FM General Equation  

Let the carrier be x_{c}(t) = X_{c}·cos (w_{c}t), and the modulating signal be x_{m}(t) = b·sin (w_{m}t) 
Then x(t) = X_{c}·cos [w_{c}t + b·sin (w_{m}t)] 
Modulation Index  


Narrowband FM (NBFM) 

Narrowband FM is defined as the condition where b is small enough to make all terms after the first two in the series expansion of the FM equation negligible. Narrowband Approximation: b = Dw/w_{m} < 0.2 (could be as high as 0.5, though) BW ~ 2w_{m} 
Wideband FM (WBFM) 

Wideband FM is defined as when a significant number of sidebands have significant amplitudes. BW ~ 2Dw 
Carson's Rule 

J.R. Carson showed in the 1920's that a good approximation that for both very small and very large b, BW ~ 2 (Dw + w_{m}) = 2*w_{m} (1 + b) 
Modulation Index (b) = 1  

Here, the maximum frequency (f_{max}) causes a maximum deviation of 1*f_{max} in the carrier. From the modulation index formula:

Modulation Index (b) = 5  

Here, the maximum frequency (f_{max}) causes a maximum deviation of 5*f_{max} in the carrier. From the modulation index formula:

Modulation Index (b) = 25  

Here, the maximum frequency (f_{max}) causes a maximum deviation of 25*f_{max} in the carrier. From the modulation index formula:
