Armstrong (indirect) FM generation; pre-emphasis

Armstrong (indirect) FM modulator

In the indirect method, a narrowband FM wave is first generated using a crystal-controlled phase modulator, and then frequency multipliers and a mixer convert it to wideband FM.

m(t) +----------+   +----------+  NBFM
---->|Integrator|-->| Balanced |--+
     +----------+   | modulator|  |
                    +----------+  v
                         ^      +---+  f1, df1
          +--------+  +------+  | + |---------+
          |Crystal |->|-90deg|  +---+         |
          | osc f1 |  +------+    ^           v
          +--------+--------------+       +-------+
                                          | x n1  |
  WBFM   +-------+   +-------+            +-------+
 <-------| x n2  |<--| Mixer |<---------------+
 fc, df  +-------+   +-------+
                         ^
                     [LO fLO]

Working:

1. NBFM generation: for small $\beta$,

$$s(t) \approx A_c\cos\omega_1 t - A_c\,\phi(t)\sin\omega_1 t, \quad \phi(t) = 2\pi k_f\int_0^t m(\tau)d\tau$$

The message is integrated, multiplied with the 90°-shifted crystal carrier in a balanced modulator (giving the DSB-SC term), and the carrier is added. Deviation is kept very small (e.g. $\Delta f_1 \approx 25$ Hz, $\beta < 0.3$) to avoid distortion. 2. First multiplier ($\times n_1$): a non-linear device and a BPF multiply the instantaneous frequency, so the carrier becomes $n_1 f_1$ and the deviation becomes $n_1\Delta f_1$. 3. Mixer: shifts the carrier to $|f_{LO} - n_1 f_1|$ without changing the deviation. This lets the final carrier be set to the required value. 4. Second multiplier ($\times n_2$): gives the final deviation and carrier:

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

Example: $f_1 = 200$ kHz, $\Delta f_1 = 25$ Hz, $n_1 = 64$, $f_{LO} = 10.8$ MHz, $n_2 = 48$ gives $f_c = 48 \times 2$ MHz $= 96$ MHz and $\Delta f = 76.8$ kHz.

Merits: excellent carrier stability (crystal oscillator, so no AFC is needed). Demerits: many multiplier stages, more complexity, and phase noise that is multiplied along with the deviation.

Why pre-emphasis and de-emphasis are used

• After FM detection, the output noise power spectral density is proportional to $f^2$, so noise is heaviest at high audio frequencies.
• High-frequency audio components have low amplitude, so their SNR is poor.
• Pre-emphasis (high-pass RC, 75 μs) at the transmitter boosts high audio frequencies before modulation; de-emphasis (low-pass RC, 75 μs) at the receiver restores the original balance and at the same time cuts the high-frequency noise. The overall SNR improves by about 10–13 dB without distorting the message.