{n}t+....)cos&space;2\pi&space;f_{c}t$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+k_{a}A_{1}cos&space;2\pi&space;f_{1}t&space;cos&space;2\pi&space;f_{c}t&space;+k_{a}A_{2}cos&space;2\pi&space;f_{2}t&space;cos&space;2\pi&space;f_{c}t+.......$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+A_{c}\mu&space;_{1}cos&space;2\pi&space;f_{1}t&space;cos&space;2\pi&space;f_{c}t&space;+A_{c}\mu&space;_{2}cos&space;2\pi&space;f_{2}t&space;cos&space;2\pi&space;f_{c}t+.......$

$S_{AM}(t)=A_{c}cos&space;2\pi&space;f_{c}t+\frac{A_{c}\mu&space;_{1}}{2}cos&space;2\pi&space;(f_{c}+f_{1})t+&space;\frac{A_{c}\mu&space;_{1}}{2}cos&space;2\pi&space;(f_{c}-f_{1})t&space;+&space;\frac{A_{c}\mu&space;_{2}}{2}cos&space;2\pi&space;(f_{c}+f_{2})t&space;+&space;\frac{A_{c}\mu&space;_{2}}{2}cos&space;2\pi&space;(f_{c}-f_{2})t&space;+&space;..........$

from the above signal the total power can be obtained as

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{8}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{8}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1}&space;^{2}}{4}+\frac{A_{c}^{2}\mu_{2}&space;^{2}}{4}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{1}&space;^{2}}{2}+\frac{\mu_{2}&space;^{2}}{2}+......)$

This expression can further represented in terms of effective modulation index $\mu&space;_{eff}$  as   $P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{eff}&space;^{2}}{2})$ where  $\mu&space;_{eff}&space;=&space;\sqrt{\mu&space;_{1}^{2}+\mu&space;_{2}^{2}+\mu&space;_{3}^{2}+...}$

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