Effective Modulation index of a Multi-tone AM signal

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 …..fn … 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 and currently working as a faculty.