I have a question regarding frequency modulation and the modulation index. I know that the modulation index can be given by
$$\beta = \frac{\Delta \omega}{\omega_m}$$
So the value of the modulation index is highly dependent on the value of \$\omega_m\$.
When we calculate the coefficients for Bessel functions, we need to get \$J_n(\beta)\$, which is a function of \$\beta\$. So that means, whatever \$\omega_m\$ we choose, will affect the value of \$\beta\$.
So then my question is how can I do a frequency sweep then with \$s = j\omega\$? The \$\beta\$ value is always changing and thus so is the amplitude of my signals. Can I just choose the lowest value of \$\omega_m\$, and therefore, the largest \$\beta\$ as my worst case, and thus have "constant" J values? Hope this is making sense.
Best Answer
If we go back to basic theory, we have a carrier signal of the form :-
\$E_c\cos\phi_c\$
... and a sinusoidal modulation signal of the form ...
\$E_m\cos(\omega_mt)\$
and if we let the frequency deviation be proportional to the modulation amplitude, so
\$\Delta\omega\propto E_m\$
the instantaneous frequency is given by ->
\$\dot{\phi_c}=\omega_c+\Delta\omega.\cos(\omega_mt)\$
Integrating this to get the instantaneous phase ->
\$\phi_c=\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\$
So the modulated output is ->
\$E_c\cos\Big[\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\Big]\$
As you say, the modulation index is dependent upon \$\omega_m\$ so the relative amplitudes of the spectral components will vary with \$\omega_m\$, but the modulation index is also a measure of the peak phase deviation, so if you want the spectral amplitudes to be independent of \$\omega_m\$ you must have \$\omega_m\propto \Delta_\omega \propto E_m\$, ie phase modulation.
One technique of producing phase modulation is to use a frequency modulator with pre-emphasis of the modulating signal to get the amplitude proportional to the frequency.