Maximum modulation factor of a two tone signal

In summary, the maximum amplitude of the modulating signal is equal to the amplitude of the carrier which has been set to 1V, and the two possible values of ##~V~## are either -1.33 or 1.33.
  • #1
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Homework Statement
Finding the maximum modulation factor for a two tone signal that can be employed without over modulation occurring
Relevant Equations
(Mt) = sqrt(m1^2+m2^2)
I'm stuck on this because v is a 2 tone signal, so it's not as simple as Am/Ac. The teacher said I will need to differentiate it and equate it to zero, which I thought made sense. Differentiating v gives me: [- mwsin(wt) - mwsin(2w*t)], so there's still unknown variables. I don't know how I'm supposed to equate that to zero and find m when there are still variables that will not cancel.

I have seen another (looks easier) method where the Total Modulation Factor (Mt) = sqrt(m1^2+m2^2), where m1 = Am1/Ac and m2 = Am2/Ac. It does make sense, but I've spent a long time looking for where this, or something similar, is written down and haven't found anything so far.

Could anyone provide some help or point me towards some guidance please?

Thanks
 

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  • #2
As far as I know, the maximum modulation factor of Traditional Amplitude Modulation should be 1. When the modulation factor >1, the transmitted wave will be distorted, so that the signal wave cannot be exactly reproduced. This is independent of whether the modulating signal is is single tone or multi-tone. So I was wondering if you are sure you are looking for the maximum modulation factor and not the maximum amplitude of the modulating signal.
 
  • #3
I don't think I worded the post very well. I am looking for the maximum amplitude of the modulating signal (v).
 
  • #4
I agree that the following question is inappropriate, what we are really asking for is the maximum amplitude of the modulating signal.
equ1.jpg

So all we have to do is differentiate this equation, which you've already done.
$$ \frac {dv}{dt} = -V\omega~sin(\omega t)-V\omega~sin(2\omega t)$$
The next step is to find what ## \omega t ## is when it is equal zero, that is to say we need to solve the following equation.
$$ sin(\omega t)+sin(2\omega t)=0$$
Note that this equation has three solutions. :smile:
 
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  • #5
We are actually looking for what ##~\omega t~## is when the amplitude of the modulating signal v is maximum, then the value of ##~V~## can also be determined accordingly.

$$sin(\omega t)+sin(2\omega t)=0~~~\rightarrow~~~sin(\omega t)+2sin(\omega t)cos(\omega t)=0~~~~ \rightarrow~~~~sin(\omega t)\left( 1+2cos(\omega t) \right)=0$$
$$ sin(\omega t)=0~~ \text {when} ~~ \omega t =0~~ \text{or}~~ \pi ~~~~~~\text{and}~~~~~~1+2cos(\omega t)=0~~\text{when}~~\omega t = 2.094$$
Because the maximum amplitude of the modulating signal must be equal to the amplitude of the carrier which has been set to 1V, then the maximum value of ##~V~## seem to have the following possibilities.

$$\mathbf{v}= -1=Vcos(\omega t)+\frac{V}{2}cos(2\omega t)~~~~~\rightarrow ~~~~ V=\frac {-2} {2cos(\omega t)+cos(2\omega t)} = ~-\frac {2}{3}, ~ 2~~~\text{or}~~1.33 $$
But the value of ##2## is actually invalid, so in the end only ##~-\frac{2}{3}~## and ##~1.33~## are really applicable.
 
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  • #6
Thanks, that's really helpful. Also matches with the plot I've made.
 
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What is the maximum modulation factor of a two tone signal?

The maximum modulation factor of a two tone signal refers to the highest possible value that the amplitude of a two tone signal can reach. It is typically represented as a percentage, with 100% being the maximum modulation factor.

How is the maximum modulation factor calculated?

The maximum modulation factor is calculated by taking the amplitude of the higher frequency tone in the signal and dividing it by the amplitude of the lower frequency tone. This value is then multiplied by 100 to get the percentage value.

Why is the maximum modulation factor important?

The maximum modulation factor is important because it determines the quality of the signal. A higher modulation factor means a stronger and clearer signal, while a lower modulation factor can result in a weaker and distorted signal.

What factors can affect the maximum modulation factor?

The maximum modulation factor can be affected by various factors such as the frequency and amplitude of the two tones, the type of modulation being used, and any external interference or noise in the signal.

How can the maximum modulation factor be optimized?

The maximum modulation factor can be optimized by carefully selecting the frequencies and amplitudes of the two tones, using efficient modulation techniques, and minimizing any external interference or noise in the signal.

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