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A formula for sinusoidal graphs of this form? 
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#1
Jan2413, 06:33 AM

#2
Jan2413, 06:35 AM

P: 2

BTW, in the excel version, the only reason my graph doesnt reach an amplitude of 1 consistently is due to a low sampling rate.



#3
Jan2413, 06:41 AM

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Something like [itex]x(t)= A sin(\omega (sin(kt))[/itex] should work.



#4
Jan2413, 07:16 AM

P: 905

A formula for sinusoidal graphs of this form?
He was smart enough to realize that you have to add a constant offset to the inner sin function so that it's always positive. But, as I recall, that still gives an answer that's wrong. You want to use the _integral_ of the function that you are trying to modulate onto the carrier. That way the frequency of the function that you are graphing varies as the derivitive of that integral  i.e. as the signal that you're trying to carry. 


#5
Jan2413, 07:54 AM

HW Helper
P: 6,187

Welcome to PF, Carla17!
Your inner time function should increase monotonously. ##\sin(time)## does not do that. Try for instance ##x + \sin(x)## instead. That one will just be increasing monotonously. The complete function would be something like: $$amplitude = \sin(\omega (t + \frac 1 k \sin(kt)))$$ where ##\omega## is the average angular frequency of the high frequency wave, Here's an example: http://m.wolframalpha.com/input/?i=s...%29%29&x=0&y=0 It graphs ##\sin(11 (t + \frac 1 3 \sin(2t)))##. 


#6
Jan2413, 12:20 PM

P: 824

Why don't you guys use the obvious form
[tex]f(t)= A\sin\left( \left( \frac{\omega_\text{max}\omega_\text{min}}{2} \left( \sin\left( \omega_\text{mod}t+\phi_\text{mod} \right)+1 \right)+\omega_\text{min} \right)t + \phi_\text{carrier} \right)[/tex] with [itex]\omega_\text{carrier}>\omega_\text{mod}[/itex]? Am I missing something? 


#7
Jan2513, 12:04 PM

P: 905

Look at your the argument to the outer sine function. Is it monotone increasing? Are you sure? Have you tested it with some sample values? Have you graphed it? If you don't believe that it needs to be monotone increasing, have you actually picked some test parameters and graphed the function you propose? "I like Serena" has pointed out that the argument to the outer sin function must be monotone increasing. I have pointed out that it should be the integral of the [hopefully positive] function that you were trying to modulate onto the carrier. 


#8
Jan2813, 02:39 PM

P: 824

Oh you're right the frequency should be integrated to get the phase, it cannot simply be multiplied by t. Sorry for that. This should be better. And it plots ok ;)
[tex] f(t)= A\sin\left( \frac{\omega_\text{max}\omega_\text{min}}{2} \left( \frac{ \sin\left( \omega_\text{mod}t+\phi_\text{mod} \right)}{\omega_\text{mod}}+t \right)+\omega_\text{min} t + \phi_\text{carrier} \right) [/tex] 


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