NBFM spectrum vs AM; Armstrong NBFM generation

NBFM spectrum compared with AM

For single-tone modulation:

$$\begin{aligned} s_{AM}(t) &= A_c\cos\omega_c t + \frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t + \frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t \\ s_{NBFM}(t) &\approx A_c\cos\omega_c t + \frac{\beta A_c}{2}\cos(\omega_c+\omega_m)t - \frac{\beta A_c}{2}\cos(\omega_c-\omega_m)t \end{aligned}$$

Similarities: both have a carrier and one pair of sidebands at $f_c \pm f_m$; both have bandwidth $2f_m$; sideband amplitudes are proportional to the modulation index.

Differences: in NBFM the lower sideband is inverted (180° phase shift). In AM the sideband phasors add to the carrier in phase, changing its amplitude; in NBFM they add at 90° to the carrier, changing mainly its phase. NBFM also has small amplitude variation and some higher-order terms that AM does not have.

Generation of NBFM by Armstrong's method

From the NBFM expression with general message:

$$s(t) \approx A_c\cos 2\pi f_c t - A_c\,\phi(t)\sin 2\pi f_c t, \qquad \phi(t) = 2\pi k_f\int_0^t m(\tau)\,d\tau$$

The second term is a DSB-SC wave of the integrated message on a carrier shifted by 90°. This leads to the block diagram:

m(t) +------------+     +----------+  -A.phi.sin wct
---->| Integrator |---->| Balanced |----------+
     +------------+     | modulator|          |
                        +----------+          v
                             ^              +---+  NBFM
                       +----------+         | + |------>
                       | -90 deg  |         +---+
                       |  shift   |           ^
                       +----------+           |
                             ^                |
     +-------------+         |    Ac cos wct  |
     | Crystal osc |---------+----------------+
     |    fc       |
     +-------------+

Working:

1. The message is integrated (so that the phase modulator produces FM rather than PM).
2. A crystal oscillator gives a very stable carrier $A_c\cos\omega_c t$.
3. The carrier is shifted by 90° and fed to a balanced modulator with the integrated message, producing $-A_c\phi(t)\sin\omega_c t$ (DSB-SC).
4. The carrier is added to this output, giving NBFM.

The deviation is kept small ($\beta < 0.3$, about 25 Hz in practice) to limit distortion. Wideband FM is then obtained by frequency multipliers and a mixer. The main merit is crystal-controlled carrier stability.