Finding the times when one cycle occurs for FM wave

[Rick66]

Hello everyone,

I'm trying to find the times when one cycle occurs for a FM wave. For instance, given

y(t) = sin(2∏f_{c}*t_{1} + sin(2∏f_{m}*t_{1}))

at an arbitrary time t_{1}, I wish to find the time t_{2} such that

(f_{c}* t_{2} + sin(2∏f_{m}* t_{2}) – (f_{c}*t_{1} + sin(2∏f_{m}*t_{1}) = 1 (cycle).

Now I've tried plotting the wave argument in n-t "space" (i.e. n is cycles),

n = f_{c}*t + sin(2∏f_{m}*t)

to see if some solution presents itself. For instance, we can see that it gives oscillations about the regularly increasing line n = f_{c}*t so that t_{2} ≈1/f_{c} + t_{1} gives an approximate solution. But as to an exact solution, I've hit a brick wall. So if somebody could point me in the right direction it would be greatly appreciated.
Thank you
Rick66

 

[Rick66]

Hello everyone,

I'm trying to find the times when one cycle occurs for a FM wave. For instance, given

y(t) = sin(2∏f_{c}*t_{1} + (1/∏)sin(2∏f_{m}*t_{1}))

at an arbitrary time t_{1}, I wish to find the time t_{2} such that

(f_{c}* t_{2} + sin(2∏f_{m}* t_{2}) – (f_{c}*t_{1} + sin(2∏f_{m}*t_{1}) = 1 (cycle).

Now I've tried plotting the wave argument in n-t "space" (i.e. n is cycles),

n = f_{c}*t + sin(2∏f_{m}*t)

to see if some solution presents itself. For instance, we can see that it gives oscillations about the regularly increasing line n = f_{c}*t so that t_{2} ≈1/f_{c} + t_{1} gives an approximate solution. But as to an exact solution, I've hit a brick wall. So if somebody could point me in the right direction it would be greatly appreciated.
Thank you
Rick66

PS: I have made the correction to y(t) above.