# How to generate WBFM from NBFM

A narrowband FM is approximately given by:

$$x(t)= A[\cos w_ct - k_f a(t)\sin w_ct]$$

where

$$a(t)= \int _{ -\infty } ^t m(\alpha) d\alpha$$

How does frequency multiplying $$x(t)$$ by a desired multiplier result in WBFM?

• what is "frequency multiplying $x(t)$"? – Marcus Müller Apr 20 '20 at 16:34
An FM signal modulated by $$m(t)$$ is $$x(t)=A_c\cos(\omega_c t+(k_m\int m(\alpha)d\alpha) t)$$. This can be expanded as $$x(t)=A_c\cos(\omega_c t)sin(\theta_m t)-A_c\sin(\omega_c t)\sin(\theta_m t)$$ , where $$\theta_m = k_m\int m(\alpha)d\alpha$$.
For NBFM, $$k_m << 1$$, so $$\theta_m(t) << 1$$, so we can approximate $$x(t) \approx A_c\cos(\omega_ct)-A_c\theta_m(t)\sin(\omega_ct)$$ For generating WBFM, frequency multiplier are used to generate NBFM at $$n\omega_c$$ and $$n\Delta f$$ (frequency deviation). For example, for 8x multiplication, the signal is passed through square law device 3 times, with each stage having a BPF centered around 2 times center frequency after squaring. For squaring, $$x(t)^2 = A_c^2\frac{1}{2}\Big[1+\cos(2\omega_c t)\Big]+A_c^2\theta_m^2(t)\frac{1}{2}\Big[1-\cos(2\omega_ct)\Big] -A_c^2\theta_m(t)\sin(2\omega_ct)$$ After BPF with center frequency around $$2\omega_c$$, $$\tilde{x_2}(t) = \frac{A_c^2}{2}(1-\theta_m^2(t))\cos(2\omega_ct)-A_c^2\theta_m(t)\sin(2\omega_ct)\\ \approx \frac{A_c^2}{2}\cos(2\omega_ct)-\frac{A_c^2}{2}\Big(2\theta_m(t)\Big)\sin(2\omega_ct)$$ You can see that not only carrier frequency has doubled but modulation sensitivity has also doubled, resulting in doubling of frequency deviation $$\Delta f$$.