Condition for multi tone AM for proper envelope detection

Question

Given a multi-tone amplitude modulated wave as follows :

s(t) = A_c[1+2k_asin(w₁) + k_asin(w₂)] cos(w_ct)

where w_c is the carrier frequency and w₁ and w₂ are the modulating frequency components. My question is : what is the largest value of amplitude sensitivity k_a for which envelope detection can be performed on s(t) to recover the message signal without distortion ?

Method 1 (calculate effective modulation index)

Effective modulation index for multi-tone AM signal is given by \$u_{eff}=\sqrt{u_1^2+u_2^2}\$ (u_eff = sqrt(u₁² + u₂²) where u₁ and u₂ represent modulation index for single tone AM signal and sqrt stands for square-root. By definition, u = maximum value of message signal after modulation.So u₁ = 2K_a and u₂ = K_a. In the case of single tone AM wave, for envelope detector to work without any distortion, the condition is u<=1. Analogously for multi-tone AM wave the same condition becomes u_eff<=1.

4k_a² + k_a² <= 1

k_a<=0.2

Method 2

Here again we take analogy from single-tone AM for envelope detection condition. However, instead of u_eff<=1, we do

u₁ + u₂ <=1

2k_a + k_a <= 1

k_a <= 0.33

Please tell me which of one of these two methods is correct ?

(Sorry for poor formatting. I am a novice to MathJax )

Answer 1

Do not take the modulation index to mess this, because you already have the time domain expression for the modulated output s(t).

I guess you only forgot to insert t. I mean you planned to write this:

s(t) = A_c[1+2k_asin(w₁t) + k_asin(w₂t)] cos(w_ct)

Distortion (see NOTE1) occurs if the amplitude term goes momentarily to negative. To prevent it means you keep 1+2k_asin(w₁t) + k_asin(w₂t) non-negative.

The worst case occurs when the sines happen to be at the same time =-1.

You want 1-2k_a- k_a >0 i.e. k_a < 1/3 as you have already written.

NOTE1: The distortion gets its birth in the common envelope detector(= diode detector). Synchronous detector (=mixing the signal back down) doesn't have that problem.