NBFM time domain expression; NBFM to WBFM

Time-domain expression of NBFM

An FM wave modulated by a single tone $m(t) = A_m \cos 2\pi f_m t$ is

$$s(t) = A_c \cos\left[2\pi f_c t + \beta \sin 2\pi f_m t\right], \qquad \beta = \frac{k_f A_m}{f_m} = \frac{\Delta f}{f_m}$$

Expanding with $\cos(A+B) = \cos A\cos B - \sin A \sin B$:

$$s(t) = A_c \cos 2\pi f_c t \,\cos(\beta \sin 2\pi f_m t) - A_c \sin 2\pi f_c t\, \sin(\beta \sin 2\pi f_m t)$$

For narrowband FM, $\beta \ll 1$ (in practice $\beta < 0.3$). Then

$$\cos(\beta \sin 2\pi f_m t) \approx 1, \qquad \sin(\beta \sin 2\pi f_m t) \approx \beta \sin 2\pi f_m t$$

so

$$s(t) \approx A_c \cos 2\pi f_c t - \beta A_c \sin 2\pi f_c t \,\sin 2\pi f_m t$$

Using $\sin A \sin B = \tfrac12[\cos(A-B) - \cos(A+B)]$:

$$s_{NBFM}(t) \approx A_c \cos 2\pi f_c t + \frac{\beta A_c}{2}\cos 2\pi (f_c+f_m)t - \frac{\beta A_c}{2}\cos 2\pi (f_c-f_m)t$$

So NBFM has a carrier and one pair of side frequencies at $f_c \pm f_m$, like AM, but the lower side frequency has a negative sign (180° phase shift). Its bandwidth is $2f_m$, the same as AM.

Generating WBFM from NBFM (Armstrong / indirect method)

WBFM is obtained from NBFM using frequency multipliers and a mixer:

m(t) -> [Integrator] -> [Balanced ] -> (+) -> NBFM
                        [modulator]     ^     f1, df1 (b1 < 0.3)
                             ^          |
                         [-90 deg]      |
                             ^          |
         crystal osc f1 -----+----------+

NBFM -> [x n1] -> [Mixer] -> [x n2] -> WBFM
                     ^                fc, df
                  [LO fLO]

1. A stable crystal oscillator, a −90° phase shifter, a balanced modulator and an adder (which adds the carrier back) produce NBFM with small deviation $\Delta f_1$ at carrier $f_1$.
2. A frequency multiplier (a non-linear device + band-pass filter) multiplies the instantaneous frequency by $n$. Both carrier and deviation are multiplied: $f_c \to n f_1$, $\Delta f \to n \Delta f_1$, so $\beta \to n\beta_1$.
3. Because the required deviation needs a large $n$, the carrier would become too high. A mixer with a local oscillator shifts the carrier down ($f_{LO} - n_1 f_1$) without changing the deviation.
4. A second multiplier $n_2$ raises the deviation (and carrier) to the final values:

$$\Delta f = n_1 n_2 \Delta f_1, \qquad f_c = n_2\,|f_{LO} - n_1 f_1|$$

Example: $\Delta f_1 = 25$ Hz needs $n_1 n_2 = 3000$ to give 75 kHz deviation for broadcast FM. The method gives a very stable carrier (crystal controlled) and is used in FM broadcast transmitters.