Non-synchronous FM demodulation

Non-synchronous (non-coherent) FM demodulation recovers the message without generating a local carrier in phase with the received carrier. The standard method is differentiation followed by envelope detection (the discriminator method).

FM  +-------+ +-----+ +------------+ +--------+ m(t)
--->|Limiter|>| BPF |>|Differentia-|>|Envelope|---->
    +-------+ +-----+ |tor j2pi af | |detector|
                      +------------+ +--------+

Received FM (after limiter and BPF):

$$s(t) = A_c\cos\left[2\pi f_c t + 2\pi k_f\int_0^t m(\tau)\,d\tau\right]$$

Step 1: Differentiation. A circuit whose gain rises linearly with frequency, $H(f) = j2\pi a f$, acts as a differentiator:

$$v_1(t) = a\frac{ds(t)}{dt} = -2\pi a A_c\left[f_c + k_f m(t)\right]\sin\left[2\pi f_c t + 2\pi k_f\int_0^t m(\tau)\,d\tau\right]$$

$v_1(t)$ is still FM, but its amplitude now varies as $f_c + k_f m(t)$, i.e. it is an AM–FM wave.

Step 2: Envelope detection. Since $f_c \gg k_f|m(t)|$, the envelope is always positive:

$$v_2(t) = 2\pi a A_c\left[f_c + k_f m(t)\right]$$

Step 3: DC removal. A blocking capacitor removes $2\pi a A_c f_c$:

$$y(t) = 2\pi a A_c k_f\, m(t) \;\propto\; m(t)$$

Practical circuits:

• Slope detector: a tuned circuit with resonance slightly above $f_c$; the carrier sits on its sloping skirt, so amplitude varies with frequency. Simple but non-linear.
• Balanced slope detector: two tuned circuits at $f_c \pm \delta f$ with diode detectors connected back to back; the output is the difference, giving an S-shaped, more linear response and no DC.
• Foster–Seeley discriminator and ratio detector: use the phase shift of a double-tuned transformer.
• Zero-crossing detector: counts zero crossings; the average of a pulse train triggered at each crossing is proportional to the instantaneous frequency.

Role of the limiter: the envelope detector also responds to amplitude noise. The limiter makes the input amplitude constant so that only frequency variations reach the output.