Effective Modulation index of a Multi-tone AM signal

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In a single-tone AM, message signal is a single-tone $i.e, m(t) = A_{m}cos 2\pi f_{m}t$ 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 …..f_n … being modulated by a carrier signal to generate an Amplitude Modulated signal.

i.e, Multi-tone message signal is

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

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

the Multi-tone modulated signal can be obtained as

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

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

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

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

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

from the above signal the total power can be obtained as

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

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

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

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

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Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems. View all posts by Lakshmi Prasanna Ponnala

Effective Modulation index of a Multi-tone AM signal

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